Chunmao Huang

Quansheng Liu

Huang, Chunmao

Liu, Quansheng

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1199–1202Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comProbability TheoryHarmonic moments and large deviations for a supercritical branchingprocess in a random environmentMoments harmoniques et grandes déviations pour un processus de branchementsur-critique dans un environnement aléatoireChunmao Huang a,b, Quansheng Liu a,ba Université de Bretagne-Sud, LMAM, campus de Tohannic, BP 573, 56017 Vannes, Franceb Université Européenne de Bretagne, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 29 July 2011Accepted 3 October 2011Available online 22 October 2011Presented by the Editorial BoardLet (Zn) be a supercritical branching process in a random environment ξ , and W bethe limit of the normalized population size Zn/E[Zn|ξ ]. We show large and moderatedeviation principles for the sequence log Zn (with appropriate normalization) by finding anequivalence of the moments of Zn and a criterion for the existence of harmonic momentsof W .© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoient (Zn) un processus de branchement sur-critique dans un environnement aléatoire ξ ,et W la limite de la population normalisée Zn/E[Zn|ξ ]. Nous montrons les principes degrande déviation et de déviation modérée pour la suite log Zn en trouvant un équivalentdes moments de Zn et un critère pour l’existence des moments harmoniques de W .© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. Introduction and resultsLet ξ = (ξ0, ξ1, ξ2, . . .) be a sequence of independent and identically distributed (i.i.d.) random variables taking values insome space Θ , whose realization determines a sequence of probability generating functionsfn(s) = fξn(s) =∞∑i=0pi(ξn)si, s ∈ [0,1], pi(ξn)  0,∞∑i=0pi(ξn) = 1.A branching process (Zn)n0 in the random environment ξ can be defined as follows:Z0 = 1, Zn+1 =Zn∑i=1Xn,i (n  0),where given the environment ξ , Xn,i (i = 1,2, . . .) are independent of each other and independent of Zn , and have the samedistribution determined by fn .E-mail addresses: sasamao02@gmail.com (C. Huang), quansheng.liu@univ-ubs.fr (Q. Liu).1631-073X/$ – see front matter © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2011.10.0011200 C. Huang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1199–1202Let (Γ,Pξ ) be the probability space under which the process is defined when the environment ξ is given. As usual, Pξis called quenched law. The total probability space can be formulated as the product space (Γ × ΘN,P), where P = Pξ ⊗ τ ,and τ is the law of the environment ξ . The total probability P is usually called annealed law. The quenched law Pξ can beconsidered to be the conditional probability of the annealed law P given ξ . The expectation with respect to Pξ (resp. P)will be denoted by Eξ (resp. E).For n  0, definemn = m(ξn) =∞∑i=0ipi(ξn), Π0 = 1 and Πn = m0 · · ·mn−1 if n  1.It is well known that the normalized population size Wn = Zn/Πn is a nonnegative martingale under Pξ (for each ξ ) withrespect to the filtration Fn = σ(ξ, Xk,i,0  k  n − 1, i = 1,2, . . .), so that the limit W = lim Wn exists almost sure (a.s.)with EW  1.We consider the supercritical case where E logm0 ∈ (0,∞) and E Z1m0 log+ Z1 < ∞. For simplicity, we write pi := pi(ξ0)and assume that p0 = 0 a.s., so that W > 0 and Zn → ∞ a.s.It is known that log Znn → E log m0 a.s. (cf. e.g. [7]). We are interested in the convergence rate of the correspondingdeviation probabilities. We shall show that log Zn and log Πn satisfy the same large and moderate deviation principlesunder suitable conditions.We first consider large deviations. Let Λ(t) = log Emt0 < ∞ for all t ∈ R and Λ∗(x) = supt∈R{xt − Λ(t)} be the Fenchel–Legendre transform of Λ. We introduce the