# Deviations for Jumping Times of a Branching Process Indexed by a Poisson Process

Authors: Yanhua Zhang; Zhenlong Gao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Consider a continuous time process{Yt=ZNt,t≥0}, where {Zn} is a supercritical Galton–Watson process and {Nt} is a Poisson process which is independent of {Zn}. Let τn be the n-th jumping time of {Yt}, we obtain that the typical rate of growth for {τn} is n/λ, where λ is the intensity of {Nt}. Probabilities of deviations n-1τn-λ-1>δ are estimated for three types of positive δ.


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## Body

## 1. Statements of the Main Results

The model of Poisson randomly indexed branching process (PRIBP){Yt=ZNt,t≥0} was introduced by [1] to study the evolution of stock prices and its statistical investigation was done in [2].In a recent manuscript [3] the authors there consider the asymptotic properties of log⁡Yt. Let {pk,k≥0} be the offspring distribution of the branching process with mean m=∑kkpk∈(1,∞); we distinguish between the Shröder case and the Böttcher case depending on whether p0+p1>0 or p0+p1=0. In Böttcher case, it was proved in [3] that log⁡Yt have similar asymptotic results to the Poisson process {Nt}. But differences appeared in Shröder case; see [4]. For subcritical and critical PRIBP, one can see [5, 6] for details.In this paper, we deal with the asymptotic theory for the jumping times of PRIBP defined as follows. For anyω, define (1)τ∞ω=inft:t>0;lims↑tYsω=∞, where inf⁡∅=∞; then (2)τ∞<∞=⋃l=1∞τ∞<l. Note that {Zn} is independent of {Nt}; one has (3)Pτ∞<l≤PYl=∞=∑nPZn=∞PNl=n=0,and thus P(τ∞=∞)=1. Define τ0=0 and {τn,n≥1} as the successive times of jump of the PRIBP {Yt,t≥0}.In Böttcher case, the jumping times of{Yt} coincide with that of {Nt}. Let {Tn} be the successive times of jump of {Nt}; then both τn and Tn have a gamma distribution with parameters n and λ. But when p0+p1>0, at the jumping time of Nt, PRIBP can have no jump, since an individual can replicate himself at this time. So the jumping times of {Yt} are likely to be delayed; see Figure 1 for example. In the path of Figure 1, τ1=T1 and τ2=T3,⋯.Figure 1
A path of Shröder case branching process.Althoughτn≥Tn for all n, the growth rate of τn is not too fast as that of Tn. In fact, the typical growth rate of Tn is n/λ by the law of large numbers and we can show that the typical growth rate of τn is (4)∑k=1n1λ1-p1k≤nλ+1λ1-p12,and see the proof of Theorem 1. Thus, for almost all the path of Shröder case PRIBP, τn/n has a limit λ-1 when n→∞.In the rest of this paper, we always assume that our branching process belongs to the Shröder case,p0=0 and Z0=1.We are interested in the decay rates about the probabilities of(5)ω:τnωn-1λ>δfor some positive δ. Typically, there are three classes of δ to be chosen.The first one is thatδ=an for some fixed a>0. In this case, the event in (5) is said to be a normal deviation event. The decay rate of its probability can be characterized by the following central limit theorem.Theorem 1.
{ τ n } satisfies the law of large number and the central limit theorem; that is, τn/n→a.e.λ-1 and λn(τn/n-λ-1)→dN(0,1) when n→∞, where N(0,1) is standard normal distribution.Next, ifδ=a for some fixed a>0, the event in (5) is said to be a large deviation event whose probability has an exponential convergence rate by the following large deviation principle.Theorem 2 (LDP).
For any measurable subsetB of R, (6)-infx∈BoΛ∗x≤liminfn→∞1nlog⁡Pτnn∈B≤limsupn→∞1nlog⁡Pτnn∈B≤-infx∈B¯Λ∗x, where Bo denotes the interior of B, B¯ its closure, and (7)Λ∗x=λ1-p1x+log⁡p1,x≥λp1-1;λx-log⁡λx-1,λp1-1>x>0;+∞,x≤0.Remark.By Cramér’s theorem (see Theorem 2.2.3 of [7]), Tn/n satisfies the large deviation principle with rate function (8)Ψx=λx-logλx-1,x>0;+∞,x≤0.By Theorem 2, the rate function of τn/n coincides with that of Tn/n for x≤(λp1)-1, but differences appeared for large x; see Figure 2 for example.Figure 2
λ = 1 ,  p 1 = 1 / 2.Ifδ=δn→∞ and δn=o(n) as n→∞, we call the event in (5) a moderate deviation event. Let {an,n≥0} be a family of positive numbers satisfying (9)ann→0andann→∞asn→∞. As in the case of large deviation principle, based on the Gärtner-Ellis theorem (see [7], page 44), we have the following moderate deviation principle.Theorem 3 (MDP).
For any measurable subsetB of R,(10)-infx∈Boλ2x22≤liminfn→∞nan2log⁡Pτn-λ-1nan∈B≤limsupt→∞nan2log⁡Pτn-λ-1nan∈B≤-infx∈B¯λ2x22.The rest of the paper is organized as follows. In Section2, we prove the law of the large number and the central limit theorem. Section 3 is devoted to the proof of the large deviation principle. In Section 4, the moderate deviation principle is obtained. Basic facts on Gärtner-Ellis theorem are given in the Appendix.


