Regularly Varying Functions

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Lecture Notes in Mathematics Edited by A. Dold and 13. Eckmann

508 Eugene Seneta

Regularly Varying Functions

i!

Springer-Verlag Berlin.Heidelberg 9New York 1976

Author Eugene Seneta Department of Statistics The Australian National University P.O.Box 4 Canberra, A.C.T. 2600/Australia

AMS Subject Classifications (1970): 26A12, 26A48, 60E05 ISBN 3-540-07618-2 ISBN 0-387-07618-2

Springer-Verlag Berlin 9 Heidelberg 9 N e w Y o r k Springer-Verlag New York 9 Heidelberg 9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9Heidelberg 1976 Printed in Germany Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr.

PREFACE The main purpose

the basic real-variable

stated assumptions, functions,

of these notes is to present,

in self-contained

reader wishing

to acquire

tool, irrespective

theory of regularly varying

Thus they may be used by any

a user's knowledge

of this valuable

of his field of mathematical

these aims in mind, where possible;

manner.

With

to keep proofs simple

have been provided

the theory as well as to yield practice

analytical

specialization.

the author has endeavoured

and exercises

under precisely

to show the scope of

in the use of the material pre-

sented. The author's probabilistic

in the subject matter was stimulated by

own interest

applications.

theory of regularly

varying

functions

suggested by the book of Gnedenko to be widely

recognized

among probabilists

Applications which contained elements theory.

Unfortunately,

edition9 clear.

in probability

and Kolmogorov.

2 of Feller's An Introduction

of Volume

other hand,

the papers

with precise

difficult

of their non-existence. de Haan's

material

of the Karamata

(and remains

assumptions

in the newer

and conditions reader.

un-

On the

theory has been progressively

contributions

in the early 1950's,

that there is a general

impression

modest hope that these notes

in a manner somewhat

different

from

(1970a).

Apart from the presentation discern

was

It is the author's

these gaps,

came

in 1966

to Probability Theory and Its

in which Karamata's

refined and extended since the original

will help to bridge

with the publication

for the non-expert

are so little known to prohabilists

theory was already

It subsequently

of an exposition

this presentation

highly personal, It thus proves

role played by Karamata's

The fundamental

of the basic

an attempt by the author to provide e.g.

It needs

w

the reader will of less standard

and the Appendix.

to be mentioned

only to the material

theory,

a selection

also that the references

presented,

given pertain

and so cannot in any sense he regarded

as complete. The bulk of these notes was prepared early in 1973 in the course of an academic year spent at the Department University.

of Statistics,

(The author takes this opportunity

G.S. Watson and D.R. McNeil

Princeton

to thank Professors

for their kind hospitality.)

The motivation

for the work was a proposed book with N.H. Bingham and J.L. Teugels, which the present material The author wishes

was to form the first two chapters.

to express

also his indebtedness

to

in

IV

Professor Ranko Bojani~ in regard to materials and stimulating correspondence, and more generally, to the strong Yugoslav school of mathematicians founded by Karamata. Finally, the author is indebted to Ms Helmi Patrikka for her careful typing of the manuscript.

Canberra

E. SENETA, 1975.

CONTENTS CHAPTER 1. 1.1

FUNCTIONS OF REGULAR VARIATION

Introduction.

1.2

Fundamental Theorems.

1.3

Refinement of Definition of Regular Variation. Characterization of Regular Variation.

1.4

The Structure of Slowly Varying Functions and Alternative Proofs.

13

1.S

Further Properties of Regularly Varying Functions.

17

1.6

Conjugate and Complementary Regularly Varying Functions.

2S

1.7

The Definition of a Regularly Varying Function.

29

1.8

Monotone Regular Variation.

37

1.9

Bibliographic Notes and Discussion.

43

Exercises to Chapter i.

47

CHAPTER 2.

SOME SECONDARY THEORY OF REGULARLY VARYING FUNCTIONS

2.1

Necessary and Sufficient Integral Conditions for Regular Variation.

53

2.2

Tauberian Theorems Involving Regular Variation.

59

2.3

A Class of Integrals Involving Regularly Varying 63

Functions. 2.4

A Class

of Functions

Related

to Regularly

2.5

Varying 69

Functions. Bibliographic Notes and Discussion.

8S

Exercises to Chapter 2.

86

APPENDIX.

GENERALIZATIONS OF REGULAR VARIATION

A .1

R=O V a r y i n g

Functions.

92

A .2

S-O V a r y i n g

Functions.

97

A.3

Monotonicity;

A.4

Bibliographic Notes and Discussion.

Dominated Variation.

99 104

REFERENCES

100

SUBJECT INDEX

111

CHAPTER 1 FUNCTIONS OF REGULAR V A R I A T I O N i.I.

Introduction. Regular v a r i a t i o n of a function is a one-sided,

tic p r o p e r t y of the function, which logical

local and asympto-

arises out of trying to extend in a

and useful manner the class of functions whose asymptotic be-

haviour near a point is that of a power such asymptotic b e h a v i o u r factor which varies

'more slowly'

Being a local property, point.

i.i.

R

~ > 0

(I.I)

= ~

for some

p

is defined relative

to a

is said to be regularly varying at in-

positive

A > 0 , and if for each lim ~

than a power function.

is taken to be as follows.

A function

finity if it is real-valued,

to functions where

function m u l t i p l i e d by a

regular v a r i a t i o n

The defining p r o p e r t y

Definition

function,

is that of a power

in the interval

and m e a s u r a b l e

-- < p < ".

(0

on

[A,-),

for some

is called the index of

regular variation). A function

R(.)

is said to be regularly varying at zero if R(i/x)

is regularly varying at infinity. at any finite point point.

a

by shifting the origin of the function to this

It is thus apparent

that it suffices

regular v a r i a t i o n at infinity, the words "at infinity" tion of results at

0

to develop

the theory of

which we shall do, frequently omitting

in the sequel.

Some exercises

in the transla-

from regular v a r i a t i o n at infinity to regular v a r i a t i o n

are given later. Let us write

form

Regular v a r i a t i o n can now be d e f i n e d

xPL(x).

urable on

It follows that

[A,~)

(1.2)

a regularly varying function with index L(x)

is real-valued,

p

in the

positive

and meas-

and from (i.i)

lira ~

= 1

X+~

for each index

~ > 0

Thus

L(.)

is also a r e g u l a r l y varying function,

~ = 0 .

D e f i n i t i o n 1.2.

A function

index of regular v a r i a t i o n

L(.)

which is r e g u l a r l y varying,

~ = 0 , is called slowly varying.

with

of

The notation

L(.)

is customarily

used for such functions

of the first letter of the French word "lentement" the foundation + Karamata.

papers

Thus a function be written

which means

of the theory having been written

R(.)

is regularly varying

because "slowly",

in French by

if and only if it can

in the form

RCx) = xPLCx) where

-- < p < -

and

L(.)

is slowly varying.

This

is the product

form alluded to in the opening paragraph. Any eventually limit as example

x + ~

positive

is clearly

of a slowly varying

log log x

measurable function

is

regularly varying (others

possessing

a positive

The simplest non-trivial

log x ; any iterate

of it e.g.

is also slowly varying.

On the other hand the exponential 2 + sin x

function

slowly varying.

functions

at all; and undampened

are similarly

less obvious

not regularly

oscillatory

varying.

is involved

functions

These

are given in the exercises)

intuitive notion of what

e x , e -x , are not such as

few examples

should provide

in the concepts

of regular

some and slow

variation. It should also be clear that to study regular variation, to study the properties 1.2.

Fundamental

of slowly varying

functions functions

Theorem

i.I.

pertaining

follow readily

1.2.

(The Representation

such that for all

x > B

Theorem). [a,b],

If

L(.)

is a slowly

0 < a < b < ~, the rela-

h~[a,b].

Theorem).

If

L(.)

defined on

then there exists a positive number we have X

L(x) = exp { q ( x ) + ~

(1.3)

B

where

n

is a bounded measurable

c(t)

Notes

dt }

t

function on

§

See Bibliographic

can

of slowly

from them.

[1.2) holds uniformly with respect to

Theorem

of slowly

in that either

and most other properties

then for every fixed

A > 0 , is slowly varying,

to the properties

they are fundamental

from the other,

(The Uniform Convergence

varying function, tion

theorems

in the theory;

be obtained readily varying

it suffices

for most purposes.

Theorems.

There are two basic varying

functions,

and Discussion.

[B, |

such that

[A,~).

B ~ A

n(X)

+ c

that

([C[

e(x)

< ~),

+ 0

and

(as

E

is

a continuous

function

We shall proceed by first proving Theorem via a sequence Theorem

[B,~)

on

such

x + ~).

of lemmas.

The converse

i.I and then Theorem

deduction,

of Theorem

1.2

I.i from

1.2 is left to an exercise. +

For the following

lemmas

itself but a function formed by

f(x)

We shall

f

= log L(e x)

thus assume

is real and measurable

it is rather

easier

of a kind to which

L

to work not with can be readily

.

that we are dealing with a function

on

L(.)

trans-

[y,|

for some

f

y, and satisfying

which

the con-

dition (1.4)

f(x + u)

f(x) +.0

The relation

Lemma 1 . 1 .

as

x + ~, for each

hoZd8 uniformly for

(1.4)

~

~ .

in a n y fixed

finite closed interval. Proof.

We first prove

[0,I].

Suppose

the assertion

the assertion xn § |

E > 0, {x n} such that each n, satisfying

[f(x n + ~n)

(1.S) Define

sets

Un,

Vn

with

(l.6b)

Vn={X:XE[0,2],

[f(Xm+~m+X )

tone

are clearly measurable

increasing

such that

interval

Then

3

~n r [0 ,I]

for

.

- f(Xm) [ < ~1 e

[f(Xm+~)

Vn

in the particular

by

Un={~:~e[0,2],

and

~

n , {~n }

- f(Xn) I ~ c

(l.6a)

Un

for

is not true for this interval.

sequence

of sets,

,

~g/m k n }

f(Xm+~m) I < ~1 , ~/m _> n } . and each of

and such that

{Un},

{V n}

is a mono-

Un, V n § [0,2]

in

virtue of (1.4). Hence

if

sufficiently that

m(V~)

m(.) large

= m(VN)

is used to denote the measure, m(Un)

> 3/2, m(Vn)

> 3/2.

Let

it follows

that

V~ = V N + ~N

VN

' so

> 3/2, and note that I

u N C [o,z] C [0,3] v~ c [o,31 it follows set). +

Thus

that for any J~

e UN

See Exercise

1.3.

; N

sufficiently

such that

large,

u - UN ~ VN

"

UN nV~

~ ~

(the empty

For this (l.7a)

If(xN+~ )

(1.7b)

If(xN+~N+ ~ -~N)

f(xN)

1

I

<

~E

I f(xN+~N) [ < ~ e

by

(l.6a);

by

(l.6b);

or

equivalently

i f(xN+~N) I < ~ c

If(xN+~) Putting

(l. Ta) and

inequality,

(l.?b]

together

I f(x N + u N) a contradiction

to

For the case by

f(x)

U

~

=

[a,b]

Lemma

[X,X'],

- f(x] v =

~=~ . e [0,I] ~X(X

~ ~)

By L e m m a

taking

- f(y)

(~-a)/(b-a),

i.i,

JX

for

x

- f(x)

so

f

b > a, define

f(-)

+ f(x-a) that

- f(x)

y § -

~

x § -

;

i8 bounded on every interval

[X + k - I, X + k]

is b o u n d e d

for any

y

in the

interval

[X,X+I]

,

+ 1

and c a r r y i n g

i IfCx+z)1

for p o s i t i v e

f

V~[O,l]

this

argument

further

we

[X + i, X + 2]

+ 1 i IfCX) l + 2

integer

If(x) I i If(x)I

Corollary.

that

I < 1 , x ix,

If(x) l

inequality;

on

Ifcx)l have

such

x = X, X + ~ = y

by an e l e m e n t a r y

on

[a,b],

.

If(y) l ~

We thus

= f(y+~)

sueh that

IfCx+~]

obtain

interval

X' > X .

Proof.

Thus,

< c ,

Then

(x-a)/(b-a),

1.2.

- f(x N]I

of an a r b i t r a r y

= f((b-a)x)

y

side of the t r i a n g l e

(I.S).

f(x+~) where

as the d o m i n a n t

we o b t a i n

that

+ k

and so on

is integrable ~ver

and m e a s u r a b l e

k

thereon).

[X,X + k] [X,X']

D for any

X'

> X,

(since

it

Lemma

1.3.

if

X

is

as

in

1.2,

Lemma

then

for

x > X

,

X

f(x)

= c(x)

r

+ f X

where and

c

and

c(x)

~

are

measurable

~ c(]c I < =),

Proof.

For

x ~ X

~(x)

write,

and

§ 0 using

x+l f(x)

= f

x

~

Lemma

on

[X,X'],

any

(f(x)-f(t))dt

if we i n t r o d u c e

> X,

1.2, X+I

+ f (f(t+l) X

new n o t a t i o n

X'

|

x

x Then

bounded

as

f(t))dt

+ f

f(t)dt X

by p u t t i n g

respectively

X

= ~(x)

+ f

~(t)dt

+ c

X it f o l l o w s

that r

= f(t+l)

6(x)

= f

- f(t) § 0

as

t § ~

from

(1.4)

,

and x+l

1 (f(x)

f(t))dt

= f

X

-~

in v i r t u e

of L e m m a

1.4.

For

0

as

i.I.

c(x)

Lemma

(f(x)

f(x+~))d~

O

:

all

X

Hence

a(x)

x

+

c

> X~

-~

o~

the p r o o f .

is c o m p l e t e

if we put

m

, for

some

X~ > X

,

n

X

(1.8)

f(x)

= c*(x)

+ f

r X~

where and,

ca

and

moreover,

~*

have

~

is

the

properties

Let

f*(x)

= f

Take

f(x)

~

in

Lemma

X

~(t)dt

= f

X (1.9)

and

c

continuous. X

Proof.

of

(f(t+l)

f(t))dt

, so that

X

- f*(x)

= c(x),

+ C

as

X

+

oD

> 0 ; then X+~

f~(x § ~)

- f*(x)

= /

(f(t+l)

f(t))dt

X

= ;

(f(y+x+l) O

Now for

y

in

[0,u]

f(y§

.

1.3~

f(y+x+l) and,

by L e m m a

i.I,

- f(y+x)

+ 0

uniformly

f*(x+u) This

is true

argument

some

for for

Clearly,

all

X*

> X

f*(x)

any

true

= f(x+y+l)

~ > 0

v < 0

for

; trivially

of Lemmas X

such

y

- (f(x+y)

; hence

as

- f(x))

x +

+ 0

; hence

replacing

f(x)

true

1.1-1.3

so for

for

~ = 0

every

are now

; and by a s i m i l a r

~.

applicable

to

f*

, with

, X

f*(x)

= ~*(x)

+ f

~*(t)dt

+ c*

X* where

we

can

take e*(t)

which

= f*(t+l)

is c o n t i n u o u s ,

since

-

f*(t)

f*(t)

is.

Hence

from

(1.9)

X

f(x)

= c(x)

+ f*(x)

= c(x)

+ 8*(x)

r

+ f

+ c*

X* which

gives

Remark.

the

result

By r e p e a t i n g

of times,

we

far along,

All

"undesirable"

into

c*(x),

bounded

on

Theorems vely

by

for

x > 0

Theorem

the

we

has

about

finite

I.i

the

which

1.2 now

/

exp

still

with

(1.8)

stage

already

is

where

that

from Lemmas

so that

e*(t),

x +

I.i

and

f(x)

order.

accumulated

it is m e a s u r a b l e

as

in the

number

suffi-

specified

increasingly

limit

mentioned:

x)}

an a p p r o p r i a t e

of any

say only

a finite

follow

{f(log

lemma

derivative

at any

we may

transformation

can

representation

behaviour

and

of this

a continuous

intervals,

, L(x)~

9

the p r o c e d u r e

can o b t a i n

ciently the

required.

1.4

and

respecti-

= log L(e x)

i.e.

Representation

take = c*(log

n(x)

x)

,

= c*(log

E(x}

x)

since log x

x e, c*(t)dt

X* where

B = exp

Corollary

tion

(1.3)

X*

f B

(log y) Y

dy

.

to T h e o r e m

where

=

n

Any function

1.2.

and

c

defined and having representa-

have the properties

stated is slowly

varying. The proof is simple and is left to the reader; one consequence worth noting, large

x

(1.3) states

that for sufficiently

we may write a slowly varying function in the form

L(x) where

in that

there is however

M(x)

= M(X)Lo(X )

is positive,

along and approaches

measurable,

a positive

a particularly well-behaved

b o u n d e d in intervals

limit

M

as

far enough

x + ~ ; while

slowly varying function,

L (x) is o so that as x +

L(x) ~ M L o(x) where

~dt}

X

L ~ (x) where have

e(t) the

is

continuous

representation

(1.10)

~(t)

where

the

prime

There which

is

is

an e l e m e n t a r y positive,

xg'(x)/g(x)

and integrate

but

t § ~

In fact

we

)

important

and has

converse

continuous

to

this:

derivative

any for

function x > B ,

~ 0

for

To see this put the left hand side

g , finally using the corollary above.

(i.ii)

will be regularly varying, Refinement

as

B, and satisfies

the right hand side of

1.3.

zero

itself:

a derivative.

x § | , is slowly varying.

= e(x)

g

e(t)

0

indicates

defined,

(1.11)

and approaches

for

= x L'(x)/Lo(X

for some positive

as

e x p { fB

=

is, more generally, of index

If

p , -~ < p < ~ , then

p

of D e f i n i t i o n of Resular Variation.

Characterization

of Regular Variation. The defining relations

(i.i) and

(1.2) of regular and slow v a r i a t i o n

can be much w e a k e n e d without changing the theory. sometimes

of use in applications;

ring them is to demonstrate restrictive

Such refinements

however the chief purpose

that the r e l a t i o n

are

in conside-

(i.I) is not nearly as

as it at first appears.

A p r e l i m i n a r y result,

of w h i c h we shall have need in the sequel

is the following. Lemma 1.5.

Suppose

a function

R , defined,

measurable

and positive,

[A,~)

on

, for

(1.12)

A > 0

some

, satisfies

lira R R @ = r X-~m

for

each

X

is f i n i t e > 0

in a c l o s e d

and positive

, and for

Proof.

Let

some

finite

u > 0 .

it follows,

for

any

and

The

left-hand

the

Repeating

limit

Lemma

1.6.

that

(I.12)

measure,

Proof. f(x) exp

x+|

RR@

r

(1.13) for

9 a S~

.

, since

that

>

be

then,

for

[a/b,b/a],

k-i

since

times,

a/b

y < xla < bla

and

say,

< I, b / a

by this

for a set

limit

for

such

y,

, de-

it

follows

that

~(X)

any p o s i t i v e

value

interval.

of Lemma S

> 1

1.5 are p r e s e r v e d

of positive

is f i n i t e

except

X , of positive

and positive.

Then

the

1.5 p e r s i s t s .

easier x

then

have

, a set

o

0

covered

, all

f(x

in

the c o n d i t i o n s

the

We

defined,

y

>

,

and

It is s l i g h t l y

~ S

each

> o

merely

of Lemma

for

r

eventually

holds

,

argument,

+

Suppose

r162

is well

~(y)

= log R(e x) {~}

X/y ~ b

for

R(u165 R(Xx/y)

= lim

by

on w h i c h

conclusion

X > 0

X~[a,b]

a ~

; i.e.

the o r i g i n a l

will

fixed

satisfying

limit

for y c [ ( a / b ) k , ( b l a ) k ] , ~

each

R(Xx/y) R(~'J

y

R(u

of

for

~(x)

, where

holds

is

y ~ xlb ~ a/b

noting

taking

(1.12)

Then

R(Xx/~)

=

and

> 0

t(~(Xx/~))

lim R ( ~ x ) x+~ R ( x J exists,

, 0 < a < b < ~

interval.

~(X)

Then,

R(xx) = R-K-CKY-

[a,b]

interval

on this

+ ~)

to w o r k

, ~(z) that

f(x)

of p o s i t i v e

= log as

with

the

~(e T)

transformed

for

T

such

x +

+ ~(~) measure.

Then

if

v E S*

forms that

where

f(X+T+~)-f(x)

[] f ( X + T + ~ ) - f ( x + ~ ) + f ( x + T ) - f ( x ) + r

as

x + -

; and

so

(1.13)

D = {~; ~ = ~ + ~, according

T , ~ g S *}

interval

I .

for

all

mises

of L e m m a The

shows must

1.5,

r

Theorem

Lemma

1.3. 1.6,

Proof:

r

and

so its

have

varying

has

positive

defined

, where measure.

in this

(if n e c e s s a r y ) the

conclusion,

way

Now contains

of this the

section

form

10

to L e m m a

lim ~

1.6 we

=

r (x)

following,

so the

defined

R

since

p

have

~ > 0

for e a c h

it

considered

sense.

~P , for some

> 0

the pre-

Under the conditions

Theorem).

the form

~(V)+r

I

is the

, and

as

transformations,

hold.

in the p r e v i o u s l y

assumes

According

u E D

Inverting

(Characterization

~(x)

D

is d e f i n e d

theorem

must

be r e g u l a r l y

S*

for

r

§

, ~ , T ~ S*

fundamental

that

where

t h e o r e m +, a

f(x)

~ c I , where u = v + T

defined

Hence

f(x+~)

where

is w e l l

to a w e l l - k n o w n

a closed

+ ~(~)

of

satisfying

o

X-~

Then

for any

y > 0

,

R(~x) and

so,

letting

for each positive

x, y > 0 . real

numbers,

limit

of m e a s u r a b l e

these

conditions

last

proposition, proof

we

of Lusin's

R-r(D--

= r

This for

shall

Theorem

is the H a m e l a function

solutions

instructive

since

are

to give

it is done

give will

functional

r > 0

also

in the p r e s e n t

of the

a simple

as an

setting,

being

It is known* form

infrequently serve

equation

, which,

is measurable.

functions,

the only

It is, h o w e v e r

The

-

x § r (x)r (~)

(1.14)

R(x~x)

_

" R--T~-

k p,

direct

(1920, T h 6 o r ~ m e VII) and R o s e n t h a l (1948,

that

proof

in e l e m e n t a r y

which

with

is the o r i g i n a l pp. 1 1 6 - 1 1 8 ) .

a

of this text

books.

of the use

Egorov's

memoir.

under

-" < 0 < |

illustration

+

, Steinhaus e.g. H a h n

on the a pointwise

i0

T h e o r e m + and S t e i n h a u s ' s theoretic

tools

(already used),

for the p r e s e n t

theory

t i c e d by several

authors.

u s e d in n u m e r o u s

other probabilistic

given

in p r o b a b i l i s t i c

Theorem

fying

(1.14) for

for all mary

X > 0

functional

(i.15)

r

The

i8 necessarily of form

x,y

, where

equation

of m e a s u r a b i l i t y do so initially.

is p a r t i c u l a r l y

it solves

we shall (1.15)

x k = x, k = l , . . . , n

(1.16)

r

r

(1.17) for p o s i t i v e

easy

to solve

if the a s s u m p t i o n and we shall

show that the m e a s u r a b i l i t y

i m p lies

it c o n t i n u i t y .

+ r (x n)

, we o b t a i n

, we have

(1.16) = me(y)

integers

r y=l

It is custo-

= r

nr

Putting

transforms

= he(X)

x = (m/n)y

for p o s i t i v e

and m e a s u r a b l e .

+ Xn) = @(Xl) + . . . . .

and if we put

whence from

, (1.14)

implies

@(Xl+ . . . . .

