Branching processes in a random environment with immigration stopped at zero

Abstract

A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

Critical branching processes in random environment with immigration: survival of a single family

“…The tail distribution of the life-periods of BPIRE's were considered in [4] and [9] under weaker assumptions than the ones we impose here. An (i, n)-clan of a BPIRE is the set of individuals alive at generation n and being descendants of the immigrant which entered the population at generation i.…”

Limit theorems for the critical Galton-Watson processes with immigration stopped at zero

“…The distribution of the so-called life periods is important and interesting from not only theoretical but also practical points of view, see [2], [3], [10], [19]. Information on the length of such periods may be used in epidemiology, ecology and seismology, see [5], [7], [9].…”

Exact convergence rate in the central limit theorem for a branching process with immigration in a random environment

“…This means that the population size at every generation is influenced by both the environment and immigration populations. In recent years, many results were obtained for BPIRE, including moments, limit theorems, convergence rates and so on; see for example [4,5,7,17,18]. In this paper, we are interested in the central limit theorem associated to the process.…”