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October, 1989 A Limit Theorem for a Class of Inhomogeneous Markov Processes
Tzuu-Shuh Chiang, Yunshyong Chow
Ann. Probab. 17(4): 1483-1502 (October, 1989). DOI: 10.1214/aop/1176991169


Let $\{X(t): t \in R^+ \text{or} I^+\}$ be an (aperiodic) irreducible Markov process with a finite state space $S$ and transition rate $q_{ij}(t) = p(i, j)(\lambda(t))^{U(i, j)}$, where $0 \leq U(i, j) \leq \infty$ and $\lambda(t)$ is some suitable rate function with $\lim_{t \rightarrow \infty}\lambda(t) = 0$. We shall show in this article that there are constants $h(i) \geq 0$ and $\beta_i > 0$ such that independent of $X(0), \lim_{t \rightarrow \infty}P(X(t) = i) \div (\lambda(t))^{h(i)} = \beta_i$ for each $i \in S$. The height function $h$ is determined by $(p(i, j))$ and $(U(i, j))$. In particular, a limit distribution exists and concentrates on $\underline{S} = \{i \in S: h(i) = 0\}$.


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Tzuu-Shuh Chiang. Yunshyong Chow. "A Limit Theorem for a Class of Inhomogeneous Markov Processes." Ann. Probab. 17 (4) 1483 - 1502, October, 1989.


Published: October, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0687.60070
MathSciNet: MR1048941
Digital Object Identifier: 10.1214/aop/1176991169

Primary: 60J27
Secondary: 60F05 , 60F10

Keywords: convergence rate , cycle method , Forward equations , inhomogeneous Markov process , Perron-Frobenius theorem

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 4 • October, 1989
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