On large deviation probabilities for empirical distribution of branching random walks: Schr{ö}der case and B{ö}ttcher case

Abstract

Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $Z\_n(\cdot )$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \cite{B90} that for any interval $A\subset \mathbb{R}$, $\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}$ converges a.s. to $\nu (A)$, where $\nu$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\mathbb{P}\left(\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}-\nu (A)>\Delta \right),$$ for $\Delta \in (0,1-\nu (A))$, in both Schr{ö}der case and B{ö}ttcher case.

References in corpus (1)

• Large Deviations for the Empirical Distribution in the General Branching Random Walk

Cited by in corpus (1)

• Large deviations for the maximum of a branching random walk