Large deviations for the maximum of a branching random walk with stretched exponential tails

Abstract

We prove large deviation results for the position of the rightmost particle, denoted by $M_{n}$, in a one-dimensional branching random walk in a case when Cramér’s condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e. both tails decay as $e^{-\Theta (|t|^{r})}$ for some $r\in ( 0,1)$. It is known that in this case, $M_{n}$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_{n}$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde{M} _{n}$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde{M} _{n}$ as well.