Large Deviation Rates for the Continuous‐Time Supercritical Branching Processes with Immigration

Abstract

Let {Y(t); t ≥ 0} be a supercritical continuous‐time branching process with immigration; our focus is on the large deviation rates of Y(t) and thus extending the results of the discrete‐time Galton–Watson process to the continuous‐time case. Firstly, we prove that Z(t) = e−mt[Y(t) − ((em(t + 1) − 1)/(em − 1))ea+m] is a submartingale and converges to a random variable Z. Then, we study the decay rates of P(|Z(t) − Z| > ε) as t⟶∞ and P(|(Y(t + v)/Y(t)) − emv| > ε|Z ≥ α) as t⟶∞ for α > 0 and ε > 0 under various moment conditions on {bk; k ≥ 0} and {aj; j ≥ 0}. We conclude that the rates are supergeometric under the assumption of finite moment generation functions.