Central limit theorems for a branching random walk with a random environment in time
نویسندگان
چکیده
منابع مشابه
Central limit theorems for a branching random walk with a random environment in time
We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be stationary and ergodic. For A ⊂ R , let Zn(A) be the number of particles of generation n located in A. We show central l...
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A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.
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Chen [Ann. Appl. Probab. 11 (2001), 1242–1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen’s results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen’s by relaxing the second moment condition used by Chen to a moment condition...
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We consider branching random walks in d-dimensional integer lattice with time-space i.i.d. offspring distributions. When d ≥ 3 and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase tran...
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For a supercritical branching process (Zn) in a stationary and ergodic environment ξ, we study the rate of convergence of the normalized population Wn = Zn/E[Zn|ξ] to its limitW∞: we show a central limit theorem forW∞−Wn with suitable normalization and derive a Berry-Esseen bound for the rate of convergence in the central limit theorem when the environment is independent and identically distrib...
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ژورنال
عنوان ژورنال: Acta Mathematica Scientia
سال: 2014
ISSN: 0252-9602
DOI: 10.1016/s0252-9602(14)60023-0