Let $X_1$, $X_2$,... be a finite state, irreducible and aperiodic Markov chain with initial state $X_0=i$. It is known that \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{t=1}^n I\{X_t=j\}=\pi(j) \Big\} = 1, \end{equation} where $I(\cdot)$ is the indicator function and $\pi(j)$ is the stationary distribution of state $j$.
Question: Is the above also true if the state of the Markov chain is observed at arbitrary time instances (for example recording only every second state transition of the chain $X_1, X_3, X_5,...$)? To put more formally, I'd like to know if the following is also true : \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n I\{X_{\tau(k)}=j\}=\pi(j) \Big\} = 1, \end{equation} where $\{\tau(k)\}$ is any infinite strictly increasing sequence of natural numbers.