## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 4 (1989), 1483-1502.

### A Limit Theorem for a Class of Inhomogeneous Markov Processes

Tzuu-Shuh Chiang and Yunshyong Chow

#### Abstract

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 $\underline{S} = \{i \in S: h(i) = 0\}$.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 4 (1989), 1483-1502.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991169

**Digital Object Identifier**

doi:10.1214/aop/1176991169

**Mathematical Reviews number (MathSciNet)**

MR1048941

**Zentralblatt MATH identifier**

0687.60070

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J27: Continuous-time Markov processes on discrete state spaces

Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

**Keywords**

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

#### Citation

Chiang, Tzuu-Shuh; Chow, Yunshyong. A Limit Theorem for a Class of Inhomogeneous Markov Processes. Ann. Probab. 17 (1989), no. 4, 1483--1502. doi:10.1214/aop/1176991169. https://projecteuclid.org/euclid.aop/1176991169