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 $j$.

Properties

$$\begin{array}{rl}{\pi }_j& =\sum _k{\pi }_k{P}_{kj}\\ \sum _j{\pi }_j& =1\\ {\pi }_j& =\frac{1}{{\mu }_{jj}}\end{array}$$

Related

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