Chernoff bounds for branching random walks
Changqing Liu
TL;DR
The paper introduces a generalized BRW framework that treats branching random walks as a random walk augmented by a displacement point process, enabling a direct link to concentration phenomena. Under a position-independent branching factor and assuming a Chernoff bound for the underlying displacement walk, it proves a Chernoff-type tail bound for BRW leaves, showing subgaussian decay of deviations of the quantiles from their mean. The work formalizes the spine decomposition and connects BRW tails to RW deviations via quantities like $Q_n(\alpha)$ and $p_{(\tau)}(\lambda)$, with $\mathbb{E}[p_{(\tau)}(\lambda)] = \Pr(S_n - na \ge \lambda)$. It further establishes that the bound holds in both subcritical and supercritical regimes and discusses broader implications and extensions to more general concentration inequalities beyond the stated assumptions.
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.