Ergodic properties of a subsquence of an ergodic diffusion process

Question

Let $X_t$ be an ergodic diffusion process with invariant measure $\mu(dx)$ on $I$ and satisfying, say for bounded $f$ \begin{equation} \lim_{T\rightarrow \infty}\frac{1}{T}\int^T_0f(X_t)dt= \int_If(x)\mu(dx) \end{equation} Let $t_k, k=1,2,\ldots$ be as subsequence $0\leq t_k \rightarrow \infty$. What conditions on the $t_k$ are necessary for \begin{equation} \lim_{T\rightarrow \infty}\frac{1}{n_T}\sum_{k=0}^{n_T}f(X_{ t_k})=\int_If(x)\mu(dx) \end{equation} where $n_T=\#\{t_k\leq T\}$?