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)03-0261-0...
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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
~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 ~ 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~. 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) /no, X. is constant with positive probability. {HQKOn the other hand, notice that for any n>~no ,P(X.+l = X. [X.o )~''>' '~;'~/Var(]'-[f,(W,,,))Var( 11 f,(w,,,))
'>~',e/6 E%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~~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)]0. Then ,=l P q there exist constants C1 ,Cz>O such that
C EZ p
E W p n-z]