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\noindent Statistica Sinica {\bf 13}(2003), 227-242
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\centerline{\large\bf POISSON STYLE CONVERGENCE THEOREMS FOR}
\vspace{2pt}
\centerline{\large\bf ADDITIVE PROCESSES DEFINED ON MARKOV CHAINS}
\vspace{.25cm}
\centerline{Y. H. Wang and Lingqi Tang}
\vspace{.4cm}
\centerline{\it Tunghai University and University of California}
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\begin{quotation}
\noindent {\it 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.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Compound Poisson distribution, convergence theorem,
interarrival time, limiting distribution,
Markov chain, Markov renewal process,
sum of random variables.
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