2019-11-01

# Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching

## Publication

### Publication

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Statistics and Probability Letters
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Volume 154
p. 108550
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Motivated by seminal paper of Kozlov et al. Kesten et al. (1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability A and immigration equals 1 in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is log A

^{−1}(1 − A) is regularly varying with a parameter α > 1, that is that P (log A^{−1}(1 − A) > x) = x^{−α}L(x) for a slowly varying function L. We will prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of distribution of the population at nth generation which gets even heavier with n increasing. Precisely, in this work, we prove that asymptotic tail P (Z_{l}≥ m) of lth population Z_{l}is of order (log^{(l)}m)^{−α}L (log^{(l)}m) for large m, where log^{(l)}m = log…log m. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behavior of the first passage time T_{n}of the state n ∈ Z by the walker in a neighborhood random walk in random environment created by independent copies (A_{i}: i ∈ Z) of (0, 1)-valued random variable A.Additional Metadata | |
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doi.org/10.1016/j.spl.2019.06.026 | |

Statistics and Probability Letters | |

Rare events: Asymptotics, Algorithms, Applications | |

Organisation | Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands |

Bhattacharya, A., & Palmowski, Z. (2019). Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching. Statistics and Probability Letters, 154. doi:10.1016/j.spl.2019.06.026 |