Convergence of martingale and moderate deviations for a branching random walk with a random environment in time

We consider a branching random walk on \(\mathbb{R}\) with a stationary and ergodic environment \(\xi=(\xi_n)\) indexed by time \(n\in\mathbb{N}\). Let \(Z_n\) be the counting measure of particles of generation \(n\) and \(\tilde Z_n(t)=\int e^{tx}Z_n(dx)\) be its Laplace transform. We show the \(L^...

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Bibliographic Details
Published inarXiv.org
Main Authors Wang, Xiaoqiang, Huang, Chunmao
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.04.2015
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Summary:We consider a branching random walk on \(\mathbb{R}\) with a stationary and ergodic environment \(\xi=(\xi_n)\) indexed by time \(n\in\mathbb{N}\). Let \(Z_n\) be the counting measure of particles of generation \(n\) and \(\tilde Z_n(t)=\int e^{tx}Z_n(dx)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde Z_n(t)/\mathbb E[\tilde Z_n(t)|\xi]\), and establish a moderate deviation principle for the measures \(Z_n\).
ISSN:2331-8422