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)_kN be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence T :\!= (T_k)_kN defined by T_k :\!= ξ_1+…+ξ_k-1+η_k for kN . 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 EN_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.