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.

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, is a double sequence of independent and nonnegative integer valued random variables, where for fixed , have the same distribution with mean and variance .

Definition 1. Assume and for any , define then is said to be a GWVE.

Define ; it is well known that and there exists a nonnegative random variable such that , as (see [8]).

For any fixed , let be the size of the th generation of GWVE starting with the th particle at time ; then are i.i.d. with mean and variance (see (4) and (5)). For each , define where is the largest integer that is less than . Our main result is a weak limit theorem for GWVE, which is an extension of Donsker’s theorem.

Theorem 2. Suppose that and ; then , where is the standard Brown motion on .

Let be the space of functions defined on and having discontinuities of at most the first kind. For any , define ; it turns out that , where is the Wiener measure on . Note that ; by Theorem 2 one has the following.

Corollary 3 (CLT). Suppose that and ; then for any fixed , where 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 .

Proposition 4. are independent and identically distributed with

Proof. According to the definition of definition of GWVE, are independent and identically distributed.
Denote the generating functions of and by and , respectively; then it can be proved that Therefore, So (4) is proved. In addition, the first and second derivatives of are as follows: By (8) one has Thus, Since , , , we complete the proof of (5) by (10).

For any , define

The proof of Theorem 2 depends on the following proposition.

Proposition 5. , where is standard Brown motion on .

Proof. It lose no generality if we assume that are integers. The proof is divided into two steps. We first show that the finite-dimensional distributions of the are convergent to those of . Consider first a single time point . We must prove that .
Since have the same distribution, we can set Note and , according to of [11] P101; one obtains For any fixed and large enough, Since , for large enough, we have
This means that the characteristic function of is convergent to that of ; by Lévy continuous theorem we complete the proof of single point case.
Consider now two time points and with ; we are to prove Note that By Corollary 1 to Theorem  5.1 in [12], it is only needed to prove Since the components on the left are independent by the independence of the . 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 is tight. According to Theorem 15.6 of [12], it is enough to establish the inequality Since are i.i.d. with and ; by the definition of , we have If , then there exist such that Hence, So (19) is true when . Next, if , then either and lie in the same subinterval or else and do. In either of these cases 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 ,
We assume at first that is bounded, so that there exists a constant such that with probability 1. We may adjust the so that they are integer and so that .
If we define Since converges in probability in the sense of the Skorohod topology to the elements of , where consists of those elements of that are nondecreasing and satisfy for all . Define where is a sequence of nonnegative integers going to infinity slowly enough that as . Define ; then By Minkowski’s inequality and the fact that , one has So that by Chebyshev’s inequality . By Proposition 4, . Since , where is the metric in which generates the Skorohod topology, it follows by Theorem 4.1 of [12] that . So, if is a -continuity set in , we have
Let be the field of cylinders sets; that is, consists of the form with , the Borel -field of .
If , since and are independent, then for large ,
It follows by (29) that Since in the sense of the product topology on and every is measurable, it follows by Theorem 4.5 of [12] that is relative to the product topology in , where is independent of and has the same distribution as , . By the fact that , Now the mapping that carries the point to is continuous at that point , and . By Corollary 1 to Theorem 5.1 of [12], Since and are independent, has the same distribution as . Moreover coincides with if , the probability of which goes to 1 since . Thus if is bounded.
Suppose is not bounded. For , define Then for each , and by the case already treated if then . Since , follows Theorem 4.2 of [12].

Conflict of Interests

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

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).