# Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random... We consider a branching random walk on $${\mathbb {R}}$$ R with a stationary and ergodic environment $$\xi =(\xi _n)$$ ξ = ( ξ n ) indexed by time $$n\in {\mathbb {N}}$$ n ∈ N . Let $$Z_n$$ Z n be the counting measure of particles of generation n and $$\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)$$ Z ~ n ( t ) = ∫ e t x Z n ( d x ) be its Laplace transform. We show the $$L^p$$ L p convergence rate and the uniform convergence of the martingale $$\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]$$ Z ~ n ( t ) / E [ Z ~ n ( t ) | ξ ] , and establish a moderate deviation principle for the measures $$Z_n$$ Z n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

# Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

, Volume 30 (3) – Jan 23, 2016
35 pages

/lp/springer_journal/convergence-of-martingale-and-moderate-deviations-for-a-branching-xVPsrFEpqu
Publisher
Springer US
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-016-0668-6
Publisher site
See Article on Publisher Site

### Abstract

We consider a branching random walk on $${\mathbb {R}}$$ R with a stationary and ergodic environment $$\xi =(\xi _n)$$ ξ = ( ξ n ) indexed by time $$n\in {\mathbb {N}}$$ n ∈ N . Let $$Z_n$$ Z n be the counting measure of particles of generation n and $$\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)$$ Z ~ n ( t ) = ∫ e t x Z n ( d x ) be its Laplace transform. We show the $$L^p$$ L p convergence rate and the uniform convergence of the martingale $$\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]$$ Z ~ n ( t ) / E [ Z ~ n ( t ) | ξ ] , and establish a moderate deviation principle for the measures $$Z_n$$ Z n .

### Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Jan 23, 2016

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