Abstract
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.
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Acknowledgement
This work is substantially supported by the National Natural Sciences Foundations of China (No. 11771452, No. 11971486).
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Li, J., Cheng, L. & Li, L. Long time behaviour for Markovian branching-immigration systems. Discrete Event Dyn Syst 31, 37–57 (2021). https://doi.org/10.1007/s10626-020-00323-z
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DOI: https://doi.org/10.1007/s10626-020-00323-z