Irreducible Markov Chain

Definition

State space $E$ which has only one Communication Class. where $\{X_t\}$ is a HMC defined on the state space $E$ and the $\mathbf{P}$ is a Transition Matrix of $\{X_t\}$

$\forall x,y,\in E:\exists \text{ path }x\to \cdots \to y\text{ or }y\to \cdots \to x$ where $x\to y\iff \exists T_0\in {\mathbb{N}}_0:p_{ij}^{(T_0)}>0$ and $p_{ij}^{(T_0)}$ is the probability of departing from $x$ and reaching $y$ after $T_0$

Facts

If HMC $\{X_t\}$ has finite state space and is irreducible, then the unique Stationary Distribution $\mathbf{\pi }$ of $\{X_t\}$ exists, and the probability of every state is positive.

Since Communication is a Equivalence Class, the reducible Markov chain can be partitioned into the irreducible Markov chains.