A NOTE ON EXACT CONVERGENCE RATE IN THE LOCAL LIMIT THEOREM FOR A LATTICE BRANCHING RANDOM WALK?

Zhiqiang GAO(高志強)

Laboratory of Mathematics and Complex Systems(Ministry of Education of China),School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China

E-mail:gaozq@bnu.edu.cn

Abstract Consider a branching random walk,where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z.Denote by Zn(z)the number of particles in the n-th generation in the model for each z∈Z.We derive the exact convergence rate in the local limit theorem for Zn(z)assuming a condition like“EN(logN)1+λ< ∞”for the offspring distribution and a finite moment condition on the motion law.This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.

Key words lattice branching random walks;local limit theorem;exact convergence rate

1 Introduction

Consider a branching random walk on the integer lattice Z.At time 0,an ancestor particle? is located at the S?=0.At time 1,? reproduces N?new particles of generation 1,and each particle?i(1≤ i≤ N?)moves to S?i=S?+L?i.In general,at time n+1,each particle u=u1u2···unof generation n is replaced by Nunew particles of generation n+1,with displacements Lu1,Lu2,···,LuNu.This means,for 1 ≤ i≤ Nu,each particle ui moves to Sui=Su+Lui.Here all Nuand Lu,indexed by finite sequences of integers u,are independent and identically distributed with random variables N and L on some probability space(?,F,P),satisfying the following conditions(H).

(H1)N is an integer-valued random variable such that

(H2)L is an integer-valued random variable satisfying that the lattice span of the distribution of L is 1,which means that there is no pair h ∈ {2,3,···},a ∈ Z such that all possible values of L are contained in the arithmetic progression a+hZ.

Denote by Zn(·)the counting measure of the number of particles in the n-th generation.The central limit theorems for suitably normalised Zn(·), firstly conjectured by Harris[16,Chapter III§16],were extensively investigated by many mathematicians(see,for instance,[3,7,13,17–21,24]and references therein).The reader may refer to[10–12,14]for more recent developments.See also[23]and[25]for other aspects of branching random walks.

Révész[19]started the research on the convergence speed in the central limit theorem for Zn(·),he treated two kinds of branching random walks where the moving mechanism is governed by simple random walks or Wiener processes.Later on under the condition EN2<∞,Chen[9]improved Révész’s results on these two cases by giving the explicit convergence rates.Furthermore,under some mild conditions on N and L,Gao and Liu[12]extended Chen’s result for branching Wiener processes to the strongly non-lattice branching random walk on the real line(strongly non-lattice means that the characteristic function of L satisfies the Cramér conditionOn the other hand,when the step size L is a lattice random variable satisfying condition(H2),Grübel and Kabluchko[14]obtained the Egdeworth-type expansion for Zn(·)implying the convergence rate in a local limit theorem.However,they imposed some rather strong moment conditions on both offspring distribution and moving law(roughly speaking,like EN1+?< ∞ and Ee?|L|for some ?>0).Naturally,one would like to find weaker moment conditions on N and L in the lattice case,which entail the exact convergence rate in the local limit theorem for Zn(·).This is the main object of the present article.

To be precise,let us first recall the local limit theorem for Zn(·).Throughout the article,we write Zn(z)=Zn({z}),which is the number of the n-th generation individuals located at z∈Z.According to Theorem 7 in[7]and the paragraph following it,the following result holds.

Theorem A(see[7])Assume conditions(H),EN logN< ∞ and σ2:=VarL< ∞.Then

uniformly in z∈Z,where l=EL and W is the limit of the usual branching process martingale{Wn}defined by Wn=Zn(Z)/mn.

We are interested in the rate of convergence in(1.2).To state our main results,we shall need some notation.

By convention,let T be the genealogical tree with{Nu}as defining elements.Let

be the set of particles of generation n,where|u|denotes the length of the sequence u,thereby representing the number of generation to which u belongs.

The condition P(N≥1)=1 in(H1)is only for technical simplicity and can be removed,but then our results hold conditionally on the survival event.With condition(H1),we have P(Zn(Z)→∞)=1.By the famous Kesten-Stigum theorem[2,4],EW=1 and P(W>0)=1.Put α3:=E(L ? l)3,α4:=E(L ? l)4.Then our main result can be stated as follows.

Theorem 1.1 Assume conditions(H)and EN(logN)1+λ< ∞ for some λ >9,together with E|L|η< ∞ for some η >6.Then

(I)for each z∈Z,

where,respectively,the real-valued random variables V1and V2are the almost sure(a.s.)limits of the sequences{N1,n}and{N2,n}defined by

Remark 1.2 When L is a random variable with P(L=1)=P(L=?1)=1/2 or with the standard normal distribution,the quantities{N1,n}and{N2,n}were firstly introduced by Chen[9],where their convergence was proved under the second moment condition EN2<∞.In the general case,the convergence to V1and V2of the sequences{N1,n}and{N2,n}was proved in[12,Propositions 2.1 and 2.2]under some mild moment conditions on N and L.We recall the results in Section 2.1.

Remark 1.3 Our results are not covered by[14],as the moment conditions assumed here are weaker.As will be seen in the proof,the two numbers 9 and 6 in the conditions are due to technical reasons.It will be interesting to find out the best values of λ and η,which seems difficult.

Remark 1.4 As an example,we consider a special case in which L is a random variable with the law

In this case,(1.3)and(1.4)are consistent with a specialization of the results in[10]for the case d=1,but there the stronger condition EN(logN)1+λ< ∞ for some λ >16 was assumed.

