A Weak Limit Theorem for Galton-Watson Processes in Varying Environments

Abstract

We extend Donsker's theorem and the central limit theorem of classical Galton-Watson process to the Galton-Watson processes in varying environment.

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Academic Editor:Chong Lin

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 28 March 2014; Accepted 22 July 2014; 12 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

There has been a lot of interesting works on Markov chains in random environments, which is mainly concentrated in branching processes in random environments and random walks in random environments (see [1]).

The study of branching processes in random environments dates back to late 60s or early 70s in the last century (see [2-5]). Our paper deals with a Galton-Watson branching process in the varying environment (GWVE) which is a special case of branching processes in random environments. The main concern is the weak convergence for a GWVE, which is an extension of Donsker's theorem (see [6, 7]).

In the following context, {_Xni ,n...5;0,i...5;1} is a double sequence of independent and nonnegative integer valued random variables, where for fixed n , {_Xni ,i...5;1} have the same distribution {_pni ,i=0,1,2,...} with mean _μn >0 and variance ^σn2 >0 .

Definition 1.

Assume _Z0 ...1;1 and for any n...5;1 , define [figure omitted; refer to PDF] then {_Zn ,n...5;0} is said to be a GWVE.

Define _mn =E(_Zn ) ; it is well known that _mn =_μ0 ·_μ1 ,...,_μn-1 and there exists a nonnegative random variable V such that _Zn /_mn [arrow right]a.e.V , as n[arrow right]+∞ (see [8]).

For any fixed r , let _ξnj ...=^Xn,r(j) be the size of the r th generation of GWVE starting with the j th particle at time n ; then {_ξnj ,j...5;1} are i.i.d. with mean _mn,r and variance ^σn,r2 (see (4) and (5)). For each n , define [figure omitted; refer to PDF] where [x] is the largest integer that is less than x . Our main result is a weak limit theorem for GWVE, which is an extension of Donsker's theorem.

Theorem 2.

Suppose that _mn [arrow right]∞ and P(V=0)=0 ; then _Yn [arrow right]dB , where B is the standard Brown motion on [0,1] .

Let D be the space of functions defined on [0,1] and having discontinuities of at most the first kind. For any α∈R , define Aα ={x∈D:x(1)...4;α} ; it turns out that W(∂Aα )=0 , where W is the Wiener measure on D . Note that Zn+r =∑j=1Znξnj ; by Theorem 2 one has the following.

Corollary 3 (CLT).

Suppose that _mn [arrow right]∞ and P(V=0)=0 ; then for any fixed r , [figure omitted; refer to PDF] where N(0,1) is the standard normal random variable.

So, Theorem 2 is an extension of the central limit theorem for classical Galton-Watson process (see [9, 10]).

2. Auxiliary Results

Let us begin with a result of _ξnj .

Proposition 4.

{ _{ξ n j} , j ...5; 1 } are independent and identically distributed with [figure omitted; refer to PDF]

Proof.

According to the definition of definition of GWVE, {_ξnj ,j...5;1} are independent and identically distributed.

Denote the generating functions of _Xn1 and _ξn,1 by _[varphi]n (s) and _gn,r (s) , respectively; then it can be proved that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] So (4) is proved. In addition, the first and second derivatives of _gn,r (s) are as follows: [figure omitted; refer to PDF] By (8) one has [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Since _mn,1 =_μn , ^σn,12 =^σn2 , _ξn1 =^Xn,r(1) , we complete the proof of (5) by (10).

For any n , define [figure omitted; refer to PDF]

The proof of Theorem 2 depends on the following proposition.

Proposition 5.

_{X n} [arrow right] d B , where B is standard Brown motion on [0,1] .

Proof.

It lose no generality if we assume that {_mn } are integers. The proof is divided into two steps. We first show that the finite-dimensional distributions of the _Xn are convergent to those of B . Consider first a single time point s . We must prove that _Xn (s,·)[arrow right]d_Ws .

Since {_ηnj ,j...5;1} have the same distribution, we can set [figure omitted; refer to PDF] Note E(_ηnj )...1;0 and Var...(_ηnj )...1;1 , according to (3.8) of [11] P101; one obtains [figure omitted; refer to PDF] For any fixed t and k large enough, [figure omitted; refer to PDF] Since _mn [arrow right]∞ , for n large enough, we have [figure omitted; refer to PDF]

This means that the characteristic function of _Xn (s) is convergent to that of _Bs ; by Lévy continuous theorem we complete the proof of single point case.

Consider now two time points s and t with s<t ; we are to prove [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] By Corollary 1 to Theorem 5.1 in [12], it is only needed to prove [figure omitted; refer to PDF] Since the components on the left are independent by the independence of the {_ξni ,i...5;1} . Equation (16) follows from the case of one time point and Theorem 3.2 of [12].

