A central limit theorem of branching process with mixing interactions

Wuhan University Journal of Natural Sciences

Vol. 7 No. 3 2 0 0 2 , 2 6 1 ~ 2 6 6

Article ID: 1007-1202(2002)03-0261-06

A Central Limit Theorem of Branching Process with Mixing Interactions Liu Yan-yan School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China

Abstract: In this paper, we discuss a class of branching processes which generalize the clsumical GaltonWatson l~'Ocemes, we permit some mixing dependence between the offspring In the same generation. A central limit theorem is established and the Hausdorff dimension on such kind of branching process is given. Key words: mixing-dependent, Galton-Watson process; Hausdorff dimension

C L C n u m b e r : O 211.65; O 192

I

Introduction and Statement of Results

Consider a Galton-Watson branching process {X.,n>~0} with offspring distribution (Pk)k >to. We start with a single particle in generation 0 and each particle independently produces k offspring with probability p~, in other words, we can decompose X. to X

Xo = 1, X,,+I = ~

Z.., (n>/0, i/>1)

i--I

where Z.. denotes the number of the offspring of the i-th particle in generation n. The collection of all individuals form the vertices of a tree, with edges connecting parents to their children. In the classical case, {Z.,,, n>~0, i>~1} are required to be independent. This seems a very strong requirement. For the background material on classical Galton-Watson processes, we refer to Athreya It3. For formulating the geometric models of B. Mandelbrot CzJ which appeared, for example, in percolations and polymers, J. Peyri~reC33 introduced and studied a branching process with neighbor interactions, and then Z. Y. Wen E43 studied this process more deeply. For this process, dependence between Z.,, and Z.,~ are

permitted when I i - j l >/l, for more motivations and examples, see Ref. [3]. In this paper, we introduce a more general branching process, we allow dependence between all Z., of the same generation n but require that mixing coefficients converge to zero sufficiently fast. A central limit theorem on this mixing process is given. Using the same method of Hawkes [s], we get the Hausdorff dimension of our branching tree. Throughout this paper, we suppose that: 9 {Z..,, i ~ O , n ~ l } is a family of random variables taking values in the set of non-negative integers and having the same distribution (Pk),>_.o. The mean number of children per particle i s p =

EZ.. = ~ k p k


paper,we require Po = 0, Pt < 1, so the case we studied is supercritical. X

9 Xo = 1 , X~I = ~ Z . . , ,

n>~0

9 For every n>/0, the sequence {Z.., i>/1} is strictly stationary and the a-algebras {o{Z..,i >tl}, n>/1} are independent; If for each n, the family Z..(i=l,2,'") are independent, then we have the classical case. The

Received dates Z001-11-12 Foundation item, Supported by the National Natural Science Foundation of China(19971064) Biographys Liu Yan-yan(1968-), female, Ph. D, research direction~ branching process, random fracta[. E-mails yy[iuala(~163,

net

262

Wuhan University Journal o f N a t u r a l Sciences

/-dependent process c43 are obtained if Z.,, and Z.. i are independent whenever t i - j l > ~ l . Our more general approach requires only that certain correlation coefficients among the Z.. converge to zero. To this end, we need some definitions: Let (O,Y-,P) be a probability space, ~ ' a n d Y"two a-algebras contained in Y.. We define the mixing coefficients which measure the dependence of ft"and Y"by: = su 7(Y,Y'),

P(Y'Y") Yec,(~,~eL,(s,~ a(Y~Y') = sup [ P(AB)-P(A)P(B)[,

Definition 1.1 {X, }.>to is called a p(resp. a,i~,~b)-mixing branching process provided p(r) (resp. a(r),!~(r),~b(r))---~0 as r--~oo. X, Define Z=Zo., and W, = - - . Let F be the distribution function of Z and Y. the o-algebra generated by {Z~.,,k=O,1,...,n-1; i=1,2,...}. Here are the main results: Theorem 1.1 ( Central limit theorem ) Suppose {X, },>/o is a p-mixing process with EZ z , ( c o and ~-]p(2") < co

ACzY ; B E ~r"

~-

SU

su

Then the spectral

n--I

[ P(B I A)