If we put

x)

is r e p l a c e d by one of c o n t i n u i t y ,

Subsequently,

(1.15)

= log r

is finite v a l u e d

~

and the fact that First,

~(x)

X p , -~ < p < ~

form.

(1.15) on

m,n

.

Thus

= ~((m/n)y)

= (m/n)~(y)

, we o b t a i n

r

= re(l)

rational

r .

Putting

x = y = 0

+

See B i b l i o g r a p h i c

are

are n e v e r

= r

~

this

but p r o o f s

equation

+ r

to w o r k w i t h

of the p r o p o s i t i o n

connections,

measure-

as has b e e n no-

finite, measurable, positive, and satis-

r

With the t r a n s f o r m a t i o n

to C a u c h y ' s

versions

to be the n a t u r a l sight,

texts).

A function

1.4.

Proof.

(Further,

appear

at first

Notes

and D i s c u s s i o n .

in

(i.15) y i e l d s

of

,) , s o x)

at

(I.17) any

holds

point,

it

~,(x)

.oh

is

evidently

:ore ( 1 . . t 5 ) ,

Further,

it. any

is

that

t

for

that

~

readily

m(I-F)



number t:

c is

exists -2

Fn

is

nn > 0

, such

satisfying

r

is

It

negative

0).

x

compact,

exists

that is

follows

a closed

and

the

closed restricted

uniformly

Lusin set

of

= x~

but to

is

merely

any

closed

now i m p t i e s F ~

measure

~p

r

of

t

such

I-F

subsets

,

{Fn;'

to

Fn

continuous.

of

~s con-

Hence

there

that

and

clearly

(1,.t5), of

(1.15);

case

relative

The theorem

there

solves

satisfies

cor~tinuous , such

x

in which

measurable

!~!

n-1

<

< !~n

Let

0 < ~n < ~n = ~!~n(nr~ ' n - ' ~ ' ) "

+ ~n ~ Fn < 2n -2

c Fn

form;

a sequence

> # -- n

since

~+6

for

= const,

this

I@(~a+8) @(~)1 providing

also

.

where

to

m(Fn)

and

x = O;

is

Thu.s t h e r e

r

(1.17j

problem

length

positive

that

for

solution

restricted

tinuous, is

to

, whose

any

from

= -~.(--x)

measurable,

interval,

a_

y = -.x

co.r, t i n u o u s

~'e now p a s s assumed

= 0

follows

also

r

hence

r

= x,(~)

true

putting

for

measurable,

The

and

of

6n set

be of

measure

a fixed ~ g Fn

at

re.tuber such

most

n-2

that + '~n '

that -1

I '~"( ~ + 6 n ) when Now

~

is

let

number

contained

G = of

the

U

in

{7

*(~)l

a set

E i , the

j=! i:j ~Si s.

EnC set

Putting

< n Fn

of

Ki

~

such

that

belonging

I - Ei

, H =

m(En) to

all

I - G

> ~ - 3n -2 but

, it

a finite

follows

e~

that

H =

I - G =

~

~

Ki,

the

set

of

points

in

I

belonging

to

in-

j=l i=j finitely

many

of

the

re(H)

Ki

i m

Thus

(0) Ki

i=j

for

each

Hence

for

j = l , 2 , . .... ; so any

seauence '

that

~n} ~

< 3i -2 ~Z - i=j

finally,

, where

re(H) 6n

= 0

, so

satisfies

that 0 < ~n

re(G)

= Z

<

'

~n

:

12

that

follows

(1.18)

lim ~(m+6n)

~((~)

=

n.+~

for a l m o s t Now,

every let

I

fixed numbers x = ~o

- m

and if

~

from

w

in

I

be the i n t e r v a l

satisfying

and let

[ml,~2]

Then

ml < m < mo < ~2

~ and ~ be o from (1.15), t a k i n g

' Y = m + ~n

is t a k e n o u t s i d e

a subset

of m e a s u r e

= ~(eo-e)

+ lira r

= q,(%-~)

+ r

I , it follows

zero of

(1.18) lim r

)

i.e. (i.19)

lim ~ ( ~ o + ~ n ) = ~(~o) n-~

for e v e r y

mo

satisfying

sequence which I

and must be c h o s e n

any null

sequence

in a c c o r d a n c e

lim sup r n§

It is p o s s i b l e

to s e l e c t

n.

< ~i

not q u i t e

of p o s i t i v e

(1.20)

0 < @

ml < ~o < ~2

is a p p a r e n t l y

with

numbers,

Here

{6n }

is a p o s i t i v e

since

it d e p e n d s

0 < 6 n < ~n

Let

on

{@n }

be

and s u p p o s e

) > lim inf r n§ a subsequence

) .

{en}

' and

{e n}

from

such that

i

1

lim r i§

1

) = lim sup r n§ =

b y (1.19).

A similar lira inf

so we have Since

"

arbitrary,

{0 n}

argument

(%)

gives

the result

that

~(~o+Sn ) = ~(~o ) ,

a contradiction is a r b i t r a r y ,

right-continuous

r

) ,

to

(1.20);

except

at any p o i n t

mo

and c o n c l u d e

e a c h limit

that

@n > 0 , it follows

' as

I

can be a r b i t r a r i l y

is r

~(mo). is

chosen.

13

To obtain

left continuity, (x)

and h e n c e we o b t a i n tinuous ~o = -xo 1.4.

' which

x~

entails

first,

alternative

rather

w

strong

these

; then from

(1.15)

that ~

xo

Functions

that

~

right

is arbitrary.

conat

9

and A l t e r n a t i v e

the definition

properties.

is

is left continuous

Proofs.

of a slowly varying

The p u r p o s e

and s e c o n d l y

of this

section

to consider

some

proofs.

have the

to

One p o s s i b i l i t y

for

e(t)

as r e q u i r e d

However,

clear

it

is

we may s t i l l s(t)

the

function

as c o n t i n u o u s ,

with

that

since

more d e e p l y ,

We b e g i n b y c o n s i d e r i n g desirable

implies

the proof,

seen in

to explore

= ~(-x)

m a n n e r as b e f o r e

o f S.lowl?' Va r y i n ~

We h a v e a l r e a d y is

~(x)

: ~(x+y)

This

completes

The S t r u c t u r e

function

+ ~(y)

in similar

at any point

put

, gotten

(1.3).

expressed

that

from Lemma 1 . 4 ,

i f we d o n ' t

get

a (simpler) measurable

insist

p r o o f o f Lemma 1 . 3 s l i g h t l y ,

and b o u n d ed ; by w r i t i n g

-i < x+~ = m f (f(x) x

(1.3) in fact

for

is

allows

sometimes L(-)

itself.

us to t a k e

it

Theorem's statement.

on t h e

representation

It

in terms of

by t h e R e p r e s e n t a t i o n

still

f(x)

representation ~(t)

continuity

by u s i n g let

of

s(t),

Lemma 1 . 3 ,

us g e n e r a l i z e

x ~ X , for

any

the

~ > 0

x f(t))dt

+ f

{f(~+t)

- f(t)}dt

X X+O)

+

It follows

as before

fX

f(t)dt

that we can take ~(t)

;

{f(e+t)

c(x)

=

~(log

- f(t)}/e

and so

using the

fact

that

f(t)

x)

= ~

= l o g L(e t )

; so t h a t

Xo (= e ~) > t

, we h a v e t h e r e p r e s e n t a t i o n

boundedness,

but not necessarily

(1.21) which

E(x) _ log 1 ~o l o g

is a simpler

expression

1 log

continuity,

(1.3), of

f o r any f i x e d

number

with measurability e(x)

g i v e n by

L(•~

-L'TiT-

than that entailed

by using Lemma 1.4,

and

14

and w i l l

be made

use

rariness

of

that

It is not

~o'

difficult

representation will

become

form

continuous

define

of r e p r e s e n t a t i o n

to see

that

even w i t h

already

is e s s e n t i a l l y from

the

derivative

fl(t)

at the

c(x)

satisfies

c(x)

E

but

[A,~)

end of

w

if a s l o w l y (i.ii).

is as follows.

, where

continuous

non-unique;

of the

is far

arbit-

f r o m unique. required,

in any

case

the

this

sequel.

discussed,

for c o n t i n u o u s

for continuous

on a c c o u n t

kind

(1.3)

t > y : log A

It is clear,

this

apparent

We have simple

of later.

that we

varying

A general

Take

f(t)

is the d o m a i n

of

can

simple

L

a with

construction

= log L(e t) L

get

function

for

, as b e f o r e ;

and

by t-n

(1.22)

fl(t)

= f(n)

+ 6(f(n+l)

f(n))

f

u(l-u)du

,

o for > ~

n ! t ! n+l .

(1.23)

for

, and

all

n ~ no

, where

is the s m a l l e s t

no

integer

Since f{(t)

n _< t _< n+l

f~(t)

:

6(f(n+l)

-

, it f o l l o w s

is c o n t i n u o u s ,

f(n))(t-n)(1-{t-n})

that

for

all

9 f~(n)

n _> n o

:

0

,

that

and

If~(t)l ~ ( 3 / 2 ) l f ( n + l )

f(n) l

for

n

< t

m

-

< n+l -

Also

]fl(t)-f(t)l for

n < t < n+l !

Now,

as

n

§

~

If(n)-f(t)I

, where

~ ~ ~(t,n)

, f(n+l)

~

n § ~ for

n

< t

< n+l

by

(1.24)

in

, f(n) the -

-

16(f(n+l)-f(n))(t-n)~(l-~)l is c o n t a i n e d

(3/2)If(n+1)

§ 0

; and

f(t)

§ 0

uniform

as

convergence

-

w.r.

Thus

f(t)

§

as

0

t

, fi(t)

+ |

§ 0

.

in

[0,t-n],

fCn) l

t + |

f(t)

Lemma 1 . 1 .

fl(t)

l +

f(n)

f(t+~)

established

f(t)

If(n )

+

, we have

to

v

in

that

[0,1]

of

15

t ~ no ,

Now, for

t = fl(no)

fl(t)

fl (u) du

+ f n

so t h a t i f

we p u t log

for

t

o

~ no

Ll(et ) = fl(t) LI(X)

= exp

, we g e t {fl(lOg

x)}

, x L exp n o

const 9 exp { flog n

x f ~ (u) du }

o

x f{Clog t) exp { f t K

const,

, = K say

dt }

Put

c(x)

=

so that that

C(x) C(x)

x > B

is

~ 1

say.

a measurable

as

x + |

function,

, from

defined

(1.24);

so

for

C(x)

is

x L K

and

bounded

such

for

Hence X

L(x) = exp {n(x) + f

e(t~

dt}

B where

n(x)

(r

~ 0

and by

(1.2s)

e(t)

(I.24), e(t)

are as

as

required

t § -),

= f~(log

t)

by t h e

Representation

Theorem

and ,

t i

B

say

This reasoning has a number of important consequences;

for first

it follows that numerous fl(t) can be constructed in similar manner, merely by replacing the integral X

6 f

u(1-u)du 0

in (i.22) by the indefinite integral of some other suitable probability density on [0,i], (suitable in that it will render f~(t) continuous). Secondly we have the following : Lemma 1.7. If L(t) ie a slowly varying function which is eventually non-decreasing (non-increasing), then the continuous E(t) in its representation for sufficiently large values may be taken as satisfying

(t) L 0 (~ 0)

16

Proof:

If

is

; and so

f(t)

whence,

L

from

fi(t)

will

at the Representation

step of Lemma

L

itself follows

directly

fact

this

boundedness

cal evolution of the theory, Representation Indeed,

0

intervals

,

on

finite

the

This last, property

if established

intervals

function

not realized 9

.

condltlons

a consequence

is

form that

in the general

for some time in the histori§

on

a substantial L(.)

obstacle,

to obtain the

prior to the above argument.

sufficiently

far) directly

from

1.2

(boundedness

(1.4) without

on

the

(which can then be deduced as a consequence@).

For

let S n = {~ > O; - ~U --< f(x+u)

From

Theorem

one can arrive quite easily at Lemma

agency of Lemma i.i >

far enough along.

and this presented

auxiliary

Theorem,

(i.e.,

the intermediate

from the uniform convergence

of a slowly varying

various

enables us to

section.

we have given, was apparently necessitating

9

that a slowly varying func-

states

or from the Representation

that

of the definition

invoking

then so

fi(t) L 0 (! 0),

c(t) ~ 0 (~ 0)

directly

1.2, which effectively

(as we have shown)

The

(decreasing), satisfy

Theorem using only Lemma I.i

without

Theorem)

in the manner of the present

a

satisfy

i8 bounded on any finite interval

of course,

finite

increasing

(1.23) will

it is clear that the above construction

Uniform Convergence

tion

given by

(1.25) , e(t)

Finally, arrive

is eventually weakly

(1.4),

there is an

it follows no

that

such that

U S = (0,~), n=l n Sn

f(x) _< a~ and since

has positive

Lebesgue

, V X --9 n} L

.

is measurable,

measure.

Now it

o is easy to check for any fixed ~i + u2 e S n .

Thus #

Sn

n

, that if

contains

Ul,U 2 g S n , then

a half-line,

(T(a),-).

Thus for

, which

is tantamount

O

all

~e(T(a),~),

f(no)

- a~ ~ f(no+~ ) ~ f(no)

+ a~

to the required. We have already mentioned tation

(1.3) where

of times.

e

Indeed more

in w

that we may obtain the represen-

is in fact differentiable is true

any specified number

in this vein as we now state

:

Such as continuity of L(.), which of course implies boundedness on closed intervals. See Bibliographic Notes and D i s c u s s i o n to this chapter. @ L4tac (1970a,b). # Steinhaus (1920) TheOr~me VII. * Adamovi6 (1966).

17

For a given slowly varying function

on

L(x)

there exists

[A,=)

another, infinitely differentiable, slowly varying function the following properties : 1~

L l(x) ~ L(x)

, as

2~

Ll(n)

, all integer

-- L(n)

with

L l(x)

x § | ; n

sufficiently

3~

If

L

is ultimately

monotone,

4~

If

L

is ultimately

convex

then so is

then so is

large

;

L1 ;

L1

II Propositions tion carried 1.6).

of this kind can be obtained

out in this section

The infinite

proposition

above

probability

density

(of which

differentiability

follow

readily

on

by the kind of construc-

a further

and parts

by choosing

1~

example and

in (1.22),

3~

is Exercise of the

in place of the

[0,i]

x f

u(1-u)du

,

o the density x Jr exp - {u(l-u)}-idu / o is here replaced

(Proposition

2~

1.5.

Prpperties

Further

These

most

and the discussion

Theorem

of useful

easily

Varying

of regularly

Convergence

imply a number

speaking,

exp-

{u(l-u)}-idu

o : L(e n) = Ll(en))

of Regularly

The basic properties in the Uniform

by

1

Functions.

varying

functions

are embodied

and the Representation

secondary

deduced with

properties,

Theorem.

which

are,

generally

the aid of the Representation

of some of these

is the purpose

Theorem,

of the present

section. Before already

(and hence tervals many

proceeding,

been deduced

a regularly

sufficiently

applications,

integrals

involving

we recall

in w varying

far along

what

regularly

important

defer

it to the next chapter.

and extensive

In the sequel

the symbols

important

that a slowly

function) the real

is of interest

ently

functions.

that one most

namely

varying to merit

is bounded line.

Also,

we mention

L, LI, L 2 , denote

separately,

in-

that in

behaviour

this topic

discussion

has

function

on all finite

is the asymptotic functions;

property

varying

of

is sufficiand we

slowly varying

18

1~ .

Vo~

y > 0

any

, xYL(x)

+ -

, x-YL(x) a8

Proof.

We give a p r o o f

the r e p r e s e n t a t i o n

of

for

L(x)

xYL(x)

X

§ 0

~

~

; the other

, we h a v e

that as

case

is similar.

Using

x §

X

XYL(x)

~ const

exp

{y log x ~ ~

E(t) it } " B X

const,

where

X

is c h o s e n

as is p o s s i b l e

exp

{y l o g

sufficiently ~(t) § 0

since

x (_~ y log x + fX

large as

(t) t

x + _~

at

}

t > X ,

so that for

t § |

I~(t) l

9 ~/z

Now

dt > y log x

x 1 f X [ dt

(y/2)

= (y/2) log x + const. -~

This

completes

the p r o o f

~

X

as

->

of the p r o p o s i t i o n .

~

R

II log

2~ 9 Proof.

L(x)/log

Using

the

x + 0

as

representation

log L(x)

= n(x)

x § | for

+ f

L(x)

x s (t)

B Now,

let

[r

~ > 0 < ~ .

be

Thus

an arbitrarily for

x

, for

dt

sufficiently

large

.

t small

number;

then

for

t !

X m X(~),

x 9 X

X

I~

~

dt I <_ ~ log

x

+

const.

X so

that

since

for

n(x)

x

> X ~ X(6)

is b o u n d e d

the p r o p o s i t i o n

30.

L~(x)

for large

follows,

,

for

any

x .

Since

6

is a r b i t r a r i l y

small,

g

a

satisfying

-~

<

a

<

~

,

LI(XJL2(x)

,

19

LI(X) + L2(x) are 8lowly varying. is slowly varying. Proof.

There

about

the

positivity

are

sum

and

and

only the

two

L2(x)

If

non-trivial

composition

measurability

propositions

of two

hold

slowly

x

as

+ ~

§ |

LI(L2(x))

to be p r o v e d

varying

here,

functions:

the

trivially.

LI(XX ) + L2(xX ) L 1 (X) + L2 (x) Ll(/X) t

Ll(X)

= 7

= (1+~1 (x, X)) where

for

t

L2(~x) f

E l ( x ) +L 2 (x]

fixed

X > 0

= 1 +

I

LI(X) L1 (x)+L2(x)

, r

§ 0

z

Lz(x)

+~

t

L1 ( x ] +L 2 (x)

1 as

I

(1+r

+

L1 (x]+L2(x)

x § |

, i = 1,2

}

,

] Li(x) }

i=l r

~Li(xl+L2(x)

Z

§ 1

as

x § |

, for

each

~ > 0

since

0 < For

fixed

Li(x) < 1 L1 (x) +L2 (x) --

~ > 0

Lz(x~) ] LI(Lz(X~)) = L 1 [L2(x)L2-L-j-~j / LI(L2(x))

Ll(L2(x))

= Ll(L2(x)(1 + r and

since

Theorem

L2(x)

applied

§ = to

as L1

x + = (since

, it

follows

1 + r

from §

i)

the Uniform that

the

Convergence

above

is

L1 (L2 (x))/L1 (L2(x)) -- 1 for each fixed

~ > 0 .

H 4~ 9

U(X) ~ L(x) , ~(x) ~ L(X)

as

x

§

|

where

for

any

fixed

20

y

9

, L

0

and

are specified for

xY~(x) =

B

x ~

by

{tYL(t)}

sup b < t < x - -

xYL ( x )

:

u

inf {tYL(t) } X < t < ~

(where B is taken sufficiently representation (1.3).) Thus,

as a c o n s e q u e n c e ,

equal

to a non-decreasing

ally

index,

for w h i c h

xY~(x)

, being

~(x)

and

Proof.

~(x) For

the

above

monotone

are

xYL(x)

, with

regularly

formulae

and

L(x)

large e.g. for

finite

are

fixed

to be given by

y 9 0

varying

constructions.

valued,

are

, is a s y m p t o t i c -

function

with

the same

(xY~(x)

clearly

and

measurable,

whence

also.)

x > B

1 < L(x)/L(x)

Sup

:

{tYL(t)}/xYL(x)

.

B < t < x Suppose

there

exists

a sequence

of p o s i t i v e

numbers

{x r}

, xr

that

such

(1.27)

1 < lim

E(Xr)/L(Xr)

.

r+~

Then

there

exists

(1.28) for the

a positive

~ > 0

such

that

1 + 2a < r ( X r ) / L ( X r ) r ! r o ~ r o ( a ).

interval

(1.29)

Now

for each

B ! Yr i Xr

such

such

r

, we

can

find

a number

Yr

in

that

sup

{ Y r Y L ( Y r ) / X r Y L ( X r )} + ~ 9

{tYL(t)}/xrYL(x r)

B ~ t ~ xr Clearly (1.28)

the and

Yr

(1.30) Since fact as

m a y be

(1.29)

chosen

it f o l l o w s

monotone

1 + ~ < YrYL(Yr)/xrYL(xr tYL(t)

+ ~

as

the m o n o t o n e r § -

, and

Representation

t § |

, by

sequence L

{yr } is b o u n d e d on

Theorem

non-decreasing

with

r

.

From

that

in

that

for the

I~

) , r _9 r o it f o l l o w s

satisfies finite

right-hand

Yr

from

§ "

intervals. side

of

(I.30)

, since

xr

Invoking

the

(1.30),

§

we m a y w r i t e

21 X

1 + ~ < exp {n(Yr)

n(Xr)

< exp { n ( y r ) - n ( X r ) - y for

r ! r I ; r I , A r e , is such

Thus

for

log

that

(Xr/Yr)

(Xr/Yr)

f r E(t)t d t } Yr

+ (y/2)log

I~(t) I < y/2

for

(Xr/Yr)} t ! Yr 1

r L rI

1 + 6 < exp { n ( y r ) as

log

y

r + |

; which

Hence such that

there

is,

no

is

(1.22)

finally,

n(Xr)}

+ 1

a contradiction.

sequence

holds.

,

of p o s i t i v e

numbers

{x r}

, xr §

as

x + ~

Thus

1 = lim L(x)/L(x) as r e q u i r e d . We leave

the p r o o f

of the p r o p o s i t i o n

an e x e r c i s e +', and b r i n g position

analogous

to

to the r e a d e r ' s 4~

holds

constructions

for a m o n o t o n e

with

index which

the same

S~ .

Corresponding

another regularly

to

~ x

(l.31b)

R2(RI(X))

~ x

x

§

that if place of

Moreover

~

R3(x )

non-increasing

R2(x )

satisfies

R2(x ) , and

For

x i C1

See E x e r c i s e See E x e r c i s e

1.7. 1.8.

varying

equal

x-YL(x).*

to

R2(X ) = x l / ~ L 2 (x)

of

function

such that

uniquely

either of the above asymptotic

R3(x ) § ~

as

x

+

|

as

x

+

|

determined relations

, then

we h a v e t h e r e p r e s e n t a t i o n

Rl(X ) = exp { n l ( X ) + *

that a pro-

, ~ > 0 , in terms

regularly

is asymptotically

R3(x ) ~ x l / Y L 2 ( x ) Proof.

the fact

as

R l ( X ) = x Y L I ( X ) , ~ > 0 , there exists

varying function

RI(R2(x))

~ ~(x)

attention

x-YL(x)

is a s y m p t o t i c a l l y

(l.31a)

as

for

L(x)

+ const.

x y+r + ~1 t

dt}

in in

22

where

is chosen

C1

sufficiently

large so that

y + r

) > 0

for

Thus

t i C1

R l(x)

fx

( exp i

= K l(x)

Y+r ----[----

dt

,

x ! C1 ,

C1 where

Kl(X)

§ KI > 0

as

r l(x)

= exp

(1.32)

which

for

infinity;

x ! C1 on

of

r2(x)

rl(x),

~f

Consider

x

y+e 1 ( t )

C1

dt

and strictly

has an inverse

x i rl(a)

now the function

t

is continuous

and therefore

properties

x § |

Thus

monotone

function

increasing

r2(x )

with

to

the same

rl(r2(x))

: r2(rl(x))

= x , and each

positive

derivative,

these being

has a continuous

related by (1.33) where

r{(rzCx))r 89 from

) :

1 = r~(rl(x))r~Cx)

(1.32)

(1.34)

r~(x)

= r l(x)

(V+r (x))

r89 r2(rl(x) (1.33)

from

rx(x) )

=

; =

from

(1.34);

so substituting tr 89

where x i

ez(t )

(y

§

x = r2(t) ) = y

is continuous

-1

-1

el(X))

we o b t a i n

+ r

and approaches

zero as

t § |

Hence

for

C2 = r l ( a ) r2(x ) = exp

which R2(.)

is thus also regularly

SC 2

t

varying with

dt

index

= xl/~exp

---i--- dt

i/~

If we now define

by R2(x ) = K2r2(x )

,

where

K2 = K1 - 1 / v

,

23

the asymptotic r2(rl(x))

relations

(l.31a)

= x , invoking

and

(l.31b)

follow from

in the second instance

rl(r2(x))

=

the Uniform Convergence

Theorem. It remains

to deduce RI(R3(x))

where

c{x)

§ 0

as

asymptotic = x(l

x § |

Convergence

(l.31b),

R3(x)

The procedure

R(x)

4~

and 5~

shows

= xYL(x)

function

weakly monotone, that

that the property

of the same index such as

and not necessarily

and

existence

x-7~(x)

What,

function

to prove? from

then,

generally

continuous

= xTL(x)

speaking, function

a

xY[(x)

The essence

R(x)

to which we are led to in relation

R2(x ) , from

Rl(X)

theoretical

it is in certain situations

able to give an explicit existence

inverse

, or is

in a

attest r(x)

question,

convenient

form for the

to

for

= XYLl(X ) , in like manner?

a rather

R2(x ) , in terms of terious

x ~ ~ ,

regularly

which merely produces

is: can we construct an asymptotic

may first appear, theory,

as

varying

to the

, as in

5~

The question result,

are obtained

increasing

equal,

7, is trivial.

