Stationary Distribution

Definition

A probability vector $\pi =({\pi }_0,{\pi }_1,\dots )$ is a stationary distribution if it satisfies:

$\pi =\pi \mathbf{P}\text{and}{\sum }_j{\pi }_j=1$

Interpretation

If the initial state $X_0$ is chosen according to $\pi$, then the probability of being in state $j$ at any time $n$ is also ${\pi }_j$ (i.e., $P(X_n=j)={\pi }_j$ for all $n$).

Properties

• A stationary distribution may exist without a limiting distribution
• For an irreducible ergodic Markov chain, the limiting distribution always exists and equals the unique stationary distribution
• Periodic chains can have stationary distributions but no limiting distributions

Example

For TPM $\mathbf{P}=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\\ 0& \frac{1}{2}& \frac{1}{2}\end{array}\right]$, solving $\pi =\pi \mathbf{P}$:

• ${\pi }_1=\frac{1}{4}$
• ${\pi }_2=\frac{1}{2}$
• ${\pi }_3=\frac{1}{4}$

Related

• Limiting Probability
• Limiting Distribution Theorem
• Ergodic State