RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy MIAN: Year: Volume: Issue: Page: Find

 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 257–287 (Mi tm3480)

Critical Bellman–Harris branching processes with long-living particles

V. A. Vatutina, V. A. Topchiib

a Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (Omsk Branch), Novosibirsk, Russia

Abstract: A critical indecomposable two-type Bellman–Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter $\beta\in(0,1]$. It is shown that, contrary to the critical indecomposable Bellman–Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment $t$ is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment $t$ given that the population contains particles of the first type at this moment.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00139 This work was supported by the Russian Foundation for Basic Research, project no. 11-01-00139.

DOI: https://doi.org/10.1134/S0371968513030199

Full text: PDF file (340 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 243–272

Bibliographic databases:

Document Type: Article
UDC: 519.218.24

Citation: V. A. Vatutin, V. A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 257–287; Proc. Steklov Inst. Math., 282 (2013), 243–272

Citation in format AMSBIB
\Bibitem{VatTop13} \by V.~A.~Vatutin, V.~A.~Topchii \paper Critical Bellman--Harris branching processes with long-living particles \inbook Branching processes, random walks, and related problems \bookinfo Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences \serial Tr. Mat. Inst. Steklova \yr 2013 \vol 282 \pages 257--287 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3480} \crossref{https://doi.org/10.1134/S0371968513030199} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3308595} \elib{http://elibrary.ru/item.asp?id=20280559} \transl \jour Proc. Steklov Inst. Math. \yr 2013 \vol 282 \pages 243--272 \crossref{https://doi.org/10.1134/S0081543813060199} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000325961800019} \elib{http://elibrary.ru/item.asp?id=21883369} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84886024707} 

• http://mi.mathnet.ru/eng/tm3480
• https://doi.org/10.1134/S0371968513030199
• http://mi.mathnet.ru/eng/tm/v282/p257

 SHARE:

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. E. Vl. Bulinskaya, “Complete classification of catalytic branching processes”, Theory Probab. Appl., 59:4 (2015), 545–566
2. Valentin A. Topchiy, “Two-dimensional renewal theorems with weak moment conditions and critical Bellman – Harris branching processes”, Discrete Math. Appl., 26:1 (2016), 51–69
3. Vatutin V., Iksanov A., Topchii V., “a Two-Type Bellman-Harris Process Initiated By a Large Number of Particles”, Acta Appl. Math., 138:1 (2015), 279–312
4. 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
5. 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
•  Number of views: This page: 282 Full text: 13 References: 44