Down/up crossing properties of weighted Markov collision processes

Abstract

This paper concentrates on considering the down/up crossing property of weighted Markov collision processes. The joint probability generating function of down crossing and up crossing numbers of weighted Markov collision processes until its extinction are obtained by constructing and studying a related multi-dimensional Markov chain. Hence, the joint probability distribution of down crossing and up crossing numbers and the mean numbers are obtained.

Keywords

Weighted Markov collision process / down crossing / up crossing / joint probability distribution

References

[1] Asmussen S, Hering H. Branching Processes.Boston: Birkhäuser, 1983

[2] Asmussen S, Jagers P. Classical and Mordern Branching Processes.Berlin: Springer, 1997

[3] Athreya K B, Ney P E. Branching Processes.Berlin: Springer, 1972

[4] Chen A Y, Li J P, Ramesh N. Uniqueness and extinction of weighted Markov branching processes. Methodol Comput Appl Probab, 2005, 7(4): 489–516

[5] Chen A Y, Pollett P K, Zhang H J, Li J P. The collision branching process. J Appl Probab, 2004, 41(4): 1033–1048

[6] Cottingham W, Greenwood D. An Introduction to Nuclear Physics. 2nd ed. Cambridge: Cambridge Univ Press, 2001

[7] Daley D, Kendal D. Stochastic rumours. IMA J Appl Math, 1965, 1(1): 42–55

[8] Ezhov I I. Branching processes with group death. Theory Probab Appl, 1980, 25: 202–203

[9] Harris T E. The Theory of Branching Processes.Berlin: Springer, 1963

[10] Kalinkin A V. On the extinction probability of a branching process with two kinds of interaction of particles. Theory Prob Appl, 2003, 46: 347–352

[11] Li J P, Chen A Y. Generalized Markov interacting branching processes. Sci China Math, 2018, 61(3): 545{561

[12] Sevastyanov B A. On certain types of Markov processes. Uspehi Mat Nauk, 1949, 4: 194 (in Russian)

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