Edgeworth expansions for profiles of lattice branching random walks

Consider a branching random walk on Z in discrete time. Denote by L-n(k) the number of particles at site k is an element of Z at time n is an element of N-0. By the profile of the branching random walk (at time n) we mean the function k bar right arrow L-n(k). We establish the following asymptotic expansion of L-n(k), as n -> infinity: e(-phi(0))n L-n (k) = e(-1/2x2/n(k))/root 2 pi phi ''(0)n Sigma(r)(j=0) Fj(x(n)(k))/n(j/2) + 0(n(-r+1/2)) a.s., where r is an element of N-0 is arbitrary,phi(beta) = log Sigma(k is an element of Z)e(beta k)EL(1)(k) is the cumulant generating function of the intensity of the branching random walk and x(n) (k) = k-phi '(0)n/root phi ''(0)n The expansion is valid uniformly in k is an element of Z with probability 1 and the F-j's are polynomials whose random coefficients can be expressed through the derivatives of phi and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers L-n(k(n)), where k(n) is an element of Z depends on n in some regular way. We also prove a.s. limit theorems for the mode arg max(k is an element of Z) L-n(k) and the height max(k is an element of Z) L-n(k) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter. phi ' (0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.