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Large deviations of branching processes with immigration in random environment

  • Dmitriy V. Dmitruschenkov EMAIL logo and Alexander V. Shklyaev
Published/Copyright: December 7, 2017
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 27 Issue 6

Abstract

We consider branching process Zn in random environment such that the associated random walk Sn has increments ξi with mean μ and satisfy the Cramér condition Eehξi < ∞, 0 < h < h+. Let χi be the number of particles immigrating into the ith generation of the process, Eχih < ∞, 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 P{Zn ≥ exp(θ n)}, θ > μ. In the case of subcritical processes analogous result is obtained for θ > γ, where γ > 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

Originally published in Diskretnaya Matematika (2016) 28, №3, 28–48 (in Russian).

Funding source: Russian Foundation for Basic Research

Award Identifier / Grant number: 14-01-31091

Funding statement: The work was supported by the RFBR grant No. 14-01-31091 mol-a.

References

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Received: 2015-11-24
Accepted: 2016-7-17
Published Online: 2017-12-7
Published in Print: 2017-12-20

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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  1. Frontmatter
  3. Estimating the level of affinity of a quadratic form
  5. Cyclic decomposition of sets, set-splitting digraphs and cyclic classes of risk-free games
  7. Large deviations of branching processes with immigration in random environment
  9. On the probability of existence of substrings with the same structure in a random sequence
  11. Linearly realizable automata
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https://doi.org/10.1515/dma-2017-0037
Keywords for this article
Large deviations; random walks; branching processes; random environments; Cramér condition; processes with immigration

Articles in the same Issue

  1. Frontmatter
  3. Estimating the level of affinity of a quadratic form
  5. Cyclic decomposition of sets, set-splitting digraphs and cyclic classes of risk-free games
  7. Large deviations of branching processes with immigration in random environment
  9. On the probability of existence of substrings with the same structure in a random sequence
  11. Linearly realizable automata
  13. On the number of labeled outerplanar k-cycle blocks
  15. Convergence of the sequence of the Pearson statistics values to the normalized square of the Bessel process
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