Exponential rate of almost sure convergence of intrinsic martingales in supercritical branching random walks

TL;DR

This paper establishes conditions under which the intrinsic martingale in supercritical branching random walks converges exponentially fast, generalizing classical results and providing new insights into renewal measures.

Contribution

It introduces sufficient conditions for exponential convergence of intrinsic martingales in supercritical branching random walks, extending known results to a broader class of processes.

Findings

Exponential convergence rates are established under new conditions.

Generalization of classical Galton-Watson results.

New auxiliary results on exponential renewal measures.

Abstract

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. The case of Galton-Watson processes is particularly included so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering. As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be interesting on its own and which correct, generalize and simplify some earlier works.