Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations
TLDR
In this paper , the authors proved counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem, and the key renewal theorem) for the number of j th-generation individuals with birth times, when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}big)$ .Abstract:
Abstract An iterated perturbed random walk is a sequence of point processes defined by the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. We prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell’s theorem, and the key renewal theorem) for the number of j th-generation individuals with birth times $\leq t$ , when $j,t\to\infty$ and $j(t)={\textrm{o}}\big(t^{2/3}\big)$ . According to our terminology, such generations form a subset of the set of intermediate generations. read more
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Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations
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References
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Journal ArticleDOI
Uniform Convergence of Martingales in the Branching Random Walk
TL;DR: In this article, the convergence of the martingale limit of a supercritical branching random walk was shown to be uniform in √ √ n, where n is the number of generations.
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Chernoff's theorem in the branching random walk
Abstract: If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗ (na))1/n , is established for Fn∗ . If Z (n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z (n).
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Growth Rates in the Branching Random Walk
TL;DR: In this article, the authors considered the branching random walk on the real line, where an initial ancestor is at the origin and has children, the first generation, and these have positions which form a point process on the line.