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

Ayan Bhattacharya and Zbigniew Palmowski

Statistics & Probability Letters, 2019, vol. 154, issue C, -

Abstract: 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 logA−1(1−A) is regularly varying with a parameter α>1, that is that P(logA−1(1−A)>x)=x−αL(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(Zl≥m) of lth population Zl is of order (log(l)m)−αL(log(l)m) for large m, where log(l)m=log…logm. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behavior of the first passage time Tn of the state n∈Z by the walker in a neighborhood random walk in random environment created by independent copies (Ai:i∈Z) of (0,1)-valued random variable A.

Keywords: Branching process; Random environment; Random walk in random environment; Regular variation; Slow variation (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0167715219301877
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:154:y:2019:i:c:5

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spl.2019.06.026

Access Statistics for this article

Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul

More articles in Statistics & Probability Letters from Elsevier
Bibliographic data for series maintained by Catherine Liu ().