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

Abstract

Let (ξ k , η k ) k∈N be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequenceConsider a general branching process generated by T and denote by N j (t) the number of the jth generation individuals with birth times ≤ t. We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for EN 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), . . . , N j (ut)) u≥0 , properly normalized and centered, as t → ∞. The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

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

“…At this point we stress that even though both Theorem 2.1 and Theorem 2.2 hold true for early generations, the assumptions of these theorems are too restrictive as far as early generations are concerned. We refer to the companion article [18] for a proper version of the elementary renewal theorem in early generations. Recall the standard notation x ∧ y = min (x, y) for x, y ∈ R.…”

“…While the present paper deals with some intermediate generations, the early generations, which admit a much simpler analysis, are treated in a separate paper [18]. The behavior of the iterated perturbed random walks in the early and intermediate generations is very different from that in the late generations.…”

“…This statement is confirmed by counterparts of the elementary renewal theorem (Theorems 2.1 and 2.2), the key renewal theorem (Theorem 2.3), and Blackwell's theorem (Corollary 2.1), which are our main results. As far as early generations are concerned, the claim is justified by results obtained in [18].…”