Geometrical and Statistical Aspects of Probability in Banach Spaces: Actes des Journées SMF de Calcul des Probabilités dans les Espaces de Banach, ... 20 Juin 1985 (Lecture Notes in Mathematics 1193)

Ù, sup( 1/2 ( 1,

p (t) =

Ilx-ty!!) -

i!xl! = IIYII = 1 ) '

30

satisfies :

P(t)

C tP

C being a positive constant. It is well known that the norm II II is differentiable away from the origin ; let's denote by D the derivative of

H.

If now one associates to D the following fonction F : B

x 0, and :

F(x) = xII F(0) = 0 ,

D(x/

B' :

)

one can check that F has the following two properties [19] :

(IIF(x)II Bt (ii)

C > 0

=

IxI 1333-1

Ilx-F y liP - Px

V (x, y) E 13 2 ,

P I F(x)(Y)

These two properties are crucial for reducing the infinite dimensional SLLN to a scalar one. Other geometrical properties of a p-uniformly smooth space that we will use are its reflexivity [5 ]and the fact that it is of type p [ 15 ] . Usually the SLLN problem in (B.

H) is stated in the following way :

"Let (X ) be a sequence of independent, centered,

B-valued r. v. and denote by k Sn = X 1 +...+X n the associated partial sums ; under what hypotheses does

(S

n

/n ) converge a. s. to 0 ?

(1) "

This point of view is very restrictive because two other asymptotic behaviours of the sequence ( S

/n ) are worth of interest : n P( sup IISn /n1I -<+ ) 1, which behaviour can be called a bounded law of large numbers (BLLN)

(2) ,

and :

P( w : S n (u.1) /n

0 weakly ) =-• 1, which is a law of large numbers in the weak topo-

logy (WTLLN)

(3) . The main goal of this paper is to study the three forms (1), (2) and (3) of the

LLN, under Prohorov boundedness conditions of the r. v. , in p-uniformly smooth spaces ; the good geometrical properties of these spaces allow to see that even if the asymptotic behaviours (1), (2) and (3) seem to be close , they happen under hypotheses which are very different. In an appendix we will also state without proof some Kolmogorov-Brunk type SLLN which can also be obtained by applying the same Dubins-Freedman comparison result.

§ 2. PROHOROV'S BOUNDED LAW OF LARGE NUMBERS. A sufficient condition for the BLLN in the classical Prohorov setting is as follows :

31 THEOREM 1 : Let (X ) be a sequence of independent, centered,

k

r. v. with values

in a real separable p-uniformly smooth ( 1

such

that :

>Q

YkE

1X k

(k / L 2 k)

a. s,

where : L x = Log(Log sup (x, e)).

2

Let's define for every integer n :

A (n) = 2 -2n

E

sup ( E f 2 (X k) :

where : 1(n) = (2 n + 1,

1)

B'

k E I(n)

,

, 2 11+1 ) ,

and , suppose that the following hold : a) The sequence ( S

) is stochastically bounded . n In

b) The sequence ( 2 -211P Log n 2. C > O :

c)

Then :

Z n 1

P( sup

Z k E I(n)

Xk

) is stochastically bounded.

exp ( e / A(n) ) < + -

<+)=1

Sn /n

REMARK : One checks easily that condition b) above holds for instance if the sequence ( n -2 P L

2 P ) is stochastically bounded.

2n 1 k n Xk

Now, let's give the proof of Theorem 1. PROOF : An easy symmetrization argument, similar to the one used in the proof

of [171 shows that it suffices to prove Theorem 1 for symmetrically distributed r, v. Xk So we only consider that case. Another symmetrization Zp argument shows that h) implies that the sequence ( 2-2np Log n E X ) k k E I(n) is L 1 -bounded. of Lemma 2. 1

For technical simplicity we make the following three assumptions - which aren't a loss of generality - : i ) K' > 0 : 486 K' (2p + 2C ) n

2, ,

Vi E I (n) ,

where of course C

)

X.

1/8 and : K' 2 n / Log n a. s.

denotes the constant involved in the fundamental inequality

recalled in Section 1. E exp ( -4K' 2 / A(n) ) <+

a)

.

n 1 -2nn

sup 2 -

Log n E k E I(n)

2P

K' 2 .

Now, let's start the proof itself. -n E If one denotes by T the r. v. 2 X k ' an easy application of the Boreln k E 1(n)

32

Cantelli lemma and of the symmetry of the X k shows that the conclusion of Theorem 1 holds if the following is true :

at>0:

Z n 1

P( T

(4)

>t) <+83.

n

By symmetry [1], hypothesis a) implies :

E S

sup

IIP

/n

n

< + CI) ,

and therefore also :

c=

<

sup E T n

.

So (4) will hold if we find V > 0 such that :

IIP

- E Tn

E P( T n 1

> t' ) <+ .

In order to find such a t', we begin by looking for good bounds for the quantities : u

= P( T

n

where x

I P > 2x ) ,

P - E Tn

n

sup (2. c) will be specified later.

Suppose that the integer n 2 is fixed and denote by Z 1 ,

,Z

the r. v.

2n

(X5 / 2 n) i E 1(n) ; we will also denote T n by T and A(n) by A. The symbol E will denote a sum taken on the set of integers (1,

For finding a bound for u

, 2 n).

, we will consider two cases :

n

First case : Log n

/ A.

By adapting a well known trick of Yurinskii [21] , Ledoux [19] has noticed that a

I

r. v. of the type T

-E

and :

= Z TIk E(

where :

can be written as a martingale :

-E

T

k

(Z

P

I

E( T , Zk )

I

k1'

, a o =(0,

For our needs this decomposition is not precise enough, so we will refine it by setting :

V k = 1,

2n ,

A

k

B

k

=(IZ+ 1

"•

+Z

k- 1

+Z

+ +Z k+1 ' • '

I

>y ) ,

2n

= A

k ' where y > 0 will be chosen later, and : k P

k

E(

TIIP

E ( T

Ak

I

73`

I

1 Bk

k

73`

k

-E( )-E(

The following inequality obviously holds : P( T P -E

TII P > 2x ) P(

In a first step we will bound P( E

ak > x ) + > x).

P(

Pk >

TI! I

A

I k

113 1 B

3

k-1

I ak-1

)

'

33 )

In order to do this we notice for beginning that ( 1 sk sj

211

is a martingale to which we plan to apply the following comparison result of

Dubins and Freedman [3] : ) be a real valued martingale and denote by (Y k ) its incre-

LEMMA 1 : Let (Sk ,

ments. If one defines for every k 1,..., n:

V k = E ( Yk 2

k- 1

) - with a — 0

✓ a >0. V b >0, P(

j = 1,

(

,

) - then :

, n :(Y 1 +...+Yi )

a (V 1 +...+v i ) + b )

1/ (1+ab).

For applying this lemma to our situation we need first a bound for the r. v. 1 this bound will be obtained by an easy change in the computations of the proof of Lemma 1 in [19] :

P IA I 3k )sE(

E (

k ) + p E ( I F( T-Z k)(Z k) IA I 1

T-Zk lIP IA I

+C E(dZ

k

A

)

) k '

k

and :

E ( T

P IA

k

1 a k-l ) E ( T-Zk

) I 1 a, I k-1 ) - P E(IF(T-Z0Z, .1() A k-1

IA

I k - C E (1lZ k IP IA

P

k

I k

a k-1 ) '

From these two inequalities one deduces : p E(

E(T-Z, k )(Z k )

IA I I

C E ( 11Zr k P IA I

3k )

+ p E ( 1 F(T-Zk)(Zk) IAk 113

C p E 1lZ k

k-1 )

The same computation can obviously be done for -

I

cek

I

p E ( I F(T-Zk)(Z) 'AI I +p E ( I F(T-Zk)(Zk) IA

TA k

k ' so :

) + C E ( Z k IP IA 1 a k ) 1

k

)

1 3 k-1 ) C p E 1l Z k

)•

lAk

From this inequality it follows easily that :

E cel2c

ak-1)

8p 2 E(F 2 (T -Z k)(Z k) IA

If one puts now a

p( 1 k 2

E E E(a

n Ak ) '

2 flc-1 ) 8C p

1 2P E

(

P k_

-

one sees that :

2 E( IT Zk 12(P-1)I I 73' ) s 8p 2 sup ) A + 8C IM2a / Log n k k-1 12) Ak 1 sk 1,r_zkl 8(p-1)) 1/4 A+ ,,„2 Kt 2 cr/Log 2 3/4 ou p s 8p a sup (E 1 k 2–

By assumption a) and symmetry there exists L > 0, such that : sup sup

n 1 sk 2n

(E 1T-Z

k 8(p-1)) 114

L.

34

Hence :

8p 2 A 013 " L + 8C 2 K' Z a' /Log n

E E E (oek Z I 3)

and so by assumption c) , one has for n large enough :

E E E ( cy2 k

I

k-1

a3/4

)

Applying now Lemma 1 for a = a - 1/2 and b = 1, one obtains : Œ -1/2 zE(ct,2 .1:= ( Ea, >xt P( E ak >x ) ) k "k 2 kk-1 ) + 2 + P( Ea >x, Ea <Œ k E Œk 1 3 k-1 ) 1 ) k 1/2 u1/4 1/4 a Ax- 1 ) ) s a( From the inequalities : Œ

P(

sup 1

E Z. > y/3 ) 1 sjs k

2n

2 P(

E Z.

>y/3 )

,

an application of Hoffmann-Jibrgensen's Lemma ( [14] Lemma 4.4 ) gives that for every y

81 :

P( Z ak >x ) s 2 (2) s 4 (2) Now , fix y

n E

3/4 3/4

1/2

P

(

P( E Z

sup ( 486, 24 c),

,

2

>y/27 ) s 32 P ( E Z

k

k

>y/81 ).

such that :

32 P( z Z k >181 ) s 1/2

- such a choice is of course possible by hypothesis a) - ; y being fixed, we put : x = y/243 . For such a couple (x, y) one has :

P(

-E

2 P(

T 1 13 > 2x )

(6)

Pk > x )

In the next step of the proof we will bound the right-hand side of (6) by using another martingale result, also due to Dubins and Freedman [3] LEMMA 2 : Let (Sk , a k) 3.11.1 be a real valued martingale and denote by (Y k ) its

increments. Suppose that : n Then:

lY

k

I

1

a. s. .

VuER.VX>O,Vv> 0, X( u+ Y 1 +... -Fy n)

E ch (

where :

e(X/v) ) ,

) s ch (Xu/v ) exp v+ V k

I

+...+V

n

1,.... n ,

Vk = E ( Y

a k-1 )

:

e(x) = exp x - 1 -x . For applying this lemma to our situation, we first notice that by the same computations as for the

r.v•k

one has :

35

p E ( 1 F(T-Zk)(Zk)

I 13 k I

I 'Bk

la k

) + C E ( Z

+ p E ( I F(T-Z k )(Z k) I B I I O

k-1

If for every k = 1, „. , 2 n one defines : 1-p Y = ( y Log n / KT(2p+2C p ) ) k

k

one has : V k = 1,

s 1

1Yk 1

, 2n ,

' Bk

) + C E ( Il Z k

I

B

I k

-K-1 ) .

,

a. s.

Now we want to bound : a

n

=P(EY

k

> xy

1-p

Log n / K 1 (2p+2C ) ) ,

by using Lemma 2. Let's notice that : V

1

(Log n) 2 (8p 2 A + 8C 2 K 12 /Log n) / K' 2 (2p+2C ) 2

+... +V

4 Log n

Hence : a

n

s P( (1 / v+V i +... +V) E Y k > xy 2

Remembering now that p E

'II, 21 and

1-p

Log n / K 1 (2p+2C ) (v+4Log

).

also the assumption i) made on K', one ob-

tains : a

n

s P( (1/ v+V + +V ) E Y > 16 Log n / (v+4Log n) ) . 1' • ' k 2n

Applying now Lemma 2 for X =17 = 4 Log n, one gets : -4 a . s exp 4Log n / ch(8 Log n) s 2 n n So for the integers n n (x) and such that Log n s K 12 / A , one has : 0 -4 u s4n n Now we go to the second case.

Second case : Log n > K 12 / A . It is easy to see that the relation (6) remains true - for the same couples (x, y) and that the only thing to do is to apply Lemma 2 in a different manner.

If one defines : V k = 1,

, 2n ,

yk

= ( KTy l-P / A (2p+2C ) ) r3 k

one has : V k = 1,

, 2n,

Yk I

a . s.

Furthermore : 2 12 2 2 V + +V /Log n)) / A2 (2p+2C p ) 2 s 4 K' / A s K 12 (8p +8C (K 1 • • 2n So un can be bounded by : 2 Pi( (1/ v+V I +... +V n ) E 5k > xlç y1 2

/ (2p-E2c. ) (Av + 4K12)

Now we apply Lemma 2 for X =y = 4K 12 / A and we obtain :

36

u

s 2 exp (4K' 2 / A) /

n

ch (8K' 2 / A)

4 exp - (4K' 2

/ A) .

Putting together the results obtained in the two cases, one gets : Vn

N(x),

un s 4 ( n -4 + exp - (4K' 2 / A)

).

Hence : n1 n and this finishes the proof of Theorem 1.

§ 3. PROHOROV'S LAW OF LARGE NUMBERS IN THE WEAK TOPOLOGY. It is natural to expect that this asymptotic behaviour can be obtained under some "weak topological refinement" of

the hypotheses of Theorem 1. The precise result

is as follows :

THEOREM 2 : Let (X ) be a sequence of independent, centered, r. v. with values k in a real separable, p-uniformly smooth ( 1

0 : Vk E1NT,

1 X0 s K (k/L 2k)

a. s.

Let's define for every integer n and every f E B' : Z

X(n, f) :-• 2 -2n k

E I(n)

E f2 (X ) k

•

Suppose that assumptions a), b), c) of Theorem 1 are fulfilled and that the following one holds also : YE> 0 , 1TfEBI,

d)

Z n 1

exp ( _ E / X(n, f) ) < + cr .

Then : P( tu : S n(tu ) /n --' 0 weakly ) = 1.

PROOF : By Theorem 1, we know that : P( sup n A standard argument then gives [13] : E

sup n

(7)

Furthermore the one dimensional Prohorov SLLN [20 1 implies : Vf E B',

f( Sn /n) -' 0

a. s.

This property and (7) show that (Sn in , a(X 1 , ... , X n) ) is a weak sequential amart of class (B) (for the definition and the main properties of weak sequential amarts, see for instance [4] V.3 ).

37

The space B being reflexive,

the conclusion of Theorem 2 follows immediately

from a well known convergence theorem of weak sequential amarts due to Brunel and Sucheston [21.

§ 4. PROHOROV'S STRONG LAW OF LARGE NUMBERS.

The sufficient condition for the SLLN in the Prohorov setting can now be guessed easily from Theorems 1 and 2 ; the statement is as follows :

THEOREM 3 : Let (X) be a sequence of independent, centered, r. v. with values r,

in a real, separable, p-uniformly smooth ( 1

,

such

that : 2K>0:

1X k l sK(k/L 2 k)

a. s.

Suppose that the following hold : S

b')

2 -2nP Log n Z k E I(n)

c')

y e >0,

n

Then : S

/n

n

0

in probability.

a')

Xk 2p

-' 0 in probability

E exp (_ C / A(n) ) <+ n 1 0 a. s.

In

REMARK : It is easy to see that b') holds for instance if : -2p n Ln Z X 1 2P 0 in probability. 2 k 1 "lcS n PROOF : By a classical argument [17 ] it is sufficient to consider the symmetrical case. By symmetrization ( see [10] proof of Lemma 1 ) 13') implies : b")

lim 2 -2 " Log n E + k E I(n)

E Xk

The ideas of the proof of Theorem 3 are the same as those used for proving Theorem 1 ; so we will keep the same notations as in the proof of Theorem 1 and we will only detail what needs to be detailed. First one notices [1 1 that from a') it follows that : E T

n

11P

S

n

In

P

0 7 and also :

O.

So the the conclusion of Theorem 3 will be true if we show : x > 0 ,

E n 1

P( T n

- E T n P > ?,x ) < +

.

Without loss of generality it suffices to check (8) only for x E 0, 243 -13 [

(8) Fix

such an x. As previously, we denote by u n the general term of the series involved in (8) ; we will bound u

n

by considering again two classes of indices n, slightly different

38

from those taken in the proof of Theorem 1. Let's suppose - and this of course isn't a loss of generality - that : ✓n

2, VjE I(n) ,

a. s.

1 X. s 2 n /Log n J

For simplicity we put

0 = exp ( -8(2p+2C ) Ix) . P

We consider first the following situation :

First case :

A Log n 0 ,.

It is easy to see that if one puts y = 243 x

1/p

, relation (6) holds for n large enough.

Now we will again apply Lemma 2 ; for this we introduce the following sequence ( Yk ) : ✓ k = 1, .... 2 n

. Y k = (Log n/(2p+2C p ) ) P k

One has : ✓ k = 1, ... , 2 11

I Y k I s 1 a. s.

and : V

+ +V n 1• • ' 2

where : d

( 2(Log n) 2 A + 2Log n dn ) , -4 0, by condition b") .

n

Therefore : ✓n

V 1 +.. .+ V n s 4 0 Log n. 2

n(x) ,

k

Applying now Lemma 2 with y = 4 e Log n and X = 4 y (2p+2C p ) /x , one gets : u

n

s ( 2/ch2Log n) exp 4 0

1 /2

Log n s 4 n

3/2 .

It remains to consider the complementary situation :

Second case : A Log n >

e.

If one chooses now :

A (2p F2C p ) )

Yk = ( 0 /

-

Ok

one has : Vk=1

2

n

,

IY

k

I

1 a. s.

and : ✓ n > n'(x) ,

V +... +V 1 zn

40

2

/

A.

2 Applying finally Lemma 2 for y = 4 0 2 / A and X = y x one gets : 4 2 s (2 / ch(x 3 0 7/8 /2A) ) exp (4x 0 /A) s 4 exp ( - x 3 0 7 /8 / 4,11) . u n Collecting the partial results we obtain : yx E

I), 243 - Pr ,

Y n N( x)

u

n

a

N(x) Er ,

s4(n

a

y (x) > 0 :

+ exp (- y

(X)

/

A ))

and (8) immediately follows from hypothesis c') .

,

39

REMARK : Hypothesis b') in Theorem 3 seems at first glance somewhat surprising and artificial. To shed light on its meaning we will now give some corollaries of Theorem 3 which will show that b') is a very weak hypothesis. COROLLARY 1 : Let (X k ) be a sequence of independent, centered, r. v. with

values in a real, separable, p-uniformly smooth ( 1

K > 0 : Vk

a. s.

Suppose that the following hold : 1)

n -13

lixk

E 1

2)

in probability

_4 0

[

n

eXp(-

V

A(n) ) <+

n> 1 Then :

S

0

In

n

a, s.

The proof of this result is very easy. First : z 2p 2 -2np -2np s (c/Log n) 11 X II k

kEI(n.)

E k E I(n)

X

k

P

and so hypothesis b') of Theorem 3 is fulfilled by applying 1). By symmetrization one has : n-P

E 1

EX P3

0;

n

the space B being of type p, it follows that the sequence ( S n /n ) converges to 0 in L(B). All the assumptions of Theorem 3 being fulfilled, this ends the proof of Corollary 1. In the case p = 2, Corollary 1 reduces to Prohorov's SLLN proved in [1l] . Let's stay for a moment in this situation p 2 for making precise the difference between the hypotheses of Theorem 3 and those of the result in [11] , One observes first that the following easy corollary of Theorem 3 holds : COROLLARY 2 : Let (X k) be a sequence of independent, centered, r e v, with

values in a real, separable, 2-uniformly smooth Banach space (B, that :

EK>O: y k E N

K (k/L 2 k)

I! X k II

a. s.