P(B) [,

P(A)~O

=

Vol. 7

P( AB )

i I

P(A) P(B) r

where )'(Y,Y') denotes the correlation coefficient of Y,Y' and Lz(5T) = {Y ff Y:[Y z dP < co}. Suppose that {Y}={Y,},~>I is a sequence of real valued random variables defined on ( f l, Y, P), and define .,r (Y) = a(Y,, 1 <~ i ~< m), (Y) = a(Yi ,i >~ n), pr(r) = sup p(~p(Y),5"+P(Y)) p>~l

Denote by p({Y,}~)={gr(r)[ r ~ l } the p-mixing coefficients of the random sequence {Y~ }~>~. Other kinds of mixing coefficients of the random sequence are defined in a similar way. By the above definitions and notations, it's easy to prove that ~] ~ ( r ) ~ O=~t~,(r) ~ O:~pr (r) ~ O~ ae(r) ~ O (asr--,-co) Remark 1. 1 Obviously, p ({ Y, }. ) only depends on the distribution of {Y,},>~. That is, when {YI, i>~ 1} is another sequence with the same distribution, then p( {Y, }~) = p( {YI }, ). By this reason, p({Z~.~},) does not depend on k and we only use {p(r), r>/1} to denote the p-mixing coefficients of {Z,., },>.~. Briefly we call it the generation p-mixing coefficients of the process {X,},>~o. Similarly, we use {a(r), r ~ l } ,{(p(r), r>~l}, {tb(r) and r~>l} to denote the generation a, to, ~mixing coefficients of { X. }.>to respectively. We define the mixing branching processes as:

density function of {Zk.~,i>/1 } ,denoted by g(a), exists and does not depend on k. If g(O)~O, then lim p ( ( W - W , ) p " ~< y [ y.,) _- r a.s., ff2ng(0)X, for every y E R x The definition of g(x) will be given in section 2. By an analogous argument to that of Ref. [5], we have Theorem 1.2 Let {X,, n>~0} be a ~mixing branching process. Suppose that EZ(lg + Z)*+2 < (0
1

rZ+a, (,I > a), the Hausdorff dimension of the mixing branching tree is log~'. (a. s. )

2

Central Limit Theorem

Let {X. }.>/o be a mixing branching process. In this section, we introduce the definitions of spectral measure and the spectral density function,and give the proof of theorem 1.3. Let {Y~},~>x be a strictly stationary sequence of random variables and R(n)=coy (Y,, Y,+.). Then by the Herglotz theorem, there exists a measure G such that

R(n)= fSi e" dG(a), nE Z The measure G (~) is called the spectral measure of the sequence {Y~),>~. If G ( a ) is absolutely continuous, its derivative g(,~) = G' (~) is called the spectral density function of the sequence {Y~ }~>~,. It is well known that if +o0 ~_jR(n) , ( c o , then the spectral density function gOD exists and

No. 3

Liu Yan-yan: A Central Limit Theorem of Branching Process-." ' I

-+,oo

R] --~Ct

Ibragimove proved the following Lemma in Rd. [7]. Lemuma2. 1 Suppose that Y = {Y,},>~l is a strictly stationary random sequence with EY~ =0 and E ~ < co and suppose that the ?mixing oo

coefficients of {Y,}, satisfy ~

Pr(2") < co. Let

r=l

S. =

~=]Y~, then {Y~ },a~ has continuous i=I

spectral density function. Moreover for any n>~l Var S. = 2~g(0)n + o(n) From Lemma 2.1, we have Corollary 2. 1 Suppose {X. }.>Io is a ? mixing process with EZ z < c o and ~-~p(2") < co. Then the spectral density function of {Zk,,i >~1}, denoted by g (,1), exists and does not depend on k. Moreover (i) Var(X.+~ l Y . ) = 2 x g ( O ) X . + o ( X . ) a.s.;

(if) ~+1 =Var (X.+llXo)=2rcg(O)c.Xo+ c.o(1);

263

converges to W almost surely, moreover 0 < W < co a.s., E(W) <~ 1; (iii) X . ~ c o a.s. as n-+co. Proof The assertions (i) and (if) can be proved by the same discussions as in Ref. [4] and we omit the details. To prove assertion (iii), we assume that X. does not converge to oo with a positive probability. Then there exists a positive integer no such that for all n>/no, X. is constant with positive probability. {HQKOn the other hand, notice that for any n>~no ,P(X.+l = X. [X.o )<~pl , so

