## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 6 (1974), 1046-1064.

### Limit Theorems for Discontinuous Random Evolutinos with Applications to Initial Value Problems and to Markov Processes on $N$ Lines

#### Abstract

Let $X(t); t \geqq 0$ be a stationary continuous-time Markov chain with state space $\{1,2,\cdots, N\}$ and jump times $t_1, t_2,\cdots$. Let $T_\alpha(t); t \geqq 0, 1 \leqq \alpha \leqq N$, be semi-groups and $\Pi_{jk} (u); u \geqq 0, 1 \leqq j \neq k \leqq N$, operators defined on Banach space $B$. Under suitable conditions on these operators, including commutativity, and an appropriate time change in $\varepsilon > 0$ on $X(t)$, we give limiting behavior for the discontinuous random evolutions $T_{X(0)}(t_1^\varepsilon) \Pi_{X(0)X(t_1)} (\varepsilon)T_{X(t_1)}(t_2^\varepsilon - t_1^\varepsilon)\cdots T_{X(t_\nu)}(t - t_\nu^\varepsilon)$ as $\varepsilon \rightarrow 0$. By considering the `expectation semi-group' of the discontinuous random evolutions, we prove a type of singular perturbation theorem and give formulas for the asymptotic solution. These results rely on a limit theorem for the joint distribution of the occupation-time and number-of-jump random variables of the chain $X(\bullet)$. We prove this theorem and with `random evolution' techniques use it to give new proofs of limit theorems for Markov processes on $N$ lines. Analogous results are obtained when the controlling process is a discrete-time finite-state Markov chain.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 6 (1974), 1046-1064.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996497

**Digital Object Identifier**

doi:10.1214/aop/1176996497

**Mathematical Reviews number (MathSciNet)**

MR368180

**Zentralblatt MATH identifier**

0323.60064

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F056

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60H99: None of the above, but in this section 47D05 35B25: Singular perturbations 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces

**Keywords**

Multipilcative operator functional random evolution semi-groups of operators singular perturbation central limit theorem

#### Citation

Kertz, Robert P. Limit Theorems for Discontinuous Random Evolutinos with Applications to Initial Value Problems and to Markov Processes on $N$ Lines. Ann. Probab. 2 (1974), no. 6, 1046--1064. doi:10.1214/aop/1176996497. https://projecteuclid.org/euclid.aop/1176996497