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