P(X.o+k = X.o+~-I p

- X.o ] X.o)

- . o , k --. oo

we obtain thus a contradiction. Let (n, i)be the i-th partide in n-th generation,let X . , . , ( k ~ l ) be the number of descendants of (n, i) in the ( n + k ) - t h generation and W..; be the correspondent limit starting from (n, i). In other words, X.,,a = ~ { (n + k,j) 9 (n + k 4 )

descend from (n,i)};W.,, = lira 4 X .... By these definitions we have X

(iii) lira V a r ( W . ) - 2~g(0)Xo .~ ~ ( p - 1) " ,u" (/1"+ l _ 1) #" where c. = and c:, = (/," - 1) #-I ~,-1

X,,+k = ~

x

W =

Proof

(i) is a direct consequence of Lemma 2.1; (ii) Note that ~+~ --EVar(X.+t 137.) +Var (E(X.+~&)). Combine it with conclusion (i), r~.+,= 2~g(0)Xo#" + ~ + # " o(1) tl

= 2

tl

g(O)Xo..

= 2=g(O)c.Xo +c.o(1) (iii) It fol['ows from (if) that 9 r2.+l 2~g(O)Xo [im VarW.+l = lira .-Y~2 . . . . . ,u ~(~, - 1) Proof of the following well-known results does not require any dependence within a generation. Lemma2. 2 Let {W.}.>1o be defined as in section 1. Then (i) E(X.+, lY.) = # X . ,E(X.+, ) =/~E(X.) =/~"+' ,E(X.+k 1.9.) = / ? X. a.s. ; (if) {W.}.~>o is a positive martigale which

X.,i,,~,

i=1

- lira-t-~op

=

/~k

x

-•/I"

W.~

(1)

i=l

define ~3..(k) =a(Wk, ,O(x i(xm) ,~P"(k) =a(W,.,, m~i/o is a ?mixing branching process. Then (i) For every rE./r p" (r)<~p(r). (if) P(W..<~al.g.)=P(W~a) a. s. x

(iii)

W

g"

W..,. i~l

264

Wuhan

University

Journal

of Natural

Vol. 7

Sciences

Proof

It is ready to verify the conclusions (ii) and (iii). To prove (i), let Cb be the set of all bounded continuous functions from R 1 to R l , from the definition of p" (r), we have cov( ]-[ f, (Wk.i), p" (r) =Pk" (r) = sup

sup

7 ( f , g ) = sup sup sup

f , (W,.,) ) 1 = r+p

i= 1

/ec, l p .~_ '>~''>' '~;'~/Var(]'-[f,(W,,,))Var( 11 f,(w,,,))

'>~',e/6 E%<,),t_,,r

~/ p

i =r+p

i=1

s

=sup sup sup lim l ~,~;,

Var(

) ) Var(

fi (

fi (

))

I=r+p

i-I "

cov(]-[ L <~liminf sup N--oo ? ~ c~ I

L(X,.,.N))

,-i

./Var(r[ f, (X,.,.N)Var( ~/

liminfp(r) = p(r)

j-,+p

P

h

.?, (X,.,.N))

1 =r+p

i=1

/a------y--). where )', (Xk.i.N)= f,(Xk.,.NRemark 2. 1 For other kinds of mixing processes, there are similar results by the similar arguments as above. Let {X,}n>~x be a a(resp, p,~ and r branching process. By Lemma 2.3, we have ( I ) P(W,.,<~yIffn)=P(z<~y); ( i l ) {Wn.,},>~l are a (resp. p,~,r with respect to 9"n; X

([1])

1 W - W. --= /a=. ~-~ ( W. . , - 1 ) . i--1

Notice that for every too , P ( W . . , < y [ ~ . ) ( % ) is a distribution function. Thus from the stationary of {W., }i>o and Kolmogorov Theorem, we can define a sequence of random variables denoting by {Wi}i>~o such that: (a) It has the same distribution as {W.., }i>~o; (b) For every n E N, E (Wn., W . a ) = E (W,W,) ; (c) For every n and fixed k,

with E Yl =0 and E y~
Y, and

k=l

assume that ~ =ES2.'-*'c~ and ~-]g,(2") < oo. n=l

Then p(S. ~ y) ~ r an

I

where ~(y)

e- r dr.

