Statistica Sinica 13(2003), 227-242

POISSON STYLE CONVERGENCE THEOREMS FOR

ADDITIVE PROCESSES DEFINED ON MARKOV CHAINS

Y. H. Wang and Lingqi Tang

Tunghai University and University of California

Abstract: Given a Markov chain $\left\{ X_{i}:i\geq 0\right\} \;$with finite state space and irreducible primitive stationary transition matrix $\mathbf{P}$, at time $n$, corresponding to each possible one-step transition $j\rightarrow k$, we associate random variables $W_{n}$ with distribution $F_{jk},$ depending only on states $j$ and/or $k$. Given $\{X_{i}\}$, $W_{1}$, $W_{2},\ldots,$ are conditionally independent and need not be integer-valued, nor positive. Define the cumulative sum $Y_{n}=W_{1}+\cdots+W_{n},$ with $Y_{0}=0.$ It is proved under certain conditions that the limiting distribution of $\{ Y_{n}\}$ is in the class of compound Poisson type distributions. Some applications of the theorem are illustrated.

Key words and phrases: Compound Poisson distribution, convergence theorem, interarrival time, limiting distribution, Markov chain, Markov renewal process, sum of random variables.