;

Suppose that the following hold : a') S n / n

0 in probability

( 1 / n 2 1, 2 n )

c`) Then :

6

> 0 ,

E 1 sk< n E n> 1

J X k 11

-4 0 in probability / A.(n) ) <

exp ( S

2

n

/n

12, .

0 a. s.

such

40

The gap between hypothesis b') in the above statement and assumption 1) in Corollary 1 is clear. It is easy to give examples of sequences of r • v. which belong to the domain of application of Corollary 2, but not to the one of Corollary 1 ; the sequence considered in the last section of [18 ] provides such an example. In [11] it has been noticed that in the special case of Hilbert spaces - which are of course 2-uniformly smooth - condition 1) in Corollary 1 is necessary for the SLLN in the Prohorov setting. More generally, is it possible to simplify the hypotheses of Theorem 3 by making additional cotype restrictions on B ? Godbole has characterized the spaces of cotype q in terms of SLLN for the q / kg -1 ) ( [6] Theorem 2.1 ) ; unfortunately his result will of no k help in our situation. sequence ( X

The ( partial ) result we are able to prove is as follows : COROLLARY 3 : Let 3/2 s p s 2 and 2, s q s 2p - 1 ; consider a sequence (X k ) of

independent, centered, r. v, with values in a real, separable, p-uniformly smooth Banach space (B, 11 1),

of cotype cl.

Suppose that the following hold : i)a- K>0 :VkEN, 11 X E n>1

ii) V e > 0 , Then :

S

n

k

11 s K (k/L k) 2

exP ( - E /

/ n ,-. 0 a. s.

MO

) S

4.)

a. s.

n

/ n -4 0 in probability.

PROOF : Of course the only thing to do is to check that the weak law of large num_ bers implies the strong one. As Sn / n -4 0 in probability, it is sufficient to consider the symmetrical case. By cotype q and assumption i), one has : lim n -ci Z E r Xk I q , n -4 + ee lskri Hence : -2p n Ln 2

E 1 sksn

E 11 X

k

o.

2 P s K 21)-(1 n -ci (L2n)1 _2p

E 1 sksn

E 11 X k d cl ,

and Corollary 3 immediately follows, by application of Theorem 3.

REMARK : Let's suppose that conditions i) and ii) of Corollary 3 are fulfilled for a sequence of symmetrically distributed r. v. X k . If the weak law of large numbers holds, then, by Godbole's result n -cl implication holds : S

n

In

' 0

in probability

Z Iskn

= n -cl

1 Xk 1

E 1 skn

q

1 X

-4 0

k

a. s. . So the following

q' 0

a. s. ;

this shows the very special geometric nature of these spaces which are both p-uniformly smooth ( 3/2 s p s 2 ) and of cotype q (2sqs 2p - 1 ) .

41

§ 5. APPENDIX: SOME LAWS OF LARGE NUMBERS OF KOLMOGOROV-BRUNK TYPE.

In [9] and [12] the SLLN of Kolmogorov and Brunk are studied in 2-uniformly smooth spaces under hypotheses requiring both assumptions on the strong and on the weak moments of the r. v. . Results stronger than the classical ones known for type 2 spaces are obtained. By applying - in a more elementary manner as above - the martingale technique developed in section 2, Kolmogorov-Brunk type SLLN can also be obtained in

p-uniiormly smooth ( 1

THEOREM 4 : Let (X ) be a sequence of independent, centered, r. v. with values

k

in a real, separable, p-uniformly smooth ( 1

1

)•

Suppose that :

) There exists a sequence of positive numbers (d.) i, J If j E rN , a. s. x.J .5 cl• J ii) S / n •-• 0 in probability . n

converging to 0, such that :

I

If one defines for every integer n :

I"(p,n) = 2 211P

E j E I(n)

J

1 2X. P

then the following implication holds :

( a k integer, k

1:

E n 1

( Mn) + lip, n) ) k <+ 113 )

=S n / n --, 0 a. s. .

As an obvious corollary of Theorem 4 one has the following result which can be compared with the well known law of large numbers in type p spaces of Hoffmann-

JSrgensen and Pisier [ 15 ] : COROLLARY 4 : Let (X k) be a sequence of independent, centered, r. v. with values in a real, separable, p-uniformly smooth ( 1

a)

S

n

/ n -• 0 in probability

b)

E k 1

c)

1,11 :

Then :

k -2 P E X E n 1

2p k

< +13

(An)) i

<+,3 .

S

n

/n -. 0

a. s. .

42

Bounded laws of large numbers and laws of large numbers in the weak topology can of course also be considered in the Kolmogorov-Brunk setting. We only give an example of such a result :

THEOREM 5 : _ Let (X k ) be a sequence of independent, centered, r. v. with values in a real separable, p-uniformly smooth ( 1

b) k c)

Ej

Then :

k -2 P E X 1 1 :

E n 1

k 1 2P ‹+

12

( A (n) )i < + CC

.

P( sup II S n / n 11 < + 0) ) n

Let's conclude by mentioning an application of the above results. It is well known that all the classical situations in which one can conclude that a Banach space valued, symmetrically distributed r. v. X - and it is always possible to reduce to that case - satisfies the LIL,

are the union of a Prohorov type BLLN

and of a SLLN ( 1-16] , [7] ) or of a law of large numbers in the weak topology [8]

. Therefore it is not surprising that Ledoux's recent necessary and sufficient

condition for the LIL in uniformly convex spaces [19] can also be obtained as a corollary of Theorem I and Theorem 5. The computations for proving this observation are left to the reader.

REFERENCES.

[1]

DE ACOSTA, A. : Inequalities for B-valued random vectors with applications to the strong law of large numbers. Ann. Prob. 9 (1981) , 157-161

[2]

BRUNEL, A. et SUCHESTON, L. : Sur les amarts faibles à valeurs vectorielles. C.R. Acad. Sc. Paris 282, Sér. A (1976), 1011-1014

[3]

DUBINS, L. E. and FREEDMAN, D.A. : A sharper form of the Borel-Cantelli lemma and the strong law. Ann. Math. Stat. 36 (1965), 800-807

[4]

EGGHE, L. : Stopping time techniques for analysts and probabilists. Cambridge University Press - Cambridge 1984

[5]

ENFLO. P. : Banach spaces which can be given an equivalent uniformly convex norm, Israel J. of Math, 13 (1972). 281-288

[6]

GODBOLE, A. : Strong laws of large numbers and laws of the iterated logarithm in Banach spaces, PHD, Michigan State University 1984

[7]

HEINKEL, B. : Relation entre théorème central-limite et loi du logarithme itéré dans les espaces de Banach. Z. Wahrscheinlichkeitstheorie very/. Gebiete 49 (1979), 211-220

43

[8]

HEINKEL. B. : Sur la loi du logarithme itéré dans les espaces ré fl exifs. Séminaire de Probabilités 16 - 1980/81 - Lecture Notes in Math 920, 602-608

[9]

HEINKEL, B. : On the law of large numbers in 2-uniformly smooth Banach spaces. Ann. Prob. 12 (1984), 851-857

[10]

HEINKEL, B. : The non 1. i. d. strong law of large numbers in 2-uniformly smooth Banach spaces. Probability Theory on Vector Spaces III - Lublin 1983 Lecture Notes in Math 1080, 90-118

[11]

HEINKEL, B. : Une extension de la loi des grands nombres de Prohorov. Z. Wahrscheinlichkeitstheorie verw. Gebiete 67 (1984), 349-362

[12]

HEINKEL, B. : On Brunk's law of large numbers in some type 2 spaces. to appear in "Probability in Banach spaces 5" - Medford 1984 - , Lecture Notes in Math

[13]

HOFFMANN-JQ5RGENSEN, J. : Sums of independent Banach space valued random variables. Studia Math 52 (1974). 159-186

[14]

HOFFMANN-JORGENSEN, J. : Probability in Banach space. Ecole d'été de Probabilités de St Flour 6 - 1976 - Lecture Notes in Math 598, 1-186

[15]

HOFFMANN-JORGENSEN, J. and PISIER, G. : The law of large numbers and the central limit theorem in Banach spaces. Atm. Prob. 4 (1976), 587-599

[16]

KUELBS, J. : Kolmogorov law of the iterated logarithm for Banach space valued random variables. Ill. J. of Math. 21 (1977), 784-800

[17]

KUELBS, J. and ZINN, J. : Some stability results for vector valued random variables. Ann. Prob. 7 (1979), 75-84

[18]

LEDOUX, M. : Sur une inégalité de H. P, Rosenthal et le théorème limite central dans les espaces de Banach. Israel J. of Math. 50 (1985), 290-318

[19]

LEDOUX, M. : La loi du logarithme itéré dans les espaces de Banach uniformément convexes. C. R. Acad. Sc. Paris 300, Sér. I, n° 17 (1985), 613-616

[20]

STOUT, W. F. : Almost sure convergence. Academic Press, New York 1974

[21]

YURINSKII, V. V. : Exponential bounds for large deviations. Theor. Prob. Appl. 19 (1974). 154-155

ON THE

SMALL BALLS

CONDITION IN THE

CENTRAL LIMIT THEOREM

IN UNIFORMLY CONVEX SPACES

Michel Ledoux Département de Mathématique, Université Louis-Pasteur 7, rue René-Descartes, F-67084 Strasbourg, France

Introduction. Let E

be a Banach space. By random variable with values in E

we mean a measurable map X from some probability space (C4T,AD) into E equipped

a(E)

with its Borel

such that the image of 1P

probability measure on S(E) . Denote by (X ) n n copies of X

and, for each n 1 , S n

with values in E if the sequence

X1

by X defines a Radon

a sequence of independent Xn . A random variable X

is said to satisfy the central limit theorem ( CLT CS

in short)

converges weakly to a Gaussian Radon probability

nvn e

measure on E . In his remarkable work on the Glivenko-Cantelli problem, M. Talagrand [T] observed the following characterization of the CLT for random variables with a strong second moment which relates the central limit property to conditions on small balls : if X

takes its values in E and 11E(

X

< oo , then X satisfies

the CLT iff, for each g > 0 , S 11m Lnf W[

n

n


> 0 .

■/171

(Actually, as detailled in [T] , this equivalence holds in the more general setting of non-separable range spaces and in the framework of empirical processes.)

Although it seems rather difficult to verify these conditions on small balls, the preceding property is intriguing since it reduces a central limit property in Banach spaces to some kind of weak convergence on the line by taking norm. This property also lies at some intermediate stage since, as we will see below, a random

45

variable X K in

E

with values in

E

satisfies the CLT iff there exists a compact set

such that

lim ink IP( --21-1 EK1 > 0 n (S

and the sequence

/fn)

n E

is stochastically bounded as soon as for some

M>0

11S lim inf IP( n-'

n <M 1171

> 0 .

M. Talagrand (oral communication) raised the question whether the equivalence

he proved holds without the strong second moment assumption which is not necessary in general for the CLT . In this note, we answer this question in a positive way in uniformly convex spaces. Precisely, we will establish the following result :

THEOREM 1

. Let

with values in

(i)

E • Then X satisfies the CLT iff

lim t 2 F( t

a uniformly convex Banach space and X a random variable

be

E

X

>t3=o

oc)

and (ii) for each E > 0 , lim inf TP( n cx)

Is

F

< C

3>o.

/71

This result will follow easily from a new quadratic estimate of sums of independent random variables in uniformly convex spaces obtained in [L2]

Preliminary results. We begin this section by a characterization of the CLT which follows easily from the concentration's inequality of M. Kanter [K] . I

am

grateful to Prof. X. Fernique for useful informations on this result.

PROPOSITION 2 . Let X

be a random variable with values in a Banach space

Then X satisfies the CLT iff there is a compact

lim ink IP n

E K

> O .

set

K in

E

E.

such that

(1)

46

Further, the sequence (S

//17)

n)

n E

is stochastically bounded iff for some M > 0

IS I < M

11m 1sf IPC fl - Oo

> 0 .

(2)

Proof. The necessity of (1) and (2) implies the stochastic boundedness of X is symmetric. There exist 8 > 0

is obvious. Let us first show why (2)

CS n /N51) n and k

o

• Assume to begin

E

with

that

such that for all integers k k

o

and n S

IPC

nk

<M

Firc

>8.

By Kanter's inequality

dsnd

3

•(1

kip(

> 1,1,./Tc ))

1

8

sin and thus

!

Is

> m sA- 3

kip(

_2_ 48

JR

2

It follows that the sequence (S

ECIlx cy <

for all

co

n

is stochastically bounded and also that

/WI) n ET

ce <2 . In the non-symmetric case, let

copy of X ; the symmetric random variable

X- X'

EC

X

<

implies that X

op

be an independent

satisfies (2) (with 2M instead

of M ) and hence the preceding conclusions apply to that

X'

X- X I

. In particular, we have

; therefore the strong law of large numbers combined with (2) must be centered. Hence the conclusion to the second part of Propo-

sition 2 holds by classical considerations involving Jensen's inequality.

The first

part is established in the same way.

Since in cotype 2 spaces, random variables X

such that (S 4/Ti) n n

is

stochastically bounded satisfy the CLT [P-z] , the previous proposition yields immediately the following corollary.

COROLLARY 3 . Let E

be a Banach space of cotype 2 and X a random variable

with values in E . Then X satisfies the CLT iff (2)

holds.

47

We now turn to the small balls condition. Since for a centered Gaussian Radon probability, each ball centered at the origin of positive radius has positive mass, it is clearly necessary for a random variable

1S 1 --E—

for each C > 0 , lim inl

<

X

CLT

to satisfy the

that

E 3 > o .

(3)

n 1121

M. Talagrand [T]

is also sufficient for

showed that when 1E(1 XH 3 < oo , (3)

X

CLT • For the sake of completeness, we reproduce here Talagrand's

to satisfy the

proof of this result ; it will illustrate the idea we will use next in uniformly convex spaces.

THEOREM 4 . Let

IEC

that

42 1

be a random variable with values in a Banach space

X

< co . Then

Proof. Let C > 0

X

satisfies the

and

be fixed

CLT iff (3)

8 . 8(e) > 0

E

such

holds.

be such that

1S 6

< c 1 >

lim inf n

Choose a finite dimensional subspace

E

map

Et

E/Ii and

T(X)11 2 1

.

H

of

E

quotient norm given by

8.E 2 . For each

n

, 11T(S n)11 -

T

such that if

Ircx) T(S)

denotes the quotient

= d(x,H) , then

11

can be written as a

martingale

!T(S)

IECHT(S n )D

= E

di

i=1

withincrementsd.,i = 1,...,n , such that, for each Efd2i 3 s lEf

IT( xi )

2

i

1

[Y] ) and thus by Chebyschev's inequality

(cf

ITC

-11 )11 _

RUT(

-2-1 )1111 > E 1 5 E -2 ECHT(X) xfil

Since

e

1 t

lim inf 13 (

cto

11T(

--11

)fJ

<

E 1 >

2

1

s ô.

48

and hence, intersecting,

lim sup TEC T( n 00 X

)d1 < 2c

AT:1

therefore satisfies the CLT

by classical arguments (cf [P2] )

A short analysis of this proof shows the central rae of the martingale trick leading to the quadratic estimate and of the integrability condition

EC

4

2

3

<

CO

providing tightness at some point. An improved version, in uniformly convex spaces, of the previous martingale argument of Yurinskii was recently obtained

in [L2]

in some work on the law of the iterated logarithm. It will allow us to establish Theorem 1 which thus characterizes in those spaces the CLT through the small balls condition (3)

and a moment condition which is necessary for the CLT .

Recall that a Banach space E is uniformly convex if for each C > 0 there is a

8 = 8 (c) > 0 such that for all x,y

Itx _ y one has 1 - ---2

with

in E

c,

Y = 1 and 1 x y

> 8 . According to a well-known fundamental result of G. Pisier

[P1] , every uniformly convex Banach space E is p-smooth for some p > 1 i.e. admits an equivalent norm (denoted again

)

with corresponding modulus of

smoothness

p(t) = satisfying

sup ( kix + ty

p(t)

1

-ty0 - 1 ,

+

Kt' for all t > 0

= 1 '}

and some positive finite constant K .

This p-smooth norm is uniformly Fréchet-differentiable away from the origin with derivative

D : E

(01

and F(0) , 0 , then 1 F (x) C > 0 (cf

P-1 D(x/I

E* such that if F (x) = =

P-1

for all x

X

) for x / 0

in E and, for some constant

DI-J] ),

F (x) -F (y)11

C x

y p -1

for all x,y in E .

(4)

The following lemma was the key point in the proof of the main result of EL2] . It will allow to achieve our wish in the next section.

49

LEMMA 5 . Let E

be a p-smooth Banach space for some p > 1 with norm 1

satisfying (4) . Let also (Y.)

be a finite sequence of independent bounded hen HSH11P

E-valuedrandomvariablesandlet

1E(

1 1)/

can be

written as a martingale

1 1)- ECI with increments d

1 = 1,...,n , such that, for each i

2 r 2 2p lEtF (S -Y1)(-Yi))

r Etd3 . 1

2c2 E rjj y. d2p 1 CH

1 11

where C is the constant appearing in (4) .

Before turning to the proof of Theorem 1 , let us point out that a quotient of a p-smooth Banach space E is also p-smooth, and, if norm of E , property (4) holds true for any quotient norm of

denotes the p-smooth with uniform

constant C .

Proof of Theorem 1 . We may and do assume that E is equipped with a p-smooth

1.1

for some p > 1

for which (4)

and Lemma 5 hold. By the previous remark,

these will also hold for every quotient norm with uniform constant C . We assume moreover p <2 .

Condition ( i)

is well-known to be necessary for X to satisfy the CLT

( EA-A-C] , EP-Z] ). Let us show the suffiency part of the theorem and assume first that X is symmetric. Proposition 2 and (ii)

imply that the sequence 2

(- /1/ 1-1 nEN

is stochastically bounded and thus, from the integrability results of LP2] , we know that

sup E C(

n ,, P ) I

KP < co .

N/71

Let

e > 0

be fixed. For each n ,define

(5)

50 X.

u. 1

= u(n)

=

-- 2--

/17_

I,t1 tilX i 1 S A 1 ' i -

n andsetUri = . Eu.;(i) and 1 1=1 number 8 = 8(e) > 0 such that Urnif IF(

n

< e

Un

Since the sequence

(1i)

combine to imply the existence of a real

>6

(S /A) n n E

does not contain an isomorphic copy of

co

[P-Z]

Theorem 5.1 of

is pregaussian, that is, there exists a Gaussian random variable the same covariance structure as

IEC

T( G )11 2 1

s

e2 P

E

and

ensures that G in

X

E with

X . The integrability of Gaussian random vectors

allows then to choose a finite dimensional subspace denotes the quotient map

(ii)

is stochastically bounded under

H of

E

such that if

T

E E/H 2 p2 ic2 P 1 )

We now apply Lemma 5 to the sum

-

(7)

T(U ) n

in

E/H ; F p will therefore denote below

the Fréchet derivative of the quotient norm of

E/H . For each

n , we have by

orthogonality,

1E01 T(un)1P - F[IT(un)P3 1 2 1 n n 2p2 E IE(F 2p (T(Un -u.1 ))(T(u i ))1 + 2 0 2 r lENT(u i )11 2P 1 1 .=.1 i=1 n 2(p-1)3 + 2 0 2 n 7E( u 1 (n)11 2P 1 S 2p2 n-1 7E (1 T(G) 2 1 E lEf U - u n il 1=1

(8 )

since by independence 2

sup 1E flx*(T(x)) I 1 lEaT(U n - ue (P-1) 1 1E(F2p (T(Un -u ))(T(u i ))1 s n-1 x* E (E/ )* H 1 (where the supremum runs over the unit ball

*

sup

)

E(Ix* (T(X))1 2 1 =

H 1 Now, by symmetry and

sup

(E/H )1- of the dual of

2 21 1E(Ix* (T(G))1 1 s laT(G))1

x* e(E/ )* I-1 1 (5) , for each

E/H ) and

i = 1,...,n ,

51

E[ n u.