4~

continuous,

whereas we may,

of a strictly

the proof of

the relation-

that a regularly

and indeed continuous,

of the sort; and is not as simple xY[(x)

to discuss

, 7 > 0 , is asymptotically increasing,

r(x)

constructive manner;

type

it is necessary

5~ .

is the point of a proposition xY~(x)

R3(x ) § ~ , and the Uniform

~ R2(x)

to a strictly monotone varying

+ E(x)))

x §

this section,

The proof of function

= R2(x(l

for the other case is similar.

To conclude ship between

+ r

the fact that as

Theorem,

If

, then

R2(RI(R3(x)))

and invoking

uniqueness.

5 ~ , as a Rl(X) This

of the

is not,

and interesting

(asymptotically

unique)

to be function

Rl(X ) , rather than to refer to a somewhat mys-

result.

as

for in probability

24

Such by u s i n g Lemma

a construction

is

4~

.

We n e e d

L

is a slowly varying f u n c t i o n defined and posi-

and

Suppose

1.8.

tive on

5~

Put for

[A,~).

R*(x)

Then

R*(x)

sense of Proof.

{y,y

y > 0 , R(x)

~ [ A , - ) I R ( y ) ! x}

, where

first

note

that,

R(R*(x) the

common

to o b t a i n

is non-dec-

= x~L(x)

.

is slowly varying and

L*

inverse f u n c t i o n of

can be

R*

R

in the

for a f i x e d

by

the m o n o t o n i c i t y

for

right

, from

> R(x-E) -- ~

Convergence

its

R

, for

x > R(A)

and

left

hand

limits.

E > 0

=

{ x-r 1Y L ( x ( I - ( ~ / x ) ) ) ~

-~ 1 by the U n i f o r m

of

- 0) < x < R(R*(x) + 0)

convention

1 _> ~R(x-0)

R*(x)

difficult

5~ We

Now

and not

the p r e l i m i n a r y

x > R(A)

(asymptotically unique)

(1.35) using

= inf

= xl/YL*(x)

taken as the

possible,

first

and for some fixed

[A,~),

reasing on

indeed

as

Theorem.

definition.

Thus

L (x(l+ (E/x))) x + ~

Since

R(x)

letting

R(R*(x)

- 0) ~ x ~ R(R*(x)

R(R ~(x)

~ x

+ ~

x + -

with

in

x

, so does

(1.35)

+ 0)

i.e. (1.36) The

remainder

follows

as

x +

f r o m the u n i q u e n e s s

part

of p r o p e r t y

5~

N Suppose y > 0 Then

and for

now L(x)

x > B

, so

R(x)

= xYL(x)

is s l o w l y , let

in

4~

Now

consider

R(x)

R~

is c o n s t r u c t e d from

R*(x)

R(x)

, defined

varying)

is not

= xYL(x)

, where

is now n o n - d e c r e a s i n g , = xl/YL*(x)

R

and p o s i t i v e

related

necessarily L(x)

and to

in a reasonably

R(x) R

on

[A,~)

(where

non-decreasing.

= L(x) ~ R(x)

as in L e m m a

or

~(x)

as 1.8.

as

x §

Thus

s t r a i g h t f o r w a r d manner.

25

Consider now R(R.*(X))

as

x + |

,

since

, '~ ~ . ( f i ' ~ ( x ) )

R*(x)

+-

and

R ~ R ;

and

R*

x from the relation between 5 ~ we may

of

sense of

1.6.

take

R

Hence using the uniqueness

as the asymptotic inverse of

R*Cx)

5~

Conjugate

and Complementary

The whole of topic

Regularly Varying

1.5.5 ~ leads naturally

pairs of conjugate slowly varying functions; varying functions.

Functions.

also into the topics of

and complementary regularly

We treat them in this order.

Let

Theorem 1.5. L*

part

in the

R(X)

L

be slowly varying.

Then there exists a function

such that (i)

L*

(ii)

i8 slowly varying ;

L(x)L*(xL(x))

(iii)

+ 1

L*(x)L(xL*(x))

as

+ 1

x § ~ ;

as

x + - ;

(iv)

L*(x)

is asymptotically unique ;

(v)

L**(X)

~

Proof. y = 1

(i)-(iv) and

L(x)

as

§

from property

L2(x ) = L*(x)

5~

of w

Property

by taking

(v) follows

from

(ii),

L*(x)L**(xL*(x)) comparing which with

§ 1 ,

(iii), and applying

L** ~ L, which The totality

-

follow directly

Rl(X ) = xL(x),

the fact that, by

x

is

of these properties

(iv)

(v) makes

The slowly varying

sensible

functions

the following

Definition

1.3.

in Theorem

1.5 are said to be a pair of conjugate

L

and

L*

definition.

referred

slowly varying

tions. It may be useful, obtained properties

before proceeding

of a conjugate

pair

to note a few of the easily :

to

func-

26

Corollary.

L(x)

If

functions,

(i)

L(ax),

L*(bx)

aL(x),

a-iL*(x)

L~(xl/~),

are a conjugate pair of slowly varying

:

(ii) (iii)

L*(x)

and

then so are

for each

a,b 9 0 ;

for each

(L*(xl/a)) ~

a 9 0 ;

for each

a 9 0

II The remarks

in w

~ , in the proof of the proposition

of Lemma 1.8, provide L(x).

two constructive

The former method

be seen from

(1.32)

"normalised"

regularly varying

component

L*

to obtain an

usefulness.

and also

L*(x)

In fact,

from it may

that it is usable basically when dealing with functions,

of form specified by

However, which

methods

is of less general

itself,

it is clearly desirable

may be expressed

that

(1.37)

1

~

to have some criteria

(asymptotically)

One such is the requirement

L(xL-I(x))

Which have their slowly varying

(1.9)-(1.10).

as

x §

in terms of

according

L

to

itself.

=

L(x) for then L(xL=I(x))L-I(x)

§ 1 L

and comparing with Theorem 1.5 with (1.38)

L*(x)

~ L "l(x)

as

replaced

by

La

, we find that

x § =

A more detailed result on these lines + is the following. Lemma 1.9.

If

lim L(xLa(x))/L(x)

= x(a)

exists as a finite p o s i t i v e

X+~

limit for all real over

~, then

(1.38) holds if and only if

Proof.

Taking

for some finite

logarithms

Cauchy's

See also

follows

equation

x

1.9.

~,8 9 0

as a measurable

(I.15),

from the content

Exercise

More-

T(~+B)

we obtain log

functional

assertion

:

y.

~ = 0

It is easily deduced that for each T(~)T(B)

+

x(a) = exp ~

finite

solution

so that the first part of the

of Theorem

1.4.

to

31

Now

let

v ~ 2

be a p o s i t i v e ]h(x)

for a r b i t r a r y

h(x-1)

fixed

h(x)

integer

T h e n for

x

n(x) Z {h(r+8(x))

=

that

for

such

that

x L v ! v(r

< c

- c]

r > 0

cn(x)

such

n(x)

> v ',

h(r-l+8(x))

- c}

h (r-l+6 (x))

c}

r--x) ~-1

Thus

using

the t r i a n g l e

+

Z {h(r+~ (x)) r=l

+

h(6(x))

.

inequality

i h x, n (_x )cncx, i

n(x)

!

+ (1/n(x))

r

r. ] h ( r + ~ ( x ) ) r=l

- h(r-l+8(x))

- c I

+ lh(~ ( x ) ) J/nCx) Thus

letting

boundedness

x § | of

, keeping

h(x)

on h(x)

(1.43)

in m i n d

[r,r%l]

the a r b i t r a r i n e s s

for e a c h

- cn(x) n(x)

0

of

s , and

r = 0,1,2,...

we

the

obtain

.

Now h(x)

h (•

c

x + as

x § |

Theorem

, on a c c o u n t

1.7.

of

c-~ (•

cnLS) nCx]

+ ~(x)

0

(1.43),

which

completes

(Weak Characterization Theorem).

finite, defined, and positive for

X > 0

the proof.

The function

<

p

<

,

lp

for some

p ,

|

Proof.

Let

R

be

log R(e x)

Then

follows

as

(1.44)

r

occurring in the definition

of a weakly regularly varying function has the form -|

9

that

a weakly

since

regularly

for e a c h

varying

x > 0

x § | p(x+~)

~ pCx)

§ log

@Ce~}

function.

, R(Xx)/R(x)

+ r

Put

p(x)

> 0 , it

=

32

for

any

fixed

intervals

~

in

beyond

-| 0

; then

putting

p(u(t+l)) as

:

t + ~

= 1

- p(ut)

, so a p p e a l i n g

to

§ log

pCx) x

§

t = x/p

+ log

r

1.12

,

Lemma

p(~t) t

, clearly

r

. , [1.44)

yields

p)

~)

i.e,

as

x § -

Hence

for

-1

log

~

-i

log

r (e ~)

v > 0

r

~)

~ const

= p

say

i,e~ r

Case

= ~P

3

< 0 . p(y)

as

y § ~

for Put

y

that

.

; thus

§ log

r

~)

i.e.

From

§

the

y = x+~

- p(y-~)

P(y+]~]) as

~ > i

limit

Case

p(Y)

+ - log

2 applied

to

r

the

~) left

hand

Plvl

; = -P~ =

-log

r

v)

i.e.

r

for

Definition to b e

1.5.

A weakly

characterized

slowly

varying

The reader elementary

than

side

of

this

we

have

is

when

by

value

of

varying the

0 < ~ < 1

function

index

(which

p) w i l l

be

is n o w

called

known

weakly

p = 0

may note the

regularly

the

= ~o

that

functional

the

method

equation

of

Theorem

methods

1.7

(such

as

is

rather

Theorem

more 1.4)

33

of g i v i n g bility that

the

h(x)

a complete

assumption

p r o o f of the e q u i v a l e n t

on

R

implication

(1.42)

only m e a s u r a b l e ,

intervals,

(Theorem

1.3).

theorem under

However

is no l o n g e r n e c e s s a r i l y

rather

as the f o l l o w i n g

then assuming

simple

example

the m e a s u r a -

he s h o u l d true

also note if one a s s u m e s

its b o u n d e d n e s s shows

on finite

:

With

h(x)

= ]cosec

~x[

, x + m

= I h(x+l) but

if we take

(m

an

otherwise

- h(x)

the s e q u e n c e

= 0

for all , where

{Xn}

integer)

, x

; -i

x n = n+n

, then

h ( X n ) / X n = ]cosec ~ n - l l / ( n + n -I) = {)sin ~n-ll (n+n -I)}-I -I

as

n§

|

Hence of

this s i m p l e p r o o f

(ordinary)

regularly

of the c h a r a c t e r i z a t i o n

varying

R

away from b o t h

zero and i n f i n i t y

has b e e n first

established.

As r e g a r d s slowly varying most

of

3~

previously

of

some of the m o r e functions, w

the U n i f o r m

on f i n i t e

intervals

obviously

apparent

we n o t e

persist,

but

theorem

c a n n o t be u s e d u n l e s s

first

Convergence

sufficiently

properties

that p r o p e r t i e s

require

in the case

its b o u n d e d n e s s

different

1~

far

of w e a k l y

2~

and

proofs wherever

or the R e p r e s e n t a t i o n

T h e o r e m was

invoked. We r e s t a t e L2

the p r o p e r t i e s

are now w e a k l y 1~

For any

2~ .

log L ( x ) / l o g

P r o o f of 1 ~ and 2 ~ , L(Xx)/L(x)

+ 1

here

slowly varying

as

p(x+~)

y > 0 , x~L(x)

x § 0

Putting x § -

-

p(x)

for c o n v e n i e n c e ,

§ |

, x-YL(x)

as

p(x)

for e a c h

~- 0

where

L, L 1

:

= log L(e x)

§ 0

x § , we have

~ > 0 , that

since

and

34

as

x + -

vals,

we

1.7}

that

, -| < u < ~ obtain

From

from Lemma

p(x)

0

§

is

tantamount

Consider

now

log

to

log

for

{proceeding

as

X

which

the b o u n d e d n e s s

1.12

§

p

on f i n i t e

in the p r o o f

inter-

of T h e o r e m

~

L(x)/log

7 > 0

{x•

X

of

as

x §

0

as

x § |

, which

proves

2~

,

= + u log x + log L(x)

=

:._, ,o,, [ , . _:v + y log x on a c c o u n t 3~ . are w e a k l y

of the

first

part

La(x)

for any

8lowly

varying.

o f the p r o o f .

a

i. The

9

-~ < ~ < ~ proofs

of

, Ll(X)L2(x),

w

apply

Ll(X)

+ L2(x)

here.

II ii

To c o m p l e t e

the d i s c u s s i o n

variation,

we

functions

R(x)

shall,

satisfying

for e a c h

where

0 < r

finite

p

bounded we

for all

.

< Thus

on all

[A,|

with

some

finite

, as

R

show

[A,~)

that

for

there

some

k > 0

.

~

will

, but

r

to c o n s t r u c t for

L(xk)/L(x)

§ 1

uniform

# k0

, ~ > 0

, for

be not m e a s u r a b l e

on

sufficiently

although

the c o n v e r g e n c e

and satisfying

functions

each

for

~ > 0

X

in

far,

in

(1.45)

L(x) , as

no

[A,-)

+

; but

, and

will

, defined

such construction

is m o r e

for the

and positive

[a,b]

difficult.

T h e i n t e r e s t e d r e a d e r s h o u l d c o n s u l t the n o t e vgn Aardenne-Ehrenfest, and de B r u i j n (1949).

R

be u n i f o r m

x + -

interval

some

, a n d not

+ 0 < a 0

x § |

intervals

construct,

being

on

of regular

§ 0(:),)

shall

convergence

the d e f i n i t i o n

preliminaries)

and positive

for e a c h

such

It is p o s s i b l e on

~ > 0

R(x~,)/R(x)

(1.4S)

which

after

, defined

as r e g a r d s

of K o r e v a a r ,

3S

We b a s e tional

our d i s c u s s i o n

~,(x)

all

real

x,y

Theorem

1.4,

where

conditions

on

+ 'I'(y)

.

~

(1.47)

solutions

of the

Cauchy

in

proof

func-

= r

We e n c o u n t e r e d we i n i t i a l l y

at

a11,

this

that

if

that,

Xl,X2,...,x

n

r@(x)

that

~(x)

non-positive

= -~(-x) r

Suppose (a,b)

is

then

hence

real

+...+

r

number

immediate for

a solution

r

numbers,

then

n) , and any

real

x

all

that

(1.46)j

of

(1.48)

rational

r and

holds

for

.

is

bounded

o n an

= x~(1)

is continuous,

~(x)

any

of

regularity

Then

.

~(x)

(8o that

is

, and

~

any

,

It

rational

Lemma 1 . 1 3 .

interval

= @(rx)

are

rational

the

without

+ @(x 2)

for any n o n - n e g a t i v e

(1.48)

equation

deduced

~(Xl+X2+...+Xn ) = r

a n d also

and

(finite)

equation

(1.46) for

on

let alone measurable and bounded on every

finite interval). Proof. also

The

bounded

r~(1)

function on

= ~(r)

g(x)

(a,b).

for

rational g(r)

for ~ a t i o n a l now

any

x r (a,b)

real

r

; and

number

such

that

that

g(y)

Finally, Then by

the

is

x+r

that

satisfies

in

(1.48),

(1.46),

we

see

and

is

that

, so that

g(x+r)

; then = y

= g(x)

there

there g

g(nXo)

+ g(r)

exists

= g(x).

a rational

Let r

us c o n s i d e r

and

an

, so that

= g(x+r)

bounded

suppose fact

r

x~(1) x = i

= 0

so y

g(y) so

= ~(x)

Putting

over

the

= g(x) entire

exists

a point

satisfies

(1.46)

= ng(Xo)

real xo

line. such

we h a v e

that

g(Xo)

+ 0

36

which lies outside

any fixed bounds g(x)

kt reals any

the

next

(there real

stage

exists

number

= 0

for

all

we n e e d

to

can

be

sufficiently

real

x

make use

a non-denumerable

x

n

set

represented

.

large;

hence

1

of

a Hamel basis

B

of

real

uniquely

as

B

numbers

a finite

for

the

by which linear

combination

(1.49)

x = rlb I + r2b 2 + ... + rnb n

with rational of basis on

rl,r2,...,rn

; the coefficients,

and the number of terms

the choice

in the r e p r e s e n t a t i o n

depends

X .)

Suppose a function

Lemma 1.14.

by

coefficients

elements,

(b i E B)

: 1)

giving it an arbitrary

subject only to the constraint and 2) defining

measurable, Proof.

-- r l ~ ( b l )

Then

(1.49).

finite value

that

+ r 2 ~ / ( b 2)

for any

~(b)

+ const,

...+

is a 8olution

~(x)

It is readily verified,

b c

fo~ each

b

b ~ B;

rnr

n)

,

of (1.46), but is neither

is assumed measurable Thus

; which

using

(1.46),

(1.47)

and

(1.48) that

We saw in Theorem 1.3 that if a solution of (1.46).

(1.46).

true, by Lemma 1.13, interval.

~(b)

nor bounded on any finite interval.

satisfies

b c B

is defined for all real numbers

in general by

~(x)

@(x)

using

r

if

then ~

@(x)

= const,

x , and that this was also

was merely assumed bounded on some finite

in either case,

in particular,

does not hold for our solution.

r

= const,

b

for all

1

II Consider

a

~

of the sort m e n t i o n e d

in Lemma 1.14.

Then,

from

(1.46), we have

(x+.) for all real we see that,

u,x

.

for each

)(x) Defining

= ~{~) R(x)

~ > 0

R(Xx)/R(x)

= R(X)

, for

x >__ A > 0

by

exp

){log

x)

,

3?

where Xp

0 < R(X)

< ~

for some f i x e d

1.8.

Monotone

for all finite

R(X)

is not of the form

p

Regular Variation.

The early developments often pertained a prior

X > 0 , but

in the theory of regularly varying

to a situation

assumption

enables many of the aspects

larly characterizations, simpler manner;

in which m o n o t o n i c i t y

to be developed

we give a few examples

functions

was assumed + .

of the theory,

Such

particu-

either more fully or in a much of this fact in the present

++

section Lemma 1.15.

L(x)

Let

be defined,

positive

and m o n o t o n e

[A,=)

on

If L(XoX)/L(x)

Xo' XO > 0 , X O + 1 , then

for some f i x e d

Proof.

Since monotonicity

the method

X e [a,b]

Suppose

if

X

Xo < 1

it suffices,

L

is increasing, 9 if

in

[I,~o]

Xo

>

i

<_ L(XoX)/L(x)

, providing

x

is sufficiently

large; whereas

similarly

~

L

Theorem 1 . 8 .

in

[~o,I]

is decreasing, Let

R(x)

~ L(XoX)/L(x)

Letting

x + |

side converges the procedure be defined,

gives

the result,

to unity,

is analogous,

positive

R(Xx)/R(x)

since in

by assumption. m

and m o n o t o n e

zf (1.50)

by

, 0 < a < b <

either case the right-hand If

varying.

+ 1

1 ~ L(Xx)/L(x) for each

is 8lowly

implies measurability,

1 <_ L(Xx)/L(x) for each

L

of Lemma 1.5 to show that L(Xx)/f.(x)

for all

§ 1

+ r

+ See Bibliogrsphic Notes and Discussion to this chapter. ++ The reader should also consult Exercise i.i0.

on

[A,~)

38

for values

kl,

k2

of

k , such that

0 < @(ki) < | ; and such that is regularly varying. Proof.

Put as usual

p(x)

= log R(e x)

p(t+u i) - p(t) where

Pi = log k i , and

0 < k i ~ 1 , i = 1,2

log kl/IOg

§ log r

Ul/~2

lira p(t+~)

k2

and

is irrational,

; we have

then that

then

for

i=1,2,

ui)

is i r r a t i o n a l ,

hence

clearly

p(t)

t~

exists

for

u r S , where

let us w r i t e

for

S = {u

; v = qul + rv2

' q,r

integers}

;

u c S

lim p (t+~)

p(t)

log r

--

~)

t-~

where

0 < r

(-=,|

, we now have

of

u) < |

Since (1.50)

by K r o n e c k e r ' s

holding

theorem + S

for each

k

is dense

in a dense

S*

(0,-) Now take

sufficiently proof

any

u e S ; since

far,

of T h e o r e m

being monotone, 1.7 to deduce

~(k) Assume Now,

= Xp

for the m o m e n t

let

k

be any and

R

{si(1)}

Thus by m o n o t o n i c i t y

letting

finite

p ,

in n o n - d e c r e a s i n g ;

i R(kx)/R(x)

x § ~ , for a r b i t r a r y

(k_ci(1)) 0 <_ lim inf ~

1.12

intervals as in the

k r S* this

implies

i ; letting lim R(kx)/R(x)

i , <_ (k+e i (2))0

X-~

i + |

gives

= XP

X~

e.g.

P ! 0 .

s R((X+si(2))x)/R(x))

<__ lim sup ~

X-~

for a r b i t r a r y

on finite Lemma

that

R((X-ei(1))x)/R(x) so that,

is b o u n d e d

I r (0,| ; then we can find p o s i t i v e null (2)} , such that k - s.i (I) ' k + E.1 (2) r S* ' {E i

each

i

R

we may apply

for some

sequences

+

in

subset

Hardy and Wright (1954) C h a p t e r X X I I I , Theorem 438.

39

for each

x e (0,-)

If

R

is non-increasing,

ments in the

inequalities

Finally, varying. /

since

o <_ 0 , a n d o n l y t h e a p p r o p r i a t e

adjust-

n e e d t o b e made.

R , being monotone, is measurable,

R

is

regularly

II Theorem 1.9. 8

n

+ |

[A, ~)

and for

some

R

A 9 0 , and such

X1, X 2

for v a l u e s

0 < r

< |

is r e g u l a r l y

Proof. {e n}

function,

R(en+l)/R(en)

last

and

be a s e q u e n c e

a monotone

(1.51) the

{8 n}

Let R

Let ; then

; and

of positive

defined,

that

as

numbers

finite

such

that

and positive

on

n ~ |

+ 1 , R(enX)/R(en)

+ r

of

0 < Xj ~ I , j = 1,2

such

~ ,'such

that

log Xl/lOg X 2

that

is i r r a t i o n a l .

