Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

TL;DR

This paper extends classical renewal theory results to the intermediate generations of a general branching process with perturbed random walk birth times, under specific growth conditions for the generation index.

Contribution

It develops renewal-theoretic results for intermediate generations in branching processes with perturbed random walks, a novel extension of classical renewal theory.

Findings

Renewal theorems are valid for intermediate generations with growth rate j(t)=o(t^{2/3})

Classical renewal results are adapted to complex branching process structures

The approach bridges renewal theory and branching processes with dependent perturbations

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 $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=\xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $j\in\mathbb{N}$ and $t\geq 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)\to\infty$ and $j(t)=o(t^{2/3})$ as $t\to\infty$. According to our terminology, such generations form a subset of the set of intermediate generations.