(

(3E(

12 C p 1) 1 ) 1/2 (p 1 ) -

-

U

K .

n

Further, n

TECdu i (n)

2P )

13 ( 7., 11> t

n i-P

dt 2P

0 1

r( X >

n

dt 2 P

so that lim

nIE( u(n)

0

n

by

(i) and dominated convergence. These observations and (8)

lim sup IP( n 00

I

therefore imply that

T(Un )d P - 1E( T(un )

e

-2p

,2 ( p-1) 2] ( 22 p I\ TEt T(c)

s

(by (7) ) and since (6) holds, by intersection,

11m sup

n

1E (

1T un) (

<

2e

cho

which easily implies that X satisfies the CLT

In the general case, let X' random variable

X - X'

(using (i)

one more time).

be an independent copy of X ; the symmetric

satisfies (i)

and (Ii) and thus the CLT and therefore

so does X .

Conclusion. It is an open problem to know whether Theorem 1 holds true in any Banach space ; since we used the fact that random variables satisfying (10 pregaussian in spaces which do not contain an isomorphic copy of c

o

are

, a general

statement should probably include the condition

(iii) X

is pregaussian

as an additional (necessary) assumption. It would also be interesting to know in what spaces, random variables satisfying (i) (and possibly (iii) ) verify the CLT iff the sequence

CS /1Ti) n n

ix is stochastically bounded (which is weaker

52

than (ii) ). At the present, only trivial situations in which this happens (such that cotype 2 spaces or spaces satisfying A-Ros(2) [L1] like L (1

p
spaces) have been described.

References.

[A-A-G]

de Acosta, A., Araujo, A., Giné, E. : On Poisson measures, Gaussian measures, and the central limit theorem in Banach spaces. Advances in Prob., vol. 4, 1-68, Dekker, New York (1978).

[H-J

On the modulus of smoothness and the G -conditions

Hoffmann-4rgensen, J.

in B-spaces. Aarhus Preprint Series 1974-75, 11 ° 2 (1975). [K]

Kanter, M. : Probability inequalities for convex sets. J. Multivariate Anal. 6, 222-236 (1978) .

[L1]

Ledoux, M. : Sur une inégalité de H.P. Rosenthal et le théorème limite central dans les espaces de Banach. Israel J. Math. 50, 290-318 (1985).

[L2]

Ledoux, M. : The 'Law of the iterated logarithm in uniformly convex Banach spaces. Trans. Amer. Math. Soc. (May 1986).

[Pl]

Pisier, G. : Martingales with values in uniformly convex spaces. Israel J. Math. 20, 326-350 (1975).

[R2]

Pisier, G. : Le théorème de la limite centrale et la loi du logarithme itéré dans les espaces de Banach. Séminaire Maurey-Schwartz 1975-76, exposés 3 et 4, Ecole Polytechnique, Paris (1976).

ER-Z]

Pisier, G., Zinn, J. : On the limit theorems for random variables with values in the spaces L

[T]

p

(2

p < 00) . Zeit. ftir Wahr. 41, 289-304 ( 1 97 8 )-

Talagrand, M. : The Glivenko-Cantelli problem. Annals of Math., to appear (1984).

[Y]

Yurinskii, V.V. : Exponential bounds for large deviations. Theor. Probability Appl. 19, 154-155 (1974).

SOME REMARKS OF

GAUSSIAN

AND

ON THE

UNIFORM CONVERGENCE

RADEMACHER FOURIER QUADRATIC FORMS

M. Ledoux

M.B. Marcus

Département de Mathématique

Department of Mathematics

Université Louis-Pasteur

Texas A & M University

67084 Strasbourg, France

College Station, Texas 77843, U.S.A.

1. Introduction ,00

Let

(

g j n n=0

be an i.i.d. sequence of normal random variables with mean

zero and variance 1 and let (Eno° nn=0

be a Rademacher sequence, i.e. a sequence

of i.i.d. random variables where P(E

0

= 1) = P(E

0

= -1) - 1 . Let

be a sequence of complex numbers satisfying E la m,n m,n and Rademacher Fourier quadratic forms as follows : X (s,t) =

Xe(s,t) =

Eagge m n m n m
i(ms+ nt)

iCins E a CCe m,n m n m
nt

,

12

< co • We define Gaussian

12

(s,t) E

[0,2i 7

E

[0,27J

(s,t)

12

.

We are concerned with the uniform convergence a.s. of the stochastic processes X (s,t)

and X (s,t) . (By convergence we mean e

N n-1 i(ms+nt) lim EEa gge m,n m n N n=1 m=0 and similarly for Xe (s,t).) It is known that if these series converge in probability then they converge uniformly a.s. and in

L P of their sup-norm. This

Professor Marcus was a visiting Professor at the University of Strasbourg when this work was initiated. He is very grateful for the hospitality and friendship that was extended to him while he was there. His work on this paper was also supported by a grant from the National Science Foundation of the U.S.A..

54

is given in

[2]

forms (cf. also

(g'3

Let

respectively for Gaussian and Rademacher quadratic

[3]

and

[9] ). and

fe , }

be independent copies of

from the decoupling inequalities that for all

p

as in (1.1) and (1.2) with finitely many non zero

(1.3)

1

and

a

m,n •

EllX g (s,t)11 P

E11 E a g g' e gms nt) II P m,n m n m
ElIx e (s,t)11 P —

Ed

[e

and

n

X (s,t)

n

.

It follows

and

X (s,t) c

and

(1.4)

E a E E'e i(Ins+nt) 11 P m,n m n m
where the constants of equivalence only depend on sup (s,t)E [0,27 ] 2

.H

and

indicates

.1 . We will show how (1.3) and (1.4) follow from the usual

decoupling inequalities in the Appendix.

f x(S

We associate with

and

(s,t) E[0,27]2

(X e (s,t)3

] (s,t)E [0,27 2

the

pseudometric 121 e i(ms+nt) d((s,t),(s',t')) = Ç E l a m,n' ' m
d

where

(s,t) , (s',t')

E

10,277 2 . As usual let

1 entropy of [0,2n 2 radius

c > 0

N([0,27] 2 7 d;c)

denote the metric

d , i.e. the minimum number of open balls of

respect to

in the pseudometric

e i(ms t +nt 1 )12)1/2

d

that covers

,2 [0,2n] . A number of people

have recognized that

(1.6)

J 1 ([0,27 ] 2 ,d) =

co I log N([0,27] 2 ,d;e) dc <

op

0 is a sufficient condition for the uniform convergence a.s. of (1.1) or (1.2). It appears for example in

[4]

(stated in a somewhat stronger form involving

majorizing measures). In fact C. Borell pointed this out to us and raised the question, can this sufficient condition be improved ? The suspicion that (1.6) can be improved is reasonable because there are some obvious cases in which

oo 217 1 2 ,d) 1 /2 (E0 7

0

(

log N ([0,27 2 ,d; e)) 1/2 de <

co

55

is sufficient for the uniform convergence a.s. of X (s,t) or X (s,t) . One of c these is when a

is a product ; (details will be given in § 2). Nevertheless,

m,n

no smaller function of the metric entropy than the one appearing in (1.6) suffices as a sufficient condition for the uniform convergence a.s. of X (s,t) or X e (s,t) . To be more precise, for all functions f : R +

1R+ such that f(0) . 0 , f is

increasing and lim f(x)/x - 0 (and also satisfies a weak smoothness condition 00

which will be given in §2), we give examples of Gaussian and Rademacher Fourier quadratic forms which are not uniformly convergent a.s. but for which oo ( 1 .7)

f Clog N C[0,27] 2 ,d;e)) de < cm

J([0 27] 2 ,d) '

.

0

These examples also show that contrary to our experience with Gaussian processes no condition on the metric

d

alone can give necessary and sufficient conditions

for the uniform convergence a.s. of (1.1).

Pertaining to our study, X. Fernique [6] has obtained an important corollary of the deep recent result of M. Talagrand [13] on the existence of a majorizing measure for bounded Gaussian processes. Fernique's corollary deals with stationary vector valued Gaussian processes ; using this corollary (in a form adapted to our needs, Theorem 3.1 of this paper) and Theorem 1.1 of [8,Chapter I] we obtain the following theorem which extends the equivalence between Gaussian and Rademacher Fourier series to quadratic forms.

6 (s,t)1 (s, t ) E Theorem 1.1. Let (X (s t g ' )/(s,t)E[0,27]2 and tX as given in (1.1) and (1.2). If only finitely many of the coefficients

0 , 2 Tr ] 2

a

m,n

be are

non zero, then for all p 1

(1.8)

EdX g (5,t)11 P

where, as above,

11.d =

EliX e (s,t)H P

sup (s,t)E [0,211J

.

and the constants of equivalence depend

only on p . In particular, X (s,t) converges uniformly a.s. iff X (s,t) does and (1.8) holds in this case.

56

Developping further the arguments of the proof of Theorem 1.1 , we characterize the a.s, uniform convergence of X (s,t) and Xe(5t) in terms of one-dimensional g entropies following Fernique's characterization of boundedness and continuity of vector valued stationary Gaussian processes. Assume for notational convenience that a

m,n

= 0

(1.9)

n and set for s, s', t, t' E [0,27] ,

if m

d (s,s') = d((s,0),(s',0))

d

and

2

(t,t') = d((0,t),(0,t'))

and for j = 1 , 2 d ) 1/ 2([0 ' 27]' '

(1.10)

Recall that the space C

=

a.s.

(log N([0,27],d .c)) 1 /2 de . 0 , of a.s. continuous random Fourier series, is defined

as the space of all sequences of complex numbers

E a g e

int

n n

a - fa 1°° n n= 0

in

4 2

such that

t E [0,27]

7

converges uniformly a.s. equipped with the norm

Fa-

= E

sup I Eag e int n n n t E [0,2n1

The dual space C * a.s.

of

Ca.s.

.

has been carefully described by G. Pisier as a

space of multipliers M(2, 2 ) ; we refer the reader to [10] , [8]

for further

details. We will not be directly concerned with this here although various interpretations of what we obtain can be given in the language of [10] .

Define for each m

(1.11)

A l = (a

and n the one-dimensional sequences

°11° m,n m=0

Set further for each T

in C* a. s.

d.(t,t') = ( E 1 J

and

A

co . (a m,n n=0

2

and j

T, A i > n

1,2

12 l e i nt I

-e

int'12 ) 1/2 7

t, t' E [0,27]

whenever it is defined. As a corollary of the results of Talagrand and Fernique we obtain :

57

Theorem 1.2. Let

X9. (5 7 t)l (s,t)c [0,2n:2 be as given in (1.1) with only

finitely many non zero coefficients am

E

,n

. Then for

( E lam,n't 2 ) 1/2 m
(1.12)

+

j = 1 or 2 /J[0,27 ],d ) 1/ 2

J /2 ([0,2u],d 1 ) + 1 sup

1 T, J , /_ ([0,27j,d) i

T E c* a. s. 11T11 ••-1 where the constants of equivalence are numerical constants and defined as in

(1.10) with

T

d_

instead of

, T, J/ ([0,27j,d) 2

is

d_ .

We note that by considering Cauchy sequences Theorem 1.2 gives necessary and sufficient conditions for uniform convergence a.s. of also of

X (s,t)

and, by Theorem 1.1 ,

X (s,t) . Actually it is easily seen that, by Theorem 1.1 of

Ea ,Chapter

]

(1.12) also implies (1.8) . It is customary to leave out the "diagonal" (i.e. the terms

a

m,m

) when

studying random quadratics forms. For one thine (1.3) and (1.4) are no longer true, in general, if amm / 0 . Furthermore, the diagonal terms can be handled separately. We write

(1.13)

a 2 e im(s+ t) = E a m,m m

a

e im(s+ t) m,m

(a2 a m,m m

E,2 e im(s +t)

By standard symmetrization argument and Theorem 1.1 [8,ChapterI] , the second series on the right in (1.13) converges uniformly a.s. if and only if

Ea g e

rn

imu

m,m m

u E [0,2rr]

converges uniformly a.s.. The first series on the right in (1.13) is a deterministic Fourier series and the dichotomy betwee n unboundedness and uniform convergence a.s. is no longer relevant. Obviously in the Rademacher case one only gets on the right in (1.13) •

e E a m m,m

im(s+t)

58 2. Random Fourier quadratic forms and entropy

The sufficient conditions that we know for the uniform convergence a.s. of (1.1) and (1.2) follow immediately from well known results and techniques. We will present them here for the convenience of the reader.

the Fourier quadratic forms (1.1) and

Theorem 2.1. If J 1 (10,2n1 2 ,d) < co (1.2) converge uniformly a.s. and

(2.1)

E

sup

C ( ( E

,1X (s,t)1

m
(s,t) E [0 , 2n1 c- g

j cr0,21112 ,d) )

lam,n 12N2 I

where C is a numerical constant. (2.1) is also valid with X replaced by X . Conversely, if either of the Fourier quadratic forms X a.s. then J

V2

([0 2n] 2 ,d) < œ

e

converge uniformly

.

'

12 < op . One can I be complex numbers satisfying E lb m,n m
Proof. Let fb

easily

or X

m,n

constant C > 0 such that

(2.2)

p(

>

gg

Eb

in
12 ) 1 /2 )

E lb m< n

C exp ( X/C) , V X > 0

m ' 11

An estimate similar to (2.2) holds when fg

n

I

is replaced by (c I n

(see e.g. [1] ).

Using these estimates Theorem 2.1 follows from the usual extensions of Dudley's theorem as presented for example in [11] or [5]

since we have that

X (s,t) - x (st,t , )

Eexpce

g

d((s,t),(s',t')) for some u > 0

and similarly with X replaced by X .

For the converse we note that by (1.3) the uniform convergence a.s. of (1.1) implies that of

E ( E a g, ) g e m n m,n n m

ims

s E [0,27]

59

in which, to simplify the notation, we take

independent from

E a gi l m,n n

a

m,n

= 0

m n • Since (g

if

m

is

and is symmetric, the uniform convergence a.s. of

(1.1) implies, by Theorem 1.1 [,Chapter I] , that of

E ( E lam,11 12 1 ; 1/2 -rn e ims n m

7

S E [0,2U]

7

which is equivalent to

J 1/2 ([0,27] 7 d 1 ) <

(2.3) where d

is defined in (1.9). Similarly we sec that

1

(2.4)

OD

J 1/2 ([0,2n],d2 ) <

OD

By the triangle inequality the metric

d((s,s 1 ),(t,t'))

d 1 (s,s 1 )

d

defined ir (1.5) satisfies

+ d2 (t,t') .

Therefore (2.5)

N([0,271 2 1 d; e)

1\1([0,27],d 1 ; e/2) N(E0 7 27] ,d2 ; c/2)

and thus we see by (2.3), (2.4) and (2.5) that the uniform convergence a.s. of (1.1)

Exactly the same argument applies if (1.1) is

implies J 1/2 ([0,2u1 2 7 d) < ai replaced by (1.2) .

In an analogous fashion one can show that

r/2 ([0 ' 271r

a)

co (log N([0,271 r :a;

=

e)) r/2 de

<

co

0

implies the uniform convergence a.s. of r-dimensional tetrahedral (i.e. restricted to the subset of a

for which m

and Rademacher Fourier forms. Here, as above, distance in

L

2

d

1

< m

2

<


r

attention is

) Gaussian

denotes the corresponding

.

We now give examples which show that if one wishes to give metric entropy conditions for the uniform convergence a.s. of (1.1) or (1.2) solely in terms of the metric

d , then the sufficient condition of Theorem 2.1 can not be improved.

60

Let f

R+ R+ , f(0) = 0 , be an increasing function satisfying lim x/f(x) =

lim f(x) = cc . Furthermore we assume that for some C > 0 and all n X

CD

op ,

f(2n )

El2n)

(2.6) n=n

o

2

o

co large enough

o

C

n

2

•

n0

(Such a condition will follow if for some positive finite numbers M

and K ,

f(2xK 1(x)forallx>14.)Let(11.1be a sequence of integers such that + N i _ l and let (1).1 be a sequence of positive numbers. Their precise

N. > N 1 +

a) 2 2 Define values will be specified later but we assume already that E b. N. < op -J J=1 J co (ms + nt ) 12 X (s,t) = (s 7 t) E 10,27] Z b. 7 (2.7) g-g,e .

j=i

3 m,n E i(j)

m
1 , I(j) - ( n : N

1

+ ... + N. J-1

n < N

+ ... + Ni 3 ,

1

N.0 =0.Notethat(j)=N..we will show that for some appropriate choice of J (N.3 and (b_l depending upon f , the process X defined by (2.7) does not g J J converge uniformly a.s. even though op J. f ([0 ' 2712,d) = where

d

12 f(logN([0 7 2rrj ,d;e)) de < op

0

is the metric associated with X

as given in (1.5).

By independence and Jensen's inequality we see that for each J

E II E j=1

g g e i(ms +nt)11

E m,n E 1(j) m
mn

1

E sup b. Ed g g e j J m,n E I(j) m n

i(ms+nt)11

m
and, by Remark 1.3 [8,Chapter V]] , for each j

E

E gg e mn m,n E I(j) m
(ms 4- nt

( E

sup

g 2 t E [0,273 n E I(j) n

eint12

- VT N. )

C N. log F.

(Throughout this example C will denote a strictly positive constant, possibly changing from line to line.) Thus we have that

61

(2.8)

sup J >

Ell E b_

E

j=1

i(ms+nt)

g g e

m,nE 1(j) m
d

C sup b J.N J:Log N. j>

m n

To obtain an upper bound for the left side of (2.8) we note that co d((s,s 1 ),(t,t 1 )) = ( E b2.

e

(ms + nt)

e

2 i(ms' + nt t) 120/

I )

3 m,n E i(j)

j=-1

m
+

5(s,s?)

where 8 denotes the one-dimensional metric given by

co .5(t,t') = ( Z

b 2 N.

j=1

Z

l

eint

eint'i2)1/2

3 n EI(j)

It follows that oo 2S f(2 log N ([0,2n],8;e)) de 0

J ([0 2712,d) f '

We now estimate this entropy. For

sup 8(u,0)

u(s) =

lu I

and each j we define

s > 0

sup

'13 .(s )

Jul

s

( le s nE 1(j)

mu

- 1 1 2 ) 1/2 •

Following [8, p.123-124 and Lemma 3.6 Chapter II] we see that Co f( 2 log N ([0,27],8 ;

de

E))

0 oo 1 1 , , 2 .1/2 C ( ( E b.2 N. ) + f f (2 log 7 ) da(s) ) j=1

0

co

cx)

+

C(. (E b2. N2

E b j=1

j=1

1

'J f (210g J O

) d&/s) )

Furthermore,

1

co f (2 log

) a& _Cs)

0

z f(21.(+2) (.& _(2 -k ) -

_(2-(1+1) ))

k=0 oo

f(2) j.(1) +

Now for each

•

and

E L(2 -k ) (f(21(+2) - f( 2k)) • k=1

s > 0 N.-1

(2.9)

13..(s)

sup ( E lp inu - /1 2 ) 1/2 +

lu]

i(N i +...+N. )u sup

lu]

VT le

'

3-1

— 11

62

(s)

Let us denote by

the first term of the right hand side of (2.9). It is easy

k ,

toseethat N .-1

(2 -k )

2 -k (

n2 ) 1/2

2 -k N 3/2

n=0 for

j

that for

j

and (2.6) we see

q'.(2 -1 )

sufficiently large. Using these estimates for sufficiently large

[log2Nj]

oo E '?.(2 -k ) (f(2k+2) - f(2k)) k=1 J

E k=1

2 A-. (f(2k+2) - f(2k)) J op

E

+

3/2 -k , .421(+2) -f (2k)) 2 J

N.

k=[log N.]+1 2 j C

k

By a similar decomposition of the sum on op E

j

k=1

[(N 1

+...+N. ) 2

-k

J-1

co

j

k=1

( N.2 J

-k

we cret

A 2] (f(2k+2) - f(2k))

A 2) (f(21+2)

2(21))

c

f (3logN.) .

