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

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 $Z_n$.

References in corpus (1)

• Branching random walk with a random environment in time