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

2023/07/14 by Fernley, John, Jacob, Emmanuel
#60J80 (Primary) 60F10 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2307.07241

Abstract

We consider a supercritical Galton-Watson process Z_n whose offspring distribution has mean m>1 and is bounded by some d∈ \2,3,…\. As well-known, the associated martingale W_n=Z_n/mⁿ converges a.s. to some nonnegative random variable W_∞. We provide a universal upper bound for the right tail of W_∞ and W_n, which is uniform in n and in all offspring distributions with given m and d, namely: P(W_n≥ x)≤ c₁ exp\-c₂ \frac m-1m \frac x d\, ∀ n∈ \mathbb 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 is nonasymptotic - it does not require x large - and valid simultaneously for all supercritical bounded offspring distributions.

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