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

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 $t\ge 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\le t$. 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 $j=j(t)\to \mathrm{\infty }$ and $j(t)=o(t^{2\mathrm{/}3})$ as $t\to \mathrm{\infty }$. According to our terminology, such generations form a subset of the set of intermediate generations.