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

Abstract An iterated perturbed random walk is a sequence of point processes defined by the birth times individuals in subsequent generations general branching process provided that first generation are given walk. We prove counterparts classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s and key theorem) for number j th-generation with $\leq t$ , when $j,t\to\infty$ $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such form subset set intermediate generations.