Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations. (2nd March 2023) - Iksanov, Alexander; Rashytov, Bohdan; Samoilenko, Igor
Abstract: Let $(ξ_k, η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\, {:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\, {:\!=}\, ξ_1+…+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the j th generation individuals with birth times $≤ t$ . We treat early generations, that is, fixed generations j which do not depend on t . In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$, and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut), …, N_j(ut))_{u≥0}$, properly normalized and centered, as $t→∞$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.