Subcritical catalytic branching random walk with finite or infinite variance of offspring number

E. Vl. Bulinskaya

Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 69-79.

Résumé

Subcritical catalytic branching random walk on the $d$-dimensional integer lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particle numbers are established. To prove the results, different approaches are used, including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated under the assumptions of finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper, for the offspring number in the subcritical regime, the finiteness of the moment of order $1+\delta$ is required where $\delta $ is some positive number.

Bibliographie

[1] 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

[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] Yarovaya E.B., “Kriterii eksponentsialnogo rosta chisla chastits v modelyakh vetvyaschikhsya sluchainykh bluzhdanii”, Teoriya veroyatn. i ee primen., 55:4 (2010), 705–731 | DOI | MR

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

[5] Yarovaya E.B., “Kriticheskie vetvyaschiesya sluchainye bluzhdaniya po reshetkam nizkikh razmernostei”, Diskret. matematika, 21:1 (2009), 117–138 | DOI | MR | Zbl

[6] Yarovaya E., “Critical and subcritical branching symmetric random walks on $d$-dimensional lattices”, Advances in data analysis: Theory and applications to reliability and inference, data mining, bioinformatics, lifetime data, and neural networks, eds. C.H. Skiadas, Birkhäuser, Boston, 2010, 157–168 | MR

[7] Bulinskaya E.Vl., “Catalytic branching random walk on three-dimensional lattice”, Theory Stoch. Processes, 16:2 (2010), 23–32 | MR | Zbl

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

[9] 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

[10] Vatutin V.A., Topchii V.A., Khu Yu., “Vetvyascheesya sluchainoe bluzhdanie po reshetke $\mathbb Z^4$ s vetvleniem lish v nachale koordinat”, Teoriya veroyatn. i ee primen., 56:2 (2011), 224–247 | DOI

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

[12] Bulinskaya E.Vl., “Predelnye teoremy dlya lokalnykh chislennostei chastits v vetvyaschemsya sluchainom bluzhdanii”, DAN., 444:6 (2012), 593–596 | Zbl

[13] Yarovaya E.B., Vetvyaschiesya sluchainye bluzhdaniya v neodnorodnoi srede, Izd-vo Tsentra prikl. issled. pri mekh.-mat. f-te MGU, M., 2007

[14] Vatutin V.A., Vetvyaschiesya protsessy Bellmana–Kharrisa, MIAN, M., 2009

[15] Wolfe S.J., “On moments of probability distribution functions”, Fractional calculus and its applications, Lect. Notes Math., 457, Springer, Berlin, 1975, 306–316 | DOI | MR

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

[17] Bulinskaya E.Vl., “Local particles numbers in critical branching random walk”, J. Theor. Probab., 2012 | DOI

[18] Vatutin V., Xiong J., “Some limit theorems for a particle system of single point catalytic branching random walks”, Acta math. Sin., 23:6 (2007), 997–1012 | DOI | MR | Zbl

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

[20] de Brëin N.G., Asimptoticheskie metody v analize, Izd-vo inostr. lit., M., 1961

[21] Klar B., “On a test for exponentiality against Laplace order dominance”, Statistics, 37:6 (2003), 505–515 | DOI | MR | Zbl