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High level subcritical branching processes in a random environment

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Abstract

A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk Θ(t), t ≥ 0, is equal to 1 for some value of the argument ϰ > 0. Let T x be the time when the process first attains the half-axis (x,+∞) and T be the lifetime of this process. It is shown that the random variable T x /lnx, considered under the condition T x < +∞, converges in distribution to a degenerate random variable equal to 1/Θ′(ϰ), and the random variable T/ ln x, considered under the same condition, converges in distribution to a degenerate random variable equal to 1/Θ′(ϰ) − 1/Θ′(0).

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References

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. I. Afanasyev

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  1. V. I. Afanasyev
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Correspondence to V. I. Afanasyev.

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Original Russian Text © V.I. Afanasyev, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 282, pp. 10–21.

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Afanasyev, V.I. High level subcritical branching processes in a random environment. Proc. Steklov Inst. Math. 282, 4–14 (2013). https://doi.org/10.1134/S0081543813060023

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  • DOI: https://doi.org/10.1134/S0081543813060023

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