The maximum of a branching random walk with stretched exponential tails

Abstract

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X\ge t]\sim a\exp (-\lambda t^r)$ for some constants $a,\lambda >0$ where $r\in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $r\in (0,2/3)$ and $r\in (2/3,1)$, with yet different limits in the boundary case $r=2/3$.

30 pages