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 $e^{-|t{|}^r}$ 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.

arXiv admin note: text overlap with arXiv:1802.03960