## 2. The Law of Large Number and the Cental Limit Theorem

Proposition 4.
PRIBP is a homogenous continuous time Markov chain in whichQ matrix satisfies qi=λ(1-p1i) for all i≥1.Proof.
For any nonnegative real numberst0<t1<⋯<tn-1<s<t+s and nonnegative integers i0≤i1≤⋯≤in-1≤i≤j, define (11)A=Ys=i,Ytn-1=in-1,…,Yt0=i0.For any nonnegative integers k0≤k1≤⋯≤kn-1≤ks≤kt+s, define (12)Bk0,…,kn-1,ks,kt+s=Nt0=k0,…,Ntn-1=kn-1,Ns=ks,Nt+s=kt+s.Since Poisson process {Nt} is independent of the Galton-Watson process {Zn}, (13)PYt+s=j∣A=∑k0,…,kn-1,ks,kt+sPBk0,…,kn-1,ks,kt+s·PZkt+s=j∣Zks=i,Zkn-1=in-1,…,Zk0=i0.Note that the Galton-Watson process is a Markov chain with n- step transition probabilities Pn(i,j), and summing k0,…,kn-1, one has (14)PYt+s=j∣A=∑ks,kt+sPNs=ks,Nt+s=kt+sPZkt+s=j∣Zks=i=∑ks,kt+sPNs=ks,Nt+s=kt+sPkt+s-ksi,j=EPNt+s-Nsi,j=EPNti,j. Similarly, (15)PYt+s=j∣Ys=i=EPNti,j=PYt+s=j∣A,which means that PRIBP is a homogenous continuous time Markov chain.
Next, note thatNt has a Poisson distribution with parameter λt>0; we have (16)EPNti,i=e-λt+p1iλte-λt+∑n≥2p1niPNt=n, which implies qi=λ(1-pi).Proof of Theorem1.
DefineXn(ω)=Y(τn(ω))(ω); then {Xn} is a homogeneous discrete-time Markov chain. Define ρn=τn-τn-1 for n≥1; then the conditional distribution of ρn relative to X1,X2,…,Xn-1 equals exponential distribution with parameter qXn-1, where qi=λ(1-pi), see page 259 of [8] for example. So (17)τn′≔τn-∑k=1n1λ1-p1Xk-1=∑k=1nρk-1λ1-p1Xk-1is a square-integrable martingale adapted to the σ- fields σ{X1,…,Xn}. Consequently, there exists a random variable Z such that τn′→a.e.Z, see page 2 of [9]. Note that Xn-1≥n; one has(18)nλ≤∑k=1n1λ1-p1Xk-1≤∑k=1n1λ1-p1k≤nλ+1λ1-p12,which implies that τn/n→a.e.λ-1 as n→∞.
Next, we proveλτn′/n→dN(0,1). Let ηn,i=λn-1/2[ρi-(λ(1-p1Xi-1))-1], Fi-1 be the σ- field generated by X1,…,Xi-1; by Hölder’s inequality, one has(19)Eηn,i2Iηn,i>ϵ∣Fi-1≤Eηn,i3∣Fi-12/3EIηn,i>ϵ∣Fi-11/3,where IA is the indicator function of A.
Note that the conditional distribution ofρn relative to ρ1,…,ρn-1 equals exponential distribution with parameter λ(1-p1Xn-1); one has(20)Eηn,i3∣Fi-1=λn-3/2Eρi-λ1-p1Xi-1-13∣Fi-1≤4n-3/2λ1/21-p1Xi-1-3≤4n-3/2λ1/21-p1-3a.s..According to Markov’s inequality,(21)EIηn,i>ϵ∣Fi-1≤ϵ-2Eηn,i2∣Fi-1=nϵ21-p1Xi-12-1≤nϵ21-p12-1a.s..By formulas (19)-(21) we have(22)∑i=1nEηn,i2Iηn,i>ϵ∣Fi-1≤Cn-1/3a.e.for some positive constant C. Similarly,(23)∑i=1nEηn,i2∣Fi-1=1n∑i=1n1-p1Xi-1-2→a.e.1.According to (22), (23), and Corollary 3.1 of [9], one has λτn′/n→dN(0,1). Note that (24)λτn′n-λnτnn-λ-1=n-1n∑i=1n1-p1Xi-1-1≤1-p12n-1→0,and the central limit theorem follows from Theorem 6 in page 39 of [10].


