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

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 Cramer'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(-circle minus(vertical bar t vertical bar r)) for some r is an element of (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 (M) over tilde (n), and we show a large deviation principle for the laws of n(-1/r) (M) over tilde (n) as well.