Limiting Probability

Definition

For a Markov chain $\{X_n,n\ge 0\}$, the limiting probability ${\pi }_j$ of state $j$ represents the long-run probability that the process will be in state $j$, independent of the initial state $i$:

${\pi }_j={\lim }_{n\to \infty }{P}_{ij}^n$

where $P_{ij}^n$ is the n-step transition probability from state $i$ to state $j$.

Relation to Mean Recurrence Time

The limiting probability is related to the mean recurrence time ${\mu }_{jj}$ (expected time to return to state $j$ starting from $j$):

${\pi }_j=\frac{1}{{\mu }_{jj}}$

Related

• Stationary Distribution
• Limiting Distribution Theorem
• Limiting vs Hitting Probability