Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Zhiqiang Gao 1 Quansheng Liu 2
2 LMBA_UBS
LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
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.
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Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2018, 24 (1), pp.772-800. 〈10.3150/16-BEJ895〉
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Soumis le : lundi 4 janvier 2016 - 07:44:33
Dernière modification le : mardi 13 février 2018 - 15:24:07

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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, Bernoulli Society for Mathematical Statistics and Probability, 2018, 24 (1), pp.772-800. 〈10.3150/16-BEJ895〉. 〈hal-01244736v2〉

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