Putting this all together we see that

(2.10)

Since

eo C ( ( E b j=1

.1 f ([0,27 1 2 ,d)

f

co

N2. ) 1/2 +

E b .N _f (3logN ) ) . J J j= 1

lim f(x)// x - 0 , we can first choose

satisfies

x

[N.1

such that

co

.2 j f(3logN.) lim IogN. and then take =

.2

\ \

N-f(31-ogN.))

j > 1 •

For these choices, we see from (2.10) that

12 J ([0,2n j ,d) < en

even though,

by (2.8) and the integrability properties of Gaussian quadratic foins, X (s,t) g does not converge uniformly a.s..

63

Thus (1.6) can not be replaced by (1.7), with any increasing function f satisfying

lim -4

f(x) //x = 0

and (2.6), as a sufficient condition for the

CO

uniform convergence a.s. of (1.1). Note that exactly the same argument applies for the corresponding Rademacher Fourier quadratic form, i.e., with (c c 3 m n (gmgril

replacing

in (2.7). 12 . In some cases J 1/2 ([0,27J ,d) < œ is necessary and sufficient for the

uniform convergence a.s. of (1.1) and (1.2). One of these, which is trivial, is when the coefficients

(am,nI

vanish outside some one—dimensional set of indices.

We write (1.1) in the form

(2.11)

E

where m < n

akgnagnice ik(ms + nt)

( s ,t) E [0,271 2

are non—negative integers. Using Gaussian decoupling we see that this

series converges uniformly a.s. if and only if z

(2.12)

e ik(ms + nt)

k "Oic mnk

,

(s,t) E [0,2171 2

converges uniformly a.s. where (g;a 3 is an independent copy of f (gmk grilk )

is an independent symmetric sequence in k

. Since

it follows from Theorem 1.1

[8,Chapter I] that the series in (2.12) and consequently (2.11) converges uniformly a-s. if and only if J holds

ifmkgnk/

1/2

([0 211] 2 ,d) < oo . Once again exactly the same argument '

in (2.11) is replaced by (e c ) mk nk *

The finiteness of

1 J1/2([0'27]2 ,d)

marginal processes formed from X (s,t)

Theorem 2.2. Let X (s,t) the processes X (s,t ) o t oE [0,27]

also implies the continuity a.s. of the and X (s,t) . c

be as given in (1.1). Then if

j 1/2 (1° ' 2712 ' d) <

and X (s ,t) are uniformly convergent a.s. for each fixed g o

E [0,27] . Conversely if X (s,t ) and X(s t) converge o g 0 12 uniformly a.s. for some t E [0,2n] and s E [0,2rr] then J 1/2 (10,27J ,d) < oo . o o and s

o

This theorem is also valid if X (n,t)

is replaced by X (s,t) .

64

be as defined in (1.9) . Let us note that

Proof. Let d l (s,s') and d 2 (t,t')

(2.13)

N([°,271,di ;20

j = 1,2 .

N([0,27 ],d;c)

We will show this for j = 1 . The proof when j = 2 is completely similar. Let 1 ,

(s k' t k ) ' k

, N([0,27: 2 ,d;C)

that cover [0,27 2 in the metric

be the centers of the balls of radius

d . For a fixed k consider

.

B c,k = U ((s,0) E [0,27 1 2 : d((s10),(skttk)) < c 1 Let (;'.

k'

0)

be some fixed element in B

c,k

. It follows from the triangle inequality

that B

,°)) < 26 e t k C U ((s,0) E [0,271 2 : d((s,0),(e k

since

d((s,0),(s k ,t k )) + Since d

1k

that if J/

(2. 1 4)

E

2

) = d((s,0),(

(To ' 277 2 ,d)

(

m n

<

k

,0)) we have verified (2.13) when j = 1 . It follows

co

then

J1/ /2

12)V2 gme ims

la

([0 27] d ) < ' 1

co

and this implies that

s E [0,217 1

m'n

converges uniformly a.s.. By Theorem 1.1 E8,Chapter I] , the uniform convergence a.s. of (2.14) is equivalent to that of

E ( E a g'e m,n n m n

int

°) g e m

ims

s E [0,2n]

which, by decoupling inequalities similar to (1.3), implies the uniform convergence = 0 for all a.s. of X (s,t ) . (To avoid confusing notation assume that a m,n o m n

in all these series.) A similar argument shows that

X (s t) g o

converges

uniformly a.s.. The converse follows from the proof of Theorem 2.1 since in the relevant part of the proof of Theorem 2.1 the only property of the continuity of X (s,t) that is used is that its marginats are continuous. All the above statements remain valid when X (s,t) is replaced by X (s,t) . c The following corollary is immediate.

65

Corollary 2.3. Assume that a £2

. Then J

/2

m,n

= a a m n

where (a

n

is a real sequence in

12 ([0,27_1 ,d) < oo is necessary and sufficient for the uniform

convergence a.s. of X (s,t) and Xe(st) .

A1 r , Proof. Since ta2 j E , and because of (1.3) , X (s,t) converges m uniformly a.s. if and only if E a g e ims converges uniformly a.s.. The result m m now follows from Theorem 2.2. The same proof is valid for X e (s,t) . Now let us consider stochastic processes of the form

(2.15)

EaE gge m,n m,n m n m< n

where

a

and

m,n

indexed

i(ms +nt)

(s,t) E [0,27: 2

,

(g 1 are as given in (1.1) but where (e .} is a doubly n m,n

sequence of random variables with P (c- 1 0,0

P( e00 . —1) . ,

,

that

which is independent of (gn } . It follows from Theorem 1.1 [8,Ghapter I

the series in (2.15) converges uniformly a.s. if and only if J 1/2 ([0,2n17 2 ,d) < oo where

d

is as given in (1.5). This observation has an interesting interpretation.

Let (0,,P)

be a probability space on which

(Emyn)

is defined. For each w E 0

we consider the Gaussian Fourier quadratic form

(2.16)

E a E (w)g g e m
The metric

i(ms +nt)

,

72 (s,t) E [0,211J .

d (as defined in (1.5)) is the same for all these processes irregardless

of w E 0 . Furthermore, if J 1/2 ([0,217]

2

d) < œ then for almost all w E

Û

the

series in (2.16) converge uniformly a.s. (with respect to the probability space supporting (g

n

). However, the series in (2.16) do not necessarily converge

uniformly a.s. for all w E 0 . This can be seen from the examples we just gave of Gaussian Fourier quadratic forms which do not converge uniformly a.s. but for which 72 J 1 /([0,27 J ,c1) < oo . Thus, unlike the random Fourier series studied in Es] 2 metric continuity of Gaussian Fourier quadratic forms is not determined by the L

d

given in (1.5). Once again exactly the same argument applies if (gn }

replaced by (En}

in (2.16).

Nevertheless, entropy still can be used to characterize a.s. uniform

is

66 convergence of Gaussian and Rademacher Fourier quadratic forms. However, as we will see in the next section, it is necessary to consider the supremum of classical onedimensional entropies over a family of metrics.

3. Gaussian random Fourier series with coefficients in a Banach space

Fernique's corollary

Theorem 3.1. Let

sequence of elements of

E

sup

t E Lc) ,2n]

(B,11.11) B

of Talagrand's theorem

[6]

[13]

be a Banach space with dual

is the following.

B* . Let

3.cril

be a

with only finitely many non zero terms. Then

dE gx eint H n n n

E!lEgnxiJI

(3.1) E

sup

Eg < x* x > e int l n [0,27] n n

sup

tE x* E B* .

where

Therefore uniform convergence a.s. of random Fourier series with coefficients in a Banach space is characterized through conditions involving

a family of

classical one-dimensional entropies. We will see in the proof of Theorem 1.2 that

(3.1) can be used to obtain a similar result for Gaussian and Rademacher Fourier quadratic forms. Using a set of one-dimensional entropy conditions instead of a single two-dimensional entropy condition to characterize uniform convergence a.s. of random Fourier quadratic forms seems to be necessary, as was shown by the class

of examples described in § 2. Theorem 3.1 sheds some light on these examples. Indeed, the quadratic form (2.7) can almost be realized as a random Fourier series with coefficients in a Hilbert space. Define a sequence

(x

n

of elements in

setting

Vj> 1 , VnEI(j)

where

(eic l

xn

=

r

denotes the canonical basis of

X(t).Egxe int , n nn

b.

/

e

) 1/2

Ni

kEI(j) k

A2 • Consider

tE[0,273 .

02

by

67

Clearly

Hx(t)11 2 = E

n

11 2 +

E < x ,x > g g

mn mn

m n

e i(ITI-11)t

(3.2)

= E

Thus (I)

11X(t)11

+ 2 E b. j

J

j

E

EI(j) m
g g cos((m -n)t) . mn

is closely related to the quadratic form (2.7). For the choices of

§ 2, X(t)

J x_ = (

)

1/2

is unbounded a.s. since for

E

e

kE TO)

( bc*.i d = 1 )

k

we have

E

e int l _ E < x44.- x > n n t E [0,27] n sup

E

b 1/2

sup

t E [0,27]

C (b iN i logN i ) for some absolute constant of the form

1/

E n E T(j)

g e inti n

2

C . Likewise the examples of §2 show that no condition

J f ([0,2171,15) <

oo

(see (1.7)) where

e l m-n)t 1 12 ) 1/2

i(m-n)t 8(t,t') = ( E l< x ,x > 1 2 le m n m,n

is sufficient for the uniform convergence a.s. of the Gaussian quadratic form in

(3.2). (The relationship between Theorem 3.1 and metric entropy conditions is clearly

E

sup

IlEgxe int H n n

EHE g x n n

t E [0,2TT1 n

+

J,/2 ([0,27],d x *) r/

sup

1

where

dx*(t,t')

= ( E< x * x n> 1 2 1e

1nt

e

int'12)1/2

.)

We now use Theorem 3.1 to obtain Theorem 1.1.

Proof of Theorem 1.1. We will give the proof in the case

(1.4) we will prove this theorem with

X (s,t) and X (s,t) g

X'(s,t) -Eaggie m,n m n g m
i ms+ nt)

E

p = 1

By

(1.3) and

replaced respectively by

(s,t) E [0,2n 2

68

and X' (s.,t) =

E

a

m,n

m < n

c c'e

i(ms + nt)

(s,t) E [0,2n 2

n

ITI

where, to simplify the notation we take a

finitQly many of the probability space

(Q,,31,p1) and

(g

are non zero. Let us define

a m,n

(W,Y,P)

and

(g1"1 1

and

(g,,, 1

(e m 1

and

m

E

w E Q

Eg

Ee

Eg i Let

(QX0',3)0 1 ,PXP') .

denote expectation on the product space

us ( E am,ngm(w)e3 as a sequence in

and consider

on the

on the probability space

and denote their corresponding expectation operators by

E c , . Let

us fix

m n . Recall that only

if

n

of elements

[0,2a1 . By Theorem

in the Bahach space of continuous complex valued functions on

3.1 we have E

g'

sup lE ( E am

s,t n m

g (w)e

,n m

ims

) g'e

int i

E

g'

) supl ( E am ,n gm (we s n m

ims. ) g

(3.3) g (w)e ims ) g' sup E sup l E ( E a g' m,nm

e inti

t n m

By Theorem 1.4 [8,Chapter I] we can replace

fe r,1 1

by

(g

(3.3). Doing this and taking expectation with respect to

E sup IX'(s,t) ! s,t

EsuplE(Ea g')ge / m,n n m s m n

ms

m

in the last term in

3

we get

I

(3.4)

sup IE(Ea

+ E sup

t m n

c'e m,n n

int

) g e m

ims

.

Let us denote the first and second term to the right of the equivalence sign in

(3.4) by I I

and II . By Theorem 1.4 [8,Chaptcr I] applied twice

12 ) 1/2 e e us — E sup 1E ( E la s m n m'n c')ce ims IE EsuplE(Ea s m n m'n n m

In analyzing

II

II

sup IX'(s,t)

.

s,t

we repeat the steps of (3.3) and (3.4) and obtain

E sup lE (

a c'e int ) s,t m n m,n n

(3.5)

—E sup IE(Eac teint) t m n m 'n n

me ims

a I m

int )ge ims 1 E, c sup- E suplE( E am,n c'e n m t s m n

69

Denote the first term to the left of the equivalence sign in (3.5) by I' second by

and the

II' . Using the same argument as above we have

g) e'e int I' = E sup lE ( E a t n m m,n m n

— E sup t

I E ( E am,n ETTI:) c ' e int n

E

l

E sup s,t

m

We use Theorem 1.4 [8 ,Chapter I]

II' — E

I

sup E

on

I X'e (s,t) I

.

and obtain

IT'

e sup 1E(Eam,n e nl e s m n

int

)e ims m

I 1

E sup s ,t

X '(s,t)

I

.

Thus we have shown that

(3 .6)

ElIX'(s,t)11

C E

The reverse inequality in (3.6) is obtained by

C

for some absolute constant

1)q(s,t)11

the contraction principle. This completes the proof of Theorem 1.1.

We finally prove Theorem 1.2. Proof of Theorem 1.2. It will be sufficient to show that

E

1IX

(s ,t)11

sup

E

1E

s E [0,2"a]

+ E

sup

1 2)V2 -

la

E

(A i l

j = 1 or 2

where as before a

m,n

1

lam 11

1 2 ) 1/2 gne int l

n m sup

TE C* sE [0,27] a.s.

for

e ims

min

IE(E

sup t E [0,271

(3.7)

E

m n

= 0

if

Eimsl < T,A > g m m

m

m > n

and where the sequences

have been defined in (1.11). Indeed, Theorem 1.4 [8,Chapter I] then clearly

implies (1.12). Following the notation of the preceding proof we show (3.7) for

j = 2

and with the left hand side replaced by

EHX 1 (s,t)11

proof of Theorem 1.1 (cf. (3.3) , (3.4) and the estimate of

by (1.3). As in the I ), we have

70

E supl Xics,t) I s,t

(3.8)

E sup l E ( E la 12) m,n n m s

II

II

m

ims ) g e int n

the second term to the right of (3.8). It is plain that

= E

2 ims E A g e

sup

m m C

where El is the norm on

then follows easily from a new application a.s. . (3.7)

of Theorem 3.1 but this time in

II

2 ge ims l

g e sup E , sup I E ( E a m,n m t n in

+ E

Let us call

1/

-E

g

C

a.s.

2 TEA g rn

. Indeed ,

sup TE C*

m m

ims 2 E sup E < T,A > g e m m s m a.E

I

and by Theorem 1.4 [8,Chapter I] ,

2 E A g m

m m -

= EsuplE(Ea g) 9'e int I t n m m,n m n E sl)-P I E Ç E t n in

am,n

2 ) -1/2_, s Int

Appendix.

Proof of (1.3) and (1.4) : the argument that we give here was shown to us by Gilles Pisier. We will first prove (1.4). It follows from Theorem 2 , 17=1 that for

p > 1

(A.1)

E 11X c (s,t)11 1) -

EH E

a

m
(e e' + e'e m,n mn mn

where the constants of equivalence depend only on

p )ems+nt)d i(

p . Therefore by the triangle

inequality

EllX 6 (s,t)H P

Ej

C

e etei(ms+nt)ilp a m
Therefore, taking (A.1) into consideration, we see that to establish (1.4) we need only show

(A.2)

, i(ms+nt) iip

ee e C'EHEa m
EH

E am,n(em e'l' + eln'e n )e m
71 Let

0

be a real valued random variable uniformly distributed on

[0,27]

and let

D I be an i.i,d. sequence. Let (e'l be an independent copy of (9 1 . We define n n n inen in e' principle for quadratic tn = e t' n = e n . It follows from the contraction forms, Theorem 1 , [7]

(A.3)

a

m< n tn

We replace

that (A.2) holds if

z a ( t t* t' t )■ e icms+ nt)!!p mn mn m,n m
m,n m n

by

tne

ins

° , t'n = tle n

expression on the right in (A.3) where

int s

°

o

t

and

(s-s ,t-t ) in the o o are fixed in [0,27] . These

(s,t)

and

H

o

by

changes do not change the numerical value of this term, i.e.

(e i(ms o +nt o ) t t E a mn m
= E =

,i(nso+mto) t , t ) e - i(ms °+nt 0 ) e “.ms+nt)HP mn

ct t , +e i[(n-m)s o + (m-n)t o l t , t )ems +nt) li p Z Eli a mn m, n mn

m
Finally we note that by Jensen's inequality

27 27 H =

47

2

f

00

H ds dt

o o

2rr 2rr (t + r e 1 l (n-m ) s0+ (111-11)t Jds dt t't m ,n m n 2 o o mn m
Ell Z

a

S

ms + nt)Ilp

= Ell Eam,n m tTn e i ( Ins -1-11 t)HP m
Thus we have obtained (A.3) and consequently (1.4). The same argument works in the Gaussian case. We introduce gill and al") — ig' for g.1) and in place of (t 1 and (t'3 in (A.3) where g = g

n

n

as defined in the introduction and take Since

(7:1 )

n

to be an independent copy of

gn is rotationally invariant the exact same argument as above gives (1.3).

References

[1]

A. Bonami : Etude des coefficients de Fourier des fonctions de

LP (G) . Ann.

Inst. Fourier (Grenoble) 20 , p. 335-402 (1970). [2]

C. Borell

Tail probabilities in Gauss space. Vector space measures and

applications, Dublin 1978. Lecture Notes in Math. 644 , p. 71-82 , Springer

(1979).

.

72

[37

C. Borell : On the integrability of Banach space valued Walsh polynomials. Séminaire de Probabilités XIII. Lecture Notes in Math. 721 , p. 1-3 , Springer (1979).

[4]

C. Borell : On polynomials chaos and integrability. Prob. and Math. Stat. 3 , p. 191-203 (1984).

[5]

X. Fernique Régularité de fonctions aléatoires non gaussiennes. Ecole d'été de St-Flour 1981. Lecture Notes in Math. 976 , P. 1 -74 , Springer (1983).

[6]

X. Fernique : Fonctions aléatoires gaussiennes valeurs vectorielles. Preprint (1985).

[7]

S. Kwapien : Decoupling inequalities for polynomial chaos. Preprint ( 1 985).

[8]

M.B. Marcus and G. Pisier : Random Fourier series with applications to harmonic analysis. Ann. Math. Studies 101 , Princeton Univ. Press (1981).

[9]

G. Pisier : Les inégalités de Khintchine-Kahane d'après C. Borell. Séminaire sur la géométrie des espaces de Banach 1977-78, Ecole Polytechnique, Paris (1978).

[10] G. Pister : Sur l'espace de Banach des séries de Fourier aléatoires presque sûrement continues. Séminaire sur la géométrie des espaces de Banach 1977-78, Ecole Polytechnique, Paris (1978). [11] G. Pisier : Some applications of the metric entropy condition to harmonic analysis. Banach spaces, harmonic analysis and probability, Proceedings 1980-81. Lecture Notes in Math. 995 , p. 123-154 , Springer (1983). [12] M. Schreiber : Fermeture en probabilité des chaos de Wiener. C.R. Acad. Sci. Paris, Série A , 265 , p. 859-862 (1967). [13] M. Talagrand Régularité des processus gaussiens. C.R. Acad. Sci. Paris, Série I , 301 , p. 379-381 (1985). [14] D. Varberg : Convergence of quadratic forms in independent random variables. Ann. Math. Statist. 37

p. 567-576 (1966).

