An $L_p$ Bound for the Remainder in a Combinatorial Central Limit Theorem

Abstract. Let (Z n ) be a supercritical branching process in a random environment ξ = (ξ n ). We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for log Z n under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W = lim n→∞ W n , where W n = Z n /E ξ Z n is the normalized population size. Introduction and main resultsA branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time. It was introduced for the first time in Smith and Wilkinson [24] to modelize the growth of a population submitted to an environment. For background concepts and basic results concerning a BPRE we refer to Athreya and Karlin [4,3]. In the critical and subcritical regime the process goes out and the research interest is concentrated mostly on the survival probability and conditional limit theorems for the branching process, see e.g. [21]. In this article, we complete on these results by giving the Berry-Esseen bound and asymptotics of large deviations of Cramér's type for a supercritical BPRE.A BPRE can be described as follows. The random environment is represented by a sequence ξ = (ξ 0 , ξ 1 , ...) of independent and identically distributed random variables (i.i.d. r.v.'s); each realization of ξ n corresponds to a probability law {p i (ξ n ) : i ∈ N} on N = {0, 1, 2, . . . }, whose probability generating function is

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“…We will use the following result, where we use the notation · for the infinity norm. [16], Lemma 1.1) given by…”

Section: 1mentioning

confidence: 99%

Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

Grama

Liu

Miqueu

2017

Stochastic Processes and their Applications

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“…From now, we use the following system giving by Ho and Chen (Ho, S.T., 1978) and Neammanee and Rattanawong (Neammanee, K. & Rattanawong, P., 2008). …”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…In the case of d = 2, we observe that this is a special case of the combinatorial central limit theorem (For more detail see Von Bahr (Von Bahr, B., 1976), Ho and Chen (Ho, S.T., 1978)). Under the finiteness of absolute third moment, Neammanee and Suntornchost (Neammanee, K. & Suntornchost, J., 2005) gave the uniform rate of convergence and obtained the rate 198 √ n .…”

Section: Introductionmentioning

confidence: 82%

Uniform Bound on Normal Approximation of Latin Hypercube Sampling

Rerkruthairat¹,

Neammanee²

2011

JMR

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“…In some cases, the probability thatW actually belongs to this set can be directly estimated. This is the so called concentration inequality approach; see [8]- [10] and [18]. Since |f | is large only on a small set, | E f (Z)| can be controlled if Z has a bounded density, in particular if Z is normal.…”

Section: )mentioning

confidence: 99%