## 3. Large Deviation Principle

Let us begin with some lemmas to show the conditions of Gärtner-Ellis Theorem (see Appendix) are satisfied. DefineΛn(θ)=log⁡E[eθτn/n].Lemma 5.
For anyθ∈R, Λ(θ)≔limn→∞1/nΛn(nθ) exists and satisfies (25)Λθ=log⁡λ-log⁡λ-θ,θ<λ1-p1;+∞,θ≥λ1-p1.Particularly, 0∈DΛ≔{θ:Λ(θ)<∞}.Proof.
Note thatτn=ρ1+⋯+ρn, where the conditional distribution of ρn relative to ρ1,…,ρn-1 is the same as that relative to X0,X1,…,Xn-1 and equals exponential distribution with parameter λ(1-p1Xn-1) (see page 259 of [8]); one has(26)Eeθτn=Eeθ∑i=1nρi=EEeθ∑i=1nρi∣ρ1,…,ρn-1=Eeθ∑i=1n-1ρiEeθρn∣ρ1,…,ρn-1=Eeθ∑i=1n-1ρiEeθρn∣X0,…,Xn-1.If θ<λ(1-p1), we have(27)Eeθρn∣X0,…,Xn-1=λ1-p1Xn-1λ1-p1Xn-1-θ.If θ≤0, note that Xn-1≥n; by (27) one has(28)λ1-p1nλ1-p1n-θ≤Eeθρn∣X0,…,Xn-1≤λλ-θ.By (26), (28) and induction, we have (29)∏i=1nλ1-p1iλ1-p1i-θ≤Eeθτn≤λnλ-θn,which means Λ(θ)=log⁡λ-log⁡λ-θ.
If0<θ<λ(1-p1), note that Xn-1≥n; by (27) one has (30)λλ-θ≤Eeθρn∣X0,…,Xn-1≤λ1-p1nλ1-p1n-θ.We can get Λ(θ)=log⁡λ-log⁡(λ-θ) similarly.
Ifθ≥λ(1-p1), note that X0=Z0=1 and the conditional distribution of ρ1 relative to X0 equals exponential distribution with parameter λ(1-p1); one has (31)Eeθτn≥Eeθρ1=EEeθρ1∣X0=+∞,for all n≥1. Thus Λ(θ)=+∞.Lemma 6.
LetΛ∗ be the Fenchel-Legendre transform of Λ; then(32)Λ∗x=supθ∈Rθx-Λθ=λ1-p1x+log⁡p1,x≥λp1-1;λx-log⁡λx-1,λp1-1>x>0;+∞,x≤0.In addition, the set of exposed points (see Appendix) of Λ∗ is E⊃(0,+∞).Proof.
By Lemma5, if x≤0,(33)Λ∗x=supθ<λ1-p1θx-log⁡λ+log⁡λ-θ=limθ→-∞θx-log⁡λ+log⁡λ-θ=+∞.
Next, if(λp1)-1>x>0, then λ-x-1<λ(1-p1) and(34)Λ∗x=supθ<λ1-p1θx-log⁡λ+log⁡λ-θ=θ0=λ-x-1θ0x-log⁡λ+log⁡λ-θ0=λx-log⁡λx-1.
Finally, ifx≥(λp1)-1, then λ-x-1≥λ(1-p1). Note that(35)dθx-log⁡λ+log⁡λ-θdθ=x-1λ-θ>0for all θ<λ(1-p1); we have(36)Λ∗x=supθ<λ1-p1θx-log⁡λ+log⁡λ-θ=θ0=λ1-p1θ0x-log⁡λ+log⁡λ-θ0=λ1-p1x+log⁡p1.
Equation (32) follows from (33), (34), and (36).
In addition, for anyθ<λ, Λ′(θ)=(λ-θ)-1, so the range of Λ′(θ) for θ<λ is (0,+∞), which is a subset of E, the set of exposed points of Λ∗; see Lemma 2.3.9 of [7].Proof of Theorem2.
Note that for anyx≤0, Λ∗(x)=+∞ and the set of exposed points E⊃(0,+∞), then for any open set G, (37)infx∈G∩EΛ∗x=infx∈G∩0,+∞Λ∗x=infx∈GΛ∗x.Consequently, Theorem 2 follows from Lemma 5, Lemma 6, and the Gärtner-Ellis theorem (see Appendix).


