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

TL;DR

This paper establishes large deviation principles for the maximum position in a one-dimensional branching random walk with stretched exponential tail distributions, extending understanding beyond classical Cramér's condition.

Contribution

It provides the first large deviation results for the maximum in branching random walks with stretched exponential tails, where classical assumptions do not hold.

Findings

Large deviation principle for n^{-1/r} M_n established

Comparison with maximum of independent random walks used in proofs

Results extend large deviation theory to non-Cramér tail distributions

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 Cram\'er'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.