RATES OF CONVERGENCE

IN THE CENTRAL LIMIT THEOREM FOR

EMPIRICAL PROCESSES by Pascal MASSART Université Paris Sud U.A. CNRS 743 "Statistique Appliquée" Mathématiques, Bit. 425 91405 ORSAY (France) -

SUMMARY :

In this paper we study the uniform behavior of the empirical brownian bridge over families of functions F bounded by a function F (the observations are independent with common distribution P). Under some suitable entropy conditions which were already used by Kolnnskii and Pollard, we prove exponential inequalities in the uniformly bounded case where F is a constant (the classical Kiefer's inequality (1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where F belongs to

L 2+6

(P) with

E 10,1] (our results

improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Cervonenkis class in the uniformly bounded case and are new in the unbounded case).

1.

Introduction

2

Entropy and measurability

3

Exponential bounds for the empirical brownian bridge

4.

Exponential bounds for the brownian bridge

5

Weak invariance principles with speeds of convergence

6.

Strong invariance principles with speeds of convergence

APPENDIX : 1. Proof of the lemma 3.1. 2. The distribution of the supremum of a d-dimensional parameter

brownian bridge 3. Making an exponential bound explicit.

Key words and 2hrases : Invariance principles, empirical processes, gaussian processes, exponential bounds.

74 1. INTRODUCTION 1.1. GENERALITIES. (X,X,P)

Let

be a probability space and

(x n ) 01 be some sequence

indepen-

of

-

dent and identically distributed random variables with law

brownian

1 n n i E_ i 6 x i

stands for the empirical measure

P

bridge relating to

and we choose to call empirical

the centered and normalized process

Our purpose is to study the behavior of the empirical

F,

where

F

rich

on a

(Q,A,Pr).

enough probability space

Pn

P, defined

is some subset of

brownian

vn - /6

bridge uniformly over

L 2 (P).

More precisely, we hope to generalize and sometimes to improve some classical

results about the empirical distribution functions on of quadrants on ]R

d

),

in the way opened by

R d (here

Vapnik, &rvonenkis

F

is the collection

and Dudley

.

In particular, the problem is to get bounds for:

(1.1.1) where I

Pr

( Hv n !! F > t)

4 F stands for the

mations of convergence,

vn

for any positive

uniform norm over

F

t,

and to build strong uniform

gaussian process indexed by

by some regular say

,

F with some speed

approxiof

(b n ) .

First let us recall the main known results about the subject in the classical case described above.

1.2.

THE CLASSICAL BIBLIOGRAPHY.

We only submit here a succinct bibliography in order to allow an easy comparison with our results (for a more complete bibliography see Concerning the real case

P

and are optimal

(d=1) ,

[26]) .

the results mentioned below do not depend on

:

1.2.1.

(1.1.1) is bounded by to

Dvoretsky, Kiefer

and

C exp(-2t 2 ),

where

Wolfowitz [24] (C < 4/2

C

is a universal constant according according to

[17]).

75

1.2.2. The strong invariance principle holds with

b n - log(n) ,

Komlos,

according to

In

Major and

Tusnady [37] .

In the multidimensional case (d

_> 2).

1.2.3.

(1.1.1) Kiefer

C(E) exp (-(2-E)t 2 ),

is bounded by

[34] .

In this expression

E

E > 0,

for any

cannot be removed (see

[35]

according to

[28]).

but also

1.2.4. The strong invariance principle holds with b

n

= n

1 2(2d-1) Log(n),

according

[8].

to Borisov

P

This result is not known to be optimal, besides it can be improved when

[0,11 d .

uniformly distributed on

1.2.5.

If

d

-

In this case we have

is

:

2:

The strong invariance principle holds with

b n - (Log(n))2 , /6

according to

Tusnady [50]. 1.2.6. If d > 3 :

—

The strong invariance principle holds with to

Csbrgb

and

1.2.5

1 3 2(d+1) (Log(n)) 2 , b n -n

Révész [14] .

and

1.2.6

are not known to be optimal.

Let us note that even the asymptotic distribution of (the case where in

according

d

- 2

and

P

ky n il F

is the uniform distribution on

is not well known

[0,1] 2

is studied

[12]). Now we describe the way which has already been used to extend the above results.

1.3. THE WORKS OF VAPNIK, hRVONENKIS, DUDLEY AND POLLARD. Vapnik

V

and

Cervonenkis

rally called V..-classes

-

introduce in

[51]

some classes of sets

for which they prove a strong

large numbers and an :exponential bound for

(1.1.1) •

-

which are gene-

Glivenko-Cantelli

law of

76

P.

Assouad

[40]

also

studies these classes in detail and gives many examples in

[3]

(see

for a table of examples).

P-Donsker

The functional

classes (that is to say those uniformly over which some

central limit theorem holds) were introduced and characterized for the first time by Dudley in

[20]

and were studied by Dudley himself in

[27]

Some sufficient (and sometimes necessary, see bounded) conditions for of entropy conditions

F

to be a

P-Donsker

[21]

and later by Pollard in

F

in case

[44].

is uniformly

class used in these works are some kind

:

Conditions where functions are approximated from above and below (bracketing, see

[20])

are used in case

F

is a

P-Donsker

restricted set restricted set of laws on respect to the

Lebesgue

X (P

F

is a

P-Donsker

of laws including any finite support law (the

-

belongs to some

is often absolutely continuous with

v .. Kol icnskil

measure in the applications) whereas

conditions are used in case

bility assumptions

P

class whenever

class whenever

V.L-classes

the classes of sets of this kind, see

are

P -

and Pollard's

belongs to some set under some measura-

[21]).

In our study we are interested in the latter kind of the above classes. Let us recall the already existing results in this particular direction. Whenever

F

is some V..-class and under some measurability conditions, we have:

1.3.1.

(1.1.1) Alexander in

is bounded by

[1]

and more precisely by

C(F) where

C(F,E) exp(-(2-dt 2 )

for any

in

E

]0,1],

according to

:

(i+t2)2048(D+1) exp(-2t 2 )

D stands for the integer density of

F

(from

,

in

[2] *

Assouad's

,

terminology in

[3]) .

1.3.2. f

(1.1.1)

is bounded by

4e'

D

E

\i=0

2 ))p(-2t ex 2 ) , according to Devroye in [16].

* Our result of the same kind (inequality 3.3.1°)a) in the present work) seems to have been announced earlier (in [41]) than K. Alexander's one.

77

1.3.3.

1 The strong invariance principle holds with b n -.= n

2700(D1), + according to

Dudley and Philipp in [23]. Now let us describe the scope of our work more precisely.

2, ENTROPY AND MEASURABILITY.

From now on we assume the existence of a non-negative measurable function F such that !f! < F,

for any f

in F.

v We use in this work Kolcinskii's entropy notion following Pollard [44] and the same measurability condition as Dudley in [21] . Let us define Kolcinskii's entropy notion. Let

p

and p (p) (X) F 2.1.

be in [1,+.[ . A(X) stands for the set of laws with finite support for the set of the laws making FP integrable.

DEFINITIONS. Let c

be in ]0,1[ and Q

N ( P ) (c F Q) F "

be in 4. 0 (X) .

stands for the maximal cardinality of a subset

G

of

F

for which:

(!f g!P) > EPQ( FP)

Q holds for any f,g c-net of (F,F)

in

G

with fg (such a maximal cardinality family is called an

11P) (.,F,Q) . relating to Q). We set NP ) (.,F) = sup QEA(X)

Log(N F(p)(.,F)) is called the (p)-entropy function of (F,F) . The finite or infinite quantities :

d(F) = inf fs>0 ; limsup 0

s

E N

(p) F

(c ' F) < col

e(F) =inf fs>0 ; limsup c s Log(N ( P ) (c,F)) < col F F 6-* 0 are respectively called (p)-entropy dimension and (p)-entropy exponent of (F,F).

78 Entropy computations. from that of a uniformly bounded family as

F

We can compute the entropy of

:

follows

- (T 1 (F>0) '

Let I

fE Fl ,

then

:

( ) N( F . ,' F) < N 1 P ( ' I) Q

For, given FP

so

E A(X)

in

A(X),

either

Q(F)=0

and so

qP ) (.,F,Q) = 1,

or

Q(F) > 0,

:

and then

Q(F P )

NPP ))(. ,I 1 ( 1 Q(F) n

N F( P ) (.,F,Q)

Some other properties of the

(p)-entropy

are collected in

The main examples of uniformly bounded classes with finite

[40]. (p)-entropy

dimension

or exponent are described below.

2.2.

COMPUTING A DIMENSION According to Dudley

:

THE V..-CLASSES.

[20]

on the one hand and to

Assouad [3]

on the other we

have

d (1 P (S) = pd whenever

[3]).

S

is some

[45] .

COMPUTING AN EXPONENT Let d

Dk

with real density

d

(this notion can be found in

Concerning V..-classes of functions, an analogous computation and its appli-

cations are given in

2.3.

V.C.-class

See also

:

HOLDERIAN

THE

a

be an integer and

[21]

for a converse. FUNCTIONS.

be some positive real number.

We write

8

for the greatest integer strictly less than a.

Whenever

x

belongs to

P„c1

for the differential operator

Let Let

II.11

be some norm on

k to A d , lk a kl k k ' Bx 1 1 ...x d d

and

stands for

f

and

Rd

Ad be the family of the restrictions to the unit cube of

13-differentiable functions

k 1 ”' +lc d

such that

:

Rd

of the

79

max sup

ID

!k!<13 x€Rd Then, according to

k “x)1 + max sup 1D k f(x)-D k f(y)! < 1 . !k!=i3 x#y Hx-y!1

[36]

on the one hand and using Dudley's arguments in

the other, it is easy to see that

[19] on

:

e (I P ) (Aa,d )

a

Measurability considerations.

Durst

M P-PE

and Dudley give in

[21]

an example of a V..-class

S

such that

1

So some measurability condition is needed to get any of the results we have in view. So from now on we assume the following measurability condition (which is due to Dudley

[21]) to be fulfilled :

(M)

.(X,X)

is a

Suslin space

. There exists some auxiliary Suslin space from Y onto F such that : (x,y) F

and we say that

T(y)(x)

(Y,y)

is measurable on

is image admissible

Suslin

via (Y,î)

and some mapping

(XxX, X

T

V)

.

This assumption is essentially used through one measurable selection theorem

[47]

which is due to Sion

(more about

Suslin spaces is given

in

[13]).

2.4. THEOREM. Let H on

X.

be some measurable subset of

Then

A

rable mapping from

XxX .

We write

A for its projection

is universally measurable and there exists a universally measuto

A

Y

whose graph is included in H .

A trajectory space for brownian bridges. We set

: l(F) =

We consider

: 1 °3 (F)

F-*

; h oT

is bounded and measurable on

as a measurable space equipped with the

(Y,Y)1 . a-field generated

80

by the open balls relating to

1 °3(F)

a-field because

11.11

(which is generally distinct from the Borel

is not separable).

P

This trajectory space does not depend on

[20 1 ) but only on the measurable representation

any more (as it was the case in

(Y,T)

of F .

From now on for convenience we set :

(Q,A,Pr) X

where on

\ (X ,Xw ,P )

(X ,X ,P )

8([0,1]), P oe

X)

[0,1] , B([0,1]) for the Borel

stands for the Lebesgue measure on

[0,1] and

duct

(ex [0,1],X°3

G-field

for the completed probability space of the countable pro-

of copies of

(X,X,P) .

The following theorem points out how

1;3 (F)

is convenient as a trajectory

space.

2.5. THEOREM. For any a in R n , Moreover , setting on

(F,p p )1, U b (F)

I a iX. i=1

is measurable from

u b (F) =

:F4- IR

is included in

l(F).

Q

to

l_T(F).

;his uniformly continuous and bounded Provided that

(F,p p )

is totally boun-

ded this inclusion is measurable.

L. 2 (P)

Where

is given the distance p (f-g), with 2 Gp : f÷P(f 2 )-(P(f)) 2 . For a proof of 2.5. see [21] (sec. 9) and [40] where it is also shown that many reasonable families (in particular

A

et ,u

and the "geometrical"

V.C.-classes) fulfill (M) . 2.6. REMARK. Since whenever

F

fulfills (M) it follows from [21] (sec. 12) that

UP n -P!! F -4-0 a.s.

(1) N F (.,F) < . and therefore : sup

(2)

NF

(E,F,Q) < NV q)

- ,F)

for any

E

in

]0,1[ .

( QEP2) F (X) This implies that the local behavior of the entropy function is unchanged when taking the sup in 2.1. over the set of any reasonable law .

81 3. EXPONENTIAL BOUNDS

FOR THE EMPIRICAL BROWNIAN BRIDGE

u

We assume in this section that for some constants any

f

in

F;

U - v-u

we set

F-u = {f-u ,

and

and

v,u
for

f E Fl .

The following entropy conditions are considered

:

a) d U(2) (F-u) < 00 b) e U(2) (F-u) < 2 Using a single method we build upper bounds for following two situations

1°) F.

Observe that

(1.1.1)

that are effective in the

: U2 ; ilap2 H F < -4-

nothing more is known about the variance over

In this case we prove some inequalities which are analogous to

quality

2°)

Hoeffding's

ine-

[30]. We assume that

H4H F

This time our inequalities are analogous to Bernstein's inequality (see Bennett

[5]) . 3.1.

DESCRIPTION OF THE METHOD.

N =mn .

We randomize from a sample which size is equal to Dudley's

[20]

or

Vapnik

following an idea from

In Pollard's

v and

Cervonenkis [51] symetrization

Devroye [16] ,

we choose a large

Effecting the change of central law type inequality, we may study

Pn -P N

: P-* P N

instead of

technics,

m=2

but here,

m.

with the help of a Paul

P n -P

[44],

where

Pn

Lévy's

stands for the

randomized empirical measure. Choosing some sequence of

-

measurably selected

-

nets relating to

PN

whose

mesh decreases to zero and controlling the errors committed by passing from a net to another via some one dimensional exponential bounds, we can evaluate, conditionally to

PN ,

the quantity

HP n -PN I! F .

82

Randomization . N=nm ( m

Setting from [1.n]

[1,N]

into

w

is an integer), let

be some random one-to-one mapping

whose distribution is uniform (the "sample w is drawn without

replacement"). The inequalities in the next two lemmas are fundamental for what follows :

3.1. LEMMA. For any

in

, we set

S = N

Sn =

i=1 N

\2

1=1 andU N -(max.)) - ( min 1 1
(.)) ; 1

10)

i=1

the following three quantities are, for any

S positive E , lower bounds for- Log

J

Pr (!lf

SN T1-1 > E)):

2n E 2 UN

nE

2°)

2 + EU N N

2 3°)

nE

2

2

2 ma N

These bounds only depend on

(U N ,aN ).

through numerical parameters

Bound 3°) is new ; concerning 1°) (due to Hoeffding [30]) , Serfling's bound is better (see [46 1 ) but brings no more efficiency when

m

is large.

The proof of lemma 3.1. is given in the appendix.

i From now we write

P n for the randomized empirical process

n

. 6 n i=1 No)

The

inequality allowing us to study the randomized process rather than the initial one is the following :

83

3.2. LEMMA. The

random elements

Besides, whenever

UPn-PNF andn-P F are measurable. Mcs2PI ! F —< p 2 , the following holds

2 19 2 2 , ) Pr( !P n -13 a E n any positive c and any a in (1

for

For a

proof

> E)
, n' F > (1-a) Tr E)

P n-PN

, where n = N-n .

this lemma see [16] using Dudley's measurability

of

arguments in

[21] (sec. 12) . Statement of the results.

3.3. THEOREM. The Pr

following quantities

( !v n !! F > t) 1°)

a)

h)

where k 2°)

are, for

any

positive

t

and

ri , upper bounds

for

:

if

d (2) (F-u) = 2d

if

2 0 1-1,F (1) (1 + L) 3(c1-1-1-1) exp (-2 U2 (2) < 2 , e (F-u) =

,

I

)

t k+n t2 0n,F (1) exp (O 1,F (1) (0) ) exp (-2 -7) U2 6-c C( 777 ) (when c increases from 0 to 2 so does k). Suppose a) if

that !!720 F < a2 , with G < U, then d u(2) (F-u) = 2d , 2 3 ( d + fl) ex p( o n,F (1) (2)- 4(d+n) (1 + L) U 02

h) if

0 n,F (1) exp(0

2(0- 2 +(3U+t)) ) in

4 2) (F-u) = c < 2 ,

11,F (1) (.g)U

where p - 2c(4-C)

t2

(when

5 (; ) 2n r

+1-, ) e Xp

t

2(0.2 + increases from 0 to 2 so does 2p) .

2 (3u ()P+fl +t)))

84 The constants appearing in these bounds depend on

N (2) (.,F-u)

F

only through

n .

and of course on

Comments.

2.2.,

From section

F

d (2) 1

the assumption

is some V..-class with real density Thus

bound

the factor

1')

1°)

0(F,) t 2nd

1°)

in

a) improves on

1.2.3. 1°)

moreover the optimality of

d.

F

but is less sharp than

on

F - A a,d Bakhvalov proves

[0,1] d

then

(2t2)i

2.3.

then, from section in

[4]

P

that if

in the real case

;

exp (-2t 2 )

we have

:

(*)

d e (2) 1 (F) = a- .

In other

stands for the uniform distribution

:

1_

Hn H

a

d

F > C n2 1°)

Thus we cannot get any inequality of the

e (2) 1

1.2.1.

i=0

Suppose that

,

a) is discussed in the appendix where we prove that

1'H- cc

respects,

in another connection

is the collection of quadrants on R d )

( Ilm n r F > t) > 2

Pr

1.3.1. ;

a) is specified in the appendix.

d-1 lim

is typically fulfilled whenever

a) is sharper than those of

In the classical case (i.e. bound

(F) <

surely

or

2°)

type in the situation where

(F) > 2 . The

border line case

:

,

For any modulus of continuity in the same way as

Ad by changing

greatest integer for which

(1)(u) u -I240

It is an easy exercise, using

we can introduce a family of functions

u÷u a

q)

and defining

as the

holds.

Bakhvalov's

!v

into

A ,d

method, to show that

:

> C (Log(n))Y

d

provided Of course

that

e 1(2)

4)(u) = u 2 (log(u -1 )) Y (Ad) 4D,

)=2

and

P

is uniformly distributed on

and we cannot get bounds such as in theorem

[0,11 d .

3.3.

(*) So, there is a gap for the degree of the polynomial factor in the bound 3.3.2 ° )a) between 2(d-1) and 6(d+n) .

85

But the above result is rather rough and we want to go further in the analysis of the families Then the

around the border line.

Acp,1

(2)-entropy

A a, 1 concerning the Donsker

plays the same role for

property as the metric entropy in a Hilbert space for the Hilbert ellipsoids concer-

pregaussian

ning the

1

1 (i)

A(1),1

:

property, that is to say that the following holds

P-Donsker

is a functional

2 (2) (Log(N 1 (E,A ))) q), 1

class whenever 10

dc

<

co

.

1 (ii) A

(1),1

X-Donsker

is not a functional

Log(N (2) (E,A (1), )) 1 1

and in this case we have

(E Log(c)Y

(i)

follows from Pollard's central limit theorem in

(ii)

follows from a result of Kahane's in

qb(u) -

In fact, if we set

e (t) n n KnLog(n)

E

t --,- E n>1

V 1 Log(u)1 u )1 ,1

belongs to

,

[32]

about Rademacher trigonometric series.

(E n !W(e n )!)

(E n )

with

being independent of

L n>1

>

A (1),1

W(e nLog(n) n )1

W

some Wiener process provided that of G

.

