Let $P$ be irreducible and recurrent. Then 1. $\gamma_{k}^{k}=1$ 2. $\gamma^{k}=\gamma^{k}P$ 3. $0<\gamma_{i}^{k}<\infty$ for each $i\in S$

Let $\{X_{n},n\ge0\}$ be $\mbox{Markov}(\lambda,P)$. Assume $P$ is irreducible. Then for any $i\in S$ $$P\left(\lim_{n\to\infty}\frac{V_{i}(n)}{n}= \frac{1}{m_{i}}\right)=1$$where $V_{i}(n)$ is the number of visits to state $i$ before the time $n$, and $m_{i}$ is the expected return time to state $i$, i.e., $$V_{i}(n)=\sum\limits_{k=0}^{n-1}\mathbb{1}_{\{X_{k}=i\}}$$and $$m_{i}=E_{i}[T_{i}^{(1)}]$$

Let $\{X_{n},n\ge0\}$ be $\mbox{Markov}(\lambda,P)$. Assume $P$ is irreducible and has an invariant distribution $\pi$. Then for any bounded function $f:S\to\mathbb{R}$ $$P\left(\lim_{n\to\infty} \frac{1}{n}\sum\limits_{k=0}^{n-1}f(X_{k})=\overline f \right)=1$$where $$\overline f=\sum\limits_{i\in S}\pi_{i}f(i)$$ i.e. “time average” converges to “space average”