Critical Bellman--Harris branching processes with long-living particles

V. A. Vatutin ; V. A. Topchii

Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 257-287.

Résumé

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.

Bibliographie

[1] Albeverio S., Bogachev L.V., “Branching random walk in a catalytic medium. I: Basic equations”, Positivity, 4 (2000), 41–100 | DOI | MR | Zbl

[2] Albeverio S., Bogachev L.V., Yarovaya E.B., “Asymptotics of branching symmetric random walk on the lattice with a single source”, C. r. Acad. sci. Paris. Sér. 1: Math., 326 (1998), 975–980 | DOI | MR | Zbl

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

[4] Bogachev L.V., Yarovaya E.B., “Momentnyi analiz vetvyaschegosya sluchainogo bluzhdaniya na reshetke s odnim istochnikom”, DAN, 363:4 (1998), 439–442 | MR | Zbl

[5] Bulinskaya E.Vl., “Kataliticheskoe vetvyascheesya sluchainoe bluzhdanie po dvumernoi reshetke”, Teoriya veroyatn. i ee primen., 55:1 (2010), 142–148 | DOI | MR

[6] Bulinskaya E.Vl., “Limit distributions arising in branching random walks on integer lattices”, Lith. Math. J., 51:3 (2011), 310–321 | DOI | MR | Zbl

[7] Bulinskaya E.Vl., “Predelnye raspredeleniya chislennostei chastits v vetvyaschemsya sluchainom bluzhdanii”, Mat. zametki, 90:6 (2011), 845–859 | DOI | MR | Zbl

[8] Erickson K.B., “Strong renewal theorems with infinite mean”, Trans. Amer. Math. Soc., 151 (1970), 263–291 | DOI | MR | Zbl

[9] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. 2, Mir, M., 1984

[10] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | MR | Zbl

[11] Sevastyanov B.A., Vetvyaschiesya protsessy, Nauka, M., 1971 | MR | Zbl

[12] Topchii V., “Renewal measure density for distributions with regularly varying tails of order $\alpha \in (0,1/2]$”, Workshop on branching processes and their applications, Lect. Notes Stat., 197, Springer, Berlin, 2010, 109–118 | DOI | MR

[13] Topchii V.A., “Proizvodnaya plotnosti vosstanovleniya s beskonechnym momentom pri $\alpha \in (0,1/2]$”, Sib. elektron. mat. izv., 7 (2010), 340–349 | MR

[14] Topchii V.A., “Asimptotika proizvodnykh ot funktsii vosstanovleniya dlya raspredelenii bez pervogo momenta s pravilno menyayuschimisya khvostami stepeni $\beta \in (1/2,1]$”, Diskret. matematika, 24:2 (2012), 123–148 | DOI | MR | Zbl

[15] Topchii V., Vatutin V., “Individuals at the origin in the critical catalytic branching random walk”, Discrete random walks, Proc. Conf., Paris, 2003, Discrete Math. Theor. Comput. Sci., electronic, Maison Inf. Math. Discrètes, Paris, 2003, 325–332 | MR | Zbl

[16] Topchii V., Vatutin V., “Two-dimensional limit theorem for a critical catalytic branching random walk”, Mathematics and computer science. III: Algoritms, trees, combinatorics and probabilities, eds. M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger, Birkhäuser, Basel, 2004, 387–395 | MR | Zbl

[17] Vatutin V.A., “Diskretnye predelnye raspredeleniya chisla chastits v vetvyaschikhsya protsessakh Bellmana–Kharrisa s neskolkimi tipami chastits”, Teoriya veroyatn. i ee primen., 24:3 (1979), 503–514 | MR | Zbl

[18] Vatutin V.A., Topchii V.A., “Predelnaya teorema dlya kriticheskikh kataliticheskikh vetvyaschikhsya sluchainykh bluzhdanii”, Teoriya veroyatn. i ee primen., 49:3 (2004), 461–484 | DOI | MR

[19] Vatutin V.A., Topchii V.A., “Kataliticheskie vetvyaschiesya sluchainye bluzhdaniya na $\mathbb Z^d$ s vetvleniem v nule”, Mat. trudy, 14:2 (2011), 28–72 | MR

[20] Vatutin V.A., Topchii V.A., “Osnovnaya teorema vosstanovleniya dlya raspredelenii s tyazhelymi khvostami indeksa $\beta \in (0,1/2]$”, Teoriya veroyatn. i ee primen., 58 (2013) (to appear) | DOI

[21] Vatutin V.A., Topchii V.A., Yarovaya E.B., “Catalytic branching random walk and queueing systems with random number of independent servers”, Theory Probab. Math. Stat., 69 (2004), 1–15 | DOI | MR