On large deviation probabilities for empirical distribution of supercritical branching random walks with unbounded displacements

Given a branching random walk on R started from the origin, where the tail of the branching law decays at least exponentially fast and the offspring number is at least one, let Z(n)(.) be the counting measure which counts the number of individuals at the n-th generation located in a given set. Under some mild conditions, it is known (Biggins in Stoch. Process. Appl. 34:255-274, 1990) that for any interval A subset of R, Z(n)(root nA)/Z(n)(R) converges a.s. to nu(A), where nu is the standard Gaussian measure. In this work, we investigate the convergence rates of P(Z(n)(root nA)/Z(n)(R) - nu(A) > Delta), for Delta is an element of (0, 1 - nu(A)). We consider both the Schroder case, where the offspring number could be one, and the Bottcher case, where the offspring number is at least two.