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

2021/05/06 by Alexander Iksanov, Iksanov, Alexander, Bohdan Rashytov +3 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2105.02846

openalex publication_date 2021/05/06 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28

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 sequence T:=(T_k)_k∈ℕ defined by T_k:=ξ₁+…+ξ_k-1+η_k for k∈ℕ. Consider 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 (N₁(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.

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