A Chernoff bound for branching random walk

Abstract

Concentration inequalities, which prove to be very useful in a variety of fields, provide fairly tight bounds for large deviation probability while central limit theorem (CLT) describes the asymptotic distribution around the mean (within scope of $\sqrt{n}$ order). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and iid displacement, population's positions in $nth$ generation approach to Gaussian distribution --- central limit theorem. This conjecture was latter proved by Stam (1966) and Kaplan and Asmussen (1976). Refinements and extensions followed. Yet little effort is known on large deviation probability for BRW. In this note, we suggest and verify a more general and probably more formal setting of BRW. Benefiting from this framework, a Chernoff bound for BRW is immediately obtained. The relation between RW (random walk) and BRW is addressed.Comment: 6 page