Open Access
February 2018 Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time
ZhiQiang Gao, Quansheng Liu
Bernoulli 24(1): 772-800 (February 2018). DOI: 10.3150/16-BEJ895

Abstract

Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.

Citation

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ZhiQiang Gao. Quansheng Liu. "Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time." Bernoulli 24 (1) 772 - 800, February 2018. https://doi.org/10.3150/16-BEJ895

Information

Received: 1 September 2015; Revised: 1 May 2016; Published: February 2018
First available in Project Euclid: 27 July 2017

zbMATH: 06778347
MathSciNet: MR3706776
Digital Object Identifier: 10.3150/16-BEJ895

Keywords: asymptotic expansion , branching random walks , central limit theorem , convergence rate , Martingale approximation , random environment

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 1 • February 2018
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