Limiting Distribution

Definition

If $\underset{t\to \infty }{\lim }{\mathbf{P}}^t={\mathbf{P}}^{\star }$ exists and the every row of ${\mathbf{P}}^{\star }$ are the same, an arbitrary row vector ${\mathbf{p}}_{\star }^{⊺}$ of the ${\mathbf{P}}^{\star }$ is called the limiting distribution of the stochastic process $\{X_t\}$.

where the $\mathbf{P}$ is the transition matrix of the finite HMC $\{X_t\}$

Facts

$\exists {\mathbf{p}}_{\star }^{⊺}\to \exists !{\mathbf{p}}_{\star }^{⊺}$

If a limiting distribution exists, it is unique.

$\exists {\mathbf{p}}_{\star }^{⊺}\to {\mathbf{p}}_{\star }^{⊺}\mathbf{P}={\mathbf{p}}_{\star }^{⊺}$

If a finite HMC $\{X_t\}$ has a limiting distribution ${\mathbf{p}}_{\star }^{⊺}$, then the limiting distribution ${\mathbf{p}}_{\star }^{⊺}$ satisfies the definition of Stationary Distribution

Ergodic Markov Chain $\to \exists !{\mathbf{p}}_{\star }^{⊺}$

Ergodic Markov Chain $\{X_t\}$ has a unique limiting distribution ${\mathbf{p}}_{\star }^{⊺}$

IRR TR HMC $\{X_t\}$ doesn’t have a limiting distribution because the converged Transition Matrix ${\mathbf{P}}^{\star }=0$

IRR NR HMC $\{X_t\}$ doesn’t have a limiting distribution because the converged Transition Matrix ${\mathbf{P}}^{\star }=0$