## Abstract

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters A_{n}, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of ξ_{n}: = log((1 − A_{n}) / A_{n}) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail ℙ(Z_{n}≥ m) of the n th population size Z_{n} is asymptotically equivalent to nF¯ (logm) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξ_{n}, we provide the asymptotics for the distribution tail ℙ(Z_{n}> m) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter A_{k}.

Original language | English |
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Journal | Extremes |

Early online date | 20 Sep 2021 |

DOIs | |

Publication status | E-pub ahead of print - 20 Sep 2021 |

## Keywords

- Branching process
- Random environment
- Random walk in random environment
- Slowly varying distribution
- Subexponential distribution

## ASJC Scopus subject areas

- Statistics and Probability
- Engineering (miscellaneous)
- Economics, Econometrics and Finance (miscellaneous)