No abstract

Given a supercritical branching random walk {Z n } n≥0 on R, let Z n ([y, ∞)) be the number of particles located in [y, ∞) ⊂ R at generation n. Let m be the mean of the offspring law of {Z n } n≥0 and I(x) be the large deviation rate function of the underlying random walk ofconverges almost surely to log m − I(x) on the event of nonextinction as n → ∞, where x * is the speed of maximal position of the branching random walk. In this work, we investigate its upper deviation probabilities, in other words, the convergence rates ofas n → ∞, where x > 0 and a > (log m − I(x)) + . This paper is a counterpart work of the lower deviation probabilities [28] and also completes those results in [1] for the branching Brownian motion.