Renewal theory for iterated perturbed random walks on a general branching process tree: early 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 $T:=(T_k{)}_{k\in \mathbb{N}}$ defined by $T_k:={\xi }_1+\dots +{\xi }_{k-1}+{\eta }_k$ for $k\in \mathbb{N}$. Consider a general branching process generated by $T$ and denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\le t$. 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 $N_j$, find the first-order asymptotics for the variance of $N_j$. Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq 0}$,properlynormalizedandcentered,as$t\to\infty$. The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.