Chernoff bounds for branching random walks

TL;DR

This paper introduces a new framework for branching random walks (BRWs) and derives Chernoff-type bounds, filling a gap in concentration inequalities for BRWs and linking them to classical random walk bounds.

Contribution

It proposes a more general definition of BRWs and establishes Chernoff bounds for them, extending the theoretical understanding of their deviation probabilities.

Findings

Derived Chernoff-type bounds for BRWs

Connected BRW bounds to classical random walk bounds

Provided a new general framework for BRWs

Abstract

Concentration inequalities, which have proved very useful in a variety of fields, provide fairly tight bounds on large deviation probabilities while central limit theorem (CLT) describes the asymptotic distribution around the mean (at the $\sqrt{n}$ scale). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and i.i.d displacements, positions of individuals in $nth$ generation approach to Gaussian distribution -- central limit theorem. This conjecture was later proved by Stam (1966) and Kaplan \& Asmussen (1976). Refinements and extensions followed. However, to the best of our knowledge, there is no corresponding existing work on concentration inequalities for BRWs. In this note, we propose a new definition of BRW, providing a more general framework. Owing to this definition, a Chernoff-type (subgaussian) bound for BRWs follows directly from the Chernoff bound for random walk. The relation between RW (random walk) and BRW is discussed.