A set of three or more time points can be treated in the same way, and hence the finite-dimensional distributions converge properly.

In the next step, we will show that {_Xn } is tight. According to Theorem 15.6 of [12], it is enough to establish the inequality [figure omitted; refer to PDF] Since {_ηnj ,j...5;1} are i.i.d. with E(_ηnj )...1;0 and Var...(_ηnj )...1;1 ; by the definition of _Xn , we have [figure omitted; refer to PDF] If _t2 -_t1 ...5;1/_mn , then there exist _k1 <_k2 such that [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] So (19) is true when _t2 -_t1 ...5;1/_mn . Next, if _t2 -_t1 <1/_mn , then either _t1 and t lie in the same subinterval [k/_mn ,(k+1)/_mn ) or else t and _t2 do. In either of these cases _Δn =0 by (20). This establishes (19) in general and proves the proposition.

3. The Proof of Theorem 2

We are now ready to prove Theorem 2.

Proof.

Note that for each n , [figure omitted; refer to PDF]

We assume at first that V is bounded, so that there exists a constant k such that 0<V...4;k with probability 1. We may adjust the _mn so that they are integer and so that k<1 .

If we define [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] _Φn converges in probability in the sense of the Skorohod topology to the elements Φ(t)=Vt of _D0 , where _D0 consists of those elements [straight phi] of D that are nondecreasing and satisfy 0...4;[straight phi](t)...4;1 for all t . Define [figure omitted; refer to PDF] where {_ln ,n...5;0} is a sequence of nonnegative integers going to infinity slowly enough that _ln /_mn [arrow right]0 as n[arrow right]+∞ . Define _δn =_sup...t ...|_Xn (t)-^{Xn[variant prime]} (t)| ; then [figure omitted; refer to PDF] By Minkowski's inequality and the fact that _ln /_mn [arrow right]0 , one has [figure omitted; refer to PDF] So that by Chebyshev's inequality _δn [arrow right]P0 . By Proposition 4, _Xn [arrow right]dB . Since d(_Xn ,^{Xn[variant prime]} )...4;_δn , where d is the metric in D which generates the Skorohod topology, it follows by Theorem 4.1 of [12] that ^{Xn[variant prime]} [arrow right]dB . So, if A is a W -continuity set in D , we have [figure omitted; refer to PDF]

Let _B0 be the field of cylinders sets; that is, _B0 consists of the form [figure omitted; refer to PDF] with H∈B(^Rmk ) , the Borel σ -field of ^Rmk .

If E∈_B0 , since _ln [arrow right]∞ and {_Xni ,n...5;0,i...5;1} are independent, then for large n , [figure omitted; refer to PDF]

It follows by (29) that [figure omitted; refer to PDF] Since (_Φn ,_Zn /_mn )[arrow right]P(Φ,V) in the sense of the product topology on _D0 ×R and every ^{Xn[variant prime]} is σ(_B0 ) measurable, it follows by Theorem 4.5 of [12] that [figure omitted; refer to PDF] is relative to the product topology in D×_D0 ×^R1 , where _V0 is independent of B and has the same distribution as V , _Φ0 (t)=_V0 t . By the fact that _δn [arrow right]P0 , [figure omitted; refer to PDF] Now the mapping that carries the point (x,[varphi],α) to ^α-1/2 (x[composite function][varphi]) is continuous at that point x∈C , [varphi]∈C∩_D0 and α>0 . By Corollary 1 to Theorem 5.1 of [12], [figure omitted; refer to PDF] Since _V0 and B are independent, (_V0^)-1/2 (B[composite function]_Φ0 ) has the same distribution as B . Moreover (_Zn /_mn^)-1/2 (_Xn [composite function]_Φn ) coincides with _Yn if _Zn /_mn <1 , the probability of which goes to 1 since k<1 . Thus _Yn [arrow right]dB if V is bounded.

Suppose V is not bounded. For u>0 , define [figure omitted; refer to PDF] Then for each u , _Zun /_mn [arrow right]P_Vu and by the case already treated if [figure omitted; refer to PDF] then _Yun [arrow right]dB . Since P(_Yun ...0;_Yn )...4;P(V>u) , _Yn [arrow right]dB follows Theorem 4.2 of [12].

Acknowledgments

The authors would like to thank the referee for his (her) valuable suggestions. They also thank Professor Xiaoyu Hu for her help. This work is supported by the Youth Foundation and Doctor's Initial Foundation of Qufu Normal University (XJ201113, BSQD20110127).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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