Then

varying.

{ 8 ( i )}

be t h e

e(i ) + |

as

subsequence

i + |

For

of the

successive

x > 0 , select

maxima o f r

~ r(x)

such

that @(r ) ~ x ! e ( r + l )

Then, if

]~

is monotone decreasing, R(r(x).X) R(r(x)+l)

and if

R

real

[~((r(x)+l)~) Et(r(•

is increasing the inequalities

(1.51), for each all

R(xX) ! R - R - ~ 7- L

fc=" positive

' Making use of

are reversed.

Xj , j = 1,2, in turn,we see that as

x

§

~

through

values,

R(xXj)/R(x)

§ r

,

j = 1,2

The result now follows from an application of the preceding theorem (Theorem 1.8).

B

It is clear from the proofs of Theorem I~8 and 1.9 that we only need require generally) log X

(1.50) and the right-hand side of (1.51) to obtain

for a set of

is dense in

(-|

X > 0

such that the corresponding

{more

set of

40

Let

Theorem

I.I0.

lim sup

en = |

tion,

defined,

every

X > 0

, en+i/e n < K (i < K < |

finite and p o s i t i v e on

(1.52)

lim R ( 0 n X ) / R ( 0 n )

for some finite Proof. Then,

be a sequence of p o s i t i v e numbers such that

{O n }

Let from

0

be the

assumptions

O(i ) § | Now,

let

such

that

R

Then

{0(i )}

the

t > 0

be

=

sequence

number,

O(r ) < x 2 O(r+l) the

extreme

on

{@(i)}

right .

Now,

hand

x

so that

I i

a(x)

varying

of s u c c e s s i v e

maxima

and

for

x > 0

following

~ ~ 8(x)

from

the

[a,b]

, where

= ecr)8

! K , and consequently converging

for a r b i t r a r y x § ~

pointwise

is u n i f o r m w i t h

0 < a 0

on a c c o u n t

x !

8(x)X !

by

a sequence

respect

to

a well

known

~

KX .

Now t h e

of m o n o t o n e function

on any c l o s e d result

from

func(X p)

interval

elementary

Hence

~

-

~ < --

RCX6Cx)8(r(x))) RCScr(x)))

providing

x

of the b o u n d s

on

R[x~ (• (r (.x),)) R(e (rfx])) +

condition

by

to a c o n t i n u o u s

real-variable

{o n } .

r ~ r(x)

density

X > 0

for

of

select

are

convergence

p ).

) < K .

R ( e n X ) / R ( g n ) , n = 1,2, ....

tions

and for

(with index

functions

hence

A > 0

~-- K e ( r )

inequality

define

be a monotone func-

{O n } ,

O(i+l)/O(i

a fixed

~

for some

X~

i8 regularly

on

and

, and [A,~)

'%"

< --

(X~(x)

is s u f f i c i e n t l y

)p +

large.

c

Thus

X6(x)

(x6(x))p

and it is e s s e n t i a l l y this p r o p e r t y and m e a s u r a b i l i t y we r e q u i r e in the p r o o f , not the m o n o t o n i c i t y itself.

which

as

:

41

and p u t t i n g

~ = I

R(~" (x)~ Cr Cx)])

so for a r b i t r a r y

fixed

which

completes

k > 0 , by d i v i s i o n ,

~

lim

= kP

the proof,

If on the right h a n d IP

~ (8 (x)) ~

place

of

r

to be p o s i t i v e ,

since

R

side of

being monotone,

(i.52) we put m e r e l y

, even t h o u g h we s p e c i f y continuous

is m e a s u r a b l e ,

0 < r

a

r

mm

, in

< ~ , and even take

and s t r i c t l y

monotone,

the p r o p o s i t i o n

+

is false.

We close

the s e c t i o n

sion of the c o n s e q u e n c e s to

w

Theorem

on m o n o t o n e

regular variation

of the f o l l o w i n g

result,

with

a discus =

and its r e l a t i o n

~.

Suppose

1.9.

L

is a positive function defined on

[A,~) ,

non-decreasing to infinity, and [.(x) = inf {y, y r [ A , |

for ++ x > L(A).

(z.53)

lim x§

Conversely if L

Then, if

=

(1.53)

i8 slowly varying

[ i

0 I ~

for for for

obtains for all

Suppose

L

1.8, by the U n i f o r m

U

(1.54)

I,(L(x)

is s l o w l y v a r y i n g . Convergence

"~ x

as

E x e r c i s e i.Ii. if L is s t r i c t l y i n c r e a s i n g function.

(at infinity),

0 _ x}

(i,|

then

,

L

or in

(0,I)

,

and

is slowly varying.

T h e n as in the p r o o f of L e m m a

Theorem,

x

=~ |

See

and c o n t i n u o u s ,

then ~ is its i n v e r s e

42

(and, o b v i o u s l y , sider

L

is n o n - d e c r e a s i n g ,

1 < ~ < |

-

Suppose

fi

~ , there

lim L ( X n V )

xn § |

Now,

let us con-

(x)

for any such fixed

n§ and

to infinity).

Then

L

with

n

=

g

is a s e q u e n c e

,

{x n}

such t h a t

(1 i g < ")

(Xn) .

Then

L

by the U n i f o r m

.

[.( x n )

1-(xn)

Convergence

/L (f ( X n )

Theorem;

§

1

i.e.

L (s (~x n) ) L (s (Xn))

But,

by

(1.54),

L(L(vXn))/L(L(Xn)

contradiction.

proof

Hence

) §

for

v

, and

1 < v < |

1 < , ; and

< |

for

, which 0 < ~ < 1

is

a the

is similar. Suppose now

Supposing

L

(I.55) Take

(1.53)

41.531

is s t r i c t l y

/(L(x))

a fixed

obtains

= x

increasing

,

= v ,

has

L(s

X > 1 , and d e f i n e

L(u)

for a f i x e d

L(Xu)

= x

v

and

= Uu v

~

satisfying

the c o n s e q u e n c e

0 < ~ < 1

that

.

~u

from

u

by

.

Then

[~(~u v]

_ [(l.(~,u)]

f~(v) by

(1.55),

so that

u ~u > 1

Now, a contradiction

to

41.53)

results,

43

unless

~u § 1

as

~u

as

=

v § ~

Thus

L(XU) ~

L(Xu) = ~

u § ~ , for arbitrary fixed

1

~ 9 1

The remainder of the proof is left to the reader. First of all, which

I

in the special case of a slowly varying function

is n o n - d e c r e a s i n g

to infinity, we have available a function

such that L(L(x))~ x

which corresponds when

y = 0.

indeed, any

to equation

Formally,

of the function

as

L(x)

then,

as "being r e g u l a r l y varying,

lim

L(~x)

x§

s (x)

D e f i n i t i o n 1.6. for some

of index =", and,

for in the obvious sense,

for

_

to make the definition:

A function

U

, positive

and m e a s u r a b l e

on

[A,=)

A > 0 , is said to vary rapidly at infinity if for all

lim U(~x) U--7-fT

X-~r

p -- ~ U

~ , in the "limit case"

to that result, one might think

,

It is therefore useful

that

(l.31a) of w in analogy

(1.53) conforms to this usage,

~ > 0

where

x §

or

p = -~

~ 9 0

p =

(In the respective

is regularly varying of index

~

cases we may also say

or index

-~ , but this may

be misleading.)

II 1.9.

Bibliographic

Notes

and Discussion§

The fun@amental theorems tions viz.

for the theory of regularly varying func-

the U n i f o r m Convergence T h e o r e m

Representation Theorem

(Theorem I.I) and the

(Theorem 1.2) and the C h a r a c t e r i z a t i o n Theorem

(Theorem 1.3) were first o b t a i n e d by J. Karamata first of these papers,

(1930b,

1933).

In the

continuity was assumed in the d e f i n i t i o n of

44

regularly varying local

functions,

integrability

while

was assumed.

The Uniform Convergence

Theorem

tions was proved by Korevaar, (who also obtained Theorem);

and by Delange

for measurable

restricted

(1955).

(1965);

to use Egorov's

(1954).

Besicovitch

(1949)

theorem

difficulties,

(an apparently

to construct

(1965) and Csiszar

proofs,

and Erdos

As a result one finds that certain

existing proofs of the Uniform Convergence given in Hardy and Rogosinski

while valid for continuous

and de Bruijn

Certain measure-theoretic

(1954), Matuszewska

see also Delange

func-

form of the Representation

tool), which have arisen in o~her attempts

are discussed by Agnew

slowly varying

van Aardenne-Ehrenfest

an apparently

partly connected with attempts natural

in the latter part of the second,

slowly varying

Theorem, (1945),

viz.

those of

and Matuszewska

functions,

(1962),

are not so in

general. The Representation established w

which avoids

ficiently

far.

van der Blij

w

1966,

Theorem.

van Aardenne-Ehrenfest

of Cauchy's

and the later measure

of a technique

(1928).

The particular

suf-

and

given in the paper of

which in turn is based substantially

theory of solutions w

given in

on finite intervals

and de Bruijn

the proof of Theorem 1.4 in its earlier

now classical variant

w

form (Theorem 1.2) was

is the note of Korevaar

follows the development (1971,

the paper of Korevaar,

Acz$1,

interest

(1948) on the Representation

and Seneta

In w

in the present

(1959), with the construction

the problem of boundedness

Of historical

The present Bojanic

Theorem

by N.G. de Bruijn

of Doob

equation

discussion

the

(e.g.

is a

(1942), based on a result of Auerbach

technique

that Doob uses it to initiate

(1949).

stages follows

functional

theoretic

on

is probabilistically

a discussion

interesting,

on continuous

in

time Markov

processes. In w

the form

while the subsequent de Bruijn subsequent In w

(1959).

(1.21) comes from Bojanic

construction,

as already mentioned,

Lemma 1.7 is due to Parameswaran

direct proof of Lemma 1.2 to L~tac virtually

two early papers.

and Seneta

all the properties

Property

l~

5 ~ is essentially

(1971);

is due to

(1961);

and the

(1970a). ~ occur in Karamata's due to de Bruijn

(1959).

Lemma 1.8 is attributed to W. Vervaat, in de Haan (1970a, pp. 22-25), his approach being quite different, and the result slightly less complete.

45

In w

Theorem

the assumptions

1.6 is due to Bingham

in Matuszewska's

that of Matuszewska's w

is strongly

(1962);

influenced 1.7,

are contained

so also Seneta

In regard

due to Karamata

variation;

(1970a,b)

such results

@n = n

where

~

p .

theorem

of similar

a result

= n

Theorem

It remains

(1911),

monotone

papers

L(x)

appears

log L(x)/log

+

~i

private

~2

"croftian (1965-6;

1971,

Slack

side of (1.51)

is taken as

~73

for some

another

references

croftian

are given.

This

is not necessarily

pre-

(1912)

It is of some occurring

slowly varying,

from G,E.H.

(1925)

(1923),

interest

2)

lOgkX

Reuter;

already

mentioned, to

"langsam wachsende"

to list here

+ 0

(1930).

work of

in relation

references,

x-YL(x)

in

from work

and I. Schur

Polya called

< - ; and

% ~k .... (lOgkX) , when

communication

(1925)

by the peripheral

in these

then

variations

to have come partly

and Polya

which

with some histori-

of regular

and Polya and Szeg~

x § 0 , f~{L(x)/x2}dx

(log2x)

criteria

~ ; and of de Haan

on functions

by R. Schmidt

functions

functions

is monotone

(log x)

(1917)

ahnehmende".

on these

and Polya

1.8 is again

and R.S.

(1971), Where

and earlier

the remarks

of Landau

slowly varying

and "langsam results

of

where monotonicity

of sequences

Polya

called

of Feller

for the definitions

two foundation

and further papers

Theorem

Reuter

, j=l,2

On the other hand he was also influenced Landau

(1911)

1.14.

(1963).

to conclude

on certain kinds

and well-known in Lemma

on the right-hand

values

~(~j)

sort occurs,

The motivation

Karamata's

of G.E.H.

I.i0 is from Seneta

see also Urbanik

cal notes.

in 1925).

are sometimes

integer

and

of Matuszewska

150, p. 67, and Soln. p. 251;

to a result

and convergence

paper deals with situations sent;

Ex.

(1933).

1.9 and I.i0 give sequential

1.9 is related

for all positive

results

the Hamel basis,

(1970,

Theorems

and generalizes

finite

by

idea of weak regular

1.13 is standard

of this book appeared

Theorem

(1971) + where

is motivated

1.15 goes back to Landau

and Szego

(1933).

for regular

is assumed

Lemma

involving

Lemma

theorems". p. 277),

as the basic

in the more general (1973b).

to w

see also Polya

the first edition

(1975) who relax

Our proof

by the first part of Karamata

as well

as is the kind of argument,

(1917);

and Teugels

w

w

Lemma 1.12 and Theorem variation,

(1962)

(as

further

viz.

I)

if

x § |

the function = lOgk_l(lOg

see Slack

(1972)

x)

Lemma 3,

46

is the

k-th

functional

iterate of

log x , is

(monotone)

slowly

varying. It is little known, Characterization

the work of Petrini functions

t(.)

(1916);

e > 0

w > w'

and deduces

(by a functional and that Faber

arbitrarily

and all

8

in

< t(Bw)/t(w)

(l-c)

-i

< t ( B -1 w ) / t ( w )

lim

interalia,

writ(w)

:

~

,

Thus Faber's

introduces positive

given

an

 0

that for

lim

n

w'nt(w)

>

0

= 0

is essentially

one of a slowly varying

property

is actually

function,

imposed as a re-

and I. Schur

(1930),

introduced by

as mentioned

is a sequence of positive

numbers.

] to mean "the integer part of"

lim

c([Xn])/c(n)

for each

X > 0 , where

for some

p

in purely

sequential

in

the requirement

for a sequence

[

:

0 < r

(whence

c(n) ~ a(n)

, where

n(l-{~(n-l)/~(n)})

-~ p , p

of positive

{~(n)}

logy with the property varying

terms

(1.9)

functions.

is not altogether The general

it follows

Another possible

- see also

and Seneta

r

p

this time (1973),

is

finite This definition

(i.I0)

The equivalent

trivial,

:

that

definition,

form, and given by Galambos

that

in analogy to

@(~)

< |

-- 1

,

The notion of a regularly varying sequence was

regularly

arbitra-

w'

in the definition.

J. Karamata {c(n)}

B' > 1

there exists

W+-~

t, where the uniform convergence quirement

at a

occur as early as

(1,8')

-I

W+~

(1917) that,

small,

(l-c)

from this,

that attempts

equation)

which have the property,

rily large and that for

on the other hand,

Theorem

and

(i. Ii) and w

of various

is established

is in ana- of

definitions,

by Bojanic and Seneta

theory of such sequences may often be deduced from

47

that of functions, Seneta

(1973)

on account of the result

: if the sequence

p , then so is the function EXERCISES TO CHAPTER i.i. A > 0

is a regularly varying of index

~ c([x])

of the following

functions

large

(i

log

+ x -I )

exp((log

(iii)

(v)

are

all

the

r-th

1 + e

where

x ) B)

x

.

, 0 < 8 < 1

real

( l o g k x ) elk

and f i n i t e

functional

where c~l'''''~k logrX = logr_l(lOg x) i s

and

iterate

of

log x

X

i + e -x

(vi)

In those cases where

the function

is slowly varying,

of (iv), obtain a representation

for

Theorem

indicated

1.2, by using the method

Equation

L(x)

with the exception

of the form specified by in the paragraph

containing

(1.11).

Show that

{-log x}

is regularly varying x = I-

1.3.

[A,~)

:

Clog x) ~1 ( l o g 2 x ) ~2 . . . . .

(iv)

at

is slowly varying

taken as

2 + sin x .

(ii)

1.2.

, x > 0

the domain of definition being

is sufficiently

(i)

and

1

Investigate which

(at infinity),

{c(n)}

c(x)

established by Galambos

at

; and that

Assuming

is slowly varying

x = l+ , and that -log(l-x)

at

x = 0+ ; that

-log x

is slowly varying

the validity of Theorem

log x

is regularly varying at

x = 1 .

1.2, deduce Theorem

I,I as a

consequence.

1.4.

In relation

tation

can be found where)

worse properties

to Theorem

of

L(x).

taken as continuous, has

n(x)

; but if

Thus

and if L(x)

1.2 deduce

the function L(x)

~(t)

l.S.

of (1.3) reflects

L(x)

is continuous,

has a continuous

in general.

in the representation

x

and

= h(x)

y , where

n(x)

derivative then

the can be

then so n(x)

can

[Note that, on the other

can be taken as "smooth"

Suppose we have given the functional f(x+y)

for all

in general,

n(x)

is merely assumed measurable,

only be said to be measurable, hand,

if

(a r e p r e s e n -

that,

as desired.]

relation

+ p(y)

the function

h

is assumed measurable

on

48

(-|

Show t h a t

Hint:

f

must be c o n t i n u o u s

Use t h e a p p r o a c h o f t h e

on

second part

(-|174

of Theorem 1.4.

1.6. Use t h e m e t h o d o f c o n s t r u c t i o n of w t o show t h a t quirement of continuity of c(t) in Theorem 1.2 is mildly

(Doob, 1 9 4 2 ) . if the rer e l a x e d , we

may t a k e (t) where

f(t)

= f~(log

= l o g L(e t ) fl(t)

for

n ! t ! n+l

t)

say

and + (f(n+l)

: f(n)

, and

, t ~ B

all

n ! no

f(n))(t-n)

, where

no

is s u f f i c i e n t l y

large;

so that

f~Ct) and

fi(n)

Hint:

= f(n+l)

c a n be d e f i n e d

Replace

- f(n)

t o be

, n < t < n+l

0

the integral X

6 f

u (1-u)du O

X

in

(1.221

by

x ~ f

l.dx,

x c [0,i]

O

1.7. X

~

4~ of w

show t h a t

L(x)

~, L_(x)

as

~

1.8.

Set d o w n

x-YL(x) 1.9.

Let

that

Ln+l(X)

and p r o v e

a proposition

analogous

to 4 ~ of

w

for

, y > 0

L2(x) Show

to P r o p e r t y

In r e l a t i o n

L(x)

be

slowly

= L1 I L l - - 7 1 , each

~ Ln(X )

Ln as

varying

L3(x)=

is s l o w l y x § ,

function.

L1 ( L 2 - - 7 ] , ' ' ' ,

varying.

, it f o l l o w s

L * ( x ) ", {Ln(X)}

-i

Put

Then that

show

Ll(X)

= L(x)

Ln+l (x) = L1 [ L n - ~ ] that

if,

for

some

9 n

,

49

where

L*

is the c o n j u g a t e

of

L . (B4k4ssy,

Obtain Question

an a s y m p t o t i c

i.i.

Repeat,

form for

with

form

L*(x)

when

L

1957)

is form

(iv) in

(iii). (Parameswaran,

I.i0.

A positive

function

b(x)

, defined

slowly varying in the sense of Zygmund, an i n c r e a s i n g ,

and

enough.

n

Take

b ( x ) x -~ fixed

Show that for s u f f i c i e n t l y

b(~x) for large

< ~b(x)

x ; and s i m i l a r l y b(~x)

Hence prove

interval.

x

for

in

[l,n -I]

x

is large

0 < n < 1

large

x , and

~

< n -~b [x) that

~ b(x)

Hence

sense of Z y g m u n d as d e f i n e d

~ > 0 , b(x)x 6 of

that

x § ~ , for any f i x e d

in this

function

will be called

O

> n~b(x)

b(~x)

as

x > x

if for any

a decreasing

to s a t i s f y

for

1961)

~

in

deduce

are a s u b c l a s s

in this

chapter.

[l,n that

-i

]

and even u n i f o r m l y

slowly varying

of o r d i n a r y

functions

slowly varying

(See also p r o p e r t y

4 ~ of

for

X

in the

functions

w

(Zygmund,

1968). i.Ii. small,

Put

@n = e n

Kl(X ) = x ~ {I + a sin(2~

-| < ~ < -

Show that

x > 0 , and w h e n that for any the form

XP

~ $ 0

X > 0 , K(SnX)/K(Bn) , is not

regularly

Theorem

i.I0).

private

communication.)

1.12.

In c o n t r a s t

the f u n c t i o n

(Example

monotone.

= K(~)

essentially

regularly

lOgkX

varying.

oscillating

be the k - t h

but

KCX)

(This e x a m p l e

i.i0, a

consequently,

still

functional

, with

is

for show

, not h a v i n g

is r e l e v a n t

Reuter

of Q u e s t i o n

(Note,

a

On the other hand,

due to G.E.H.

K1

where

and c o n t i n u o u s

, so that

varying.

to the f u n c t i o n

t i o n m a y be i n f i n i t e l y Let

is p o s i t i v e

K2(x ) = x ~ {i + a sin(2~/T6-g-~)}

-| < 6 < | , is

1.13.

K1

is s t r i c t l y

log x)}

show that small, that

a func-

an S.V.F.).

iterate

of

to

(1970),

log x .

Show

50

that

the

tion

following

L(x)

some

positive

Hint:

proposition

, such

that

L(x)

integer

Construct

k

an

such

L(x)

in s u c c e s s i o n

infinity

any

specific

R(t)

= tPL(t)

1.14. tion

Let such

that

the U n i f o r m

R(x)

R(t as

x § |

r |

Convergence +

, uniformly

that

be

any

L(x)/lOgkX

§ = ".

aid of the

functions

lOgkX

x ~ ~

increases

varying

func-

to find

lOgkX

more

slowly

to

. regularly

(Thus we must

to s h o w +

slowly

, it is p o s s i b l e

a non-decreasing

R{x)

with

: "For

x + -

eventually

function

as

-

the

which

Theorem

x)

as

with

k = 1,2,..., than

is false

+ |

that,

for

varying

have

any

func-

p ~ 0).

fixed

Use

to > 0 ,

0

respect

to all

(Cheong

t ! to

and T e u g e l s ,

1972). 1.15.

Show

x E [A,-) large,

that

,

if

L(x)

positive

varying

number

function

and

y

defined

for

is s u f f i c i e n t l y

then xL(x)/yL(y)

for

is a s l o w l y

is an a r b i t r a r y

all

1.16.

x ! y Let

.

Z

> 1 -

(Parameswaran,

be p o s i t i v e

1961).

non-increasing

on

(0,-)

and

suppose

that

X

Zo(X ) = f

Z(y)dy

, x > 0 , is

known t o

be slowly

varying.

Show t h a t

0

lim x Z ( x ) / Z o ( X ) = 0

Hint:

For

fixed

x > 1 , ~ > 0 , Zo(XX)

- Zo(X)

lx ~ f Z(y)dy

> X (x-l)Z(Xx)

X X

1.17.

If

M(x)

= f

{L(y)/y}dy

A and a b o u n d e d

varying,

is l i k e w i s e

slowly

1'4(x)/L (x)

as

x + -

1.18.

If

follows

that

on f i n i t e

varying +

is a s l o w l y l-e

L(x)

subintervals

is d e f i n e d , of

[A,-)

slowly

show

that

and

(Parameswaran, L

, where

|

1961). varying

< L ( e X ) / L ( e x-l)

function, < 1 + e

from for all

its

definition

x >_ Xo(E)

it

where

M

51

r > 0

is specified arbitrarily.

to show

: first that

10g L(x)/log x + 0 xYL(x) § ~

as

log L(e x) + 0

L(x)

some

A

as

X

x § ~

x + |

for p o s i t i v e

fixed

x-YL(x)

L(~x)/L(x)

§ 1

for v a l i d i t y of the argument,

as

x + -

for each fixed

1.19.

