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Diskr. Mat., 2015, Volume 27, Issue 1, Pages 123–145 (Mi dm1320)  

This article is cited in 2 scientific papers (total in 2 papers)

Two-dimensional renewal theorems with weak moment conditions and critical Bellman – Harris branching processes

Valentin A. Topchiy

Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences

Abstract: We consider critical Bellman – Harris processes with two types of particles. The tail of the lifetime distribution of the first type particles decreases as $o(t^{-2})$, the tail of the lifetime distribution of the second type particles is regularly varying with the index in $(-1,0)$. Such processes are connected with the matrix renewal functions of special two-dimensional renewal processes. V. A. Vatutin and the author have used the asymptotics of these matrix renewal functions and their first and second order increments in the proofs of several limit theorems for branching processes. Here we investigate the properties of such matrix renewal functions under significantly weaker conditions on the lifetime distributions and apply the results to the description of the asymptotics of several moments of branching processes and of their increments.
This work was supported by RFBR project 14-01-00318.

Keywords: two-dimensional renewal process, matrix renewal function, critical two-type Bellman – Harris processes, infinite mean lifetime, asymptotics, regularly varying functions

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00318


DOI: https://doi.org/10.4213/dm1320

Full text: PDF file (559 kB)
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English version:
Discrete Mathematics and Applications, 2016, 26:1, 51–69

Bibliographic databases:

Document Type: Article
UDC: 519.218.24
Received: 17.12.2014

Citation: Valentin A. Topchiy, “Two-dimensional renewal theorems with weak moment conditions and critical Bellman – Harris branching processes”, Diskr. Mat., 27:1 (2015), 123–145; Discrete Math. Appl., 26:1 (2016), 51–69

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. A. Topchiǐ, “On renewal matrices connected with branching processes with tails of distributions of different orders”, Siberian Adv. Math., 28:2 (2018), 115–153  mathnet  crossref  crossref  elib
    2. 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
  • Дискретная математика
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