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 Diskr. Mat., 2016, Volume 28, Issue 3, Pages 28–48 (Mi dm1382)

Large deviations of branching processes with immigration in random environment

D. V. Dmitrushchenkov, A. V. Shklyaev

Lomonosov Moscow State University

Abstract: We consider branching process $Z_n$ in random environment such that the associated random walk $S_n$ has increments $\xi_i$ with mean $\mu$ and satisfy the Cramér condition $\mathbf{E}e^{h\xi_i}<\infty$, $0<h<h^+$. Let $\chi_i$ be the number of particles immigrating into the $i^th$ generation of the process, $\mathbf{E}\chi_i^h<\infty$, $0<h<h^+$. We suppose that the number of offsprings of one particle conditioned on the environment has the geometric distribution. It is shown that the supplement of immigration to critical or supercritical processes results only in the change of multiplicative constant in the asymptotics of large deviation probabilities $\mathbf PŻ_n\ge \exp(\theta n)\}$, $\theta>\mu$. In the case of subcritical processes analogous result is obtained for $\theta>\gamma$, where $\gamma>0$ is some constant. For all constants explicit formulas are given.

Keywords: Large deviations, random walks, branching processes, random environments, Cramér condition, processes with immigration.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-31091 ìîë-à The work was supported by the RFBR grant No. 14-01-31091 mol-a.

DOI: https://doi.org/10.4213/dm1382

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English version:
Discrete Mathematics and Applications, 2017, 27:6, 361–376

Bibliographic databases:

UDC: 519.218.2
Revised: 17.07.2016

Citation: D. V. Dmitrushchenkov, A. V. Shklyaev, “Large deviations of branching processes with immigration in random environment”, Diskr. Mat., 28:3 (2016), 28–48; Discrete Math. Appl., 27:6 (2017), 361–376

Citation in format AMSBIB
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