In:
Extremes, Springer Science and Business Media LLC, Vol. 25, No. 1 ( 2022-03), p. 55-77
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 $\xi _{n}:=\log ((1-A_{n})/A_{n})$ 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 $\mathbb {P}(Z_{n} \ge m)$ ℙ ( Z n ≥ m ) of the n th population size Z n is asymptotically equivalent to $n\overline F(\log m)$ n F ¯ ( log m ) 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 $\mathbb {P}(Z_{n} 〉 m)$ 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 .
Type of Medium:
Online Resource
ISSN:
1386-1999
,
1572-915X
DOI:
10.1007/s10687-021-00427-1
Language:
English
Publisher:
Springer Science and Business Media LLC
Publication Date:
2022
detail.hit.zdb_id:
2013246-3
SSG:
11
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