Stationary Distribution

Definition

Invariant Measure satisfying definition of distribution

distribution $\mu$ that satisfies $\mu p=\mu$ where $\mu$ is a Distribution defined on Measurable Space $(E,\mathcal{B}(E))$ and $\mu p(\{x\}):={\sum }_{y\in E}\mu (\{y\})p_{yx}$

distribution $\mu$ that satisfies ${\mu }^{⊺}\mathbf{P}={\mu }^{⊺}$ where $\mathbf{P}$ is a Transition Matrix of the Markov Chain $\{X_t\}$

Facts

When $\mu$ is a stationary distribution of the finite stochastic process $\{X_t\}$, it becomes a PMF of the random variable $X_{\infty }$

For the finite states case, the stationary distribution can be obtained using Eigendecomposition. The stationary distribution is the unit eigenvector^[whose Norm is one] corresponding to the eigenvalue with value one.

Finite HMC has at least one Stationary Distribution

If we set the initial distribution ${\mathbf{\mu }}_0$ of the stochastic process to a stationary distribution $\mu$, the stochastic process $\{X_t\}$ has the same distributions for every $t$ (not independent)

$\{X_t\}$ is a IRR-HMC $\to \exists \mathbf{\pi }\iff \exists !\mathbf{\pi }\iff$ PR

If $\{X_t\}$ is a IRR HMC, then the $\{X_t\}$ has a Stationary Distribution $\iff$ $\{X_t\}$ has a unique Stationary Distribution $\iff$ Positive recurrent