Minima in branching random walks

Abstract

Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{Mn-EMn > x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89-108], our results fully characterize the possible behavior of EMn when the branching random walk has bounded branching and step size. © Institute of Mathematical Statistics, 2009.