Limit Theorems for the Simple Branching Process Allowing Immigration: A Review and New Results. I. The Case of Finite Offspring Mean.

摘要

This paper reviews known limit theorems for population sizes of Bienayme-Galton-Watson process allowing immigration. For non-critical cases it is known the limit distribution is non-defective if a logarithmic moment of immigration distribution is finite. New results of this paper concern the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and the continuous time process. Some results on rate of decay of transition probabilities and growth rate of stationary measure are given and used to obtain limit theorems for a reversed time process. (Author)