Problem 4 (10 Points)

Let $\{X_n\}_{n\ge 0}$ be an irreducible positive recurrent HMC with stationary distribution $\pi$. Let A be a subset of the state space $E$ and let $\{\tau(k)\}_{k\ge 1}$ be the sequence of return times to $A$, i.e. $\tau(k)$ is the time on kth return. Show that

$$\lim_{k \uparrow \infty} \frac{\tau(k)}{k} = \frac{1}{\sum_{i \in A}\pi(i)}$$ (4)