# CLT for a Markov Renewal Process

Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more precisely, $T_{n+1}-T_n$ is time the Markov chain $X$ stays in state $X_n$ and may depend on $X_n$ only. Let $\tau_m=\{\min\limits_{n\geq0} n: T_n>m\}$. Does there exist a result that says $$\frac{\tau_m-m\mu}{\sqrt{m}\sigma}\underset{m \rightarrow \infty} {\rightarrow}\mathcal{N}[0,1],$$ for some $\mu$ and $\sigma$? If yes, could someone point me in the right direction. Thanks