A universal right tail upper bound for supercritical Galton–Watson processes with bounded offspring

Abstract

We consider a supercritical Galton–Watson process Zn whose offspring distribution has mean m>1 and is bounded by some d∈{2,3,…}. As is well-known, the associated martingale Wn=Zn/mn converges a.s. to some nonnegative random variable W∞. We provide a universal upper bound for the right tail of W∞ and Wn, which is uniform in n and in all offspring distributions with given m and d, namely: P(Wn≥x)≤c1exp−c2m−1mxd,∀n∈N∪{+∞},∀x≥0,for some explicit constants c1,c2>0. For a given offspring distribution, our upper bound decays exponentially as x→∞, which is actually suboptimal, but our bound is universal: it provides a single effective expression – which does not require large x – and is valid simultaneously for all supercritical bounded offspring distributions.