Let

(a,8) show

f(t)

where

0

and

although in v i e w of w

anything [A,-)

A > 0 ?

p r o b l e m is included because a p p a r e n t l y "general" arguments occur in the literature+;

that

+ 0

y

except that it is w e l l - d e f i n e d and p o s i t i v e on

, and

attempting

; and c o n s e q u e n t l y

x § - ; from which deduce that

as

Is it n e c e s s a r y to assume, about

Iterate this r e l a t i o n back,

for (This

of this type

the answer is easy).

be a function Riemann integrable on every interval

< a

< 8 < ~

For

0

< x1

< x2

< ~

,

and

0

< a,b

< ~

that x2

[

bx 1

f(~t)

f(bt)dt = f t

x1

I

ax l

so that the integral

f o+

exists for fixed integral)

bx 2

f(t) d t t

(called the Frullani

f(at)-f(bt) t

a,b

f(t) at t ' ax 2

integral)

dt

(in the sense of being an improper Riemann

if and only if bx

; ax converges both as

f(t)

t

dt

x + O+

; and as

x § |

Show that the improper Riemann integral exists 0 < a,b < ~

if and only if the function

log

p(x)

= f

x

f(t)

1

t

p(x)

is the index of regular v a r i a t i o n at infinity and

of regular v a r i a t i o n at zero, show that

+

See

Seneta

(1973h)

for

references.

a, b,

dt

is regularly varying b o t h at i n f i n i t y and at zero. p

for aZZ

, defined by

In this case, T

if

is the index

,

52

f O+

[R(x)

~r~~kat/-~) t

dt

[

(~

-

p)

{log

is regularly varying at zero with index

is regularly varying at infinity with index

m

b

-

log

a}

if and only if

R(~)

~']

(Aljancic and Karamata, 1956). 1.20.

For a slowly varying function

L

4 ~ of w

implies that

where ~Y function.

is a monotone increasing and

(i) (ii)

xYL(x) ~ ~N(x)

and any fixed and ~Y

Y > 0

x-TL(x) ~ ~T(x)

property as

x§174

a monotone decreasing

Use the Representation Theorem to deduce this fact directly. Show, conversely,

that if

L

positive and measurable on

is a function defined, finite [A,|

, and for each fixed

Y > 0

xYL(x) ~ ~7(x) and x-NL(x) ~ ~y(x) as x + ~ , where ~N is an increasing and ~N a decreasing function, then L must be slowly varying.

(Compare with Exercise i.i0).

CHAPTER

2

SOME SECONDARY THEORY OF REGULARLY VARYING 2.1. _Necessary

and Sufficient

It is evident

condition

Theorem,

Theorem,

assertion

U(-)

readily

each of the

Theorem and the

as a necessary

to be regularly

follows

for Regular Variation.

of adjustment,

the R e p r e s e n t a t i o n

can be restated

for a function

the following

Integral_Conditions

that with a small amount

Uniform Convergence Characterization

FUNCTIONS

and sufficient

varying.

For example,

from the previously

developed

theory. Lemma 2.1. for some

U(-)

Suppose A > 0

is defined,

U(-)

Then

positive

is regularly

[A,=)

and finite on

varying if and only if it

can be put into the form

U(x)

X > B

for all

measurable e(x) ~ 0

= xPexp

for some

B > A ,

[B,|

functions on as

{n(x)

x ~ ~ , and

p

x g(t) dt} + _ 5B t

n(x)

where

~(x)

and

are b o u n d e d

n(x) § c ( I c I < |

such that

is a finite number;

and

in w h i c h case

p

i8

the index of r e g u l a r variation. 9

There exist various a function

U

other necessary

of the kind considered

which are easier to check in general; tion

U

as part of an integrand

comes necessary

to assume

far, as well

does not involve

statement where

necessity

of (2.1)).

Theorem able,

2.1.

subinterval necessary limit

Suppose

and positive of

on

function

as measurable

[A,=)

[A,=)

involve

interval

(by assumption)

in relation

U

U(x)

of

the funcU a priori. intervals

this condition

to that part of the

is also assumed

is definedj

(and (Lebesgue)

Then for

for

so that it be-

is bounded on finite

regular variation

the f u n c t i o n

and s u f f i c i e n t

these generally

over a finite

loss of generality

following

conditions

to be regularly varying

also some kind of integrability

Since a regularly varying sufficiently

and sufficient

above

finite and measur-

integrable

to be regularly

that there exist a number

(i.e. the

k

on each finite varying it is

such that the

54

xk*l U l~m (x) x§ x tku(t)dt A

(2.1)

exists and is finite and positive. +

ak+l , then the number variation of U Proof.

Suppose

U(x) = xPL(x)

U(x)

where

(2.2)

x

is slowly varying.

'

~

xL(x) X f L(t)dt B

L

is bounded on

L(t)d~ A

where

B

x>B,

is chosen so that

Now note that for fixed arbitrary X

I

if this limit is

is the index of regular

is regularly varying with index L(x)

xL~

.(

In this case,

p = ak-k

denoted by

y

[B,x]

x § |

1 > y > 0

X

: ;

B

so that

for each fixed

satisfying

X

L(t)dt

p

Then consider as

t-$tVL(t)dt

< sup { t Y L ( t ) } -- B
B

f

t'Ydt B

i.e.

x ( x 1 -y _B 1 -y] [B n(t)dt <_ B
(2.3a) Similarly

x fB L ( t ) d t

(2.35)

Now we know t h a t are respectively

as

x = fB t Y t - Y L ( t ) d t

x § |

the right

xL(x)/(1-y), from w

4~

! B
hand sides

ixl+Y_Bl+Y) l+y

of (2.3a)

and ( 2 . 3 b )

~ xL(x)/(I+~)

and the associated Exercise

1.8

Thus X

1

f s lira inf

B

I

L(t)dt

xL(x)

X~

+ For another proposition

~ lira sup X~

X

B

L(t)dt

xL(x)

1

i Trr~T

of similar kind, see Exercise

2.3.

$5

y > 0

and since

is arbitrary, lim

it follows

f

x § |

= 1

X~<xjTr ~

x§

that as

X

L(t)dt A

so that

(2.1) is satisfied with

Conversely,

k = -p

that

let us suppose

(2.4)

xk+iu(x) x

lim

x§174f

-

ak+l

(for some

,

a k > -I ).

tku(t)dt

A The first deduction

to be made

is that

x tku (t) dt A is a regularly varying u(x)

function with index

= xku(x)

, r

=

x

ak+l

xu(x)

, for suppose we write

-

(ak+l)

u(t)dt A Then we find that

uCx)

= c(x) + {~k §

X

X

X

u (t) dt A whence

integrating

x log

}

]'AU(t)dt

x = J'A r

dt + (ak+l) log x + const.

X"

whence Lemma

f u(t)dt A

has the form of the R e p r e s e n t a t i o n

in

2.1.

On the other hand from that

mentioned

(2.49 we deduce,

for any positive

x>O,

56

J~ A aS

X

-~

tku(t)dt

it follows that as

whence

::

tku(t)dt A

kk+Iu(xk)

§

x +

kak+l

U(x) i.e. U(xk) § kak-k

which completes

the proof

(since

As a c o n c l u d i n g comment that,

if

U(x)

U

is positive

on this theorem,

and measurable).

I

it is n e c e s s a r y to remark

is a r e g u l a r l y varying function in the context of the

theorem, with index

p , then the limit

(2.1) exists and is finite and

+

positive

for each

p = ak-k

, is independent of

k > -p - 1 .

regular v a r i a t i o n

Then

k .

is deducible from

a k = k+p

Suppose

sufficient

condition

the existence

U

(2.1) holding for a single

of a number

k

that

k

We

:

is as in Theorem

involving

indeed

By the theorem we see, however,

thus have another v e r s i o n of T h e o r e m 2.1 Corollary.

, so that,

Then the necessary

2.1.

(2.1) may be replaced

such that as

and

by one asserting

x §

1

(2.5)

f

yk(u(yx)/U(x))dy

§ (p+k+l) -I

> 0

A/x in which case

is the indez of regular variation.

0

change variable

in the integral of

In actual fact if if

U

A

(2.1) from

y , where

t=yx). I

p

, then for

k > -p - 1

~

1

yk(u(yx)/U(x))dy

of a

k

§ (o+k+l) -I

such that as

x +

> 0

O

where the left hand side is assumed w e l l - d e f i n e d for each ently large.

+

See E x e r c i s e

In this vein, we assert

2.1.

x

U

(2.5)

A = 0 ; so that the n e c e s s a r y and sufficient c o n d i t i o n

(2.5) may be replaced by the existence

(2.6)

to

this

can be taken as zero in the d e f i n i t i o n of

is regularly varying with index

will hold with

t

(To s e e

suffici-

,

57

Theorem

2.2.

If in each of the f o l l o w i n g conditions

the integral oc-

curing on the left-hand side is a priori assumed w e l l - d e f i n e d nite)

for each

ficient for varying,

1~ 2~ . where

p

3o"

where

p

Proof.

x

sufficiently

large,

U , taken as in Theorem

(and fi-

then each is necessary and suf-

2.1 with

A = 0 , to be regularly

the index of regular variation being

p :

(2.6) holds as

There exists a k such that 1 f log { U ( x ) / U C t x ) } d t + p o

x + | ;

( x + ~)

is some finite number ; "

~i

log{U(tx)/U(x)}

dt

t2

+ p

( x + ~)

is some finite number.

We have proved

To prove

l~

that 2 ~ is sufficient,

(i.e. essentially

we shall

again

a form of the Representation

invoke

Theorem).

the Lemma We have

1

log

U(x)

f

log

U(tx)dt

+ p

log

U(tx)dt

= p + r

o

i.e. 1

log U (x) o

where

e(x) + 0

(2.73

as log

x + ~ ; i.e. U(x)

_ 1_

x

x

x 2 fo

log

U(y)dy

p

+

= ~

~(x)

x

so, by integration, x -I f

x

x c(~ log U(t)dt

= p log x + f

o so that from Lemma

dt + const B

2.1 x exp {x -I f

log U(t)dt} o

is regularly

varying

(after multiplying

by

with x

index

p

and taking

On the other hand, exponentials)

from

2.

that

(2.7)

58

X

U(x)

= e l~

U(x)

= exp{x-i

~

log

U(y)dy}

exp{p+e(x)}

0

so t h a t ,

as

x §

|

,

X

U(x)

~ e p exp

{x -1 ;

log

U(y)dy}

o and

so

U

is r e g u l a r l y

Conversely, invoke

the

desire),

if

varying

U

index

is r e g u l a r l y

Representation

to o b t a i n

with

of

for

p

varying

Lemma

2.1

0 < t m< 1

and

with

[with x

index

p

, we may

oont4nuous

~(t)

if w e

large X

U(x)/U(tx)

= t "~ exp

{n(x)-n(xt)

+ ~ ~(Y) xt Y

dy}

so that X

log

{U(x)/U(tx)}

= -p log

t + n(x)-n(xt)

+ ; z(y) xt Y

dy

Now 1 log

t dt

= -i

;

O

and i f

-i in(x)

- nCxt)}dt

= nix)

x

- x

f

O

§ as

x § =

, since

fi -o

n(y)dy O

n(x)

§ c

(Icl

i x ~Cy) dy xt

0

< |

dt

=

as

x § =

-

I

Y

o

t

1 =

~Cwx)dw

; 0

= x

-i

X

r (yldy

; 0

§ as

x + |

, since

c[y)

§ 0

.

, and

dw

finally,

at

.

so

59

Proposition approach 2.2.

3 ~ is left as an exercise+;

similar

to the above. 9

Tauberian Theorems Speaking

simply,

of the asymptotic asymptotic

it can be proved using an

Involving

Tauberian

behaviour

behaviour

Regular Variation.

theorems

are concerned with the deduction

of (generally monotone)

of their transforms

transforms).

One of the most famous

probabilistic

(amongst other)

(e.g.

functions

their Laplace-Stieltjes

and very widely useful

contexts

from the

theorems

in

is the famous theorem of Karamata

which we prove first. Theorem

2.3.

(Karamata's

non-decreasing

w(x)

function

= f

Tauberian Theorem), [0,|

on

such

e-XUd{U(u)}

Let

U(x)

be

a monotone

that

i8. f i n i t e

for

all

x > 0 .

oThen

if

p 9 0 , and

(i)

w(x)

w(x)

Proof. of

x+|

as

as

as

.

§

function

~ xPL(x)/r(p+l)

,

= x-PL(x) x

varying

x § O+ =~ U(x)

O+

x § - ---->U(x) ~ xPL(i/x)/r(p+l)

We give a proof of (i) only,

(ii)

For

is a s l o w l y

= x-OL(I/x) as

(ii)

L

since a totally analogous

proof

holds.

X > 0 w(lx)/w(x)

§

t -p

,

= f

e-UXdG(u) O-

as

x §

O+

, where

G(u)

w(Ix)/w(x)

= uP/r(p+l)

= (f

On t h e

other

hand

e-lXYdU(y))/w(x) O-

= (

f~ e-kUdU(u/x))/w(x) o-

which +

is itself the Laplace-Stieltjes

Exercise 2.6. This exercise Theorem 2.2, 2 ~ , as Exercise

transform of the monotone

has the same kind of relation 2.3 has to T h e o r e m 2.1.

to

function

60

of

u , indexed by

x , Ux(U) , defined by

U*(u)

= UCu/x)/w(x)

X

Thus, by the e x t e n d e d continuity t h e o r e m + for Laplace transforms

as

x + 0+

U*(u)

+ G(u)

X

for

each

u > 0

u > 0

as

, since

G(u)

is continuous

on

[0,~), i.e.

for fixed

x § 0+ U(u/x)

as

i.e.

--- u P / r ( p + l )

~ uPw(x)/r(p+l)

v + -

U(v) ~ v O L ( v / u ) / r ( p + l )

~ vPL(v)/r(p+l)

. I

The following "density"

e x t e n s i o n of the above theorem is often

given with it, and we follow this tradition.

A much more general

++

theorem can be p r o v e d by the same method,

as the reader will realize

by e x a m i n i n g the proof of the subsequent T h e o r e m 2.4. Theorem

Let

2.4.

sufficiently

U(x)

defined

,

and positive

on

for

[A,~)

8ome

large, be given by X

U(x)

where

u(y)

i8 ultimately monotone. U(x)

Proof.

= xPL(x)

Suppose first

8 > a ~ ao

u(y)dy

= ;

u

=> x u ( x ) / U ( x )

t8

=

~

uk~J

ta that e.g. Feller (1971, Exercise 2.7.

P L 0 as

x § |

is u l t i m a t e l y non-decreasing.

- U(ta)

+ See ++ S e e

+ o

say,

U(tg~(t)

so

Then for

p.433)

dy

Then for

A

61

(2.8)

t(B-a)u(t8) U(t) Thus,

letting

U(t8) - U(ta)> t(B-a)u(t-a) U(t) U(t)

>

-

t § ~ , from the right-hand

side of

8P-a p tu(ta) > lim sup 8-a t§174 and letting

8 + a , since the right-hand

is independent

of

8

side of the present

inequality

we get

pa p-1 > l i m sup t u ( t a ) --

Similarly

t+=

from the l.h.s,

of (2.81

lim inf tu(tS) -i t§ U-U--(ET-L pBD

Since

a

and

B

are arbitrary,

lim tu(tc) t§

and so, putting tion.

x = tc

similarly.

Theorem

2.5.

our short discussion (i) and

For

=>

Proof.

= tPL(t)

completes

non-increasing

the asser-

may he proved

of the part regularly by

(ii) of Theorem

is

p

>

~

d{A(t)}

o-

A

U(t)

is ultimately

proving 2.3.

together with the method of Theorem

/

where

and using u

play in Tauberian Theorems

using both parts Theorem)

pc

II

We conclude functions

c > 0 ,

p-i

U-~l~j--:

The case where

for arbitrary

>

~

0

j

a8

x

another

(Karamata's 2.4.

§

x-OL(x)

(t+x) 0 A(x)

assumed

Since for

~ r ( o ) r (r ~( p- ~) + l )

xP-OL (x)

non-decreasing

on

[0,~)

~ > 0

(t+x) -p = (I/r(p))

f

e-tZe-XTTP-ld~ O

,

varying

such theorem, Tauberian

62

it

follows

that

[

w~ere

g(~)

d{A(t)} (t+x) ~

O-

= T0 - 1 ( f

:

g(T)e-XTdT

i o

e-tTd{A(t)})/r(o)

From t h e

given,

and by u s e

O-

of Theorem 2.3,

(ii)

(I

(Karamata's

Tauberian

Theorem),

it

follows

that

X

g(T)dT)

~ xCL(1/x)/r(o+l)

O

as

x + 0+

Writing

g(T)

out

in

full,

we o b t a i n

X

{i/r (~+i) } /

f(T)d(T ~

~ xOL(1/x)/r(c+l)

0 co

where

f(T)

= f

e-tTd{A(t)}

,

O-

so t h a t def

(2,9)

u(x)

j'

--

x

f ( T ) d { T ~} ~ x ~

0

as

x + 0+ , w h e r e

f(~),

monotone non-increasing Let

B > a

>

0

the as

.

T

f

(2.9)

and l e t t i n g

through

which holds

by

8~

o~ ~176 o

_

true

for

f(~)d{r p }

x~

u(•

L

f(x~)x~ U(x)

~')

x § 0+

- -

Dividing

transform

increases.

Then

U(xB) U(x~] tJfx) Using

Laplace-Stieltjes

X§

a p , and l e t t i n g o. ~ o ~ o--~-I- > l i m s u p x§

arbitrary

a > 0 .

B § a+ f(x~)x p

u - ~

Similarly

yields

of

A(t)

is

63

o8o-P ! lim inf P x§ for

arbitrary

S > 0

.

oc ~

--

=

P

Pu~ting

T = xc

, we

Hence

for

arbitrary

c

> 0

f(xc)x p

lim x§

obtain

that

as

T §

0+

f(T)

~

'Y-s

f(r)

~

(r(p)/r(a)ITa-PL(1/T)

(c)

p

i.e.

%aking

into

account

(2.9)

now

Karamata's

Applying

. Tauberian

A(x) ~ r ( ~ ) r?(p) (c-~§ as

x + |

2.3.

, which

is

the

In t h i s x § |

, (i),

it

follows

that

xO-OL(x)

required

A Class of Integrals

Theorem

result.

Involving Regularly Varying Functions,

s e c t i o n we i n i t i a l l y

o f t h e Lebesgue i n t e g r a l

s t u d y t h e a s y m p t o t i c b e h a v i o u r as (assumed w e l l d e f i n e d )

:

B (2.10) where 0 is

< ~

f L

f(t)L(xt)dt

is a slowly varying function

< 8 < |

Lebesgue

It

is

integrable

easy on

to

see

[~,8]

by

the

f

Uniform

if

0 < ~

, that

B

(2.11)

(at infinity

that

8

f(t)L(xt)dt

Convergence

~ L(x) f

Theorem,

by

f(t)dt

considering

as u s u a l ) < 8 < |

and

, and

f(t)

64

as

x § -

ditions tion

We

on

even

of i n t e r e s t

It is less case both

shall

f

first

if

in this

natural,

but

obtained,

connection

which

together

which

we

that

we

leave

L

still

varies

case

under

is the

mild

natural

slowly

important

do next;

for the

to the

persists,

, which

since

shall

a result

shall

(2.Ii)

, 0 < ~ < |

nevertheless

~ = 0 , 0 < B < | results

show

8 = ~

at

to c o n s i d e r

and

then

a = 0

conques-

the

by p u t t i n g

, 8 = -

may

be

reader. +

It s h o u l d the

be n o t e d

apparently

more

gularly

varying

absorbed

into

(2.10)

that

general

function f(t).

is t a n t a m o u n t

the

study

integral

R(t)

It may

of

where

= tPL(t)

also

to a study

(2.10)

of the

L(t)

, since

be w o r t h

subsumes

the

is r e p l a c e d the

factor

remarking

asymptotic

that

effect

study by

tp the

of

a remay

be

study

of

of the

functional F(x)

applied

to a c e r t a i n

transformation

the

of a r e g u l a r l y

similar

methods.

Theorem

2.6.

and the

(Lebesgue, [

i8 w e l l - d e f i n e d ~ 0 .

of f u n c t i o n s

by

L(xt)

section

L

, where

the

kernel

of this

= K(x,t)

function,

the p r o b l e m

a problem

is s l o w l y

as usual) tnf(t)dt

f

by c o n s i d e r i n g

varying

Suppose

(2.12)

K(x,t)f(t)dt

class

is g i v e n

We c o n c l u d e tion

= f

varying

on

partly

[A,|

of t o t a l tractable

variaby

for some

A > O,

integral ,

|

> ~ > 0

for some g i v e n real f u n c t i o n

f , and a given n u m b e r

Then the i n t e g r a l 5~ f ( t ) L

(xt) dt

i8 w e l l - d e f i n e d :

i) in general,

in this case

is f u r t h e r

if

n > 0 ;

2) in the ease

q = 0

if ++

In e i t h e r

+ ++

L

assumed

to be e v e n t u a l l y

non-increasing.

situation

E x e r c i s e 2.10. O t h e r a s s u m p t i o n s , l e a d i n g to the possible. (See E x e r c i s e 2.9).

same

eventual

conclusion,

are

65

f

f(t)L(xt)dt

~ L(x)

f

f(t)dt

,

(the extreme right-hand integral being well-defined). Proof.

The e x i s t e n c e

we n o t e

that

for

L(x)

integrals

= tnf(t)t-nL(xt)

is measurable

bounded for all

dominance

various

is

easy

to deduce;

e.g.

n > 0

f(t)L(xt) and since

of the

x

from

for

sufficiently

(2.12)

= xn(tnf(t))((xt)-nL(xt)) x > A , and

large,

x-nL(x)

is uniformly

the integrability

( f~ tnlf(t) Idt < =

,

follows

by

since we are using Lebesgue

integration). Now suppose

n > 0 ; then for large f

f(t)

-- J"

~

f(t)

dt

{

finite

f(t)dt

-

- 1} d t +

f

a

a n d we n e x t these

investigate

integrals.

(2.13)

y ,

f(t)

dt

,

y

the

asymptotic

behaviour,

as

x § ~

of each

First If | f(t)L(xt)dt] Y

If(t)]L(xt)dt

~ f Y

= xn f

{tn]f(t) l}{(xt)-nL(xt)}dt Y :o

< x n sup {(xt)-nL(xt)} y
f

tnlf(t) Idt y

0o

--x n

sup {w-nL(w)} / tnlf(t)[dt yx<w<| y

y-nL(yx) f

tnlf(t)l dt Y

as

x + |

1.8. (2.14)

Hence

Using

the analogue

in the case

of Property

n > 0 , we obtain

I f ~ f(t)L(xt)dt I < My-nL(yx) Y

4 ~ considered from f

(2.13)

that

tn[f(t)[dt Y

in Exercise

of

66

for x (since

sufficiently y > a > 0 ).

large,

where

We know also that whether

x § ~ ; from

(2.11).

(finite)

n > 0

fY f(t) { ~

as

M

or

does n o t d e p e n d on

v

n = 0 , that

- 1 } dt § 0

We thus have,

for

n > 0 , invoking

(2.14)

r

l i m sup

I f

X -+|

since

L(yx)/L(x) m

f(t)

~

fY f ( t ) d t m

dt

! MY-q f |

+ 1

as

f(t) ~

x § |

Now

f

f(t)dtl

dt

If

ldt

inequality;

dt - fYf(t)dt m

f(t) ~

IY f(t)dt m

I f | f(t)dt

by the triangle

tnlf(t)

Y

r

so that taking

lim sup

as

x § ~

of the

left-hand side we obtain the bound

My-"f

tnIf(t)

ldt

+

If

Y in which we may let

y + ~ , to see that

lira sup

mI

X ~>~

which proves

Theorem

(2.14) 2.7.

dt

f

for

assertion

for

Let

L of

= 0 ,

n > 0 the extra assumption

we have directly

i f| Y

.

