Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T15:22:49.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Behaviour of Extinction Probability of Interacting Branching Collision Processes

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Anyue Chen ,
Junping Li ,
Yiqing Chen  and
Dingxuan Zhou
Anyue Chen*
Affiliation:
South University of Science and Technology of China and University of Liverpool
Junping Li*
Affiliation:
Central South University
Yiqing Chen*
Affiliation:
University of Liverpool
Dingxuan Zhou*
Affiliation:
City University of Hong Kong
*
∗ Postal address: Department of Financial Mathematics and Financial Engineering, South University of Science and Technology of China, Shenzen, 518055, China.
∗∗∗ Postal address: School of Mathematics and Statistics, Central South University, Changsha, 410075, China. Email address: jpli@mail.csu.edu.cn
∗ Postal address: Department of Financial Mathematics and Financial Engineering, South University of Science and Technology of China, Shenzen, 518055, China.
∗∗∗∗∗ Postal address: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR. Email address: mazhou@cityu.edu.hk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) were very recently given by Chen et al. [4], some of these expressions are very complicated; hence, useful information regarding asymptotic behaviour, for example, is harder to obtain. Also, these exact expressions take very different forms for different cases and thus seem lacking in homogeneity. In this paper, we show that the asymptotic behaviour of these extremely complicated and tangled expressions for extinction probabilities of IBCP follows an elegant and homogenous power law which takes a very simple form. In fact, we are able to show that if the extinction is not certain then the extinction probabilities {an} follow an harmonious and simple asymptotic law of an ∼ kn-αρcn as n → ∞, where k and α are two constants, ρc is the unique positive zero of the C(s), and C(s) is the generating function of the infinitesimal collision rates. Moreover, the interesting and important quantity α takes a very simple and uniform form which could be interpreted as the ‘spectrum’, ranging from -∞ to +∞, of the interaction between the two components of branching and collision of the IBCP.

Keywords

Markov branching processinteracting branching collision processextinction probabilityasymptotic behaviour

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 219 - 234
Copyright
© Applied Probability Trust 

References

Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Athreya, K. B. and Jagers, P. (eds) (1997). Classical and Modern Branching Processes. Springer, New York.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Dover, Mineola, NY.Google Scholar
Chen, A., Li, J., Chen, Y. and Zhou, D. (2012). Extinction probability of interacting branching collision processes. Adv. Appl. Prob. 44, 226–259.Google Scholar
Chen, A., Pollett, P., Li, J. and Zhang, H. (2010). Uniqueness, extinction and explosivity of generalised Markov branching processes with pairwise interaction. Methodol. Comput. Appl. Prob. 12, 511–531.Google Scholar
Chen, A., Pollett, P., Zhang, H. and Li, J. (2004). The collision branching process. J. Appl. Prob. 41, 1033–1048.CrossRefGoogle Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Kalinkin, A. V. (1983). Final probabilities for a branching random process with interaction of particles. Soviet Math. Dokl. 27, 493–497.Google Scholar
Kalinkin, A. V. (2002). Markov branching processes with interaction. Russian Math. Surveys 57, 241–304.Google Scholar
Kalinkin, A. V. (2003). On the probability of the extinction of a branching process with two complexes of particle interaction. Theory Prob. Appl. 46, 347–352.Google Scholar
Lange, A. M. (2007). On the distribution of the number of final particles of a branching process with transformations and paired interactions. Theory Prob. Appl. 51, 704–714.Google Scholar
Otter, R. (1949). The multiplicative process. Ann. Math. Statist. 20, 206–224.Google Scholar
Sevastyanov, B. A. (1971). Branching Processes. Nauka, Moscow (in Russian).Google Scholar
Sevastyanov, B. A. and Kalinkin, A. V. (1982). Branching random processes with particle interaction. Soviet Math. Dokl. 25, 644–646.Google Scholar