Edgeworth expansions for profiles of lattice branching random walks

Abstract

Abstract. Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles at site k ∈ Z at time n ∈ N 0 . By the profile of the branching random walk (at time n) we mean the function k → Ln(k). We establish the following asymptotic expansion of Ln(k), as n → ∞:where r ∈ N 0 is arbitrary, ϕ(β) = log k∈Z e βk EL 1 (k) is the cumulant generating function of the intensity of the branching random walk andThe expansion is valid uniformly in k ∈ Z with probability 1 and the F j 's are polynomials whose random coefficients can be expressed through the derivatives of ϕ 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 Ln(kn), where kn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode arg max k∈Z Ln(k) and the height max k∈Z Ln(k) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter ϕ (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.

General Edgeworth expansions with applications to profiles of random trees

“…Remark 3.13. An Edgeworth expansion for the profile of a many-split BRW was obtained in [21]. Theorem 2.1 from the present paper can be applied to the manysplit BRW, but both the representation of the terms of the expansion and the proof given in [21] differ from ours.…”

“…Assumption B4 can be imposed without loss of generality: if ν(a * Z) = 1 for some a * ≥ 2 (chosen to be maximal with this property), then we can rescale the jumps by a * and work equivalently with the one-split BRW governed by the intensity measure ν * , where ν * ({k}) = ν({k/a * }). Note that this contrasts the situation in the many-split BRW [21] and in Section 2.4, where it was necessary to exclude measures ν concentrated on lattices of the form aZ + b.…”

“…Applying Theorem 2.1 with arbitrary β one can obtain asymptotic expansions for large deviation probabilities; see [21]. Note, however, that the moment condition which we imposed on Z 1 can be relaxed; see Theorem 13 in Petrov [34, Ch.…”

“…Remark 3.1. The difference between this model and the usual discrete-time, many splits BRW (for which the Edgeworth expansion was obtained in [21]) is that in the one-split BRW, only one particle (chosen uniformly at random) is allowed to split, whereas in the many-split BRW all particles split at the same time. We shall see that there are many differences between these models.…”

“…In the next theorem we describe the asymptotic behavior of the "occupation numbers" L n (k n ), where (k n ) n∈N is a deterministic sequence with sufficiently regular behavior. These quantities were the main object of study in Fuchs et al [20]; see also [7,9] and (for results on lattice branching random walks) [10,21,26]. It is known [9,20] (and not difficult to deduce from (4)) that if k n = 2e β log n + α 2e β log n + o( √ log n) for some β ∈ (β − , β + ) and α ∈ R, then…”