Laws of the iterated logarithm for iterated perturbed random walks

Let (ξ ₖ,η ₖ)k≥ 1 be independent identically distributed random vectors with arbitrarily dependent positive components and Tₖ:=ξ ₁+⋯ +ξ ₖ₋₁+η ₖ for k∈ N. The random sequence (Tₖ)k≥ 1 is called a (globally) perturbed random walk. Consider a general branching process generated by (Tₖ)k≥ 1 and let Yⱼ(t) denote the number of the jth generation individuals with birth times ≤ t. Assuming that Var0.1667emξ ₁∈ (0,∞ ) and allowing the distribution of η ₁ to be arbitrary, a law of the iterated logarithm (LIL) is proved for Yⱼ(t). In particular, an LIL for the counting process of (Tₖ)k≥ 1 is obtained. The latter result was previously established in the article by Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that Eη ₁ᵃ< ∞ for some a> 0. In this paper, it is shown that the aforementioned additional assumption is not needed.