Suppose

that

L

is

(2.13) that

of the proof

be slowly varying on (0,~)

from

i L(Y x) I | l f ( t ) Y

(t) lL(xt)dt

n = 0 ; and t h e r e s t

finite subinterval

f(t)dt] Ct

n = 0 , where we make

non-increasing,

II f(t)L(xt)dt] Y which is

f(t)

r

the required

In the case eventually

f(t)dt] Y

(0,~)

the integral

is

ldt

as b e f o r e . H

and bounded on each

67

f

t-nf(t)dt 0

i8

well-defined

n > 0 .

Then

for as

come

given

real

funotion

B

I

q > 0 ; and

a given

number

B

f(t)L(xt)dt

for

f(t)dt

~ L(x) I

0

for

f ~ and

x § ,

0

n = 0

providing

L

i8

non-deoreasing

on

(0,|

Proof. Since L is m e a s u r a b l e on [0,~) , i t is e a s y to check t h a t a l l i n t e g r a l s s p e c i f i e d e x i s t , as L is b e i n g assumed bounded on f i n i t e s u b i n t e r v a l s of (0,~) y (< i)

Now consider for

an arbitrary

positive

small number.

Then

n > 0 Y

f(t)L(xt)dt]

If

~ fY I f ( t ) l L ( x t ) d t

o

0

! x-n

Y

sup {wnL(w):, f t-nlf(t) Idt , O<w<_xy o

x .n sup {wnL(w)} fY t - n l f ( t )

ldt

,

0

O<W<X

LCx) /Y t-nlfCt)]dt 0

as

X

-~ m

)

ML(x) fY t - n l f ( t )

Idt

0

for

x

sufficiently

large, where

M

(finite)

does not depend on

y

Hence

(2.15)

lim sup ]fY f ( t ) X ~

The remainder decomposing

~ 1

~ M fY t - n l f ( t ) l d t

0

.

0

of the proof can be carried out as before,

first

:

8 f f(t)~

o

using the triangle

B 8 dt - f f(t)dt = f f ( t ) { ~ - I }

y

inequality,

y

etc. letting

dt + f Y f ( t ) ~

o

x ~ | ; and eventually

dt

68

letting

y § O+

In t h e

case

n = 0

, with

the

additional

assumption

on

L

, we

obtain

I/Yf(t)L(xt)dt[

i

0

so

IYlf(t) lL(xt) at

L(yx)

i

0

/Ylf(t) ldt

,

O

that y

lim sup I f

which

is

(2.15)

f(t)

dt[

fY 1fCt) ldt

<

0

X "+~176

for

0

n = 0

, and

the

rest

is as b e f o r e .

9

The r e a d e r o f t h i s and p r e c e d i n g s e c t i o n s o f t h i s c h a p t e r w i l l by now be aware t h a t many problems o f a s y m p t o t i c b e h a v i o u r o f i n t e g r a l s involving regularly varying integrals are relatively easily resolved by u s i n g t h e immensely v a l u a b l e ( t y p e o f ) p r o p e r t y o f r e g u l a r l y v a r y i n g functions that for y > 0 x 7 sup { t - T L ( t ) } ~ L(x) x
discussed perty,

in

w

Property

in c e r t a i n

4~

; and Exercise

circumstances,

t i o n on a s u b i n t e r v a l

enables

of d e f i n i t i o n

1.8.

This

us to e s t i m a t e

of a r e g u l a r l y

kind

of pro-

the t o t a l

varying

varia-

function.

+

We p r o v e Theorem

Let

2.8.

function on n > 0

and

[A,~)

y

L

be a non-decreasing

for some

sufficiently

]d(t-nL(t)]

A

continuous

sufficiently

large.

slowly varying Then for fixed

large,

~ My-nL(Y)

Y where Proof.

M

(finite)

is independent

:o

of

y

:o

Id(t-nL(t)

l < nf t-n-lL(t)dt

y

Y =

II

+

Y 12

,

say. +

For

generalisation,

see E x e r c i s e

+ f t-nd{L(t)_}

2.11.

69

Now

I 1 ~ n sup {t-n+eL(t)} ] y~t<~ where

e

is c h o s e n

t-l-r y

to satisfy

0 < r < n

: n sup { t - n + C L ( t ) } y - Z / c y!t< | Now,

as

y § ~

this

is

ny-nL(y)/e so that,

for

y

sufficiently

large,

y i Yo(n,E,Mo )

I1 ! MoY-~L(Y) where, dent

for s u f f i c i e n t l y

of

large

y , M

o

is finite,

positive

and indepen-

y .

On t h e

other

hand

12 = f ~ t - ~ d { L ( t ) } Y

= f t-qL2(t)L-2(t)d{L(t)} Y r

<

=

sup {t-qL2(t)} y < t <| sup

{t-nL2(t)

f L-2(t)d{L(t)} y } [-L-l(t)

y!t <|

=

]~ Y-

sup {t-~L2(t)}L-l(y) y
y-nL (y) so that,

for

y

sufficiently

large

12 ~ MIY-qL(Y)

where

M1

is

12 , c o m p l e t e s 2.4.

A Class

a constant. the proof. of Functions

In the present

section

Putting

together

the

estimates

for

I1

mm

Related we g i v e

to Regularly a brief

Varying

exposition

Functions. of a certain

and

70

class

of functions

Bibliographic

f

Notes

introduced

and Discussion

have been more recently retical

interest,

branches

Let [A,|

values r

for

studied theory,

of random

sufficiently

A > 0 , and

large

positive

of their

in particular

be a r e a l

value

of its

of which

of their theo-

application

in certain

in the asymptotic

We proceed

real-valued,

f

(see

aspects

from the viewpoint

samples.

be a m e a s u r a b l e , some

and Karamata

to this chapter),

both

and from the viewpoint

of probability

of extreme

by Bojanic

rather

positive valued

argument

theory

informally. function

function, ( s a y on

on

defined [B,v))

for

such

that the limit (2.161

exists of

lim f(xx)-f(x) x§ r (is finite)

(0,~)

for all

Suppose,

tive measure)

of

=

~

further,

H(X)

in a subset there

S 1 , and a number

e S 2} ~ {XlU e S 1 , and

S 1 , of positive

is a subset, ~Ie

S 1 , such that

H(XI~)-H(u I ~ 0}

measure,

S 2 , (still of posi{H(XII

The following

are then akin to the Characterization Theorem for regularly

+ 0 ,

two results varying

functions. Lemma 2 . 2 .

Under the above assumptions,

~(x)

= R(x),

a

regularly

varying function.

Proof.

Let

For

u c S2 ,

f(llUX) -f(xl $[x)

=

f(Xl~X )-f(ux) $(~x)

~

+ f(vxl-f(x) (x)

x + H(XIV ) = H(XII

where

the limit,

(2.171

clearly,

lim

lira ~

must exist,

=

+ H(v)

and in fact

H(XlU) mH(.I H(Xl )

X+~

exists

for all

u e S 2 , and being non-zero

Hence by the theory of Section tion,

i.e.

#(x I = x~

(Consequently, is

~~ .) 9

for all

1.3, ~

is strictly

is a function

for some finite u c (0,|

here, o

where

positive.

of regular L

varia-

is an S.V.F.

, the limit on the left of (2.171

'

71

Theorem 2.9. with

SI' $2' t l '

If

a finite

limit

H(X)

H(X) for

some

sumed

some

[B,|

on

Since

are

all

as

before,

X e (0,~)

(2.16)

then

obtains

.

If

~ + 0

f

is,

additionally,

) If

o = 0 , and

as-

then

= K2~n~ K2 + 0 .

constant

Proo_____ff.

.

K1 + 0

constant

measurable

HCX) for

Kt(X~

=

o,

for

S1

is a set of p o s i t i v e

measure,

from a w e l l - k n o w n

§

result

of S t e i n h a u s

number

C > 1

such that that

a k l n to one a l r e a d y

such t h a t

I = X3/x2

~2 v ~ S I

.

Thus

f[%x)-f( x )

for

k s [I,C]

for any ~

e

exist

there

X2,13

is a

~2

~ S1 such

X~(x/~2)~L(~2(x/X2) )

y = x l~ 2 ,

J f(X2Y)-f(Y) t X~Y~L(k2Y)

X~YCL (X2 y )

Letting

L(X2y ) ~ L(y) , XX2 e S 1 ,

y § | , and u s i n g

+ X~~ a f i n ite n u m b e r , this

to

)

and take

fixed

C(u

Steinhaus

X s [I,C].

It is n o w n e c e s s a r y

to

~ s (0,|

~ r [I,C]. _

~7~

H(X2) }

for a r b i t r a r y

To do this, we p r o c e e d

§

, there

~ r [I,C]

[I,C]

f(xx2y)-f(y)

extend

we can find a

_ f(ll2(x/X2))-f(k2(x/X2))

~"~ Putting

for any Thus

u s e d in w

by i m i t a t i n g

w

1.5

: let

y > 0 ,

Then

f(xfvx/x),)-f(,yx/x)

(vx/x)a(x/x)oL(Cx/v)vx/X)

(1920,

Lemma

T h 6 o r ~ m e VIII)

f~X/v~vx/X))-f(~x/X) (vxlX)~(xlv)aL((~/V)Xx/X)

is the o r i g i n a l

memoir.

72

and,

putting

y = yx/~

f(lY)-f(y)

=

f((X/y)y)-f(y)

(X/Y)~ Thus

(X/y)~

1 ~ X/y ~ C , letting

if

y ~ |

§ (X/y)-~ T h u s we h a v e

for

x/C !

. y ~ ~

i.e.

for

all

y

satisfying

C -1 !

y ~ C ,

that (2.18)

lim f ( y x ) - f ( x ) x§174 r

exists

and is finite.

obtain

that (2 918) o b t a i n s

since

C > 1 , any p o s i t i v e

= H(y)

Repeating

the a r g u m e n t

for all y

y

will

(k-l)

satisfying eventually

times,

say, we

C "k < y < C k

; and

be c o v e r e d by this

interval. Assuming using

~ + 0

arbitrary

X

, proceed in

place

H(XV) so

that,

interchanging

of

in Xl

= v~H(X)

the r o l e H(XV)

whence,

as

of

= X~

the

proof

and

arbitrary

+ H(v) X

of

Lemma 2 . 2 ,

except

~ , to

get

-- K 1

say

,

and

+ H(X)

equating k~

+ H(k]

~ ~~

+ H(V]

or H(X) ( 1 - v ~) Supposing

la,}, ~f i

= H ( v ) ( 1 - X ~)

,

H ( X ) / ( I - ~ o) = H ( p ) / ( l - ~ ~) For

k + 1

,

.

, = const.

.

73

H(A) which

is

correct

If is

a = 0

measurable

also

for

, and on

= KI(1-X ~ X = 1 ; and

f

is

(0,|

, and

H(kv)

Thus

~(X)

for

= exp H(X)

exp H(X)

It

clear

the

is

that

from Lemma

is an R.V.F. , then,

2.10, w h i c h Theorem

in p l a c e

2.10.

Suppose

on

, such that

exists (0,~) If

lim x*|

(is finite) Then

equation p

X,V > 0

.

, then

and p o s i t i v e

(1.14),

H(X)

solution

and so by T h e o r e m

satisfying

1.4

-~ < p <

U

awkwardness

concerning

S2

and

~I

to the p r o o f of the fact that

if we are w i l l i n g

f

2.9, we m ay

to a s s u m e

consider

is a real-valued

say on

[B,|

f(xx)-f(x) x%(x)

for all

(2.19)

,

and is r e l a t e d

of T h e o r e m

large values,

(2.19)

[B,|

+ 0 .

this n a t u r e

of

the h e a t e r T h e o r e m

no f u r t h e r proof.

ficiently [A,|

above

2.2,

on

measurable

follows.

However,

requires

+ H(V)

for some

result

H(Xl)

above)

functional

= kP

the s t a t e d

r r

= H(X)

Hence

originates

(as

, since

measurable

is a finite,

X > 0 to the H a m e l

is g i v e n by

assumed

K1 ~ 0

X

function,

, B > 0 , and

defined for sufan R.V.E.

x~

= H(X)

in a subset

S

obtains with a finite

,

of positive

limit

H(X)

measure,

for all

k

s

of (0,~).

~+0

H(k) for some constant measurable

on

[B,|

KI .

Theorem

2.11.

If

Kl(X~ 1 ) o = 0 , and

f

is, additionallyj

assumed

~ then H(X)

for some constant

=

= K2gnX

K2 .

Suppose

the conditions

of Theorem

2.10

are satisfied

74

for

a ~ 0 +.

bounded

and

on

if

Suppose

each

additionally

finite

interval

that,

beyond

in the case

a certain

f(x)

= KlXaL(x ) + o(xaL(x))

,

f(x)

= C + KlXaL(x ) + o(xaL(x))

a > 0 , f

point.

Thenj

(x § =)

is if

a > 0

;

a < 0

C = lim

where

f(x)

,

(x + - )

,

exista.

[In the proof of this theorem, which

is long only because

it seems

useful to go into some detail, we shall make heavy use of the result of 4 ~ of w

and the analogous result of Problem 1.8.

In addition,

we shall not be using an assumption of m e a s u r a b i l i t y of

since

f , our approach

will resemble that of the theory of weak regular variation,

evolved in

w We shall initially make use of a s s u m p t i o n = e

and

a # 0

(2.20)

in the form

lim h(x+l)-h(x) x+e X a L ( e x)

where

h(x) = f(eX).

is being taken as Case I.

(2.19), for the case

= Kl(ea_l )

,

K1 # 0

We h e n c e f o r t h refer to the function w h o s e limit

~(x+l).

a > 0

Assume w.s Recall,

that

from the sources

large fixed integer

(2.21)

K1 > 0

(otherwise replace

just m e n t i o n e d that,

D , and fixed

sup {t~L(t)} D
~ xYL(x)

,

y > 0 , as

h(x)

by

-h(x)).

for s u f f i c i e n t l y

x § |

inf {t-YL(t)} D
~ x-YL(x)

m

Select

D

such that

h(x)

is bounded on finite intervals beyond

Then

+

See Exercise

2.13 for c o r r e s p o n d i n g results

for

a = 0

D

75

h(x)

At this stage

= h(D+x-[x])

it is slightly more

~(x) = x-[x]

; so that

[xl

Z h(r+x-[x]) r=D+l

+

0 < 6(x)

convenient

h(r-l+x-[x])

to put

n(x)

= [x]

,

< 1 ; thus

n(x)

(2.22)

h(x)

= h(D+6(x))

Choose

small

arbitrary

below;

and assume

(2.23)

e-> 0

w.s

attention

in accordance

that

0 < Kl(e~

and focus

Z T(r+~(x))eV(r+8(x)-l)L(e(r+6(x)-l)) r=D+l

+

D

with the first

is such that for

) - ~ _< T(X) __< Kl(eC-l)

on the behaviour,

for large

inequality

x ~ D

+ e , x , of

n(x)z e a ( r + ~ C x ) - l ) L ( e ( r + ~ ( x ) - l ) ) r=D+l e g (x-l)L(eX-1) n(x)

<

Z e(~ r=D+l

for any positive

sup x_lYYL(y)}/ea(X-l)L(e x-l) D<=y!e

y < o , simply by using the fact that for

eYWL(e w) <_ { sup x_lYYL(y)} D<_y<e

Thus,

finally,

for

any

0

<

y

<

c

n(x) { sup (y) } D<_y<e x - IyYL (2.24)

"zy (x-~) t. (eX-'], }-

Z

e(O-y)(r+~(x)-l)

r=D+l

' e('C-y) ( x - l )

n(x)

>

Z e~(r+~(x)'l)L(e (r+~(x)-l)) r=D+l

e o ('x- l ) L ( e X - l i

D ~ w ! x=l

76

{ inf

n(x) e(O+p)(r+~(x)_l) Z

.y-~

D<__y~ex-I

r=D+l

e-o'~x-1)L(eX-1)

the

last

inequality

for any

first,

Finally,

being

0 > 0

e(C+p)(x-1)

obtained

in p r e c i s e l y

we need

to n o t i c e

that

n ( Xx ) e n ( r + 6 ( x ) _ l _ x + l )

(2.2s)

analogous

manner

to the

. for any

n > 0

= n(x) X e n (r-n(x))

r=D+l

r=D+l

= { e n _ e - n ( n ( x ) - D - 1) } / { e n _ l }

Thus,

letting

x + ~

in

(2.24),

and u s i n g

both

(2.21)

and

(2.25)

x ) eC ( r + ~ ( x ) - l ) L ( e ( r + 8 ( x ) - l ) ) / e O ( X - l ) L ( e X - l ) s u p n (X

e(O-Y)/{e(O-Y)-l}~lim

x+| > lim inf x§174

r=D+l

n(x) Z

e C ~ r + 6 ~r x j~_ l J~L ( e ~ rr + 6 ~ xr j _~l ) )~/ e C ~ X _ r lJL( ~ eX_l)r

r=D+l

> e(O+P)l{e(O+P)_l}

where

that

y

exists

(2.25),

intervals

if

p

0 < y < ~

x § | and

and

are .

and

Thus, is

together

beyond

K1 = 0

arbitrary

D

if w e

eC/{eC-l}

with , we

positive let

p,y

.

If w e

take

that

h(x)

the fact find,

finally,

h(x)(~

KleoeC(X-l)L(eX-l))

h(x)(=

o(eO(X-l)L(eX-l)))

; which

numbers,

on transformation

+ 0 , we

that

with

the

see t h a t

into

x + =

~ Kle~

;

= o(eOXL(eX))

,

the r e s u l t

for

limit

now

KI > 0

c > 0

re-

quired.

The C a s e

c < 0

As b e f o r e , we n e e d t h a t

as

assume w.s x § |

, for

that arbitrary

Kl(ea-1 ) > 0 , etc. fixed

y > 0 ,

as

(2.22)

on f i n i t e

: if

)

gives

the

account

is b o u n d e d as

restriction

This

time

77

(2.26)

inf

{tYL(t)}

~ xYL(x)

,

sup

x
Assuming

{t-~L(t)}

~ x-YL(x)

x
for the m o m e n t

that

C = lim h(x)

exists,

it follows,

since

X-~r

for p o s i t i v e

integer

m , m-1

(2.27)

h(x+m)

- h(x)

=

z (h(x§

- h(x§

r=o

that

(letting

(2.28)

m § ~)

C - h (x)

=

- h(x+r)}

Z {h(x+r+l) r=o

We shall p r o c e e d right

in a m a n n e r

the e x i s t e n c e

to i n v e s t i g a t e

analogous

to that

of the limit of

the n a t u r e for

h(x)

as

=

Z {h(x+r+l)-h(x+r)} r=o

and,

on the

x §

Z r(x+r+l)e~(X+r)L(eX+r) r=o

in v i e w of the b e h a v i o u r

(asymptotic-wise)

of the series

~ > 0 ; and then f i n a l l y p r o v e

of

~(x)

is the b e h a v i o u r

as

as

x

x + -

§

~

what matters

,

of

oo

Z e~(X+r)L(e x+r) r=o

Use o f a b o u n d i n g

method

as before

gives co

sup

{y-YL(y)}

co

Z e (o+Y)(x+r)

Z e~(X+r)L(e x+r)

X e

r:o

e - X Y L ( e x) inf

r:o

e[O§ {y~

X

e <._y
>_

eOXL(eX)

Z e (e-O)(x+r)

r=o

> --

where

-0 < 0

eXOL(e x)

and arbitrary,

e(Cr-O) x

0 > -y 9 ~ , so that

we g e t

eventually

78

Z {h(x+r+l)-h(x+r)}~{

(2.29a)

Z eO(X+r)L(e x+r)}{Kl(eo-1)}~

r=o

as

x

+

~

if

,

(2.29b)

-Kle~

x)

r=o

K1

>

0

;

r {h(x+r+l)-h(x+r)}

= o(eOXL(eX))

,

if

K1 = 0

r=o

To p r o v e

that

the

limit

as

x § ~

of

h(x)

exists,

note

first

that -h(x)

=

m-i Z {h(x+r+l)-h(x+r)

}-h(x+m)

r--o

so t h a t

lim h(x+m)-h(x)

=

m+|

exists;

and

in f a c t

= -KleaxL(eX)

we

know

that

+ o(eOXL(eX))

Z {h(x+r+l)-h(x+r)} r=o

as

x § ~

Likewise,

, the

for

right-hand

any

side

is

p > 0

m-i =

-h(x)

r. { h ( x + ( r + l ) p ) - h ( x + r , ) } - h ( x + m ~ )

.

r=o

Putting

x = yu

ceding theory, and

, and

h(y)

= h(uy)

(2.19)

for

o < 0 ,

Y§174

as b e f o r e

since

Kl(ecP-i )

§

we

merely

have

e~

taking

the p l a c e

of

that

(2.30)

the

lim h(x+m~)-h(x)

"o" referring

pre-

of

eYCPL(eY~)

obtain

the immediately

m a k i n g u s e now o f t h e fuZZ f o ~ o e o f

~ > 1 , i.e.

~(y+l)_s

we

, and applying

to "as

= -KleaxL(eX)

x § |

Put

C(x,u)

+ o(e~XL(eX))

for

l i m h(x+mp) m+~

Then C(x,p)-C(w,~)

= lim

{h(x+mu)-h(w+m~)}

m§174

-- l i m { h ( x - w + ( w + m p ) ) - h ( w + m ~ ) } m§174

e

,

79

lim m§

:

:

by (2.19) .'.

= C(~)

only,

C(~)-h(x) so that,

letting

and this

is

and from

= -Kle~

This c o m p l e t e s

If the assumptions (2.19),

we assume only

and in the case

(2.30)

,

) + o(e~

x § | , clearly

(2.28).

Corollary.

persists;

0

.

C(x,v)

place of

{f(eX-W.eW+mu)-f(eW+m~)}

C(~)

: C

is i n d e p e n d e n t

the proof,

In the preceding

(2.20),

o > 0

then the result for

= -KlX~

theory,

apart

we do s o , we o b t a i n

analogues

theory

varying

of regularly

lowing theorem,

x +

+ o(x~

Q

lity

of

f , uniformity

Theorem 2 . 1 2 .

from Theorem 2.10

of measurability

of

of c o n v e r g e n c e

valued function,

d e f i n e d for sufficiently

B > O, and measurable

thereon;

and

r

Let

large values, : x~

If

ordinary The f o l -

under measurabi-

intervals

(The U n i f o r m C o n v e r g e n c e T h e o r e m ) .

~ : 0,

[B,|

in w

ensures,

on finite

on in the

as d e v e l o p e d

of Theorem I . i ,

in the case

f

of a number of results

functions,

the analogue

also;

in that in

r:o

we d i d n o t make a n y a s s u m p t i o n

u

of the theorem are weakened,

~ < 0 , we find that as

Z {f(xer+l)-f(xer)}

of

m

an

in (2.19).

f

be

a real-

[B,~),

say on R.V.F.

on

[A,~)

such that

(2.31) as

x

fCkx)-f(x) ~

~

~ [a,h]

for each , where

The t h e o r e m Proof.

§

remains

[Similar

the now u s u a l

X e a,b

: O(r (0,|

true

O
if "0" is r e p l a c e d by "o"

to that of L e m m a

transformation

See E x e r c i s e

+, then this property holds uniformly for

are any two fixed numbers s a t i s f y i n g

h(x)

2.14 for a m o r e

I.I].

