Geodesic
Moments of multitype critical Bellman--Harris processes in which tails of life-length distributions of particles have different orders
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1248-1264.

Voir la notice de l'article provenant de la source Math-Net.Ru

A multitype indecomposable, nonperiodic, critical Bellman–Harris branching process is considered. It is assumed that the types of the process may be splitted into two classes. A particle whose type belongs to the first class has a finite expected life-length, while the expected life-length of a particle whose type belongs to the second class is infinite. Assuming that the tail of the life-length distribution of a particle with type from the second class is regularly varying at infinity with parameter depending on the type, we investigate the asymptotic behavior of the first and second moments for the number of particles of all types as well as the increments of the first moments. Our proofs are based on the asymptotic properties of some renewal matrices defined in terms of certain characteristics of the initial Bellman–Harris branching process.
Keywords: multitype critical Bellman–Harris branching process, limit theorems, regularly varying functions, asymptotics of moments. 
@article{SEMR_2017_14_a50,
     author = {V. A. Vatutin and V. A. Topchii},
     title = {Moments of multitype critical {Bellman--Harris} processes in which tails of life-length distributions of particles have different orders},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1248--1264},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a50/}
}
TY  - JOUR
AU  - V. A. Vatutin
AU  - V. A. Topchii
TI  - Moments of multitype critical Bellman--Harris processes in which tails of life-length distributions of particles have different orders
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2017
SP  - 1248
EP  - 1264
VL  - 14
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2017_14_a50/
LA  - ru
ID  - SEMR_2017_14_a50
ER  - 
%0 Journal Article
%A V. A. Vatutin
%A V. A. Topchii
%T Moments of multitype critical Bellman--Harris processes in which tails of life-length distributions of particles have different orders
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2017
%P 1248-1264
%V 14
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2017_14_a50/
%G ru
%F SEMR_2017_14_a50
V. A. Vatutin; V. A. Topchii. Moments of multitype critical Bellman--Harris processes in which tails of life-length distributions of particles have different orders. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1248-1264. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a50/

[1] Verzweigungsprozesse, Akademie Verlag, Berlin, 1974 | MR | MR | Zbl | Zbl

[2] K.B. Athreya, P.E. Ney, Branching processes, Springer, Berlin, 1972 | MR | Zbl

[3] M. I. Goldstein, “Critical age-dependent branching processes: Single and multitype”, Z. Wahrscheinlichkeitstheor. Verw. Geb., 17 (1971), 74–88 | DOI | MR | Zbl

[4] V.A. Vatutin, “Discrete limit distributions of the number of particles in a Bellman–Harris branching process with several types of particles”, Theory Probab. Appl., 24:3 (1979), 509–520 | DOI | MR

[5] V.A. Vatutin, “On a class of the critical multitype Bellman–Harris branching processes”, Theory Probab. Appl., 25:4 (1981), 760–771 | DOI | MR | Zbl

[6] V.A. Vatutin, V.A. Topchii, “Critical Bellman–Harris branching processes with long-living particles”, Proceedings of the Steklov Institute of Mathematics, 282:1 (2013), 243–272 | DOI | MR | Zbl

[7] V. Vatutin, A. Iksanov, V. Topchii, “A Two-Type Bellman–Harris process initiated by a large number of particles”, Acta Applicandae Mathematicae, 138:1 (2015), 279–312 | DOI | MR | Zbl

[8] V.A. Vatutin, V.A. Topchii, “Catalytic branching random walks in $\mathbb{Z}^d$ with branching at the origin”, Siberian Advances in Mathematics, 23:2 (2013), 123–153 | DOI | MR

[9] V.A. Vatutin, V.A. Topchii, “A key renewal theorem for heavy tail distributions with $\beta \in (0,0.5]$”, Theory Probab. Appl., 58:2 (2014), 333–342 | DOI | MR | Zbl

[10] V.A. Topchii, “Two-dimensional renewal theorems with weak moment conditions and critical Bellman–Harris branching processes”, Discrete Mathematics and Applications, 26:1 (2016), 51–69 | MR

[11] V.A. Topchii, “Renewal matrix connected with branching process having life-length tails of different order”, Mathematical Proceedings (Matematicheskie trudy), 20:2 (2017), 139–192 (Russian)

[12] F.R. Gantmacher, The Theory of Matrices, Nauka, M., 1967 (Russian) | MR

[13] W. Feller, An introduction to probability theory and its applications, v. II, Wiley, New York, 1966 | MR | Zbl

[14] V.A. Il'in, E.G. Poznyak, Linear Algebra, Nauka-Fizmatlit, M., 1999 (Russian) | MR

[15] G.M. Fikhtengol'ts, A Course in Differential and Integral Calculus, v. I, Nauka, M., 1966 (Russian)