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

We consider a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$ and is bounded by some $d\in \{2,3,\ldots\}$. As well-known, the associated martingale $W_n=Z_n/m^n$ converges a.s. to some nonnegative random variable $W_\infty$. We provide a universal upper bound for the right tail of $W_\infty$ and $W_n$, which is uniform in $n$ and in all offspring distributions with given $m$ and $d$, namely: $$P(W_n\ge x)\le c_1 \exp\left\{-c_2 \frac {m-1}m \frac x d\right\}, \quad \forall n\in \mathbb N \cup \{+\infty\}, \forall x\ge 0,$$ for some explicit constants $c_1,c_2>0$. For a given offspring distribution, our upper bound decays exponentially as $x\to \infty$, which is actually suboptimal, but our bound is $\textit{universal}$: it provides a single $\textit{effective}$ expression, which is $\textit{nonasymptotic}$ - it does not require $x$ large - and valid simultaneously for all supercritical bounded offspring distributions.