We p r o v e

= f(e x)

general

(2.31).

, we have

hypothesis.

as

x §

By m a k i n g

80

(2.32)

h(v+x)-h(x)

for each

~ c (-~,~)

respect to {Xn}

-=

, when

~ ~ [0,i] +.

O(t~(x)) r

= r

If this doesn't hold then there exist sequences

' Xn + | ' {Un } ' U n r [0,I]

(2.33)

We prove uniformity with

such that

Ih(x n + ~ n ) - h ( X n ) I / ~ ( X n ) § |

Define sets

UM, N , VM, N

by

(2.34a)

UM, N = {u:~E[0,2],

lh(Xn+U)-h(Xn)I/~(Xn)

(2.34b)

VM, N = {l:Ic[0,2],

lh(Xn+~n+l)-h(Xn+Un)I/~(Xn+~n)

UM, N

and

VM, N

are 'clearly measurable;

sequence of sets with

N § Select and n o t e t h a t

M,N

UM, N ~

M+

or

N§ ; and

l a r g e enough t h a t

[0,2] C

< M ,

~/n ~ N}

each is a monotone UM, N , VM, N § [0,2]

< M,Wn~N}. increasing as

M

and

m(UM,N) > 3/2 , m(VM,N) > 3/2 ,

[0,3]

v,M,N C [0,3] where

V'M,N = VM,N + ~N (the empty set). Thus - UN ~ VM,N

"

(so that m(V~,N) > 3/2). Hence UM, N V'M,N # ~ EUM, N such that ~ ~ VM,N + UN or

For this

~ , from (2.34a) and (2.34b)

(2.35a)

lh(x N + u)-h(XN)I/~(XN)

(2.35b)

Ih(x N + ~ ) - h ( x N + ~N) I / ~ ( x N + UN) < M .

Hence keeping

M

(since

fixed, and for the

UM,N ' VM,N § w i t h

N , same

< M

N

previously chosen or Zarger,

~

may be k e p t )

Ih(XN+~)-h(XN) I + IhCXN+~)=hCXN+~N)1 < M(O(x N) + ~(XN+~N)) so that, by the triangle

inequality

Ih(XN+~N)-h(XN) ] < M(~CXN) + 0(XN+VN))

+

See Exercise ~

[,::,. ~] C

2.15 for extension of this result for

(--,|

81

i.e.

lh(XN+VN)-h(x N) I/ ~(x N) < M(I+{r and since

~(x) = %(e x)

where

~

N § = , the uniform convergence

limN+|

since

theorem for R.V.F.'s

This contradicts

The p r o p o s i t i o n

*(XN)})

is a regularly varying

lh(XN+,N)-h(XN)I/,

UN r [0,I].

N) /

function,

as

gives

(XN)<-{2MM(I+e~

if

c~ > 0

if

o < 0

(2.33).

w i t h "0" r e p l a c e d

by "o"

in

(2.31)

can be p r o v e d

+

analogously,

m

The following result, totally analogous and proof,

links Theorems

2.11 and 2.12 in showing that measurability

implies the local boundedness Lemma 2 . 3 . that

f

property.

Under the conditions

See E x e r c i s e

Our f i n a l

relating

f

task

to

2.17. is

to obtain

r(x) by (2.36) where

h

a result

L , in the case

Define for

~(x)

We begin with a preliminary

theorem.

1

is a suitable function on

large

A , the function

f • h (t) L (t)dt A

[A,~)

e.g. measurable

and bounded on each finite interval sufficiently far.

Then

(below) holds.

Proof.

From (2.36)

(2.37)

§

See E x e r c i s e

L(x)h(x)

2.16.

such

of t h e k i n d of Theorem 2.1

o = 0

x > A , for suitably

-- h ( x )

h B)

[X,X'], X' > X .

9

lemma, then prove an appropriate Lemma 2.4.

2.12, ~X(X

of Theorem

is b o u n d e d on any interval

Proof.

to Lemma 1.2 in statement

= L(x)-t(x)

1

X

+ ~- S h ( t ) L ( t ) d t A

.

on

[A,~)

(2.38)

82

Now consider X

1 tt 1 dt JAh (Y) L (Y) dY }

f {h(t)L (t) A

= fAXh(t)L(t) ~- - (fXh(y)L(y)dYA ] d__2t_tt

so that, integrating by parts in the second part

= fXh(t)L(t) ~ A

-t -I ;t h(y)L(y)dy I t=x A t=A

-

X

f

t-lh (t)L(t)dt A

X

: x -1 J" h ( y ) L ( y ) d y A so by substitution in ( 2 . 3 7 )

L(x)h(x)

x 1 t 1 : L(X) T(X) + [A {h(t)L(t) - ~ f h(y)L(y)dy} ~ dt A -

so f r o m ( 2 . 3 6 ) X

(2.38)

L(x)h(x)

= L ( x ) T(x)

+ f

T(t)

~

dt

.

A [Note that X

(2.391

X

x-i f h(y)L(y)dy : I ~(t) ~ A A

Theorem 2.13.

In the case

for sufficiently

large

c = 0

A , and

dt

]

under the a s s u m p t i o n of Theorem

2.10,

x ~ A

X

f(x) = f d(t) L ( ~ A where

lim ~(x)

exists and is

dt + o(L(x))

H(X)/s

,

(x § |

(= K2)

X~

Proof. We apply the result (2.36)-(2.38) twice; first let h(x) = f(x)/L(x) which is measurable on [A,| for A sufficiently large; and bounded on finite intervals of [A,| by Lemma 2.3; since I/L is S.V.F.; and put correspondingly y(x) for z(x) ; then

83

X

(2.40)

f(x)

-- L ( x ) y ( x )

+ [

y(t)

L(~

dt

A and a second

application

for

t h e moment t h e

far

of

y(x)

of

(2.36),

requisite

(2.38)

boundedness

and c o r r e s p o n d i n g l y

x(x)

with

h(x)

on f i n i t e = e(x)

= y(x)

intervals

supposing sufficiently

, gives

X

(2.41)

LCx)yCx) = LCx) e ( x )

+ f

e(t)

L(-~-tt d t

A so combining

(2.40)

and (2.41) X

f(x)

a(t)

= f

L(~

dt + L(x) e(x)

A

where as

a(x)

x + |

= y(x) , for

+ r

this

It implies

r

now s u f f i c e s

also

the d e f i n i t i o n

of

, since

(2.42)

J" L ( t ) d t / x L ( x ) A

to prove

boundedness

as

of

y(x)

y

and

§ H(X)/~n~ r

+ 0

from

x § |

X

~ 1

by Theorem 2.1. We now p r o v e course).

the

From t h e

result

:

y(Xx)L(Xx)-y(x)L(x) L(X) by changing

about

definition

of

lim y(x)

(without

use of

(2.4)

I

f(kx)-f(x) L(x)

~-i

xL ~x]-

{

kx

fA

f(t)dt

x - f f(t)dt};

A

variable 1

fC~x) - f ( x ) L(x)

-

f(~x)-f(x) L (x)

_

X

X

f(kt)dt x--~{

IAI~

and hence also

- ~ f(t)dt} A

x x -1- ~ { IA (fC~t)-f(t))dt} - ~

Now

f(~x)-f(x) L[x)

+ H(k)

as

of

y(x)

x

§

1

A f(~t)dt fA/k

84

x-~

and finally

+ H(X)

I~ {f(xt)-f(t)}dt

since

(2 43) "

xL(x)

§ |

, we h a v e

y(Xx)L(~x)-$(x)L(x) '"L'fx)'

for each fixed

x > 0

for a n y

;

x § |

( x + =)

0

X

f(y)dy

;

=

~'(t)

f

A so that,

as

(2.42)

Now, from (2.39)

X

x -1

that

~

, by

also

~

dt

A

X > 0 ,

1

Xx f

~

~

v(t)

dt

Xx { X- 1 f

=

x

x f(y)dy

- f

A

f(y)dy

}

A

X

1

(f(xt)-f(t))dt}

+ o(1)

= x-L-C-~j-{ fA as

x

+

as before; so that

|

1

L-~

Xx J" y(t)

dt

= H(X)

+ o(1)

X-

as before.

Now putting 1

t = xy ; as

X y(xy)L(xy)

I1

x §

_ H(X)

+ o(1)

so t h a t X

fl [ (x § | of Theorem

~(xy)L(xy)-y(x)L(x)

Finally, 2.12

d_y_ + v ( x ) t n x

L~x)

we n o t e

by

} y (2.45)

for

conjunction

y E [I,X]

, so finally

+

0

we s e e

as

+ o(1)

with

,

y(xy)L(xy)-y(x)Ltx) I.(x) uniformly

in

= H(X)

x § |

the

second

part

85

(2.44)

y(x)

+ H(),)/~n

Note that since we have since

7

~ , (2.44)

confirms

Notes

Theorem 2 . 1 i s and s l i g h t l y

due t o K a r a m a t a

extended

proof

is

treatment

essentially

is

(1933, that

different

Theorem IV);

and K a r a m a t a ( 1 9 5 6 ) . occurs

Theorem)

given here

a generalization

first

Aijanclc,

+

apparently,

and f o r

B o j a n i 6 a n d Tomi6 ( 1 9 5 4 ) .

essentially

theorem,

extensions

simple

p > 0 , is p r o v e d by

a g a i n due t o

The results Bojanic

and that

of ;2.3 are

and Tomi6

(1954) For

the reader should consult

as well as the report of Bojanic and

(1963 a). 2.4

i s b a s e d on t h e T e c h n i c a l

(1963 b ) ;

(Lemma 1, p .

with

incorrect

by t h e

and de B r u i j n

(1949),

R. B o j a n i d a n d G . E . H . ErdOs a n d R u b e l h

themselves,

similar

to that

Reuter.

The

as g i v e n t h e r e ,

authors is based)

(1973)

R e p o r t o f Boj

some m o d i f i c a t i o n s .

8 of the reference)

t h e one we h a v e g i v e n ,

and

proof o f T h e o r e m 2 . 1 2 , was p o i n t e d

and v a l i d

proofs

o u t as (on w h i c h

of Korevaar,Aardenne-Ehrenfest

were p r i v a t e l y In a relevant

s t u d y so c a l l e d

anlc

c o m m u n i c a t e d by more r e c e n t

"C-slowly varying

paper,

Ash,

functions"

such that lim h(x+~)-h(x) x§174 r (x)

+

the

for

its proof here,

w

given

(Karamata's

2.7 and 2.8 differ somewhat).

of the ideas of this section,

Section

functions

the is

of a theorem of

Theorem 2.5 i s

are believed new.

(1967),

but

was i n e s s e n c e

taken from the paper of Aijanclc,

the paper of Vuilleumier

being

(1971)

p = 0

earlier versions;

(although the proofs of Theorems

Theorem 2.3

Theorem 2 . 4 ,

due t o F e l l e r

44-47);

(1931 b) following

of the preceding

Karamata

(1971).

w

Theorem 2.2

i n K a r a m a t a (1931 a , h )

i s due t o F e l l e r

E. L a n d a u ( 1 9 1 6 , p p .

Karamata

, and

a more r e c e n t

o f de Haan ( 1 9 7 0 , from both.

in the paper of Aljan~i6

Karamata

H(~)

and D i s c u s s i o n .

Tauberian proof

form of

the second part of

2.9 and 2.10.

Bibliographic

present

nowhere used the explicit

does not depend on

Theorems

2.5.

;~ .

= 0

b e i n g a v e r y famous e x t e n s i o n o f some t h e o r e m s o f H a r d y a n d L i t t l e w o o d ( e . g . 1929) who, w i t h L a n d a u ( 1 9 1 6 ) , c o n s i d e r o n l y the case where L(x) ~ c o n s t . > 0 i n t h e c i t e d p a p e r .

viz.

86

for each

v , where

r

is positive

and decreasing.

This corresponds

in large measure to part of the theory of Bojani6 and Karamata, w h e n K 1 = 0 , a ! 0 ; K 2 = 0 ; and it is not s u r p r i s i n g therefore that there is substantial Finally, for

~ = 0

overlap in the results. still

in relation to w

in a p r o b a b i l i s t i c

paper of M e j z l e r

(1949).

is that of de Haan

for an early use of such theory

setting the reader should consult the

A more recent p r o b a b i l i s t i c a l l y set d e v e l o p m e n t

(1970a, 1971).

EXERCISES TO CHAPTER 2 2.1.

Show that if

U(x)

is r e g u l a r l y v a r y i n g of index

(and s a r i s -

0

lies the integrability c o n d i t i o n of T h e o r e m 2.1) then xk+iu

(x) x§174fXtku ( t ) d t A lim

for each

- 0 + k + 1

k > -o - 1 . (Karamata,

Show that the assertion remains true for

k = -0

- 1

1955) (See

Exercise 1.16). (Parameswaran, 2.2.

The first assertion of Exercise

regularly varying function of index

2.1 shows that if

1961) is a

U(x)

a > 0 , then so is

X

t-lu (t)dt

(~ U ( x ) / : )

A

(Kohlbecker,

Show that this regular v a r i a t i o n remains even if is slowly varying,

a = 0

1958)

i.e. if

L(x)

then so is

X

f

t-lL(t)dt

(Parameswaran, 1961)

A

Hint: Use the f a c t t h a t t - i L C t ) ~ t - i L o ( t ) as t § | where (See e.g. Exercise 1.20; and then t'ILo(t) is monotone d e c r e a s i n g . the second p a r t of E x e r c i s e 2 . 1 . ) Why is it trivial that as

x §

87

X

t- (l+a)LCt)dt A is s l o w l y

varying

2.3.

Suppose

for

k < -i - p

for

U

6 > 0 ?

is r e g u l a r l y and

x

f

varying

sufficiently

with

large

exponent

Show

p

the

integral

that for

k <-l-p

that

tku(t)dt X

is w e l l - d e f i n e d

and

finite.

lim

Show

also

xk+iu(x)

= - k - p - 1

x § 1 7 4~ | X

if it is a s s u m e d

in the case

k = - 1 - p

Ft-l-PuCt)dt

that

<

A for some case,

A > 0 .

(If this

the p r o p o s i t i o n Conversely,

sitive

suppose

and m e a s u r a b l e

integral

is t r i v i a l l y

diverges

that for a f u n c t i o n

on

[A,-)

for e a c h

large

A

in this

true).

for some

U

, defined,

finite,

A > 0 , it is a k n o w n

po-

that

xk+lu(x) §

f|

-bk-I

(t)dt

X

where

-bk-I

index

p = bk - k

is f i n i t e

and p o s i t i v e ;

then

U

is r e g u l a r l y (de Haan,

2.4.

Let

f(x)

0 < B < ~ < B < |

varying

with

.

be integrable for

on each

some f i x e d

B .

finite Suppose

interval the

limit

2 ~ , to s h o w

that

1970)

(~,~)

where

X

lim x -1 f f(t)dt X§174 B

exists

and

is finite.

Use

Theorem

2.2,

= p

the

function

88

p(x)

, defined by x

= f

log p(x)

~

dt

,

x

>

B

B

p .

is regularly varying with index

(Aljan~i6 and Karamata, [Hint:

the problem

nition of p(x)

= 1

f(x)

is made analytically

and

p(x)

in this region;

to

for

by putting

then integrals

T h e regularly varying

2.S.

the representation,

[0,B)

simpler by extending

function

f(x) = 0

1956) the defiand

can be taken on

[0,x), x ~ B].

of Exercise

2.4 clearly has

p(x)

x > B , x

p(x)

= exp

{p l o g

x + f

/ ~

dt

+ const

is continuous,

and

}

B

where

y(t) = f(t)

- p -I

x

x

;

y(t)dt

+ 0

B

as

x § |

Compare

and contrast

given by the R e p r e s e n t a t i o n Construct

a

y

satisfying y(x)

+

2.6.

Prove part

2.7.

Extend the s t a t e m e n t

and subsequently

o

for

p(x).

the last condition,

as

3 ~ of Theorem

x

but such that

+ |

2.2.

of Theorem

x = J" u ( y )

U(x)

this result with the r e p r e s e n t a t i o n

Theorem

d{yo}

2.4 first to the case when

,

~ > 0

to the case when x U(x)

where

B(x) = x = ~ ( x )

with index

o , and

= f

u(y)

is monotone U(x)

= xPL(x)

{dB(y)}

non-decreasing , p > 0 .

and regularly varying

89

2.8.

Let

for

qn ~ 0 , n = O , 1 , 2 , . . ,

0 ~ s < i .

If

L

and s u p p o s e

is s l o w l y

varying

Q(s)

and

Q

|

=

z qk s k=o

k

converges

satisfies

QCs) = ( 1 - s ) - P L ( 1 / ( 1 - s ) ) as

s ~ i-

to s h o w

, 0 < p < ~

, for some

that

as

, use

Karamata's

Tauberian

Theorem

n § n-i

kZ_oqk ~ n P L ( n ) / r ( p + l ) and T h e o r e m monotone,

2.4

then

to s h o w as

that

if,

in a d d i t i o n ,

the s e q u e n c e

{qn }

is

n §

{qn/n~ so that,

;

+ o/r(o+l)

in p a r t i c u l a r ,

for

;

p > 0

qn ~ n P - I L ( n ) / r (P)

Hint:

Let

U

be the n o n - d e c r e a s i n g

function

defined

by

X

U(x) = f

u(t)dt

,

x >_ 0

0

where

and note

that

u(x)

= qn

U(n)

=

w(x)

- f

'

n < x < n+l

n-i z qk k= o |

,

n >_ 0

' n > I , while

e -xt d { U ( t ) }

= {(l-e-X)/x}

o

where

the

last

Feller

(1971,

sum p.

that

is

Q(e-X).

(This m e t h o d

~

z qk e k=o

of d e d u c t i o n

-kx

is g i v e n

by

447)).

2.9.

Show

q = 0

if the

assumption

the

conclusion that

placed

by the

assumption

that

of T h e o r e m L

2.6 p e r s i s t s

is e v e n t u a l l y

for some

large

for

the c a s e

non-increasing

fixed

C

, L(x)

is reis

90

uniformly 2.10. as

bounded

Use x -~

away

Theorem

from b o t h

2.6

to

zero and i n f i n i t y

deduce

a result

about

on

the

[C,-). asymptotic

behaviour

of

~

0o

f (t)R(xt)dt (I

where

R(x)

= xPL(x)

is a r e g u l a r l y

2.11.

Show that

assumed

to be n o n - d e c r e a s i n g

the c o n c l u s i o n

to be the p r o d u c t

2.12.

lim f ( ~ x ) - f ( x ) x+~ x~

of T h e o r e m

continuous,

of two m o n o t o n e

Show that,

varying

for f i x e d

= KI(kO-I)~

function

2.8 p e r s i s t s

but

Parts

continuous

that

K2 ~ 0

point;

f

Bojani6

~ > 1 , (with

o < 0)

(a)

we e n d o w

L(x)

results

L Show,

> const.

f(x)/L(x)

depending,

§

respectively,

Show first

I n t e g r al

Test

(with the h e l p

for c o n v e r g e n c e )

large

2.11 for b o t h

about

the case

1954)

for

x

< ~ =>

and

o < 0

~ = 0 , if we a s s u m e

interval

some a d d i t i o n a l

for e x a m p l e, > 0

o > 0

beyond

appropriate

that by r e q u i r i n g sufficiently

a certain property

addition-

far,

(x + ~)

L(x)

K2 > 0

or

is e v e n t u a l l y

in p a r t of L e m m a that

K2 < 0

1.7 and the C a u c h y

for any large

k > 1 ~

f|

non-increasing

fixed

(y)/y}dy

x

<

E

E ; and

f {L(y)/y}dy E

with

on w h e t h e r

|Z L(xk r) < | , some r=o for f i x e d

and Tomi~,

x§174

on e a c h f i n i t e

+ =

On the o t h e r h a n d a s s u m e x .

is not assumed

Z {f(xkr+l)-f(xkr)} =-KIXOL(x)+o(xaL(x))

is b o u n d e d

(such as m o n o t o n i c i t y ) . ally just that

L

functions.

(AIj anclc, ~'"

of T h e o r e m

to give some and

providing

p

r=o

of the p r o o f

can be a d a p t e d

if

(more g e n e r a l l y )

as

2.13.

of i n d e x

z L(x~ r) < ~ r=o

,

all

~ > 1 .

with

91

Consequently, f {L(y)/y}dy < ~ E

show t h a t and

if

: L(x)

lim f(x) = C x

is

eventually

non-increasing,

exists + and is finite,

then

{C-f(x) }/L (x) § + |

(b)

m

according as

2.14. for

K2 > 0

or

K2 < 0

respectively.

Extend T h e o r e m 2.12 by r e q u i r i n g

(2.31) to hold initially only

r S , a set of p o s i t i v e measure.

2.15.

In the p r o o f of T h e o r e m 2.12, extend the u n i f o r m i t y result re

(2.32) from

~ r [0,I]

to

u r [a,8]

, -- < a < 8 < ~

2.16.

Prove T h e o r e m 2.12 w h e n "0" in (2.31)

2.17.

Prove Lemma 2 . 3 .

2.18.

M a k i n g the assumptions of Qn. 2.13,

tion of ultimate m e a s u r a b i l i t y of two results

2.19.

Let

(a) and

f

(b) of Qn.

and the additional

assump-

f , use T h e o r e m 2.13 to deduce the

2.13.

b e an e v e n t u a l l y

is r e p l a c e d by "o"

Hint: Make use also of Qn.

measurable

function

such that

2.3.

for

each

~ > 0

l i m sup x+| is finite.

Show that

f(xx) - f ( x ) x~ f

is bounded on every finite interval suffici-

ently far. (See also Delange,

+

1954).

This further p r o p e r t y of f will now hold a u t o m a t i c a l l y (as is desirable), if a minor a priori r e g u l a r i t y a s s u m p t i o n about f , such as eventual continuity, is made, by a "croftian" theorem (Croft (1957); Kingman (1963)).

APPENDIX GENERALIZATIONS A.I.

R-O

Varying

procedure quires

of

Functions.

OF REGULAR VARIATION This

generalization

(i.i) by a two-sided boundedness

it to hold only for an interval

to some extent,

the uniformity

Definition A.I.

A function

if it is real-valued,

of

("0")

~

K

the limiting

condition,

values

of convergence

positive

replaces

but re-

(thus building

in,

property).

is said to be and measurable

R-0 on

varying

[A,-),

at infinity

for some

A > 0,

and (A.I)

m < K(~x)/K(x)

where

m,M

and

a

< M

are any constants

,

1 < ~ < a

satisfying

0 < m < 1 , I < M <

I < a < ~ Clearly in virtue

on

functions

of the U n i f o r m Convergence

by taking sitive

all regularly varying

A

sufficiently

and measurable [A,~)

large.

satisfy

Theorem,

for a fixed

More generally,

function which

satisfies the definition,

these requirements, a,m

is bounded away from both even for all

and

M

any real-valued po0

and

~ > I ; thus various r

simple oscillating varying, small,

such as

are

R-O

bounded M

function on

and

[A,-) 0

is automatically

as not being regularly

x~{l + ~ sin

(though

ex

still

|

is not).

If the measurabithen even a non-

it is evident

to monotonicity,

more generally,

requiring

and

that if

one of the bounds by

m

it will be evident

that to some extent an even more general

can be developed by, say,

with

if it were positive

Finally,

satisfied;

(27 log x)}

in the definition,

would comply,

and

is strengthened

from the sequel present

noted hitherto

were to be dropped

away from both

measurability and

2 + sin x varying

lity requirement measurable

functions,

theory than the

only the right-hand bound,

M, in (A.I). R-0 K(i/x)

variation to be

Lemma A.I.

