Classical and Modern Branching Processes pp 323-336 | Cite as

# Limit Theorems for Branching Processes with Random Migration Stopped at Zero

## Abstract

In this paper, we study a generalization of the classical BienayméGalton-Watson branching process by allowing a random migration component stopped at zero (i.e. the state zero is absorbing). In each generation for which the population size is positive, with probability *p* two types of emigration are available - a random number of offsprings and a random number of families (these two random variables can be dependent); with probability *q* there is not any migration; with probability *r* an immigration of new individuals is possible, *p+q+r = 1.* The critical case is investigated with an extension when the initial law is attracted to a stable *(p)* law, *p ≤* 1. The asymptotic form of the probability of non-extinction is studied and conditional limit theorems for the population size are obtained, depending on the range of an additional parameter of criticality.

## Key words

random migration extinction moments conditional limit theorems## Preview

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