Theorem (3.1.3)

For a Markov chain, if there exists an element $i$ such that $E_{i}[τ_{i}] < ∞$; the following is an Invariant probability measure $$\pi_{k}=\mathbb{E} \left[ \frac{\sum_{j=0}^{\tau_{i}-1}\mathbb{1}_{\{ x_{j}=k \}}}{\mathbb{E}_{i}[\tau_{i}]}\middle|x_{0}=i \right], \, k\in \mathbb{X}$$

Theorem (3.2.2)

For a μ-irreducible Markov chain for which $\mathbb{E}_{\alpha}[\tau_{\alpha}]<\infty$ for Atom $\alpha$, the following is the Invariant probability measure $$\pi(A)=\mathbb{E} \left[ \frac{\sum_{j=0}^{\tau_{\alpha}-1}\mathbb{1}_{\{ x_{j}\in A \}}}{\mathbb{E}_{\alpha}[\tau_{\alpha}]}\middle|x_{0}\in\alpha \right], \, A\in \mathcal{B}(\mathbb{X})$$

Theorem (3.3.1)

Let $\{ x_{t} \}$ be a Weak Feller Markov chain living in a compact subset of a Polish space. Then $\{ x_{t} \}$ admits an Invariant probability measure.

Theorem (3.3.4)

Let $\{ x_{t} \}$ be a μ-irreducible Markov chain which admits an Invariant probability measure. Then, the invariant probability measure is unique.