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Mat. Tr., 2017, Volume 20, Number 2, Pages 139–192 (Mi mt327)  

This article is cited in 1 scientific paper (total in 1 paper)

On renewal matrices connected with branching processes with tails of distributions of different orders

V. A. Topchiĭ

Sobolev Institute of Mathematics, Omsk Division, Omsk, Russia

Abstract: We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property. We assume that the heaviest tails are regularly varying at the infinity with parameter $-\beta\in[-1, 0)$ and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

Key words: renewal matrix and its increment, asymptotic representations, regularly varying functions, Bellman–Harris critical processes.

Funding Agency Grant Number
Siberian Branch of Russian Academy of Sciences II.2П


DOI: https://doi.org/10.17377/mattrudy.2017.20.207

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English version:
Siberian Advances in Mathematics, 2018, 28:2, 115–153

Document Type: Article
UDC: 519.218.4
Received: 25.06.2016

Citation: V. A. Topchiǐ, “On renewal matrices connected with branching processes with tails of distributions of different orders”, Mat. Tr., 20:2 (2017), 139–192; Siberian Adv. Math., 28:2 (2018), 115–153

Citation in format AMSBIB
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\by V.~A.~Topchi{\v\i}
\paper On renewal matrices connected with branching processes with tails of distributions of different orders
\jour Mat. Tr.
\yr 2017
\vol 20
\issue 2
\pages 139--192
\mathnet{http://mi.mathnet.ru/mt327}
\crossref{https://doi.org/10.17377/mattrudy.2017.20.207}
\elib{http://elibrary.ru/item.asp?id=30558048}
\transl
\jour Siberian Adv. Math.
\yr 2018
\vol 28
\issue 2
\pages 115--153
\crossref{https://doi.org/10.3103/S1055134418020037}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85048013246}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Vatutin, V. A. Topchii, “Momenty mnogomernykh kriticheskikh protsessov Bellmana–Kharrisa s razlichnoi skorostyu ubyvaniya khvostov raspredelenii prodolzhitelnosti zhizni chastits”, Sib. elektron. matem. izv., 14 (2017), 1248–1264  mathnet  crossref
  • Математические труды Siberian Advances in Mathematics
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