Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T10:02:59.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary effect in competition processes

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 October 2019

Vadim Shcherbakov  and
Stanislav Volkov
Vadim Shcherbakov*
Affiliation:
Royal Holloway, University of London
Stanislav Volkov*
Affiliation:
Lund University
*
*Postal address: Department of Mathematics, Royal Holloway, University of London, Egham TW20 0EX, UK.
***Postal address: Centre for Mathematical Sciences, Lund University, SE-221 00 Lund, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

Keywords

Markov chainbirth-and-death processcompetition processFriedman’s urn modelLyapunov functionmartingale

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 750 - 768
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, W. (1991). Continuous-Time Markov Chains: An Application-Oriented Approach. Springer, New York.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Freedman, D. A. (1965). Bernard Friedman’s urn. Ann. Math. Statist. 36, 956–970.CrossRefGoogle Scholar
Iglehart, D. L. (1964). Reversible competition processes. Z. Wahrseheinliehkeitsth. 2, 314–331.CrossRefGoogle Scholar
Iglehart, D. L. (1964). Multivariate competition processes. Ann. Math. Statist. 35, 350–361.CrossRefGoogle Scholar
Janson, S., Shcherbakov, V. and Volkov, S. (2019). Long-term behaviour of a reversible system of interacting random walks. J. Stat. Phys. 175, 71–96.CrossRefGoogle Scholar
Kingman, J. F. C. and Volkov, S. E. (2003). Solution to the OK Corral model via decoupling of Friedman’s urn. J. Theoret. Prob. 16, 267–276.CrossRefGoogle Scholar
MacPhee, M. I., Menshikov, M. V. and Wade, A. R. (2010). Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift. Markov Process. Relat. Fields 16, 351–388.Google Scholar
Menshikov, M. V., Popov, S. and Wade, A. R. (2017). Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press.CrossRefGoogle Scholar
Menshikov, M. and Shcherbakov, V. (2018). Long-term behaviour of two interacting birth-and-death processes. Markov Process. Relat. Fields. 24, 85–106.Google Scholar
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 1–79.CrossRefGoogle Scholar
Reuter, G. E. H. (1961). Competition processes. In Proc. 4th Berkeley Symp. Math. Statist. Probab., vol. II. University of California Press, Berkeley.Google Scholar
Shcherbakov, V. and Volkov, S. (2015). Long-term behaviour of locally interacting birth-and-death processes. J. Stat. Phys. 158, 132–157.CrossRefGoogle Scholar