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

中文翻译：

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马尔可夫分支移民系统的长时间行为

让 { X ( t ); t ≥ 0} 是一个分枝率{ bk ；k ≥ 0, k ≠ 1} 和移民率 { ak ; k≥1}。我们假设 b 0 = 0, m = : ∑ k = 1 ∞ kbk < ∞ $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ 和 a = ： ∑ k = 1 ∞ kak < ∞ $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $ . 在本文中，我们首先讨论 W ( t ) = e − mt X ( t ) − m − 1 a (1 − e − mt ) 的鞅性质，并证明它有一个极限 W 。此外，我们证明 X ( t + s )/ X ( t ) 收敛到 ems 的概率，W ( t ) 收敛到 W 的概率和 X ( t + s )/ X ( t ) 收敛到 ems 的概率条件为W ≥ α（这里 α 是一个正常数）作为 t → ∞ $t\rightarrow \infty $ 。以上三种收敛速度的显式估计是在 { bk ; 上的各种矩条件下获得的。k≥0，k≠1}。结果表明，第一个的比率是几何的，而其他两个是超几何的。