following assumption:(H) There exist constants δ > 0 and A > A1 > 1 such that a.s. A1  m0 , m0(1 + δ)  A1+δ ,where m0 = ∑∞i=0 ipi(ξ0) and m0(1 + δ) = ∑∞i=0 i1+δ pi(ξ0).Notice that m0 = Eξ Z1, m0(1 + δ) = Eξ Z 1+δ1 and that the above condition implies that m0  A a.s.The theorem below shows that log Zn and log Πn satisfy the same large deviation principle:Theorem 1.1 (Large deviation principle). Assume (H). If EZ s1 < ∞ for all s > 1 and p1 = 0 a.s., then for any measurable subset B of R,− infx∈Bo Λ∗(x)  lim infn→∞1nlog P(log Znn∈ B) lim supn→∞1nlog P(log Znn∈ B) − infx∈B̄Λ∗(x),where Bo denotes the interior of B, and B̄ its closure.Corollary 1.2. Assume (H). If EZ s1 < ∞ for all s > 1 and p1 = 0 a.s., thenlimn→∞1nlog P(log Znn x)= −Λ∗(x) for x < E logm0,limn→∞1nlog P(log Znn x)= −Λ∗(x) for x > E logm0.This result was proved by Bansaye and Berestycki (2009, [1]) when (H) holds with δ = 1. As shown in [1], if P(p1 > 0) >0, the rate function for the large deviation is no longer Λ∗ .Notice that the Laplace transform of log Zn is Eet log Zn = EZtn . Therefore, Theorem 1.1 is a consequence of the Gärtner–Ellis theorem (see [2, p. 52, Exercise 2.3.20]) and Theorem 1.3 below:Theorem 1.3 (Moments of Zn). Let t ∈ R. Suppose that one of the following conditions is satisfied:(i) t ∈ (0,1] and Emt−10 Z1 log+ Z1 < ∞;(ii) t > 1 and EZt1 < ∞;(iii) t < 0, Ep1 < Emt0 , ‖p1‖∞ := esssup p1 < 1 and (H) holds.Then as n → ∞, EZtn/(Emt0)n → C(t) for some constant C(t) ∈ (0,∞).For t < 0, Theorem 1.3 is an extension of a result of Ney and Vidyashankar [6] on the Galton–Watson process.A key step in the proof of Theorem 1.3 is the study of the moments of W . For the moments of positive orders, Guivarc’hand Liu [3] showed that for p > 1, EW p ∈ (0,∞) if and only if E(Z1/m0)p < ∞ & Em1−p0 < 1. For the moments of negativeorders, we have the following criterion:C. Huang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1199–1202 1201Theorem 1.4 (Harmonic moments of W ). Assume (H). Then there exists a constant a > 0 such that EW −a < ∞. If additionally‖p1‖∞ < 1, then for a > 0, EW −a < ∞ if and only if Ep1ma0 < 1.Theorem 1.4 reveals that under certain conditions, the number a0 > 0 satisfying Ep1ma00 = 1 is the critical value for theexistence of the annealed harmonic moments EW −a(a > 0). Hambly [4] proved that under an assumption similar to (H),the number α0 := − E log p1E logm0 is the critical value for the a.s. existence of the quenched moments Eξ W −a(a > 0). By Jensen’sinequality, we see that a0  α0.We then show that log Zn and log Πn also satisfy the same moderate deviation principle.Theorem 1.5 (Moderate deviation principle). Assume (H) and let σ 2 = var(log m0) ∈ (0,∞). Let (an) be a sequence of positivenumbers satisfying ann → 0 and an√n → ∞. Then for any measurable subset B of R,− infx∈Box22σ 2 lim infn→∞na2nlog P(log Zn − nE log m0an∈ B) lim supn→∞na2nlog P(log Zn − nE log m0an∈ B) − infx∈B̄x22σ 2,where Bo denotes the interior of B, and B̄ its closure.2. Sketch of proofsTo prove the results about the harmonic moments of W , we consider the Laplace transform of W . Set φξ (t) = Eξ e−tWand φ(t) = Eφξ (t) for t > 0. The following lemma gives uniform upper bounds for φξ (t):Lemma 2.1. Assume (H). Then there exist constants β ∈ (0,1) and K > 0 such that a.s. φξ (t)  β for all t  1/K . If additionally‖p1‖∞ < 1, then for some constants a > 0 and C > 0, we have a.s. φξ (t)  Ct−a for all t  1/K .Proof. We obtain the upper bound β by an argument similar to [4, Proof of Lemma 3.1]. For the special case where ‖p1‖∞ <1, notice that φξ (t) satisfies the functional equationφξ (t) = f0(φT ξ (t/m0)), (1)where T nξ = (ξn, ξn+1, . . .) if ξ = (ξ0, ξ1, . . .) and n  0. By iteration, we have a.s.