Laws of the iterated logarithm for iterated perturbed random walks

Abstract

Let $({\xi }_k,{\eta }_k{)}_{k\ge 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:={\xi }_1+\dots +{\xi }_{k-1}+{\eta }_k$for $k\in \mathbb{N}$. We call the random sequence {T_k, k=1,2...} a (globally) perturbed random walk. Consider a general branching process generated by {T_k, k=1,2...} and let Y_j(t) denote the number of the jth generation individuals with birth times less or equal t. Assuming that Var ξ_1 is finite and allowing the distribution of η_1 to be arbitrary, we prove a law of the iterated logarithm (LIL) for Y_j(t). In particular, a LIL for the counting process of {T_k, k=1,2...} is obtained. The latter result was previously established in the article Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that Eη^a is finite for some positive a. In this paper, we show that the aforementioned additional assumption is not needed.