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 \(T_k:=\xi_1+\ldots+\xi_{k-1}+\eta_k\) 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 \(\leq t\). 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 \(N_j\), find the first-order asymptotics for the variance of \(N_j\). Also, we prove a functional limit theorem for the vector-valued process \((N_1(ut),\ldots, N_j(ut))_{u\geq 0}\), properly normalized and centered, as \(t\to\infty\). The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.