Abstract
A general model of a catalytic branching process (CBP) with any number of catalysis centers in a discrete space is studied. The asymptotic (in time) behavior of the total number of particles and of the local particle numbers is investigated. The problems of finding the global and local extinction probabilities are solved. Necessary and sufficient conditions are established for the phase of pure global survival and strong local survival. Under wide conditions, limit theorems for the normalized total and local particle numbers in supercritical CBP are proved in the sense of almost sure convergence, as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well.
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References
E. Vl. Bulinskaya, Teor. Veroyatn. Primen. 50 (4), 639–666 (2014).
V. A. Vatutin, Branching Processes and Their Applications (Mat. Inst. Ross. Akad. Nauk, Moscow, 2008) [in Russian].
V. A. Vatutin and V. A. Topchii, Sib. Adv. Math. 23 (2), 123–153 (2013).
B. A. Sewastianow, Verzweigungsprozesse (Nauka, Moscow, 1971; Akademie, Berlin, 1974).
K. L. Chung, Markov Chains with Stationary Transition Probabilities (Springer, Berlin, 1960; Mir, Moscow, 1964).
E. B. Yarovaya, Math. Notes 92 (1), 115–131 (2012).
S. Albeverio, L. V. Bogachev, and E. B. Yarovaya, C. R. Math. Acad. Sci. Paris 326 (9), 975–980 (1998).
D. Bertacchi and F. Zucca, Adv. Appl. Probab. 46 (2), 400–421 (2014).
E. Vl. Bulinskaya, Stat. Probab. Lett. 85 (1), 15–19 (2014).
E. Vl. Bulinskaya, J. Theor. Probab. 27 (3), 878–898 (2014).
S. Bocharov and S. C. Harris, Acta Appl. Math. 134 (1), 201–228 (2014).
Ph. Carmona and Y. Hu, Ann. Inst. Henri Poincare Probab. Stat. 50 (2), 327–351 (2014).
L. Doering and M. Roberts, Lect. Notes Math. 2078, 305–322 (2013).
N. Kaplan, J. Math. Anal. Appl. 50 (1), 164–182 (1975).
Ch. J. Mode, Math. Biosci. 3, 231–247 (1968).
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Published in Russian in Doklady Akademii Nauk, 2015, Vol. 465, No. 4, pp. 398–402.
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Bulinskaya, E.V. Strong and weak convergence of the population size in a supercritical catalytic branching process. Dokl. Math. 92, 714–718 (2015). https://doi.org/10.1134/S1064562415060228
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DOI: https://doi.org/10.1134/S1064562415060228