R-0

An

at

0

of a function

R-O

varying function

on any finite subinterval Proof.

K

may be defined by requiring

at

It follows

from

of

(A.I) that,

m p < K(kx)/K(x) - -

K

is bounded away from

0

and

[A,~).

< Mp w

for

I E [ap-I,

ap]

and all

x_>A,

93

Writing

f(x)

= l o g K(e x)

log

p

Putting

x o = log A f(Xo) V~

C o r o l l a r T. (since

m

f(x+v)

<

+ p log m ! f(Xo+V)

e [0,p log a] K(x)

and

~ ~ log X e [0,p

~ p log M ,

l o g a]

x ! log A .

on each finite

positive

bound

! f(x o) + P log M ,

m

is i n t e g r a b l e

it is measurable,

T h e o r e m A.I.

f(x)

for

,

true even if the s

K(.)

, we h a v e t h a t

and bounded

in (A.I) were

(Representation

Theorem)

subinterval

[A,|

of

which would

above,

remain

dropped). For an

R-0-

varying

function

X > A ,

all

m

X

CA.2)

KCx) = exp

{nCx)

s(t)~ dt}

+ f B

n

where

and

manner

s

K(.)

function

R-O

is

are b o u n d e d having

on a

Let

missible

from Lemma A.I),

K

be

(~) which is measurable follows

on

[A,|

Conversely,

, A > 0 , a representation

R-0

varying,

1

=

and for

x ~ A , write

a

- ogY~J'

log

[ K(tx)

1

and bounded on

[A,|

that l o g K(x) = 6 ( x )

+ l o g1 a Sa1 l o g K ( t x )

d_.~t t "

Since a

I

i

in this

varying.

Proof.

It

and m e a s u r a b l e

[A,|

ax

log K(tx) ~- =

I

x

log K(t) dtt

ax = ~

A

log

K(t)

dt T

x

-

]" A

log K(t)

d_.~t t

(as is per-

a

94

aA

S

=

K(t) ~- + f

log

A

ax

K(t) -~- - f

log

aA

aA

x

A

A

x 10g

K(t)

A

dt

7-

= f log K(t) -.~ + f iog ( ~ ) ~ whence log K(x) = 6(x) + log1 a

whence

SAaA log

(A.2) follows by defining

.~_

K(t)

n(x)

x

1

SA l o g

+ log a

(KK~)~

-

as the sum of the first two

terms, and r

1

= ~

K(at) log ( ~ ) ,

t ~ A .

Indeed h < r

< H

where

(A. 3)

h = -log m/log A

,

H = log M/log a .

The converse assertion is trivially verified by checking Theorem A.2. (a)

For

There

R-O

an

exist

varying

positive

numbers

(A.4.a)

y-aK(y)

< Mx-~K(x)

(A.4.b)

mxSK(x)

< ySK(y)

for

y

> x > A

; indeed

we

function

may

take

a

a = H

K and

and

(A.I). g

. 8

such

that

8 = h , as

defined

in

(A.3) (b)

(A. S)

*

For

any

k

> h

xk+IK(x) /

-

1

~

,

ykK(y)dy

This remains true, as the proof will reveal, even without measurability in Definition A.I; further (A.4.a) remains true with only the r.h. bound in (A.I).

[(A.4) shows t h a t x-aK(x)

is "almost" decreas-

ing, while xSK(x) is "almost" increasing, on [A,| version of (b) was obtained by R. Mallet.

A one-sided

95

is b o u n d e d

away

from

Conversely,

[A,|

for

some

terval

of

[A,|

or

(A.5)

and

suppose

|

K

[B,|

on

is finite,

bounded

Let

away

implies

from

K

that

, for any

positive,

integrable

(A.4) for some

Then

assumed

(a)

and

A > 0 , and Lebesgue

B > A , each

Proof.

0

m,M

0

and

|

is

R-O

on

y > x > A , and choose

p

B > A

.

and measurable on every

finite

~-1)

< M ,

such

[B,|

as is

that

fAx y k K ( y ) d y "

xa p ~ y < xa p+I

v = 1,...,p

and m _< K ( y ) / K ( x a

whence,

on multiplication

p) _< M ,

of these

m p+I <__ K ( y ) / K ( x ) where

P Z log

(y/x)/log

p+l

inequalities,

<__ M p+I It follows

a .

m(y/x)log

m/log a < mP+l

,

m(y/x)log

m/log a Z K ( y ) / K ( x )

that

MP+l < M(y/x)log

M/log a

whence

(b)

From (A.4.a) with y-Hxk+HK(y)

whence

integrating

both

I M(Y/x)l~

a -- H , < MxkK(x)

sides

over

x r [A,y]

y-HKfy ].. [ y-k+H+lk+H+l - A.k+H+l 9 I ~ M fY xkK(x)dx A SO

yk+iK (y) j"y x k K ( x ) d x

9

<

M(k+H§ 1.'(~/B) k+H'l

A

for

y > B > A .

Similarly

mxkK(x)

from (A.4.b) with

< xk-hyhK(y)

subin-

0 < m < i, 1 < M < | , on [B,| for some k

Then m < K(xa~)/K(xa

on

8 = h

M/log

a

96

50

m fy xkK(x)dx A

Y k-h+l Ak-h+l k-h+l

< yhK(y) -

"

! ykK(y) / (k-h+l) To prove the converse, bounded away from 0 and u(x) = xkK(x) , and put

we start with the assumption |

x 9 B 9 A

for some

of (A.5)

k ; write

x

r whence,

. fA u ( t ) d t

= xu(x)

integrating

< x log

f

} u(t) dt

=

~

A

dt

A

X

so

f u(t)dt A

bounded on

has the representation

[B,|

of Theorem A.I, as

On the other hand,

for

is

1 < ~ < a , for arbitrary - -

fixed

~(x)

D

a > 1

xXu(x~)/ xXUCx) Xu~ (t)dt fA

f u(t)dt A

is bounded away from both U(X~)/u(x)

0

is also, whence

Secondly,

assuming

and

~

K(x)

for

is

initially

x > B , so it follows

R-O

(A.5) is bounded away from both

and

x > B > A , whence by the immediately K(x)

and

SA xkK(x)dx

By comparison that the derivation

are

0

and

~ , for any fixed

preceding

R-O varying,

of Theorem A.2 with Theorem of many results

to those for R.V.F.'s

are possible,

for

the result analogous

it follows

2.1, it becomes

R-0 varying

functions

and by not dissimilar

to Exercise

that

and merely

evident analogous

proofs.

We

state,

for

2.5.

Theorem A.3.

For an R-O varying function

K , for any

(the integral

f~ ykK(y)dy

and)

is w e l l - d e f i n e d

k > 8 -1

a

shall not, therefore pursue other such results, comparison,

that

[B,|

(A.4), by the proof of (b), it follows

that both

on

k < -H - i

97

(A. 6)

f

xk+IK(x)

ykK(y)dy x

i8 b o u n d e d

away

from both

Conversely,

[A,~)

suppose

for some

and b o u n d e d

R-O

as is

from

~

=

since

away from

S-O

and an "index" of

that,

and

|

on

k , (A.6) is

[A,|

on

already,

f~ ykK(y)dy

is

(A.6)

, then

K

is

R-O

it needs .

It is evident

variation.

of (b) of

only be bounded

A similar remark applies

.

of isolating a concept

R-O

and m e a s u r a b l e

for some

[A,=)

is positive

Varying Functions.

lacks the capability

0

on

to Theorem A.3 in connection with A.2.

positive,

that in the proof of the converse

(A.5)

to yield that

[A,|

on

is finite,

fx ykK(y)dy

It is worth noting Theorem A.2,

K

and

A > 0 , and it i8 known

well-defined varying

0

that the above theory

of "slow"

R-O

This may be overcome

variation,

to an extent

in the following manner. Definition A.2. varying

An

(at infinity)

R-O

varying

J

(A.7)

c <__ lira sup K ( I x ) / K ( x )

1 < c

c

-1 C

and

K

satisfying

(A,8)

+

are constants

is actually

K(x)

two results Lemma A.2.

is said to be

S-O

< C ,

1 < l

(independent

of

~),

satisfying

<

When but when

C

K

s

X-+~

where

function

if

= xPL(x)

an S.V.F.,

serve to manifest For a positive~

(with

c = C = i),

the analogy with S.V.F.'s and m e a s u r a b l e

(A.7) , for arbitrary l -E <_ K(~x)//K(x)

See Bibliographic

(A.7) holds

, p ~ 0 , this is no longer true.

Notes

function

K

The following

further. + on

r > 0, x ~ Xo(C ) , % ~ ~o(r ! l~

and Discussion.

[A,=)

,

98

Proof.

As i n

Lemma A . 1 ,

Sn = { ~ ; ~

let

.> y ,

f(x)

= log

- e ~ . < f ( x + u ).

-

K(e x)

f(x) .

.

Define +

< e~

,

V x > n}

/

where

~ y(e)

> 0

= {6 + m a x

for

since

some f

then

c,

fixed

log C)} / e , for a r b i t r a r y

Clearly

n:l so,

(-log

no

,

Sn :

[y,|

Sn

must

o measurable.

is

Pl § ~2

c Sn

.

be

a set

Now i t

Thus

is

Sn

of

easy

contains

positive

Lebesgue

measure,

to

that

Ul,U2

check

a half-line

if

e Sn

(T(e),|

,

Thus

O

for

any

~ e

(T(e),|

and

-e~

which

yields

the

(A.8)

desired

~

conclusion.

Such a function

An

A.3.

S-O

Proof. Theorem

varyingj and

K

is bounded away from both

(A.7) A.I

holds

that w e

from Lemma -a

a

is t a k e n

Following Steinhaus

with could

:

and

9

for prior fixed

(A.I)

for all

c = m,

C : M.

~ > 0 , and

We saw

X L 1

l o g1 a l o g

( ~ )

,

t

> A --

e(t)

sufficiently

L ~ t a c (1970a) (1920) T h ~ o r ~ m e

<

a

,

large

VII.

t >_ t o ( a ) (a

>_ a o ( a ) )

t ~ to(~)

in the p r o o f

take

A.2 <

0

then

in its exponential representation may be

e(t)

le(t) l ~ a

e(t)

so that

o n l y on t h e r i g h t ,

varying function satisfying

R-O

taken to satisfy

+ *

<

on any finite interval sufficiently far.

Lemma

if

f(x)

1)

Z)

is

f(x+.)

<

If inequality (A.7) is satisfied is satisfied on the right also.

Corollary.

|

x _> n o ~ n o (~)

all

"

of

99

The notion Suppose

of

a n index may b e

a positive

(A.9)

measurable

introduced

function

in

on

XPc i lim sup K ( X x ) / K ( x )

a manner

[A,~)

akin

satisfies,

! C XP ,

to for

(A.7). some

p :

I _< X

X+~

where

c

and

C

are as before:

then the function

xPK(x)

satisfies

(A.7).

Again, some more

a one-sided

general

Definition

A.3.

for e a c h

X > i

constraint

definitions Let

H

along these

be

Indeed

in (A.9) may be preferred.

a positive

lines have

function

$(X) ~ lim sup H ( I x ) / H ( x )

occurred.

defined

on

[A,~).

If

<

X+~

then

H

is called

Lemma A.4.

ficiently

Proof. 1.7.

If a far,

See

~

A

function

then

w

$(~)

H

> XD

is bounded on finite for some

intervals

of Lemma

[Under the appropriate

condition,

lim sup h(x+l)-h(x)

< s.]

--

Monotonicity;

generalized lication

Dominated

regular

of

Definition

name of

A.4.

Variation.

arisen

R-O

results

theory,

[nor indeed

and consequently

variation.

A positive

monotone

(at infinity)

lim sup K(XoX) / K(x)

function

(A.10.b)

lim sup K(x/k o) / K ( x )

< ~ , if < ~ , if

varyrefe-

to other notions monotone

K

on

if for some fixed

(A. 1 0 . a )

R-O

without

with considerable

for the specialized

dominated

to vary dominatedly

= s

Study of monotone

in probability

variation

variation,

of the general

the

1.12 and then Theorem

l

has recently

rence to the concept

suf-

p , ~ > 1

the proofs

ing functions

under

function.

and adapt

lim sup h(x)/x A.3.

a

K

[A,=) ~o > 1

dup-

setting],

is

said

is non-decreasing; K

is non-increasing.

X+~

Lemma A.5.

A function

if and only if it is

R-O

K

defined on varying on

[A,| [B,~)

is dominatedly for

some

B ~ A

of

varying .

i00

Proof.

If

K

sufficiently

is n o n - d e c r e a s i n g

large,

and v a r i e s

and a r b i t r a r y

then

for

B

!

!

0 < m 0 , and all

I < X < X --

--

0

In the case of n o n - i n c r e a s i n g and a s i m i l a r

argument

The c o n v e r s e Note entirely

that

is t r i v i a l

In this

lim inf K ( X o X ) /

f r o m the d e f i n i t i o n K , s(t)

for monotone

positive

K , we have

K(x)

> 0

then s u f f i c e s . of

R-O

variation. i

A.I m a y be t a k e n as

in T h e o r e m

or negative.

monotone case,

also,

Theorems A.2 and A.3 take

an e l e g a n t

form. T h e o r e m A.5.

(A)

Let K

If

(a)

(A.II) [indeed,

K

be p o s i t i v e

and monotone

is n o n - i n c r e a s i n g ,

then

A necessary

and sufficient

dominatedly

is the

8 > 0

that

such

mxBK(x) we m a y

[A,~)

on

condition

existence

for

o f an

for y > x > B > A

m,

it

to v a r y

0 < m 0

[B,| (c)

,

~

ykK(y)dy

varies

dominatedly,

on

B > A A.

for

A necessary

and sufficient

dominatedly

is

condition

for

K

to vary

x (A.12)

0 < ~ ~ lira inf x k + i K ( x )

f

x§

for some f i x e d

k

(B)

is n o n - d e c r e a s i n g ,

If

K

, k+l

(a) A n e c e s s a r y is

> 0 .

for

then

and sufficient

the e x i s t e n c e

that

ykK(y)dy A

o f an

Y > x > B > A

M,

condition

for

i < M < ~

, for

some

and B

,

it an

to v a r y a > 0

dominatedly such

i01

y-~K(y)

< Mx-aK(x)

[indeed we may take

(b)

For

k+l < 0 , f

~ = H = log M/log Xo]

ykK(y)dy

varies dominatedly,

if it exists.

X

(c)

(A.13)

A necessary and sufficient condition for dominatedly is

0 < ~_ - limx§174 xk+IK(x) / ~

for some fixed

Proof. We have

k, k+l < 0

K

to vary

ykK(y)dy

(given the denominator exists).

We shall prove only (A); the only really new feature is (b).

/ x (0 <_) xk+lK(x) ~ ykK(y)dy <_ xk+lK(x) {K(x) ;A =

(k+l)

ykdy ]

/{I-(A/B) k+l}

for x > B > A . We may now proceed essentially as in the proof of the converse of Theorem A.2 to obtain f~ yk+IK(y)dy R-O varying on [B,-), hence dominatedly varying. In almost precisely the same manner as in Theorem A.2 we may also obtain sufficiency in (c), using (b). K varying dominatedly then implies (A.II) by Theorem A.2, which in turn implies (A.12), which completes the circle of implications. Similar techniques can be used to prove Theorem A.6.

(A)

Under the prior condition of Theorem A.5.(B)

necessary and sufficient condition for

~ > ~ -- limx§

for some fixed

K

xk+iK(x)/f~

, a

to vary dominatedly i8

ykK(y)dy

k, k + 1 > 0

(B)

Under the prior conditions of Theorem A.5.(A)

necessary and sufficient condition for

K

to vary dominatedly is

, a

102

co

> ~ -: limx§

for some

k, k + 1 < 0

x k + i K ( x ) / f x ykK(y)dy

(given the denominator

exists).

The conditions are evidently sufficient for dominated variation of ;~ ykK(y)dy

and

fx ykK(y)dy

respectively.

Thus far the dominated variation theory has strengthened the results of the

R-O

variation theory only in quite minor ways, and this

may be expected to be generally the case.

Nevertheless,

the probabilis-

tic applications tend to give its overall further development a new direction.

In a nutshell, considerable attention is lavished on the

case when quantities

K

is non-decreasing on

[A,|

and how the behaviour of the

X +

Up(X) : ;x-Y-PK(dy)

for

,

Vp(X) = fA+yPK(dy)

p > 0 , the first quantity being assumed to exist (be finite), to that of K . Our own approach hitherto, has considered, in

relates

place of these quantities,

Sp(X) =

y-p-lK(y)dy x

to which

U

P

and

V P

,

T (x) = f y P - l K ( y ) d y , P A

are simply related

(A. 1 4 . a )

pSp(X) = Up(X) + x-PK(x -)

(A. 14 . b )

pip(X)

:

= -Vp(X) + xPK(x+) + c o n s t ,

as follows from integration by parts.

From these last relations it is

to be expected that the new interrelationships may be deduced without difficulty from the preceding results. It is of value to note tkat here

Theorem A.7. on that

[A,-)

(A)

Let

For

K

K

be positive

lim y-PK(y+) = lim y-PK(y-) = 0 u247 y~| non-decreaslng

to vary dominatedly

and left-continuous

it i8 necessary

and sufficient

103

(A.15.a)

r

lim inf x - P K ( x ) / U p ( X )

=

9 0

X-+~

for some

p > 0

(providing the denominator exists).

vary dominatedly

it is necessary and 8ufficient

(A.15.b)

~

r

lim sup x-PK(x)/Up(X)

For

Up(X)

to

that

<

X+~

on

[A,|

(B)

Let

For

K

K

be positive non-increasing and right-continuous

to vary dominatedly

it is necessary and sufficient

that

I ~_ - lim inf xPK(x)/Vp(X)_ > i

(A.16.)

for some

Proof.

p > 0

(A)

From (A.14.a) pSp (x)

(A. 17.)

= x-PK(x)

U (x) P x-PK(x)

+

i .

Taking lim sup (x § =) throughout (A.17) we find that dominated variation of K is equivalent to (A.15.a) by Theorem A.5 (B)(c). To deal with Up(X) , it is convenient to extend the definition of K(x) to [0,A] by putting K(x) = K(A) in this interval; this does not change Up(X) for x ~ A . From integration by parts X

yp-iUp(y)dy = xPU (x) + K(x)

p f o

P

whence X

P ;

yp-Iup(y)dy o

=

i

+

x PK(x)

xPUp(X) SO X

fA yP-IUp(y)dy lim sup x§

lim sup x-PK(x) xPU (x) P

x§

-Up(x)

104

which proves

that

~ < -

vary dominatedly,

(B) (A.14.b)

Up(X)

and sufficient

in mind

K

lim inf x§

p T (x) P xPK (x)

= i

A.4 Bibliographic

we have from

lim sup Vp(x) x§ xPK(x)

(A) a necessary

and sufficient

is (A.16.).

level of generality

(1971).

For S.V.F.'s,

made substantial

Lemma A.2 occurs

(1962, esp.

use previously

and Theorem A.7

in probabilistic example,

- though R-O

varia-

2 and 3 in Bojanid and

in Pitman

(1968) and

occur in the sadly neglected but profoundly

paper of Matuszewska K

for

Definition A.3 and Lemma A.4, with much related and in-

teresting material,

monotone

(1935)(1936)

- as is the concept of

Lemma A.I and Theorem A.I occur as Lemmas (1970a).

condition

9

Theorems A.I and A.2 are both due to Karamata

Seneta

to

Notes and Discussion.

not at the present

L~tac

Up(X)

is still non-decreasing,

Bearing

to vary dominatedly

tion.

for

by Theorem A.5.(A)(c).

that

Hence from Theorem A.6 K(x)

is necessary

being non-increasing,

Dominated variation

See, for

esp. Chapter i, w

in a different

et seq.,

context.

(1956) also contains material

for

(1966)(1969)

itself is earlier.

(1961),

may be found,

paper of Bari and Ste~kin

of which we have already

the concept

and Rutickii

where further properties

w

in this book.

(A) were developed by Feller

connections;

Krasnoselskii

w

important

on

The

R-O

varia-

tion. It is of some general in Zagreb,

Yugoslavia

in mathematics

interest

on Feb.

in Belgrade,

to note that J. Karamata was born

ist 1902, but began his tertiary

in 1922.

W. Feller was born in Zagreb on

July 7, 1906, and studied at the University leaving Yugoslavia.

Feller

of Zagreb,

(1965),(1969),(1971)

full credit for the theory of R.V.F.'s, variation),

although

Ironically,

Feller's

fuller development

regular variation"

appears

then

tends to accord Karamata (that is, basically,

the above circumstances

stage Feller may have been aware of Karamata's of Karamata's

1923-1925,

but does not appear to connect

him at all with the idea of dominated variation R-O

studies

indicate

that at some

(1935)(1936)

(1969) of "One-sided

contributions. analogues

in the Karamata memorial volume

105

of

L'~nseignement Mathematique, w h e r e i n t h e o b i t u a r y V

.

Bari and Steckln forbeing unaware of Karamata's death quickly s

Karamata's.

R-O

M. Tomi~ e x c u s e s

work.

Feller's

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Jovan Karamata

(1902-1967)

L'Enseignement

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(1968)

Trigonometric

Series.

Vol. i, Cambridge University

SUBJECT INDEX asymptotic behaviour of integrals 63ff, 90 asymptotic inverse of an RVF 23ff Bojani6-Karamata class 69ff, 90 boundedness on finite intervals 16, 29ff, 44, 81, 92 Cauchy's functional equation 10ff, 26, 35ff, 44 Characterization Theorem 9, 31, 43ff complementary RVF's 25ff composition of SVF's 19 conjugate SVF's 25ff, 46 croftian theorems 45, 91

differentiability (in Representation Theorem) 6, 16ff dominatedly varying function 99ff dominated variation 99ff, 104 integral conditions for 100ff A-function 99 Egorov's Theorem 9, 44 extreme values in random samples

70, 86

Feller, W. ii, 104 Frullani integral, Sl fundamental theorems for RVF's C-slowly varying functions 85 Hamel basis 36, 45 Hamel functional equation

9, 73

index of regular variation I, 54 index of R-O variation 98 integral conditions for regular variation 53ff integral conditions for R-O variation 95ff iterate of a slowly varying function 2, 46ff Karamata, J. ii, 104 Karamata's Tauberian Theorem 59ff, 85, 89 density version of 60, 85, 88 Kronecker's Theorem 38 Lusin's Theorem

9, II

Markov processes 44 monotone R-O varying function 99ff monotone RVF 20ff, 37ff, 49, 59ff, 88ff. monotone SVF 15, 45ff non-measurable RVF-like functions normalised RVF's 26 oscillatory

functions

2, 49, 92

34ff

112

probability density for Representation Theorem rapidly varying function 43 regular variation 1 integral conditions for 53ff regularly varying function (RVF) 21, 29ff regularly varying sequence 45ff Representation Theorem 2, 16ff, 43ff extended 93, 98 R-O varying function 92ff, 104 one-sided 92ff RVF - see regularly varying function sequential criteria for regular variation slowly varying function (SVF) 1 slow variation in sense of Zygmun d 49 S-O varying function 97 one-sided 98 Steinhaus' Theorems, 9, 10, 16, 71, 98 sum of SVF's 19 SVF - see slowly varying function Tauberian theorems 59ff total variation of RVF's on subintervals Uniform Convergence Theorem extended 79

2, 43ff

Weak Characterization Theorem 31 weakly slowly varying function 32 weakly regularly varying function 29, 45 weak regular variation 29ff, 74 Yugoslav school,

iii

45

68

17, 48