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

Authors: Zhenlong Gao; Yanhua Zhang
Journal: Journal of Applied Mathematics
 (2014)

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## 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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## Body

## 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≥0,i≥1} is a double sequence of independent and nonnegative integer valued random variables, where for fixed n, {Xni,i≥1} have the same distribution {pni,i=0,1,2,…} with mean μn>0 and variance σn2>0.Definition 1.
AssumeZ0≡1 and for any n≥1, define
(1)Zn+1={∑j=1Zn‍Xnj,ifZn≠0;0,ifZn=0,
then {Zn,n≥0} is said to be a GWVE.Definemn=E(Zn); it is well known that mn=μ0·μ1,…,μn-1 and there exists a nonnegative random variable V such that Zn/mn→a.e.V, as n→+∞ (see [8]).For any fixedr, let ξnj∶=Xn,r(j) be the size of the rth generation of GWVE starting with the jth particle at time n; then {ξnj,j≥1} are i.i.d. with mean mn,r and variance σn,r2 (see (4) and (5)). For each n, define
(2)Yn(t,ω)=1σnZn(∑j=1[Znt]‍ξnj-mn,r[Znt]),t∈[0,1],
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 thatmn→∞ and P(V=0)=0; then Yn→dB, where B is the standard Brown motion on [0,1].LetD 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)≤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 thatmn→∞ and P(V=0)=0; then for any fixed r,
(3)1Znσn,r2(Zn+r-mn,rZn)→dN(0,1),
where N(0,1) is the standard normal random variable.So, Theorem2 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 ofξnj.Proposition 4.
{
ξ
n
j
,
j
≥
1
} are independent and identically distributed with
(4)mn,r=E(ξnj)=μnμn+1⋯μn+r-1,(5)σn,r2=Var⁡(ξnj)=(mn,r)2∑j=nn+r-1‍σj2μj2mn,j-n.Proof.
According to the definition of definition of GWVE,{ξnj,j≥1} are independent and identically distributed.
Denote the generating functions ofXn1 and ξn,1 by ϕn(s) and gn,r(s), respectively; then it can be proved that
(6)gn,r(s)=ϕn(ϕn+1(⋯ϕn+r-1(s)⋯)).
Therefore,
(7)mn,r=E(ξn1)=gn,r′(1)=∏j=nn+r-1‍ϕj′(1)=∏j=nn+r-1‍μj.
So (4) is proved. In addition, the first and second derivatives of gn,r(s) are as follows:
(8)gn,r′(s)=gn,r-1′(ϕn+r-1(s))ϕn+r-1′(s),gn,r′′(s)=gn,r-1′′(ϕn+r-1(s))(ϕn+r-1′(s))2+gn,r-1′(ϕn+r-1(s))ϕn+r-1′′(s).
By (8) one has
(9)Var⁡(Xn,r(1))=(Var⁡(Xn,r-1(1))-mn,r-1+mn,r-12)μn+r-12+mn,r-1(σn+r-12-μn+r-1+μn+r-12)+mn,r-mn,r2=Var⁡(Xn,r-1(1))μn+r-12+σn+r-12mn,r-1.
Thus,
(10)Var⁡(Xn,r(1))mn,r2=Var⁡(Xn,r-1(1))mn,r-12+σn+r-12μn+r-12mn,r-1.
Since mn,1=μn, σn,12=σn2, ξn1=Xn,r(1), we complete the proof of (5) by (10).For anyn, define
(11)ηnj=ξnj-mn,rσn,r,Xn(t,ω)=1mn∑j=1[mnt]‍ηnj,t∈[0,1].The proof of Theorem2 depends on the following proposition.Proposition 5.
X
n
→
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,·)→dWs.
Since{ηnj,j≥1} have the same distribution, we can set
(12)φn(t)=E(exp⁡(itηnj)).
Note E(ηnj)≡0 and Var⁡(ηnj)≡1, according to (3.8) of [11] P101; one obtains
(13)φn(s)=φn(0)+φn′(0)s+φn′′(0)2!s2+o(s2)=1-s22+o(s22),(s⟶0).
For any fixed t and k large enough,
(14)φn(tk)=1-t22k+o(12k).
Since mn→∞, for n large enough, we have(15)Eexp⁡(itXn(s))=E(exp⁡(it1mn∑j=1[mns]‍ηnj))=[1-t22mn+o(12mn)][mns]⟶exp⁡{-st22}asn⟶∞.
This means that the characteristic function ofXn(s) is convergent to that of Bs; by Lévy continuous theorem we complete the proof of single point case.
Consider now two time pointss and t with s<t; we are to prove
(16)(Xn(s),Xn(t))→d(Bs,Bt).
Note that
(17)(Xn(s),Xn(t))=(Xn(s),Xn(t)-Xn(s))(1101).
By Corollary 1 to Theorem  5.1 in [12], it is only needed to prove
(18)(Xn(s),Xn(t)-Xn(s))→d(Bs,Bt-Bs).
Since the components on the left are independent by the independence of the {ξni,i≥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
(19)Δn∶=E{|Xn(t)-Xn(t1)|2|Xn(t2)-Xn(t)|2}≤4(t2-t1)2,∀0≤t1≤t≤t2≤1.
Since {ηnj,j≥1} are i.i.d. with E(ηnj)≡0 and Var⁡(ηnj)≡1; by the definition of Xn, we have
(20)Δn=E{|1mn∑j=[mnt1+1][mnt]‍ηnj|2·|1mn∑j=[mnt+1][mnt2]‍ηnj|2}=([mnt]-[mnt1])·([mnt2]-[mnt])mn2≤([mnt2]-[mnt1])2mn2.
If t2-t1≥1/mn, then there exist k1<k2 such that
(21)t1∈[k1mn,k1+1mn),t2∈[k2mn,k2+1mn).
Hence,
(22)[mnt2]-[mnt1]mn=k2-k1mn=k2-(k1+1)mn+1mn≤(t2-t1)+(t2-t1).
So (19) is true when t2-t1≥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 Theorem2

