Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations. (2nd March 2023)

• Record Type: Journal Article
• Authors: Iksanov, Alexander; Rashytov, Bohdan; Samoilenko, Igor
• Is Part Of: Journal of applied probability. Volume 60:Number 1(2023)
• Page Start: 45
• Page End: 67
• Publication Date: 2023-03-02
• Subjects: Functional limit theorem -- general branching process -- key renewal theorem -- perturbed random walk -- renewal theory -- strong law of large numbers; 60K05 -- 60J80 -- 60G05; 519.2
• DOI: 10.1017/jpr.2022.26
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract: Let $(\xi_k, \eta_k)_{k\in\mathbb{N}}$ 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 $T_k\, {:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ 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 $\leq t$ . 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 $(N_1(ut), \ldots, N_j(ut))_{u\geq0}$, properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.