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The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

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Abstract

The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant n of time for a process that started at the zero instant of time from one particle of any type is of the order of λnn−1/2, where λ ∈ (0, 1) is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.

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Acknowledgments

The authors wish to express gratitude to the referee whose remarks have led to the improvement of the exposition of the results presented in this paper.

Funding

This work was supported by the Russian Science Foundation under grant 17-11-01173.

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Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, 630090, Russia

    V. A. Vatutin & E. E. D’yakonova

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  1. V. A. Vatutin
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  2. E. E. D’yakonova
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Correspondence to V. A. Vatutin or E. E. D’yakonova.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 2, pp. 163–177.

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Vatutin, V.A., D’yakonova, E.E. The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment. Math Notes 107, 189–200 (2020). https://doi.org/10.1134/S0001434620010198

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  • DOI: https://doi.org/10.1134/S0001434620010198

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