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

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) - ((e(m(t+1)) - 1)/(e(m) - 1))e(a+m)] is a submartingale and converges to a random variable Z. e(n), we study the decay rates of P(|Z(t) - Z| > epsilon) ast ? infinity and P(|(Y(t + v)/Y(t)) - e(mv)| > epsilon|Z >= alpha) ast ? infinity for alpha > 0 and epsilon > 0 under various moment conditions on {b(k); k >= 0 and {a(j); j >= 0} . We conclude that the rates are supergeometric under the assumption of finite moment generation functions.