The maximum of a branching random walk with stretched exponential tails

We study a 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 P[X > t] ∼ a exp{−λt^r } for some constants a, λ > 0 where r ∈ (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 a growth condition dichotomy. The limit theorems reveal interesting differences betweens the two regimes r ∈ (0, 2/3) and r ∈ (2/3, 1), with yet different limits in the boundary case r = 2/3.