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–nU/(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.