Suppose I have a Markov process $X_t$ with exponentially distributed waiting times $T_i$ between the jumps, then with $\tau_k = \sum_{i=1}^k T_i$ one can describe the time until the $k$-th jump. Since $T_i\sim \text{exp}(\lambda)$, I know that $\mathbb{E}(\tau_k)=\frac{k}{\lambda}$ and Var$(\tau_k)=\frac{k}{\lambda^2}$.
Now I asked myself if it was possible to apply the CLT to the $\tau_k$, if the $T_i$ are independent?
For me it seems like it should be possible, but I don't know if it makes any difference, that the $\tau_k$ are part of a Markov process?