So

(*) We write

A

f

(lb, 1

is not

g

, when

brownian

W(1)

pregaussian

0 < lim

that

1,

where

(E n )

we may write (We n ))

1 nLog(n)

bridge G for

N(0,1)

(ii)

is proved.

(fg 1 ) <ï

:

•

is some and

:

(IW(e n )!) -

is almost surely unbounded on

The same property holds for any

66

p K -1K4-

L 2 ([0,1]),

E IW(e)

By the three series theorem the series almost surely and therefore

p.

p K , the following holds

So that, with probability more than NI

[32]

(27nt) .

Let us consider a standard Wiener process on as

[44].

we have from

cos

(*)

22 .

with some probability

e n (t) = /2

is a Rademacher sequence and

.ci , (u)

class whenever

u )2 ILog(u)I

(

diverges to infinity

A

1

f-)-G(f) + fW(1)

is

random variable independent

(fg -1 ) <

86

"An upper bound in s?:tuation 2°) is also an oscillation control". If we set

= {f-g ; G (f-g) < o , f,g E Fl , it is not difficult to see that:

G

N

314. U

thus changing

2U

into

situation 2°) hold with

(2) G +U) 2U (" a and d

G

U 2d

into

' if necessary the upper bounds in

the constants being independent of

instead of F ,

a

because of 3.4. . In particular if

F

d, we set :

is a V.0.-class with real density

A(a,n,t) = Pr ( Hy n !! Ga >

.

At it is summarized in [231 Dudley shows in [20] that is small enough, o = 0

n > 0(t

and

(!Log(0

-r

)

A(a,n,t) < t

whenever t

r > 8 .

with

Applying 3.3.2°) a) improves on this evaluation for then

A(c,n,t) < t

t

whenever

t

is small enough, o

and

- "iLog (t)1 ) t

n > 0

-4 bound 2°)a) depends on

In order to specify in what way the constant in

F,

we indicate the following variant of 3.3.2°)a) .

3.5. PROPOSITION. If we assume that o

E

1

]0,1[

in

Ha 2PII

]0,1[ that

in

N

(2)

(E,F-u) < C (Et E) -2d for any

< a2

with

F—

depending only on

E

o

g

not exceeding

and a constant

K

E

]0,1[

in

and some

U, then there exists some depending only on

C such

that : Pr ( Hy n !! F >

From now on

L

t 2 14d ( -d o -4d
t2 2 U(31.1+t),) 2(a +

x+-Log(xve).

stands for the function

3.6. COROLLARY. Let

(D n ) .

(F ) n

be some sequence of V.. -classes fulfillino(M) with entire densities

Then (with the above notations) Pr (

t ; whenever

G

2 n

= o(1/(D n L(D n )))

and

an 2

0

Gn - 0 (070 .

> t)

0

for any positive

87 (Provided that

in D n = o(--) such a choice of o Ln ' n

does exist).

Comment. { v n (f), f E F n } [38] (Lemma 2) and applying 3.6. the process i admits finite dimensional approximations whenever D n = o (Lon g(n)) and provided According to Le Cam

that Le Cam's assumption (Al) is fulfilled. This result improves on Le Cam's corollary of proposition

< 21 is needed.

for some

y

Proof of

3.6. F

Let

4, V.C.-class with entire density

be a

w > d (or w > d if d

show that, for any

(2)

in

]0,1[ ,

(2) (E,F) N1 C

C 31 e 5D

2 3D

in

w-D , K =

d

.

it is easy to

:

2w

(2D) D .

: e

for any

c

Hence, for any

1'

is "achieved"), we have

with in particular when

So from Stirling's formula we get

tant

and real density

1+(112 LogE) exp (2w) (1 + 2!LogE )w (E,F)
N1

E

D

[20] (more details are given in [401)

Using Dudley's proof in

for any

D n = 0(n -Y )

3 where

]0,1[

10, 1-1 i/-

in

we have

and some universal cons-

:

N 1(2) (E,F) < C 3 (2e) 5Dc -4D — 1 thus, applying

3.5. to the class

G

yields

3.6.

œn We propose

below

another variant of

inequality

3.3.2°)a), providing an

alternative proof of a classical result about the estimation of densities.

3.7. PROPOSITION. --
If we assume that some positive bound of Pr

d U(2) (F -u) = 2d < co and

( Ily n H F > t)

o V,F,r1W

t 2 3(d+I)

(n(1 + 7))

In the situation where

3.3.2°)a) .

is, for any positive

U

t, given by :

a _4(d+n) exp ( (0)

t2 2 U 2 (o + -- (CLLn (IL + c5)+t)))

VT' in is large this inequality may be more efficient than

88 Application to the estimation of densities : minimax risk.

K

Let

K q)

where and

M

= 11)(y'M )

(y)

kxk

is some

V.L-class

[45]

(.-x),

for any

P

Now if we assume that

Rk ,

measure on

is a

We set

K

E

ME

Rk

2

1 1 E- 2,2]

, xE R k

: 10,1[

in

C

where

w

and

depend only on

is absolutely continuous with respect to the

f

the classical kernel estimator of its density

n

K

into

that the class

of functions and so

N 1(2) (c ' K) < CE -w

(x) = h -k P n

with fixed

tp

Lebesgue

:

h

M

and

is

k.

so that

f 2 (x)dx <

.

T = E(f n )

Proposition

3.7.

gives a control of the random expression

F = th -k/2 K(' -x )

x E Rk 1 , G IL Ln

So, if we assume that

Pr

R

matrix.

K

where

Rk

y in

for any

is some continuous function with bounded variation from

Pollard shows in

is a

Rk :

be the following kernel on

= C

> h -k —

(vçj D n > t) < 0(n a ) tI3 exp

and

c2'

U = h - k/2

C2 >

where

we get, setting

2 (c 2

fn -T

by choosing

:

il .[K2(x)dx. j

D -n stip!f f(x)1 : - x (x)n

t2 0 (J_Ln )

t

)

41" for any

t

in

[1 + 04

and some positive a and

E (vrj D n ) < T + 0(n œ )

T

exp

2(c 2

13 .

T2 0 ( LLn ) %KJ

for any

T

in

We choose

[1, + .[,

provided that

T = 0(/g) ,

thus

nh k >

:

E(D) n

0 ((-15) 2 )

Hence, after an integration:

T In7

)

89 f

Provided that

T-f can be evaluated so that the minimax risk associated to the uniform R k and to 9 can be controlled with the same speed of convergence as

expression distance on in

[29] ,

belongs to some subset of regular functions G , the bias

via an appropriate choice of h

.

3.8, SKETCHES OF PROOFS OF 3.3., 3.5., 3.7.

= (fT?, fEF1

First, by studying the class G

u=0

and

v=1.

s,13

and positive

We set

3.3..

Let us proof theorem

parameters such as

: N =

:

p

a,

(in

F,

instead of

[42]) .

we may assume that

All along the proof we need to introduce

]0,1[) ; r, m

; a

(in

]1,+.1); q

(in

(in

10,21)

which are all chosen in due time.

and y

,

mn

(More details are given in

and

E = --

E' ' (

1 - 1)

(1-a)

E .

1/171

We write Pr (N)

(x

(.)

The

IP n

(

r

( Iv n

- P N II > EI)

> t)

for short.

will follow, via

3.2., from

a bound for

which is at first performed conditionally on

(x

1'

...,xN ) .

chain argument. Let

(T i ) i>1 be

a positive sequence decreasing to zero.

For each integer j of

.4

instead of

and

' xN )

A bound for Pr Pr

for the probability distribution conditional on

2.4).

A projection

2 < T. — J

2

a

T.-net F. can be measurably selected (with the help

7i may be defined from

F onto

Fi so that

holds.

Then

Il(P n - P N ) ° (Id - 7 r )I

<

n

j>r+1 So,if(.)is na

- P N ) ° (7. j

a positive series such that

j>r+1 Pr

where

A and

B

(N) (P nN

are the

I

>

E')

<

A

n. < p —

+ B

(x1''"'xN)-measurable variables

:

- Nr Pr (N) (r( P n - P N ) 7 r > (1 - P) C) B = E N2 Pr ( '11J' n -P N) 0 (7,-7J-1 J - )r > fi. J s') j>r+1

A

J-1 we get

:

90

stands for NV ) (TF))

N-

(where

is the principal part of the above bound and

A

B

is the sum of the error

terms.

1 ° ) or 2°) of Lemma 3.1. are needed to control

Inequalities whether case Bound

A

according to

1°) or 2°) is investigated.

3°) in Lemma 3.1. is used to control

B ,giving : ,2 2

2. exp ( 3.8.1. 4m 1 j2 - 1) j>r+1 N J 1 I1 7-71 Choosing fl- = ( j - 1) -a and r - 2 + '(- ) I, (so E fl- < p holds whenever Lu j>r+1 J > 2), the control of the tail of series 3.8.1. is perfb-rmed via the following B < 2

E

-

:

elementary lemma 3.8.2. Lemma. Let

tp

[r, +0,4 -* R. Provided that

following inequality holds

where

q)

is an increasing convex function, the

:

1 E exp (- 11)(j)) < exp (-t ( r)) j>r+1 cl (r) tpi 'd stands for the right-derivative of tp . We choose

13

1

13 -

under assumption h)

.

3.3. in case 1°).

Proof of theorem

We choose

under assumption a) and

a

= t -2 , m = [t 2 i and

T.

A

and apply

.

J

VF

< 2 M r e 10 exp (-2t 2 (1-2p))

3.1.1°), then :

P EIN -a.s.

Under assumption a). Considering the type of inequality we are dealing with we may assume that .

J

N < C t2d j 2((1+1)d (instead of N. < C' t 2d' j2(a+l)d'

J

We choose

, p = t -2

and a A

and

for any

d'

> d) .

—

- Max (2, 1 +

4d

, so :

< GnF(1) ( 1 +t 2 ) 3(d+n) exp (-2 t 2 ) — ,

P NN -a.s.

91

whenever

oN

0 n,F (1) exp (- 2 t 2 )

B t 2 > 7+4d(a+1) .

Now the above estimates are deterministic, so using

3.2., theorem 3.3. is proved in situation 1°)a) .

Lemma

3.5. note that, setting a

With the idea of proving proposition

, under the hypothesis in 3.5. , that

method gives

with

Pr(

= 2 , the above

v n !>t) is bounaed by :

-1 4d (2+t 2 ) 12d exp (- 2t 2 ) K 1 (E o t) K 1 depending only on C , whenever t 2 > 7+12d.

Under assumption b).

We may suppose that N J We set for y

< exp (C t

p = t-2 Y , where

> y(d -

and

tion when a = +.0

13 > 1

(1+2y(P.:1Hf ) ) = 2(1-y) , then we choose a large enough y( C)

to hold, where

: 2(1-y( ))= k).

(Namely

- 0

A

j (a+5) )

So

is the solution of the above equa-

:

(1) t k+n ) exp (-2 t 2 ) n,F (1) exp (0 n,F

and

B = o (1) exp (-2 t 2 ) n,F t 2 >+5+ C t C 2 213+2 —

whenever

So theorem

Proof

3.3. is proved in case 1').

of tneorem 3.3. in case 2 0 ).

We set

(4)- a and choose m=(,0 (1 , a =

The variable A

P - (P -G1 and

is this time controlled with the help of

2

problem is to replace

3.1.2°), so now the

2

GN by up

In fact, let O N be the

(x l ,...,xN )-measurable event :

2 2 2 aN2 - a 2p '' > sl , where a(f) =P N (f ) - (P N (f))

N

Each term of the following estimate is studied in the sequel Pr where

A'

= E (A II

j - (erh3)

T

c)

( and

rP n -P N

> E') < Pr

B' = E (B 11

c)

(a N )

+ A' + B'

for any

:

f

in

F.

99

Bounding

is a prColeT of

Pr (ON )

F2

For, setting

_

type

1')

f E F} , we have :

a2p < IPN - Pll 9 + 2 11PN - P F(2) 2 (2) Since N (.,F ) —1 ( 12 ,F) and F 2 fulfills (M), we may use the bounds < N 1 2 in 3.3.1°), so, choosing s , we get : Pr (ON ) < C o exp V2nm

IIGN2

The evaluation of

kJ12\1 A' <

A'

< a2 +s

and .8'.

holds on

2 N r exp (5 (42-q ) exp (

Moreover

-t

:

gives

2

,

0 > 2 .

whenever

2 (0 2 + ( t +j -c1/2

: B

Now

Orcl , thus applying 3.1.2°)

< 2

, 22 (j...1) 23 ) N.2 exp (- nc 4 a j>r+1

the proofs are completed as in case

a = Max (2,1 + 4d T1 ) large enough for

under assumption a) and

1-

< p + n and (3 > 1

To prove proposition

q-2

11, choosing this time o = (2-c)

and

)+ 1) • 1,,,itn

+

- 1

et

to hold.

3.5 we choose a = 2, so

A' < 2 C c o-2d e ' a-2d (4) 2d (2qt,o26d ) exp (

t

2

02+

2 1.6.1

B' < 2 C 2 c-4d a-4d (4)4d (2+T 2) 12d exp ( o whenever

4

)

cp 2 > 8 + 12d .

Besides, using

3.8.3., we get :

Pr (ON ) whenever

2 2 (cp -8)

2 4 —> 7+12d,

(1-

2 2 I2d < 2 K1 4 ) -4d () 4d (2 + TiF) exp (- 2;2 ) which completes the proof of proposition

3.5

via Lemma

Proof of proposition 3.7. We assume that

u=0

and

v=1 .

Inequality

3.3.2°)a) may

be written

:

3.2.

93

t

o

Pr (

:

v

> t) < K a

n

t2

2

-a

M

1 ( t ) OE2 exp ( -2

2 (02

a

(3+t) ) ) in

2

whenever -- > 5 . G

2 —

Defining the following sequences by induction : a. = 5 J+1 b- = with a

1

o

Pr (

and b

0

/6

,

v ri ll > t) < Ka i a

44d+

( 1

ii,71

/171

1 a

1-,

G

1 )

j , 1-

no

AT)

we call - 1

2b

(2

M. the following inequality :

,2 t2 t 2 '1 2 , ) whenever L'.> 5 (7) exp ( 2 — \ 2(c2 +b. + -11.-)/ G c

in

J

from M. by the The, assuming that M- holds, it is possible to deduce M j+1 J J same way as 3.3.21a) from 3.3.1°)a) (technical details are given in [421) . Then inequalities (M.) hold by induction. J =

LLn + [77]

Using inequality Mo , where

and a few calculations yield proposition 3.7. .

/4.

EXPONENTIAL BOUNDS FOR THE BROWNIAN BRIDGE

2 We assume that P(F ) < section still hold

. We want to show that the bounds in the preceding

for the brownian bridge.

4.1. THEOREM If e

(2)

(F) < 2 , then there exists some version G

of a brownian bridge rela-

ting to P whose trajectories are uniformly continuous and bounded on (F,p p ) Moreover, setting c = e Pr ( G 0

p F

fl,F

(2)

(F) ,

2 2 2 if o II < a < P(F ),

an upper bound for

> t) , is, for any positive t and 11 , given by :

(1) exp (0

) fl,F

2 t ( p(F 2 ) p2+ -n ( t ) 2p-ç+n + ( I)2p+n,) e xp H --2) 2c

(2) or, if more precisely d F (F) = 2d <

, by :

4.1.1.

94

2

0 fl,F (1) (P(F )) p

where

2d +ri

2 t2 t—2)2d+fl exp (- --2) 2a a

(1

a

is defined in the statement of theorem

4.1.2.

3.3.

Comments.

4.1.

In the framework of theorem

brownian

bridge is an easy consequence of the proof of

well known result (see than the more general written

:

Pr

(

G

F

[18 ] ) .

Moreover the bounds in

Fernique-Landau-Shepp

t 2 ,) , --2 2o

> t) < C(u) exp

regular

the existence of a

4.1.1., 4.1.

>

a

but is of course a

are in this case sharper

[25])

inequality (see

for any

version of a

f OP

that can be

F

Proof of theorem 4.1.

If

F

is countable :

The calculations are similar to those of the proof of theorem

(F,F) relatng to P

of course a sequence of nets in

is directly given

the following single inequality is used instead of Lemma

4.2.

then

V

gaussian

be a real and centered

Moreover

3.1.

random variable with variance

> s) < 2 exp (-

,2 2v 2

for any positive

The choice of parameters being the same as in the proof of

2

.

v2

: Pr (V

T-

but here

:

LEMMA Let

3.3.

2 a

--2(a+)

j

P(F )

),

4.1.1.

4.1.2.

and

are proved.

Since

s.

3.3.2°)

4.1.1.

oscillation control, the almost sure regularity of Gp follows from

(except

is also an

Borel-Cantelli.

The general case.

(F,p r )

Since

constructa

is separable

regular version of

the familiar extension principle may be used to

brownian

on a countable dense subset of F. this version.

,

bridge on

Inequalities

F

from a regular version defined

4.1.1.

and

4.1.2.

still hold for

95

Comment. The degrees of

The optimality of bound 4.1.2. is discussed in the appendix.

and in 4.1.2. ; the reason is

the polynomial factors are different in 3.3.1.2°)a)

that bound 3.1.3°) is less efficient than bound 4.2. .

5, WEAK INVARIANCE PRINCIPLES WITH SPEEDS OF CONVERGENCE

We assume from now that

P(F

2+6

) < co

6

for some

in 1 0,1] .

Using the results in sections 3 and 4, we can evaluate the oscillations of the empirical brownian bridge and of a regular version of the brownian bridge over F ,

E k -valued processes

so we can control the approximations of these processes by some (where

E

k

is a vector space with finite dimension

The Prokhorov distance

k ).

between the distributions of these two processes is estimated via an inequality from

Dehling [15] allowing reasonable variations of

k

with

Oscillations of the empirical brownian bridge over F

from 3.3.2°)a)

F

■,),1 over

The oscillations of

n

.

are controlled with the help of a truncation

(the proof in this case is straightforward ) on the one hand and of

a slight modification in the proof of 3.3.2°)b) (truncating twice) on the other hand. We shall not give any proof of the following theorem (the reader will find it in [421) .

5.1. THEOREM. = P(F

We set

2+6

then an upper boundfor

) .

If we assume that

Pr ( Iv rj F > 0

H52! P F — <

2 6

is, for any positive

2 < P(F ) t

given by : a) If

42) (F) = 2d < co ,

0 (1)n7d

( t ) 8d ex _ ( P

t2

802 ,, ,,,LLna, ■ ) + 128 n -6/2 o-26 t-2+6

whenever the following condition holds :

with

such that

AG

>

1,

t 2 /02>1,

96

n 5/2 cy 2+26 t -6 (2) (F) = eF

b) If

> 64 p 5

5.1.1.

—

< 2 2 exp

OF(1) exp (O n,F (1) ( -10 ) 2P+n (1+

240+ 0 11,F (1) exp (- 116 q ) 2-2p+fl )+ 011 (1) (1+p5)2 ( n () 2-2 13- fl) -6/ 2 ( G-2-6+a-5) + 0(1) 1_1 5 n -5/2 CT -26 t -2+6 (p

for any positive

is defined in the statement of

n

3.3.) whenever 5.1.1. and the following hold :

n 5/4 CT 2+5

5.1.2.

Remark.

Note that

Yukich in [54] also used KolCinskii-Pollard entropy conditions to

prove analogous results to theorems

3.3. and 5.1., but our estimates are sharper

because of the use of randomization from a large sample as described in section

3 .

Speed of convergence in the central limit theorem in finite dimension. We recall below a result that is due to same direction is due to

Dehling [151 (the first result in the

Yurinskii [53]).

5.3. THEOREM. Let

(X.)

We write and Let

G

be a sample of centered

1
n

for the distribution of the normalized sum of these variables

gaussian distribution whose covariance is that of X 1 . k be an euclidian pseudo-norm on R and 7 2 be the Prokhorov distance

for the centered I

2

that is associated to

7

where

P k -valued random variables.

