Large and moderate deviations for a -valued branching random walk with a random environment in time

Abstract

We consider a R d -valued discrete time branching random walk with a stationary and ergodic environment in time. Let Zn be the counting measure of particles of generation n. With the help of the uniform convergence of martingales and multifractal analysis, we show a large deviation result associated to the measures Zn as well as the corresponding moderate deviations.AMS 2010 subject classifications. 60J80, 60K37, 60F10.

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

“…This model of BRWre has been introduced by Biggins and Kyprianou in [8]. Huang and Liu [18] proved that the maximal displacement in the process grows at ballistic speed almost surely, and obtained central limit theorems and large deviations principles for the counting measure of the process. Additional results on the behavior of the so-called bulk of the BRWre may be found in [13,17], where the growth rate in the bulk of the process is studied, and a central limit theorem and a large deviation principle are derived.…”

Limit theorems on counting measures for a branching random walk with immigration in a random environment

“…[6] gave the sufficient and necessary conditions for existence of quenched moments as well as weighted moments of BRWRE. [3] established large and moderate deviations of BRWRE in high-dimensional real space. As an extension of BRWRE, a branching random walk with immigration in random environment in time (BRWIRE) also has attracted extensive attention.…”

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

“…We mention that some central limit theorems of different nature on the intrinsic martingale (Biggins martingale) in branching random walks have been established by Grübel and Kabluchko [16], Iksanov and Kabluchko [22], and some large deviations type results for branching random walks have been discussed by [21,36]. For other aspects on branching random walks, see for example, [1,18,20,23,30,34,37,38,40] and references therein.…”