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

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  18 February 2022

Vladyslav Bohun ,
Alexander Iksanov Open the ORCID record for Alexander Iksanov [Opens in a new window] ,
Alexander Marynych Open the ORCID record for Alexander Marynych [Opens in a new window]  and
Bohdan Rashytov
Vladyslav Bohun*
Affiliation:
Taras Shevchenko National University of Kyiv
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv
Alexander Marynych*
Affiliation:
Taras Shevchenko National University of Kyiv
Bohdan Rashytov*
Affiliation:
Taras Shevchenko National University of Kyiv
*
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 64/13, Kyiv, 01601 Ukraine.
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 64/13, Kyiv, 01601 Ukraine.
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 64/13, Kyiv, 01601 Ukraine.
*Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Volodymyrska 64/13, Kyiv, 01601 Ukraine.
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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 jth-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.

Keywords

Convolutionkey renewal theoremperturbed random walkrenewal theory

MSC classification

Primary: 60K05: Renewal theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G05: Foundations of stochastic processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 421 - 446
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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