Now we give the proof of Theorem 1.1. Proof of Theorem 1.1 From (2) we have for a.s.% X

~-'] (W.. - I) p ,-x ~ylSr.

(%)

42,tg(0) x. X. (,,,o )

=

P

~'] (W, - 1) ] ~-~ ~
a.s.

(3)

X(%)

k

P(

(w..,-1)

y l

=

and P[~/2--~-~,(~

i=l k

P(~(W~-l)~y)

a.s.

(2)

~< y

is a sub-sequence

II

i-I

To prove Theorem 1.1, we also need the following result. Lenuna 2. 4 (see Ref. [7]. ) Suppose that {Yn },~>~ is a strictly stationary random sequence

.

From Lemma 2. 2, we have E W. = 1. For any % , combine Corollary 2.1 with Lemma 2.4,

No. 3

Liu Yan-yan, A Central Limit Theorem of Branching Process...

we have X=(%)

~ y ] = O(y)

and Co = {i E I: io = 1 } and then define {C. } and {C. } inductively by taking C.+~ = {i E C.: i,,+, ~ Z""} and C,+l = {i [ (n + 1) : i E C,+l } K = ~ C. is called the branching set. Let Z.

together with (3),

rim0

X

(W.. - 1) lira P ~=l ~ y [ ~ ] = O(y) a.s. "-'| 42~g(O) X. Combine ( m ) and (4), we finish the proof of Theorem 1.1.

3

265

Hausdorff Dimension

In this section, we give another careful definition of a branching set to meet our present needs. Some results on the Hausdorff dimension are obtained. Let N+ (resp. N) be the set of positive (resp. non-negative) integers with the discrete topology. Let I = N ~ be the set of all infinite sequences of positive integers with the product topology. For every iEl, nEN, we write (/In) = (io,il ,...,i,). For i,jEI, define

d(i,j) = 2-I"'"~ where k(i,j)=max{n:(i[n)=(jln)}. It is easy to verify that d is a metric on I and is compatible with the product topology. Moreover, for all points i,j,kE I

d(i,j) ~ max{d(i,k),d(k,j)} So (I, d) is an uhrametric space. Let F, =NC.~''~ be the set of all sequences (il

n), iE I and let F = ~ F. be the set of all finite sequences. If i = (io, 6 , ' " , i, ) E F, we denote 1i1 = n. For i E F a n d j E F U I , we write ij for the sequence obtained by juxtaposition of the terms of i and j. If f E F,, the descendants and closed descendants of f are defined by D ( f ) = {g E F: (g I n) = f} and /5 (f) = {i E I: (i[ n) = f} respectively. Note that the balls in (I,d) are of the form 13 (f). Suppose that {pk, k>~0} is a sequence of non-negative numbers with ~

Pk = 1. Let {Z/,

1,--O

f E F} be i. i. d sequence distributed according to the law P{Z = k}=pk. Let Co={gCzFo:go=l}

denote the number of sequences in C, , then {Z,, n = 0,1,... } is a simple Gahon-Waston process, Xo = 1

and X,,+l = ~ Z / IEC,

We refer to Athreya and NeyE~3 for more properties of branching processes. In this section, we suppose {X, } be a p-mixing process. For each f E C,, define

Z/, i =

~_a

Z*

gE C,_~IqDCD

Then for each k > n, Zs, k is the number of descendants of f in the generation k. Define

mid ( f ) ] = k~,~,o lira Z/'~ /.fit if fEC,, and m[13 ( f ) ] =0 if f~Clll. For any A ~ I , let /~" (A) = inf{ ~ m[D(f,)], A c__U,13(f,)} i

(5) Then/1" is a metric outer measure on I and hence the Borel sets are measurable. Let m denote the corresponding measure. P{m[D(i [ n)]<~ z} = P{W <~I:x} (6) Let B(i,r) be the open ball with center i and lg tl radius r, and let o = It follows from (6) lg 2" that, conditioned on ( i [ n ) E C., m[B(i,2-")] and 2 - " m[B(i,1)] are identical in distribution. We start with the following Theorem. Theorem 3.1 [-Ref.8,Theorem 3.1.5] Suppose that { X. }.>~o is a ~mixing branching process with ~ fl,(r) < ~ . Then the following assertions are equivalent: L1

