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

TL;DR

This paper studies the convergence properties and moderate deviation principles of a branching random walk in a time-dependent random environment, providing new insights into its probabilistic behavior and convergence rates.

Contribution

It establishes $L^p$ convergence rates, uniform convergence of a key martingale, and a moderate deviation principle for the branching random walk in a random environment.

Findings

Proves $L^p$ convergence rate of the martingale

Shows uniform convergence of the martingale $ ilde Z_n(t)/ ext{E}[ ilde Z_n(t)|\xi]$

Establishes a moderate deviation principle for the measures $Z_n$

Abstract

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 $Z_n$.