Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree*

Abstract

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, …, where (ξ1, η1), (ξ2, η2), … are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.