2 Proofs of Main Results

2.1 Two Martingales and Their Limits

In this subsection,we recall some facts on the sequence{N1,n}(resp.{N2,n}),and its a.s.limit V1(resp.V2).The following results were proved in[12,Propositions 2.1 and 2.2].

Lemma 2.1(see[12]) The sequences{N1,n}and{N2,n}are martingales with respect to the natural filtration

Then

a)EN(logN)1+λfor some λ >1 and E|L|ηfor some η >2,entail thatexists a.s.in R;

b)EN(logN)1+λfor some λ >1 and E|L|ηfor some η >4,entail thatexists a.s.in R.

2.2 Asymptotic Expansion for Sums of i.i.d.Lattice Random Variables

Denote by Hm(·)the Chebyshev-Hermite polynomial of degree m(m ∈ N={0,1,2,3,···}):

where ?x?denotes the largest integer not bigger than x.More precisely,we shall need the following polynomials

Lemma 2.2 Assume that E|L|4<∞and the span of the distribution of L is equal to 1.Then

2.3 A Decomposition

As usual,we write N?={1,2,3,···}and denote bythe set of all finite sequences,where(N?)0={?}contains the null sequence?.

For all u∈U,let T(u)be the shifted tree of T at u with defining elements{Nuv}satisfying 1)?∈T(u),2)vi∈T(u)?v∈T(u)and 3)if v∈T(u),then vi∈T(u)if and only if 1≤i≤Nuv.Set Tn(u)={v∈T(u):|v|=n}and denote by|Tn(u)|the cardinality of Tn(u)(i.e.,the number of descendants of u in the n-th generation).

For u∈(N?)k(k≥0)and n≥1,define Zn(u,z)by

which counts the number of descendants of u in the n-th generation located at z+Su∈Z.

Observe for k≤n,

and for u∈Tk,

By the conditions on γ,λ and η,we can choose a real number β satisfying

Set kn= ?nβ?,the largest integer not bigger than nβ.On the basis of(2.3),we obtain the following key decomposition

with

2.4 Proof of Theorem 1.1

By using(2.5),we may divide the proof of Theorem 1.1 into the following lemmas.

Lemma 2.3 Under the conditions of Theorem 1.1,for each z∈Z,

Lemma 2.4 Under the conditions of Theorem 1.1,for each z∈Z,

Lemma 2.5 Under the conditions of Theorem 1.1,for z(n)defined in(II),

Lemma 2.6 Under the conditions of Theorem 1.1,for z(n)defined in(II),

Here we only give the proofs of Lemmas 2.3 and 2.4.Lemmas 2.5 and 2.6 can be handled in the same way after a slight modification,and the proofs are omitted.

Proof of Lemma 2.3 We start by introducing some notation.For u∈Tkn,set

It is easy to see the following fact We remind that{Wn?kn(u):u ∈ Tkn}are mutually independent and identically distributed as

Wn?kn.

The lemma will be proved if we can show the following

For this purpose,we shall need the following result.

Lemma 2.7(see[8])Let W?=supnWn.Assume(1.1)and EN(logN)1+λ<∞.Then

To prove(2.11),it suffices to show that

Observe that

Then(2.15)follows from the choice of kn,the fact λβ >1 and(2.14).

Now we turn to the proof of(2.12).To this end,we will need the following inequality(by(5.3)in[6]).For 1<α<2,

Note that in the above formula and throughout the article,K denotes all constants,and thus its value may vary even in a single inequality.Thus by taking expected value of the above,we deduce that

which is finite,since(3α+2)/(2β)?1< λ provided that α is sufficiently near one and E(W?+1)Hence(2.12)follows by the Borel-Cantelli lemma.

It remains to prove(2.13).Since EDknXn,u=0,we see a.s.

which yields

This implies the a.s.convergence of the seriesand accordingly(2.13)follows.Lemma 2.3 is proved.

Proof of Lemma 2.4 To estimate the quantity Bn(z),we write

with

where q1(x)and q2(x)are defined by(2.2),and ?n,uare infinitesimals satisfying0.

By using Taylor’s expansion and through tedious calculation,we have that as n tends to infinity,

where ?i(n,y)(i=1,2,3)are infinitesimals satisfying

Taking(2.17)–(2.20)into account,we deduce that

Next,we intend to prove

Observe that

By[1,Theorem 2],we see that under the condition EN(logN)1+λ<∞,

We only need to prove that

which is equivalent to the following

Observe that

Due to Theorem 3 in[5],the finiteness of the last series in the above is equivalent toE|L|1/β+2< ∞,which is valid since 1/β +2< η and E|L|η< ∞.Thus,(2.26)is proved,as well as(2.25).

Combining(2.23),(2.24)and(2.25)gives the first estimate in(2.22),Wkn?W=o(1/n).

Now we turn to the proof of the remaining two estimates in(2.22).By Lemma 2.1,we establish them by showing that

Observe that

Then,by using the Markov inequality and the moment inequality of sums of independent random variables(see,for instance,formula(2.3)on p.227 in[15]),we have

where 2?η/2< ?1 by η>6.Thus,it follows that N2,j?N2,j=o(1)as j→ ∞.Similarly,we can prove that?N1,j=o(1)as j→∞.Combining these with Lemma 2.1,(2.22)follows.

On the other hand,we see that by Lemma 2.2,

and then by(2.25),

Thus we can deduce the desired(2.7)from(2.21),(2.22)and(2.28). ?