φξ (t)  φT nξ (t/Πn)n−1∏j=0(p1(ξ j) +(1 − p1(ξ j))φT nξ (t/Πn)).Since φT nξ (t/Πn)  β for t  An/K , it follows that a.s. φξ (t)  βαn for t  An/K , where α = ‖p1‖∞ + (1 −‖p1‖∞)β ∈ (0,1).For t  1/K , taking n0 = [ log(Kt)log A ] yields the upper bound Ct−a for suitable a > 0 and C > 0. Proof of Theorem 1.4. We first consider the special case where ‖p1‖∞ < 1. For the necessity, notice that W = 1m0∑Z1i=1 W(1)i ,where given ξ , (W (1)i )i1 are (conditionally) independent, each has the distribution Pξ (W(1)i ∈ ·) = PT ξ (W ∈ ·). SinceP(Z1  2) > 0, we haveEW −a > Ema0(W (1)1)−a1{Z1=1} = Ep1ma0EW −a.Thus Ep1ma0 < 1. For the sufficiency, the upper bound Ct−a in Lemma 2.1 implies that ∀ε > 0, there exists a constant tε > 0such that a.s. φξ (t)  ε for t  tε . Therefore, by (1), we have φξ (t)  (p1 + (1 − p1)ε)φT ξ (t/m0) for t  Atε . Taking theexpectation givesφ(t)  E(p1 + (1 − p1)ε)φ(tm0)= pεEφ( Ãεt),where pε = E(p1 + (1 − p1)ε) < 1 and Ãε is a positive random variable whose distribution is determined by Eg( Ãε) =1pεE(p1 + (1 − p1)ε)g( 1m0 ) for all bounded and measurable function g . Since Ep1ma0 < 1, we can take a1 > a and ε > 0small enough such that pεE Ã−a1ε < 1. Then by Lemma 3.2 of Liu (2001, [5]), φ(t) = O (t−a1 )(t → ∞). Therefore EW −a < ∞(cf. e.g. [5, Lemma 3.3]).Now consider the general case without the assumption ‖p1‖∞ < 1. Notice that by Lemma 2.1, a.s. φξ (t)  β for allt  tβ = 1/K . It suffices to repeat the proof of sufficiency above with β in place of ε. 1202 C. Huang, Q. Liu / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 1199–1202Proof of Theorem 1.3. Denote the distribution of ξ0 by τ0. Fix t ∈ R and define a new distribution τ̃0 as τ̃0(dx) =m(x)tτ0(dx)/Emt0, where m(x) = E[Z1|ξ0 = x] =∑∞i=0 ipi(x). Consider the new BPRE whose environment distribution isτ̃ = τ̃⊗N0 instead of τ = τ⊗N0 . The corresponding total probability and expectation are denoted by P̃ = Pξ ⊗ τ̃ and Ẽ. ThenEZtn/(Emt0)n = ẼW tn . We distinguish three cases: t ∈ (0,1), t > 1 and t < 0. For each case, under the given moment condi-tions, ẼW tn → ẼW t ∈ (0,∞). Proof of Theorem 1.5. LetΛn(t) = log E exp(log Zn − nE log m0ant)and Γn(t) = log E exp(logΠn − nE log m0ant).By the classic moderate deviation principle, na2nΓn(a2nn t) → 12 σ 2t2. Applying Jensen’s inequality and Hölder’s inequality, wecan prove that Λn(a2nn t)/Γn(a2nn t) → 1 for all t = 0, so that na2n Λn(a2nn t) → 12 σ 2t2. This together with the Gärtner–Ellis theorem[2, p. 52, Exercise 2.3.20] implies the desired result. AcknowledgementsThe work has been partially supported by the National Natural Science Foundation of China, Grant No. 11101039 andGrant No. 11171044.References[1] V. Bansaye, J. Berestycki, Large deviations for branching processes in random environment, Markov Process. Related Fields 15 (2009) 493–524.[2] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, Springer, New York, 1998.[3] Y. Guivarc’h, Q. Liu, Propriétés asymptotiques des processus de branchement en environnement aléatoire, C. R. Acad. Sci. Paris, Ser. I 332 (2001) 339–344.[4] B. Hambly, On the limit distribution of a supercritical branching process in a random environment, J. Appl. Probab. 29 (1992) 499–518.[5] Q. Liu, Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stochastic Process. Appl. 95 (2001) 83–107.[6] P.E. Ney, A.N. Vidyashankar, Harmonic moments and large deviation rates for supercritical branching process, Ann. Appl. Probab. 13 (2003) 475–489.[7] D. Tanny, Limit theorems for branching processes in a random environment, Ann. Probab. 5 (1977) 100–116.

Harmonic moments and large deviations for a supercritical branching process in a random environment

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2011.10.001.pdf

Harmonic moments and large deviations for a supercritical branching process in a random environment

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique