## The Annals of Probability

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

Robert P. Kertz

#### 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

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
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