K

2

HI 2

(F n' G)
.

If

E ( IIX 1 II 2 +6 ) = p
r3 1 4 1/4 ,i1/2, (1 + IL(n -612 k-1 IA)! k 1 p )

is a universal constant.

Weak invariance principles for the empirical brownian bridge. In order to build some regular versions of

brownian bridges with given projection

97

on a finite dimensional vector space (or further in section 6 on a countable product of such spaces), we need two lemmas.

LEMMA

5.3.

(Berkes, Philipp [6 ] ) .

Let R 1 ,R 2 ,R 3 be Polish spaces,

Q 1 and Q 2 be some distributions respecti-

vely defined on R 1 xR 2 and R 2 xR 3 with common marginal on R 2 .

Then there exists

a distributions Q on R 1 xR 2 xR 3 whose marginals on R 1 xR 2 and R 2 xR 3 are respecQ 1 and Q 2 .

tively

Remember that l(F)

The following lemma is fun-

is generally not separable.

damental to avoid this difficulty (see [23]) .

The space

Q

to be mentionned below

is defined in Section 2.

LEMMA

5.4.

(Skorohod [48])

Let R 1 ,R 2 be Polish spaces and Q ginal is

q

on R 2 .

If

V

be some distribution on R 1 xR 2 with mar-

is a random variable from

q , then there exists a random variable Y

tribution of (Y,V)

is

from

Q

to R 2 whose distribution

Q

to R i such that the dis-

Q .

Concerning our problems of construction the point in the sequel is that the

distribution on

l oe (F)

T

of a regular version

of a

brownian bridge is concentrated on

a separable space. Now we can state some weak invariance principles for the empirical brownian

bridge with speeds of convergence. 5.5. NOTATIONS.

From now

[0,1] x IR +

y and 13 are positive functions that are respectively defined on and

[0,2]

by :

x

Y(x,Y) - 8 + 2y (4 1 x)

and

13(z) -

--

where, as in the statement of theorem 3.3. , p(z)

2(1-p(z)) z(2-2p(z)+z)

2z(4-z) 4 + z(4-z)

98

5.6.

THEOREM Under each of the following assumptions there exists some continuous version on

(F,p p)

of a

where

e (2)

brownian (B n )

(a n ) and and a)

d (2) are defined (2) (F) = 2d < If d

T <

a')

If

in Section

W ) (E,F) <

42)(F) =

d

T

FEL 2+6 (P)

and that

2) :

= 5n = 13(n-1)

C E-2d

(LE-

)

1,d

E

for any

in

]0,1[ .

= 13n = 0((Ln) (1/ 2)+d

<2 un = 0((Ln) -T )

for any

( lk) n -G (pn) ii p > a n )gn

y( 6,d)

an h) If

that Pr

G

are defined hereunder (we recall that

an for any

P,

bridge relating to

and

0((Ln) -5 )

13n

s.

and any positive

Proof of theorem 5.6.

be an oscillation rate (depending on

Let a

F

of

on a

a-net

We approximate Setting d

into

ted.

2d

vector space Writing

uniformly over

, pp (f,g) <

irco

,

by

GG

and

[49] ,

B = ucc (Fn, ,GG ) .

5.1.

liv n -vnaHa il F <

to

G

on the

ofvnIF(G)

be the corresponding

gaussian

G

a

(changing

can be

evalua-

G(a)

distribution.

with respective distributions

: ( 11 v n

11 F ( G ) > B) <

k-dimensional

1.1i F(G) and applying (Q' ,A' ,Pr' ) and two

there exists a probability space and

(o)

such that

be a projection

vn 0110.

we may apply theorem

be the distribution

Pr' where

GI

F

for the Prokhorov distance associated to

theorem

l oe (F(a))

7 0.

a

F n,a

1 °3 (F(a))

random variables on

vn

if necessary), hence the quantity

Besides, let

Strassen's

relating to

a

and

P

F(a)

G= {f -g

n)

B

F n,G

and G

99

5.3. ,

So, using lemma of a

brownian

5.4.

applying lemma

Q

with

Pr(

Gp

bridge

we may ensure the existence of some continuous version

with

V:

CO-*

\) nF(G) 1

=

,

( HvO G > 0

Pr


5.2.

and

C-

is used to control

a k < N F(2) (2,F)

noticing that

C

Moreover

Pr

B

is constructed on

.

( dG p l G > a

(with

we get

:

and

F(a) )

2.6. .

is evaluated with the help of theorem

t

.

t)

1 4 F(cy) < 114 2

according to remark

are completed via an appropriate choice of

6,

Gp

( Hyll -G p F > 2t+B) < A+B+C

a Theorem

we may assume that

and then,

t v n F(a) -G pIF(g) 1 F( a ) - d(v il -GF)0 110 11 F , Pr

A

G(a) - G piF(G)

such that

lry niF(0) -G p F(o rt F(a) > B)

Hence, noticing that

where

P

relating to

a

4.1.,

so the calculations

.

STRONG INVARIANCE PRINCIPLES WITH SPEEDS OF

CONVERGENCE.

The method to deduce strong approximations from the preceding weak invariance principles is the one used in

[43]

to prove theorem

2:

the weak estimates are used

locally, giving strong approximations with the help of maximal inequalities and via

Borel-Cantelli

lemma.

Maximal inequalities. As was noticed in from the one given in

[23], [10]

the proofs of the following inequalities may be deduced

and in

[32] .

Notation. We set

X. J

= 6 X. - P J

for any integer

j

100

6.1.

LEMMA (Ottaviani's inequality).

Sk

We set

•E j.1

j .

X.

Then, for any positive a

(1-c)

Pr (max

ilso F

>

the following inequality holds:

,

( 1%11

2a) < Pr

F> a)

k
where

max

Pr

lis k r F

(

>

a)

.

k
:

available

6.2. LEMMA Let

bles

(Paul Lévy's inequality).

be independent and identically distributed

(Yi)1
(B,

where

trical

then

where

k
:

normed

is a

)

11S 1,11 > "

Pr (max

vector space.

a) <

2

Pr

B valued -

If we assume that

( 11 S n i>

random varia-

Y1

is

symme-

a) holds for any positive

,

Strong approximations for the empirical brownian bridge.

6.3.

THEOREM.

(Y i ) j>1

Under each of the following assumptions some sequence versions of

brownian

defined on

0

a) if

such that

<

a')

1

H En j=1

(X.-Y.)

- 0(n -(1 )

(X -.)I

a.

s.

F

, N F(2)(e,F) < CE -2d (1+LE -1 ) d

F e F(2)„ = < 2 , n E i=1

y(.,.)

and

13(.)

for any E in

((Ln) (1/2)+d

1

- 0(Ln)

(X.-Y.)11

F

13 < (3() .

Where

may be

2(14--y(777 '

1 0-1 for any

(F,pp ),

d(F) = 2d <

if, more precisely

h) if

that are continuous on

:

1 1 En -VT1 j=1 for any a

P

bridges relating to

of independent

are defined in

5.5.

a.s.

](),1[

(Ln) (5/4)+(d/2) ))a.s. ;

101

For a proof of 6.3., see [42].

Comments. When passing from weak invariance principles to strong ones, the speeds of convergence are transformed as follows within our framework : (i)

n -y ,n

2(1+-y))

in case a). (ii)

Ln -/2

Ln

in case h) . Transformation (ii) appears in theorem 6.1. (under 6.3 .) from [23], it is not

(i) in the same theorem (under 6.4.).

the case for transformation

On the contrary transformation (i) is present in finite dimensional principles and appears to be optimal in that case : more precisely, the rate of weak convergence towards the gaussian distribution for 3-integrable variables is ran-

ging about

n-1/2

when the rate of strong convergence is ranging about

n-1/6

(see

[39] for the upper bound and [9] for the lower bound), in the real case .

Application to V1% classes. -

1 =1

Applying theorem 6.3. with real density 0(n -a )

in the case where F is a V..-class with

d, we get a speed of convergence towards the brownian bridge that is

for any

a

<

1

18+20d

.

This improves on

in the classical case of quadrants in

6.4. INVARIANCE PRINCIPLES IN

1.3.3.

but is less sharp that 1.2.4.

Rd .

C(S).

Following an idea from Dudley in [21] (sec.11), the study of the general empirical processes theoretically allows one to deduce some results about random walks in eeneral Banach spaces.

As an application of this principle

C(S)

metric space (SK) and the space ped with the uniform norm

1.

CO

equipped with the Lipschitz-norm:

.

Let

let us consider a compact

of real continuous functions on

X

S,

equip-

be the space of Lipschitz-functions on

S

102

N(c,S,K)

We write

K(s,t) > c

cardinality

for the maximal

for any st in

ix(t)-x(s)

+ sup t#s

' X -*

L

K( s ,t )

F = {65 ,sES}

fills

1.

(X,

(M) .

L. ) is a

Suslin

F =

and

Q

Moreover, for any distribution

N F(2) (.,F) < N(.,S,K) .

S

such that

:

H L

.

space (but is not Polish in general), so

2 Q((6 s - 6 t ) )< so

of

R.

We may apply our results through the following choices

Then

R

of a subset

2 K

(2)

in p 2 (X) we have

:

2 (5,t) Q(F )

11.11 ,0 =

Besides

F ful-

.

Therefore, considering a sequence (X)>i of independent and identically distributed

C(S)-valued

X l (s)-X l (t)1 with

6.3.

E(M 2+6 ) < .

:

random variables such that <M

E(X 2+6 1 (t o )) < .

and

K(s,t)

,

for one

to

Lipschitzian

N(.,S,K)

S,

in

theorem to get speeds of convergence towards the

structure depends on

s,t

for any

S.

in

we can apply some

gaussian

5.5.

or

distribution, whose

(the central limit theorem for such uniformly

processes as above is due to Jain and Marcus in

[31])

APPENDIX

1.

3.1.

PROOF OF LEMMA

First let us recall

Hoeffding's

lemma (see

(291) .

Noeffding's Zemma. Let

S

be a centered and

E(exp(tS)) < exp We may assume that

.

drawing

for each

i

w

[u,v] -valued ( t2,or-u)‘ 2 ■

[1,n] .

,

for any

is chosen as follows

with uniform distribution in

\ )

8

random variable, then

-

t

in

:

R .

:

a partition I

(J 4 ),,,,_ '

such that

iJ i l =m

103

J i - with uniform distribu-

independently in each

w(i)

• then, drawing an index

J

The following evaluations are conditional on

tion - •

J ,,,.,

not depend on We set

giving 3.1. .

S S n N Z - -- - -N n

place transform of

Then setting

but the last bound will

and we write A

for the logarithm of the conditional La-

Z . z

. = I

in R :

c. , we have , for any s

m iEJi

1

J

n A(s)

Log

-

7

(1

eXp (.-n (C.j - Z•))) i

m jEJ i

i =1

then, since the logarithm is a concave function :

SN A(s) < n Log (,1

fl) - n AN ()

exp q ( ,i j=1

QN for the uniform distribution on

where, writing

the logarithm of the Laplace transform under

Cramer-Chernoff transform of

Z

Q N of

fc.1 ,...,41 1 ,

AN stands for

x _.- x-E n (x) .

S'El

is larger than that of 7

'N

Therefore the

S'n

(-i ) under e n

-

N

Entgn

'N where S

n

stands for the sum of

n Li.d. random variables with common distribution

QN • Then, Hoeffding [29] and Bernstein [5] inequalities yield 3.1.1°) and 3.1.2°) .

In order to prove 3.1.3°) we may assume that S N = 0 (otherwise changing.i

nto

SN j - Tr) • Then, applying Hoeffding's lemma to the conditionally centered random variables

.. and setting u. = min

1 l i (t) - LogE

J

jEJ i

. J

and

v i - max . , we get : jEJ.I J

(exp (t( x(i) -y)) <

(v.-u.) 2 2 18 1 t , for any t in R.

Hence

n 2 s, s A(s) - E l() < --i n — 2 i-1 8n

n

2 s (v.-u.)2 < --1 1 — 2 1.1 4n

and therefore :

2 A(s) < 4s4.1 m 4 yielding 3.1.3°) via Markov's inequality.

n E i=1 jEJ.1

2

104

2. THE DISTRIBUTION OF THE SUPREMUM OF A d-DIMENSIONAL PARAMETER BROWNIAN BRIDGE. We use Goodman's work in

[28] to give a lower bound of the probability for the

brownian bridge to cross a barrier.

supremum of a

Notations. We set of

11'

I

= [0,1] and write for any integer s

Moreover, for any

.

in

I

d

d,

, we set

l d for the element

(1,...,1)

p(s) = s l ...s d .

A.1. Theorem. Let d

process, then, on the one hand

(1)

Wd be some standard d-dimensional parameter Wiener

be an integer and

WAS) < t II4d (1 d ) = at) < h d (a,t)

Pr (sup sEI

:

d

for almost any real number

a (in

Lebesgue sense) and any positive

h d (a,t) = 1 + exp(2t 2 (a-1)) and on the other hand

Proof

d-1 141 (2t2(a-1))i Z (-1)' i!1 1-c.,11 (a) i=0

:

Wd (s)-p(s) W d (1 d ) > t) < d-1 (2t 2.1 d 1 =0 1' sEI

(ii)

Pr (sup

of

t, where

. exp(-2t 2 )

.

theorem A.1.

If d-2 the whole proof is contained in

[28] .

[28] yields the following inequality : 0 Ind(s) < t 1Wd (1 d )=at) < j (1-exp(2t 2 r)) dF t,d _ 1 (a,dr) A.2.

Otherwise, proceeding exactly as in

H d (a,t) = Pr(sup sEI d where

a-1

d-1 IW F t,d-1 (a ' r) = Pr(W at ) d-1 (s)-rtp(s) < t, vsEI d-1'(1 d-1 )'---" We want to proceed by induction.

It is enough to notice that

:

A.3. Lemma. Pr(W k (s)+ap(s) < t, VsEIk 1Wk( 1 k)+= r3) for any integer

k , any positive a

Pr(W k (s) < t, vsEI k W k (1 k ) =

and almost every 8

in

R

(in

Lebesgue sense).

105

Proof of

A.3.

Since

W

is a regular

k

gaussian

process, it is enough to show that the expec-

tation and covariance functions of the processes

W k (1 k )+a

same conditionally to respectively Since

W k is gaussian,

E(W k (s)

W k (.)+p(.)

W k (.)

are the

W k (1 k ) .

and

W k (1 k )=y)

and

and

E(W k (s)W k (s l ) W k (1 k )=y)

are

respectively linear and quadratic functions of y, then the knowledge of

E(q(1 k ) W(s) q(sI))

with

l+m+n <4, yields :

E(W k (s)W k (1 k )-y) -, p(s) y , E(W k (s)W k (sTW k (1 k )=y) = p(s)p(s I )y where

sns' = (s l As il )...(s k Aq) ,

+ (s A s') - p(s)p(s')

A.3. . A.1.(i) .

Let us return to the proof of

Using lemma

2

A.3., we get : F t,d-1 (a ' r) = H d-1 (a-r t) '

SO

:

F t,d-1 (a ' r) < h d-1 (a-r ' t) Then, integrating by parts, inequality

A.2. becomes :

0 H d (a,t) < F t,d _ 1 (a,0) - exp(2t 2 r)Ft,d_1(a,r) l 0 a-1 + 2t 2 f

exp(2t 2 r)F t,d _ 1 (a,r) dr

a-1 hence H (a d '

t) < 2t 2 .r°

exp(2t 2 r) h d i (a-r,t) dr :

But an easy calculation yields

0 2t 2 r1 exp(2t 2 r) hd-1(a-r't) dr = h d (a ' t) Ja-1 Therefore inequality

A.I. (i) is proved by

A.1.(i) holds when

induction (it is shown in

[33]

p.

284

that

d=1) .

In order to proof (ii), we notice, following pr (

converges weakly in

.

C(I d ) towards

wd E.I o < wd ( 1 d )

[7], p.84, that : <

E)

the distribution of the

brownian

bridge

:

106 W d - p(.)W d (1 d )

whenever

c

converges to

0.

(i) gives :

So, inequality

Wd (s) - p(s)Wd (1 d ) > t) > 1 - h d (0,t)

Pr (sup

sEI

d

(ii) is proved.

therefore

Comment. Theorem in

A.1. was proved by ourself (see [40] and [41 1 ) but also by E. Cabana

[11] * . In another connection, inequality

t 2h(d)

with

h(d) 7 d-1

A.1. (ii) ensures that some polynomial factor

cannot be removed in bounds

-

3.3.1°)a) and 4.1.2. .

3, EXPLICITING AN EXPONENTIAL BOUND. The calculations yielding

3.3.1°)a) are slightly modified here, where the entro-

py condition a) is replaced with a more explicit one.

A.4.

Theorem.

If we assume that

F

is

[0,1]-valued and that

E 2) m(2)( c,F), < ,1+1/Log( N (1+Loq(c -2 )) d C -2d ) "1 then, an upper bound for Pr ( nl F 7 t) is, for any \) a

l ■

4H(t) exp(13) exp(-2t 2 )

+

for any

t

in

E

in

]0,1[

[1,+.[ , given by :

4H 2 (t) exp(-(t 2 -5)(Lt) 2 )

where

H(t) = K6/5 exp(16d) (1+Lt2) 5d t6d Proof of

A.4.

In the proof of

Lt 22 + 1 , then LLt 2 oN A < 2H(t) exp(13) exp(-2t )

3.3.1°)a) we choose a

=

B < 2H 2 (t) exp(-(t 2 -5)(Lt) 2 ) whenever *

P 01 -a.s.

t 2 _> 6+4d , yielding A.4. via lemma 3.2. .

Thanks to

M. Wchebor and J. Leon for communicating this reference to us.

107

Comment.

Assumption a') is typically fulfilled whenever case d of

F

is a V.L-class.

In that

may be the real density of F (if it is "achieved") or the integer density

F (see the proof of 3.6.).

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[1]

ALEXANDER, K.

[2]

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[3]

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DUDLEY, R.M. Metkie ent/Lopy o4 orne cZa6u g zet with die/Lentiat boundaAia. J. AppAoximation Theoky (1974), 10, 227-236.

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DUDLEY, R.M. CentAat timit theotem6 4oA. empiAicat mea6une6. Ann. PfLobabitity (1978), 6, p. 899-929 ; co/L/Lection 7 (1979), 909-911.

[21]

DUDLEY, R.M. ccivit Ftoun 1982

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n Mathematin n° 1097 .

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[54] YUKICH,

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oA the

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J.E. UnigoAm exponentiat boundz AoiL the notmatized empirLicat pfLocezz 0985) . PiLepnint.

MEAN SQUARE CONVERGENCE OF WEAK MARTINGALES

Mariola B. Schwarz air Mathematische Statistik

Institut

Universitgt Göttingen Lotzestr. 13, D-3400 Geittingen

[4]

In

it was shown that the mean square convergence of vector-valued martin-

gales in spaces of Rademacher type or cotype 2 is closely related to the following property of a Banach space valued martingale Radon measure space

on

'Y f

(B,B) (8

the Borel

f = (f ) n

there exists a Gaussian

a-algebra of subsets of the Banach

B) such that

(*)

(dx)

x*f1'22 = r lx* (x)1 2

for every

x E B . (*)

We give a characterization of the class of martingales satisfying

1

not containing

uniformly and having an unconditional basis

convergence of the series

where E 1 1 S(ef) 1 ' e n 2 n ' n=1

(e)

n

for spaces

(e ) in means of n S

is the dual basis and

is the standard square function. Furthermore we give necessary (resp. sufficient) conditions for the of martingales in spaces of type

L -convergence 2

2 (resp. cotype 2).

