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 $mathbb{P}[X geq t] sim a exp(-lambda t^r)$ 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 $r = 2/3$.