(I)W. -W; (H) E ( W ) = l ; ( m ) P(W>0) = 1; (IV) E (Zig z ) = f ? z l g + z d F ( z ) < o o . Theorem 3.2

[Ref. 8,Theorem 3.1.5] Let {X, }.>~o be a p-mixing branching process. Suppose EZp < o o ( 1 < p ~ < 2 ) and

Wuhan University Journal of Natural Sciences

266

~']~ (p(2"))88 < oo ,where 1+1__= 1 ,q>0. Then ,=l P q there exist constants C1 ,Cz>O such that

C EZ p

E W p <<.C2EZ p

Theorem 3. 3 Suppose { X. } be C-mixing process satisfying the condition of Theorem 3.2, let E[Z(lg + Z)Z+*]<~, (a>~l), then with probability 1,

rn[i3(i l n)]

lim lg

= - - lg ,u

First we note that P ( W > O ) = I which means our process is non-degenerate. Proof

EI g {l~"m[D(i l n)]}-~d re(i)

1

SO,

Ef:m{i:t

"m[D(i [ n)] < 1}clx = 1 3g

Thus, by a double use of Marcov's inequality (aP([yl>~a)<~ElYI), we have

n' Ern{i:l~"rn[D(il n)] n-'} 1 nZP[m{i:l~"m[D(il n)-] • n-4} 1> n-z] <

1 A double application of the Borel-Canteli Lemma shows that, for sufficiently large n, m{[-O(i[ n)] <~n-4}
liminflglmLD(i[n)jl" " ' "'~--lg~ n ~

n

Vol. 7

except on a P• rn negligible set. In the following Theorem, we determine the Hausdorff dimension of K. Theorem 3. 4 Let the space 1 is given the metric d,{X. } is ~mixing process satisfying the condition in Theorem 3.2, then the Hausdorff dimension of K satisfies dim K = lg,u lg 2 almost surely on K r Proof The lower bound for dimK follows from (3) directly. Suppose 0 < a < l I -~

and that

re(K)>0, we can find s compact subset K' of K such that m(K')>O, and lg{m[D(i l n) f] K']} ~ - n a for all i in K' and all n>/N. Let {Sj } be any open cover of K' with Si~<2-". Then

0 <m(K') ~qm(U S~) <~ 2rn(Si) <~ ~ (diam S,)* so dimK)dimK'>~aalmost surely when a
rn[i:l~"rnD(i[ n)-]> x] is distributed the same as X.

the sum

~ Y , Iir,>_.fi~ ,

References: where {Yi} are mixing

i=l

W dependent r.v. distributed the same as ~-;. Then

Elm{ i:mCD(i l n, q ~ ~. } } = I: tP(W E at) From the fact E{Z (lg+Z) *+z }0, and l < a < l + ~ ,

~=of~tP(W E dt) = f? ~< IP(W E dt) P

= O(1)Jt(lg

So i f l < a < l + ~ , E

t)P(W E

dt) <

CX)

m{i:rn[D(iln)-]>~

} is

summable. A Borel-Canteli argument on m and P shows that :

limsuplg{mLD(i l - ~-lg/~ n~oo

n

I'1-] Athreya K B, Ney P E. Branching Processes. New York: Springer, 1972. [23 Mandelbrot B. Colliers aleatoires et une alternative aux promenades au hasard sans boucle, lea cordommets discrets et fractals. C R Acad Sci Paris Set A , 1978, 286:933-936. ['3] Peyri~re J. Processus de naissance avec interaction des voisins. C R Acad Sci Paris, 1979, 287..223-224 and 557. 1"4] Wen Z Y. Sur quelques the~'oremes de convergence du processus de nsissance avec interaction des voisins. Bull Soc Math France, 1986, 114:403-429. 1"5] Hawkes J. Trees Generated by a Simple Branching Process. J London Math Soc, 1981, 24..373-384. 1"6] Lu C Y, Lin Z Y. The Limiting Theory o f MixingDependent Random Variables. Beijing Academic Press, 1997(Ch). [-7] Ibragimov I A. A Note on the Central Limit Teorem for Dependent Random Variables. Probab Theory Appl, 1975. 20:134-139. [-8] Liu Y Y. Some Studies on Random Fractals. [-Ph. D. Thesis], Wuhan.. Wuhan University, 2001.