## 4. Moderate Deviation Principle

In this section, we deal with the proof of Theorem3. Define(38)Λnθ=log⁡Eexpθ·τn-λ-1nan.Lemma 7.
For eachθ∈R, one has,(39)Δθ=limn→∞nan2Λnan2nθ=θ22λ2.Particularly, 0∈DΔ≔{θ:Δ(θ)<∞}. In addition, let Δ∗ be the Fenchel-Legendre transform of Δ; then(40)Δ∗x=λ2x22,and the set of exposed points of Δ∗ is F=R.Proof.
For anyθ∈R, we have(41)Λnan2nθ=log⁡Eexpan2nθ·τn-λ-1nan=log⁡Eexpanθnτn-∑i=1n1λ1-p1Xi-1·exp⁡anθn·∑i=1np1Xi-1λ1-p1Xi-1.For any θ≥0, note that Xi-1≥i; one has (42)0≤anθn·∑i=1np1Xi-1λ1-p1Xi-1≤anθn·∑i=1np1iλ1-p1i≤anθn1λ1-p12.Similarly, for θ<0, the above inequality should be reversed. Thus, by (41), if Δ(θ) exists, one has(43)Δθ=limn→∞nan2log⁡Eexpanθnτn-∑i=1n1λ1-p1Xi-1.Define βi=ρi-(λ(1-p1Xi-1))-1; then(44)τn-∑i=1n1λ1-p1Xi-1=∑i=1nβi.Note that the conditional distribution of ρn relative to ρ1,…,ρn-1 is the same as that relative to X0,X1,…,Xn-1 and equals exponential distribution with parameter λ(1-p1Xn-1); one has(45)Eexpanθn∑i=1nβi=Eexpanθn∑i=1n-1βiEexpanθnβn∣Fn-1,where Fi-1 is the σ- field generated by X1,…,Xi-1. After a simple calculation,(46)Eexpanθnβi∣Fi-1=exp-θannλ1-p1Xi-1·λ1-p1Xi-1λ1-p1Xi-1-anθ/nFor θ>0,(47)exp-θannλ1-p1i·λλ-anθ/n≤Eexpanθnβi∣Fi-1≤exp-θannλ·λ1-p1iλ1-p1i-anθ/n.According to (45)-(47) and induction, we obtain(48)Inθ≔∏i=1nexp-θannλ1-p1i·λλ-anθ/n≤Eexpanθn∑i=1nβi≤∏i=1nexp-θannλ·λ1-p1iλ1-p1i-anθ/n≕Hnθ.Similarly, for θ<0, the above inequality should be reversed. According to (18) and log(1+x)=x-x2/2+o(x2) as x→0, one has (49)log⁡Inθ=-θannλ∑i=1n11-p1i-nlog⁡1-anθnλ=-θanλ1+bn+θanλ+θ2an22nλ2+oan2n=-θanbnλ+θ2an22nλ2+oan2n,where bn belongs to [0,1/(n(1-p12))]. Hence,(50)limn→.∞nan2log⁡Inθ=θ22λ2,and, similarly,(51)limn→∞nan2log⁡Hnθ=θ22λ2.Equation (39) is followed by (43), (48), (50), and (51). Consequently, (52)Δ∗x=supθ∈Rθx-Δθ=supθ∈Rθx-θ22λ2=λ2x22.
In addition, for anyθ∈R, Δ′(θ)=θ/λ2; so the range of Δ′(θ) is R, which means F=R; see Lemma 2.3.9 of [7].Proof of Theorem2.
Note that the set of exposed points ofΔ∗ is R; Theorem 3 follows from Lemma 7 and the Gärtner-Ellis theorem.



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