Naslov
Limit theorems for branching processes with immigration in a random environment

Autori
Basrak, Bojan ; Kevei, Peter

Izvornik
Extremes (1386-1999) 25 (2022); 623-654

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Branching process in a random environment ; Regularly varying stationary sequences ; Tail process ; Implicit renewal theory

Sažetak
We investigate branching processes with immigration in a random environment. Using Goldie’s implicit renewal theory we prove that under a generalized Cramér condition the stationary distribution of such processes has a power law tail. We further show how several methods familiar in the extreme value theory provide a natural and elegant path to their mathematical analysis. In particular, we rely on the point processes theory and the concept of tail process to determine the limiting dis-tribution for the corresponding extremes and partial sums. Since Kesten, Kozlov and Spitzer seminal 1975 paper, it is known that one class of these processes has a close relation with random walks in a random environment. Even in that well studied con-text, the method we follow yields new results. For instance, we are able to i) move away from the conditions used by Kesten et al., ii) provide precise form of the limit-ing distribution in their main theorem, and iii) characterize the long term behavior of the worst traps a random walk in random environment encounters when drifting away from the origin.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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