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 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^−Θ(|t|r) for some r ∈ (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/rM_n . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted byM_n, and we show a large deviation principle for the laws of n^−1/r_Mn as well.