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

This article is cited in 5 scientific papers (total in 5 papers)

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

Bibliographic databases:

Document Type: Article
UDC: 519.218.24
Received in November 2012

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
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\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
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\crossref{https://doi.org/10.1134/S0371968513030199}
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    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  elib
    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  mathnet  crossref
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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