Limiting Distribution Theorem

Theorem

Let $\mathbf{P}$ be a transition probability matrix (TPM) on finite state space.

Then there exists $\pi =({\pi }_0,{\pi }_1,\dots ,{\pi }_N)$ such that:

${\pi }_j={\lim }_{n\to \infty }{P}_{ij}^{(n)},\forall i,j$

The limiting distribution satisfies: $\pi =\pi \mathbf{P}\text{and}{\sum }_j{\pi }_j=1$

Interpretation

For regular chains, the limiting distribution exists and equals the unique stationary distribution. The long-run behavior is independent of the initial state.

Related

• Limiting Probability
• Stationary Distribution
• Regular TPM