The maximum of a branching random walk with stretched exponential tails

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 $r\in (0,2\mathrm{/}3)$ and $ r \in (2/3, 1)$,withyetdifferentlimitsintheboundarycase$r = 2/3$.