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Linear competition processes and generalized Pólya urns with removals

Author

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  • Popov, Serguei
  • Shcherbakov, Vadim
  • Volkov, Stanislav

Abstract

A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Pólya urn model with removals.

Suggested Citation

  • Popov, Serguei & Shcherbakov, Vadim & Volkov, Stanislav, 2022. "Linear competition processes and generalized Pólya urns with removals," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 125-152.
    • Handle: RePEc:eee:spapps:v:144:y:2022:i:c:p:125-152
    DOI: 10.1016/j.spa.2021.11.001
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    References listed on IDEAS

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    1. Champagnat, Nicolas & Villemonais, Denis, 2021. "Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 51-74.
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