We are now ready to prove Theorem2.Proof.
Note that for eachn,
(23)Yn(t,ω)=1σn,rZn(∑j=1[Znt]‍ξnj-μn[Znt])=1Zn∑j=1[Znt]‍ηnj,t∈[0,1].
We assume at first thatV is bounded, so that there exists a constant k such that 0<V≤k with probability 1. We may adjust the mn so that they are integer and so that k<1.
If we define(24)|Φn(t)-tV|={tZnmn,ifZnmn≤1;tVotherwise.
Since
(25)|Φn(t)-tV|≤|Znmn-V|→a.e.0,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 φ of D that are nondecreasing and satisfy 0≤φ(t)≤1 for all t. Define
(26)Xn′(t,ω)={1mn∑ln≤i≤mnt‍ηni(ω),ift≥lnmn;0,otherwise,
where {ln,n≥0} is a sequence of nonnegative integers going to infinity slowly enough that ln/mn→0 as n→+∞. Define δn=sup⁡t⁡|Xn(t)-Xn′(t)|; then
(27)δn≤1δmn∑i=1ln‍|ηni|.
By Minkowski’s inequality and the fact that ln/mn→0, one has
(28)E1/2{δn2}≤1δmn∑i=1ln‍E1/2{ηni2}=lnδmn⟶0.
So that by Chebyshev’s inequality δn→P0. By Proposition 4, Xn→dB. Since d(Xn,Xn′)≤δn, where d is the metric in D which generates the Skorohod topology, it follows by Theorem 4.1 of [12] that Xn′→dB. So, if A is a W-continuity set in D, we have
(29)P{Xn′∈A}⟶W(A).
LetB0 be the field of cylinders sets; that is, B0 consists of the form
(30){ω;(ξij(w),0≤i≤m,1≤j≤k)∈H}
with H∈B(Rmk), the Borel σ-field of Rmk.
IfE∈B0, since ln→∞ and {Xni,n≥0,i≥1} are independent, then for large n,
(31)P({Xn′∈A}∩E)=P{Xn′∈A}P(E).
It follows by (29) that
(32)P({Xn′∈A}∩E)⟶W(A)P(E).
Since (Φn,Zn/mn)→P(Φ,V) in the sense of the product topology on D0×R and every Xn′ is σ(B0) measurable, it follows by Theorem 4.5 of [12] that
(33)(Xn′,Φn,Znmn)→d(B,Φ0,V0)
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)=V0t. By the fact that δn→P0,
(34)(Xn,Φn,Znmn)→d(B,Φ0,V0).
Now the mapping that carries the point (x,ϕ,α) to α-1/2(x∘ϕ) is continuous at that point x∈C, ϕ∈C∩D0 and α>0. By Corollary 1 to Theorem 5.1 of [12],
(35)(Znmn)-1/2(Xn∘Φn)→d(V0)-1/2(B∘Φ0).
Since V0 and B are independent, (V0)-1/2(B∘Φ0) has the same distribution as B. Moreover (Zn/mn)-1/2(Xn∘Φn) coincides with Yn if Zn/mn<1, the probability of which goes to 1 since k<1. Thus Yn→dB if V is bounded.
SupposeV is not bounded. For u>0, define
(36)Vu={V,ifV≤u;u,ifV>u,Zun={Zn,ifV≤u;mnu,ifV>u.
Then for each u, Zun/mn→PVu and by the case already treated if
(37)Yun(t,ω)=1σn,rZun(∑j=1[Zunt]‍ξnj-μn[Zunt])
then Yun→dB. Since P(Yun≠Yn)≤P(V>u), Yn→dB follows Theorem 4.2 of [12].



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*Source: 364372-2014-08-12.xml*