Renewal theory for iterated perturbed random walks on a general branching process tree: early generations
| Autor: | Iksanov, Alexander, Rashytov, Bohdan, Samoilenko, Igor |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: submitted for publication, 19 pages |
| Databáze: | arXiv |
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