These conditions characterize Banach spaces of Rademacher type or cotype 2 . Throughout, Borel

B

denotes a separable Banach space,

a-algebra of subsets of

A family

[f

n

respect to the filtration an

and

(0,3,P)

the dual space,

8

the

a probability space.

B-valued random variables forms a weak martingale with

1 n E NI of

Pettis integrable on

B

B

n and

n E NI (Pettis-)

A

B-valued weak martingale

M

, such that the inequality IX f !

I'Lf 1 n

E N1 M

if for every

E(f

n E N , f

1 a ) = f k

n

for

is

measurable,

n n ksn (see e.g.

[3]).

is uniformly bounded if there is a constant

*

holds a.e. for every

* * x E B and every

n E N . Note that if

f ,--- [f

1 n E N1

is a

B-valued (weak) martingale then n * * * * x f= (x f 1 n E 1\11 is a real martingale for each x E B . n

n E 1\11 is a n I with difference sequence Pi 1 n E 1\11 . In the following

f = [f

* x f=

B-valued uniformly bounded weak martingale

n

IOWA 1. Let

(e ) be an unconditional basis in B n nE N basis. Assume that B does not contain l n uniformly. Then the following conditions are equivalent

and

(e n )

nE N

the dual

111

a) the series EllS(e

n

2

e

is convergent in

n

b) there exists a Gaussian measure

r

yf

on

B ;

(B,B)

such that for each

x

E B

ix(x)12 B * * * * Proof. x f = (xf ) is for eachxEB areal martingale with increments n * Thus the square function S of x f is given by * 1/2 S(x 0 ...= (E 1 x* (dk )1 2 ) . k * * Let the functional T:B X13-, 12 be given by * * * * * * * T(x ,y ) = E E x (dk ) y (dk ) , x ,y E B . k T

*

.

is well defined everywhere in the Cartesian product

B x B since f is of * * T is positive (i.e., T(x ,x ) > 0 for every B*) * * * * x E and symmetric (i.e., T(x * ,y* ) T(y ,x ) for all x ,y E B* ) Since f is uniformly bounded we have * * * * T(x ,y ) (E E x () 2 ) 1/2 (E y (dk) 2 ) 1/2

weak second order. Furthermore,

= IS(x1S(y f) 2 2

= b[*6 2 1 H*0 2 M1 x which means that T

,

l y*

is bounded.

** B . Using a theorem of Banach (see * e.g. [1 ] ), which states that an element u of B belongs to the image of B in * ** the natural embedding of B into B , if the functional u(v), v E B , is * * is contained in B . continuous in the B-topology on B , we show that TB T

We can consider

Assume that

E

* B

as a map of

into

a) is satisfied. Since

* 1/2 S(e:f) 11 2 en =E (E(E(e* (111. )) 2 ) 1/2 e = E (Te *(e)) en n k n n n

the series

*, *, 1/2 n le n ))

E

By theorem 2.1. in

B , i.e., * * Tx (x ) =

lx* (x)1

T 2

B .

is a covariance operator of a Gaussian measure

Y(dx) .

B

Since

11x* f1 2

[2]

is convergent in

11S(x* f)11 = Tx* (x* )

the implication a) Conversely, if

b)

follows.

b) is satisfied, the series

)) 1/2 e E 11S(e* 011 e = E (Te*n(e* n n n n n n 2 is convergent in

B

as

T

is a covariance operator of

'y

'y

on

112

We recall that a Banach space does not contain

n

uniformly if it is of certain

• Banach spaces of Rademacher type 2 are of some Rademacher

Rademacher cotype r <

cotype r <03

1

(see, e.g., F5]).

Using Theorem 5.3. and 5.4. of THEOREM 1. Let

(e ) nE N n

[4] , Lemma 1 and the above remark we get B

be an unconditional basis in

and

(e ) n

N

the dual

basis. Then

i) B series of

f

is of cotype 2 if and only if for every

E HS(e

e 2 n

n

B

in

the convergence of the

is a sufficient condition for the L

2

convergence

f . ii) B

E 1 1 S(e n n

the convergence of the series

is of type 2 if and only if for every f

B

e n in —

4

is a necessary condition for the

L

2

convergence of

f •

COROLLARY 1. In Hilbert space (and only in this space) the convergence of the series

E11S(e f)I e n 2n

is a necessary and sufficient condition for a martingale

N

be the family of all

for each

x

*

lx* (x)1 2

B

E

THEOREM 2. Let

such that

y f (dx)

(e ) n

F2] N

p > 0 • Assume that

constants

(B,S)

for which there

* B .

Using Theorem 2.2. in

and

f = (f)

B-valued weak martingales

y f on

exists a Gaussian measure

1x* f1 1, 2 = 2

to

L -norm.

converge in the Let

f

2

c l (p)

and

and the techniques of the proof of Lemma

be an unconditional basis in

B

c 2 (p)

does not contain

c2 ( P )

(e)

flE N

the dual basis

l n uniformly. Then there exist

such that for each

c i (p) 11E S(e*f) 2 ePsr n n n • B

B ,

1 we get

f

E

(dx) e Z11 S(e*n f)1 2n

•

REFERENCES

[1]

S. BANACH :

Théorie des opérations linéaires, Warszawa 1932.

f- 2 1 S.A. CHOBANYAN AND V.I. TARIELADZE : Gaussian characterizations of certain Banach spaces. J. Mult. Anal. 7, 1977. [3]

K. MUSIAL :

Martingales of Pettis integrable functions. Lecture Notes in Math.

794, 1979. F4]

NGUYEN DUY TIEN : On Kolmogorov's three series theorem and mean square convergence of martingales in Banach spaces. Theor. Prob. Appl. 24(2), 1979.

[5]

W.A. WOYCZYNSKI : Geometry and martingales in Banach spaces. Advances in Prob., Dekker, 1978.

METRIC ENTROPY AND THE CENTRAL LIMIT THEOREM IN

BANACH SPACES

Institut

§1.

J. E. Yukich Recherche Mathématique Avancée Université Louis Pasteur 7 rue René Descartes 67084 Strasbourg, France de

INTRODUCTION

The intent of this article is to study the relationship between (i) the central limit theorem in Banach spaces, (ii) the Donsker property for unbounded classes of functions, especially subsets of the Banach dual, and (iii) metric entropy with LP bracketing, p > 1 . Before exposing the main results, let us first set in Place the framework for empirical processes, the setting of this paper. Throughout, take (A,A,P) to be a probability space and x i , i> 1 , the coordi-

nates for the countable product (A ,A ,P ) of copies of (A,A,P). nth empirical measure for P is defined as

The

n P(B) = n

7 1

, BA

j=1 fx4131 Given a class F<7.1 L 2 (A,A,P) of real-valued functions with envelope F F (x) = sup If(x)1 , let S:=t (F) be the space of all bounded realf.GF

valued functions on

F;

equip

S

with the sup norm, i.e., for s

S

let := sup fcF

We note that When F

Is(f)1 •

(S, H H) is a Banach space, non-separable for F infinite. F is finite P a.e. the function-indexed emnirical process

v(f) (to)

n 1/2

f(dP n -dP) (w) , w

e Acc ,

f e F ,

that F is a is a random vector v n with values in S . Recall [7,8] P-Donsker class if v n converges in law in S to a Gaussian process G P ' indexed by F , with a.s. bounded p r -uniformly continuous sample paths (abbreviated BUG). Here, 2

*

p P fg) :=

(f-g) 2 dP - (f(f-g)dP)

2

.

Current eldress:Lehigh University,Math.,Bethlehem,Penn.18015-USA

114

G

necessarily has mean 0 and the same covariance as

cov(G p (f), G p (g)) = f fg dP

f f dP

v

n

g dP .

Clearly, a necessary condition for F to be P-Donsker is that F be G p BUC , i.e., that the limiting Gaussian process Gp can be chosen to be BUC. Define the mapping

h:

S

by setting

h(x)(f) = f(x) - f f dP lefillingtherandollivariablesXj=h(x.), Dudley foreach has shown [6] that if F is Gp BUC then the P-Donsker property for F means that X 1 satisfies the central limit theorem in (s, H 11) abbreviated X i ( CLT . (Recall that if X is a random variable with values in a Banach space B and if (Xn ) 11cN is a sequence of independent copies of X , then X satisfies the CLT if L( 1=1

X./VT) 1

B . See [12,16] for an exposiIn this way we see that central tion of the CLT in Banach spaces.) limit theorems for the empirical process \) 1 (f) , f eF , can be viewed as central limit theorems with respect to the norm H h on the Banach space S . converges weakly to a Radon measure on

Throughout this paper let (B, H 11) denote a not necessarily separable Banach space , B * the dual space, and )4, the unit ball of B * . Now a subset H of B *1 is called a forming subset (see [7]) for B if and only if H x11 = sup Ih(x)1 for all xc. B . Clearly, hrH by the Hahn-Banach theorem * is always a norming subset, a fact ' B1 which we shall exploit in the second section. In this way, limit theorems in B can be viewed as limit theorems for empirical measures on B , uniformly over a class of functions, * such as B i , since for .CgB and x(1),...,x(n) E B +6

x (n)

)(f) = f(x(1)+...+x(n) ) .

In particular, We note that X (with L(X) = P) if and only if B1 is P-Donsker.

satisfies the CLT in

B

115

The relationship between limit theorems for empirical processes and limit theorems for Banach space valued random variables has been studied by several authors. For example, Dudley has shown [7] that the Jain-Marcus CLT for C(S)-valued random variables [13] is in fact a conDudley has also sequence of Pollard's CLT [17] for empirical processes. shown [7] that the Fortet-Mourier strong law of large numbers [15] in separable Banach spaces is a consequence of the DeHardt-Dudley law of large numbers for a class of functions [3,7]; we will return to this implication later on and will also provide a short and simpler approach.

(f) often involve a metric entropy condition on the index set F . In this article we will consider metric entropy with LP bracketing, defined as follows [7]: Limit theorems for

n

Definition 1. Given f,g : A-0- 1R, define the bracket [f,g] := Let p 1 , such that f(x)
FCL P (A,A,P) .

Let NM

:= Nr1 (c,F,P)

>0

mirdm:3

and such that Vf cF 3 l<j<m such that fr [f i ,f;] [fm ,f;1 ] (f is referred to as metric entropy - f.) 1) dP<E P 1 . Now log N ( P ) (c) [ J J with LP bracketing.

f

We recall that metric entropy with

1, 1 bracketing has been espe-

cially useful in the study of empirical processes; see e.g. the works of DeHardt [3], Dudley [4,7], Dudley and Philipp [8], Alexander [1], Borisov [2], Gin é and Zinn [11], Pyke [18], Koreinskii [14], and Yukich [20,22, In particular 23]. See also the recent monograph by Gaenssler [10]. 8 N (1) has been used to describe the weak convergence of n (f) [4,7, ]. [ 1 bracket One of the main ideas running through this article is that L ing is not a natural condition for describing weak convergence; L 2 On the other hand L 1 bracketing bracketing is a more natural choice. is a more natural condition for describing laws of large numbers. These points will be discussed in the following sections. Finally, we will also use metric entropy without bracketing which has enjoyed wider use, and which is defined as follows:

Definition 2. Let ff F 3 1<j

f(f-f.) P dP <E P } . ferred to as the metric entropy of F . < m

with

such that for all ,f such

N ( P ) (c,F,PI := Now

logN4 P ) (E,F,P)

is re-

116

Having set in place the framework for this paper, We now proceed Section to the main results, a summary of which was announced in [21]. two discusses the relationship between (i) and (iii) of the first paragraph, section three studies the relationship between (ii) and (iii), and section four considers metric entropy with L 2 bracketing.

§2.

METRIC ENTROPY AND THE CLT FOR IIII UNIT BALL OF THE BANACH DUAL

Our starting point is the following result of Dudley [7], which adds to Mourier's well known law of large numbers [15]. Theorem 1.

(cf. section 6.1 of [7]I

X., X 1 ,... he a sequence of i.i.d. random variables with values in a separable Banach space (B,11 t1). The following are equivalent: (i) X satisfies the strong law of large numbers (SLLN),

(ii) ElH < (iii) M(1) [

]

Let

and

(E B * P)
VE > O .

'

It is reasonable to ask whether analogous results hold for those X satisfying the CLT . The following theorems answer this query in the affirmative, showing that the study of the CLT in separable Hilbert spaces

11

N ([ 2 ] (E,F1,P) , where

is conveniently studied through the use of

*

15 the unit ball or the Hilbert dual.

In fact, in the same way that the equivalence of

X CLT

and EX}

0 , I1X1! 2 < co

characterizes, modulo an isomorphism, separable Hilbert spaces, the following results show that the same is true for the equivalence of the conditions B

1

is P-Donsker and inf N (2) (E B * P) ' s >0 [ 1

c°'

P

centered.

117

This is contained in the following results.

X be a random variable with vaiues in (B, 11 ) which need not be separable; L(X) = P Then for all p >1 the following are equivalent: (i) EIIXIIP < , and Theorem 2. Let

,

(ii)

If

inf N ( P ) (a , B 1* ,P) < oc. c>0 P is tight then the following are equivalent:
N ([ P )

(E,q,P) <

Vs > 0 .

Remarks Using the equivalence of (iii) and (iv) witn p = 1 we directly obtain the double implication of Theorem 1 without using (1)

the SLLN property of (2)

X .

As is well known

[12,16] the moment condition El general neither necessary nor sufficient for X e CLT ;

X11

2

is in

it may also be

inf N ED) (a,F,P) 0 necessary nor sufficient for F to be P-Donsker. The interest of Theorem 2 stems from the fact that when F is B 1* these are in fact equivalent conditions and they consequently share the same properties. easily seen that the entropy condition

The next result is essentially a consequence of Theorem 2.

Theorem 3. Let

X

Banach space (B,I[ of P: (i)

be a centered random variable with values in a Consider the following properties P . 1) ; L(X)

X eCLT ,

(ii) B 1* is a P-Donsker class of functions,

(iii)

l XI I

Co

(iv)

N (2) (E * P) [ ] 'B
(v)

inf N 2 L I s>0

Va > 0 ,

c B,P) < Co .

We have: (a)

P is tight and all equivalent. If

B

is a Hilbert space then (i)-(v) are

118

(h)

If B is separable, then the equivalence of (i), (ii), (iii), and (v) is equivalent to the fact that B is isomorphic to a Hilbert space.

For the proof, we need only observe that when B is a separable Hilbert space, the equivalence of (i), (ii) and (iii) stems from the introductory remarks. For separable type 2 Banach spaces B we have sharp metric entropy conditions insuring the P-Donsker property for B 1* .

Theorem 4. Let (B,11 II) a centered law. If

be a separable type 2 Banach space and

P

(2) inf N [ ] (E ' B * P) < c>0 * Conversely, if P is a law on any then B 1 is a P-Donsker class. separable Banach space B and if Iq is P - Donsker, then for all p<2 , N (p) [

(E B

P) < cc' VE >0 .

For the The proof of Theorem 4 follows at once from Theorem 2. first part we need only use implication (ii)-4(i) with p = 2 ; for the second we use implication (iii) (iv) with p < 2 together with the fact that if X cCLT then EIXI <œ Vp< 2 , see e.g. [16],

Proof of Theorem 2. We will first show (ii) (j) Let c> 0 and m := m(c) =N ([ P ) (c, B *1 , P) . By definition there is a collection of brackets [fj ' with

' j = 1,

f(x) < f(x) < fi(x)

,m , such that given any

f EB *1

1 < k < m

VX E B

and

f

(f I-' (x)-fk (x)) P dP(x))< E P .

* * and thus for fixed xc B we B 1 iNow s a naming subset of B f(x) runs over all of the values between - NI and rxd as See that f runs over B *1 . This implies that for any fixed x , the sum of the

119

y

differences of the brackets

(f -.'.(x)-f.(x))

must be at least as large

j=1 Indeed, suppose this were not the case.

as 2 1k11 that

ji =

Then

3

xo

e

B

such

f - (x 0 ) - f (x ) < 2 lixol 0

Thus there exists an

-11x011 < cï < rx01

and a

6> 0

such that the open

interval (a-6, a -1- 6) is disjoint from the union of closed intervals

M

_

,U 1 [f.(x )X(x )] . J 0 J 0 j= an f B7 such that

Moreover, by the Hahn-Banach theorem, there exists

f(x 0 ) = a

there exists 1< j<m

This leads to a contradiction since

such that

f.(x) < f(x) < f(x)

vxc B .

Thus, we have shown that

HX11 <

e(X) - f(X)

a.s.

j=1 For all

p > 1

there exists a constant

HxdP <_

j=1

Ellx11 13

<

such

((e-f.)(X))P a.s. J

Integrating this with respect to P

N ([ P )] (c ' B * P)

C := C(p,m)

and using the definition of

shows that

c.m. EP <

completing (ii)=0(i) . it suffices to consider the single bracket def i (x) = - 114 and f(x) = ilxil and to take E = 2 fIlx11 P dP(x).

To show (i)--4(ii) fined by

P is a tight measure then we may show the stronger implication (iii)-(iv); our proof is inspired by the proof of Proposition 6.1.7 of [7]. Let p> 1 be fixed and note that for all E> 0 there If

is a compact set KB such that

f

11x1I P dP < c 13 / 4

The elements of

BI , restricted to

K , form a uniformly bounded

120 equicontinueus family and hence this family is totally bounded for the sup norm on K by the Arzel- Ascoli theorem. Take f , , fm e B m < , such that Vf B i 1 ' I!f - f 11 for some

K

j .

g5. +Ill = f.3 + c/ 4


Then for every

ilgi ,m

gi = fi

on

K , g i+m (x) = Ilxil

f B i* ,

g5

E/4

K , g i (x) = - 11x11 for

if Hf -5 11 K < c/ 4,

< (2c/4)P + 2 j BK

I1x P

for

x K , for all then

x K , j = 1,

gi < < gi

,m .

and

d.p

< (c/2)P + EP/2 < c P . Thus, NIP/

(,14,P) < 2m <

proof of Theorem 2.

§3.

, proving (iii)=-4 (iv) and completing the Q.E.D.

THE P-DONSKER PROPERTY AND METRIC ENTROPY The results of the above section show that

N (2) [

i s often an

appropriate tool for describing the P-Densker property for above results also suggest that

N (p) [

(c F P)

Bi ;

the

13+2 , will not in _

general be a satisfactory tool to describe the P-Donsker eroeerty for LP(A,A,P) . unbounded classes F This is indeed the case: the following propositions show even in the presence of an envelope condition on F , that N EP) (c,F,P) , [ + 2 , can not possibly give sharp results. This may be seen by considering suitably chosen subsets of the dual to (r i L), the Banach space of bounded functions on IN equipped with the sup norm. More

ll

precisely, these subsets are classes of functions of the form

121

F

fyj

:=

a,$

a.s.1 :s.= 0 j 3 Ai 3

or

1) ,

where A j , j >1 , is a sequence of disjoint subsets of A with p. := P(A) =j -5 for some $> 1 and a j ..-j c4 for some 0< a < $ - 1 . Since elements of Fag define measures on 11\1' , we may view Fa, $ as a subset of the dual to (C,Ir

.

We will consider the cases p =1 , p >2 and l< p< 2 in this order; much of what follows may be found in [19]. Our first proposition shows that a theorem of Dudley, which is recalled below, is far from being the "best possible".

Theorem (cf. Theorem 3.1 of [S]). Suppose that F has envelope F F c LP(A,A,P) for some p> 2 . Suppose that there exists y , 0<',( < 1- 2/p and M< such that N(1) (e" F P) < exp(ME — [ for E

small enough. Then F is a P-Donsker class.

Proposition 1.

For all 2 < p< 4 there exist P-Donsker classes F with

FF e LP(A,A,P)

(1) and N [ ] (E,F,P) > 2 E -Y

where y is any number less

than 1/2. Proof. Fix 2