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

John Fernley, Emmanuel Jacob

Abstract

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.

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

Abstract

We consider a supercritical Galton-Watson process whose offspring distribution has mean and is bounded by some . As well-known, the associated martingale converges a.s. to some nonnegative random variable . We provide a universal upper bound for the right tail of and , which is uniform in and in all offspring distributions with given and , namely: for some explicit constants . For a given offspring distribution, our upper bound decays exponentially as , which is actually suboptimal, but our bound is : it provides a single expression, which is - it does not require large - and valid simultaneously for all supercritical bounded offspring distributions.
Paper Structure (2 sections, 3 theorems, 19 equations, 1 figure)

This paper contains 2 sections, 3 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:tail_GWBall']}

Key Result

Theorem 1.1

There are universal constants $c_1, c_2>0$ such that for any $m>1$, any $d\in\{2,3,\ldots\}$ and any offspring distibution on $\{0,\ldots, d\}$ with mean $m$, for all $n\in \mathbb{N} \cup\{+\infty\}$ and $x\ge 0$, we have

Figures (1)

  • Figure 1: On the left we represented the first three generations of an infinite GW process which has $d=4$ and $m=1.75$. We also choose $a=0.5$, so an individual is fecund if it has at least $\lceil adm\rceil=4$ descendants in the first generation, or $\lceil adm^2\rceil=7$ decendants in the second generation, or $\lceil adm^3\rceil=11$ in the third, etc. Here only the ancestor $\circ$ is fecund, with $n_\circ=2$. This can be observed by revealing only the individuals drawn in black, with in particular $B_\circ=7$ individuals of generation 2 and $A_\circ=1$ individual of generation 1 with unrevealed offspring, all represented by squares. On the right, the corresponding multitype tree $T$, with $\circ$ as single internal vertex, as the other vertices with type 1 and 2 correspond to non-fecund individuals of the GW process.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:InternalNodes']}
  • proof : Proof of Lemma \ref{['lem:Bernstein']}