Long time behaviour for Markovian branching-immigration systems

Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $ . In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $ . The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric.