Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean

Abstract

This paper presents some limit theorems for the simple branching process allowing immigration, {Xn }, when the offspring mean is infinite. It is shown that there exists a function U such that {e –n U/(Xn )} converges almost surely, and if s = ∑ bj , log+ U(j) < ∞, where {bj } is the immigration distribution, the limit is non-defective and non-degenerate but is infinite if s = ∞. When s = ∞, limit theorems are found for {U(Xn )} which involve a slowly varying non-linear norming.