DES - Definition of 'Limit State Probability Vector'

Limit Probability:

Let:

$${\Pi }_i\triangleq \underset{t\to +\infty }{\lim }{\Pi }_i(t)$$

Where:

$${\Pi }_i(t)\triangleq P(X(i)=i)$$

If the limit exists, ${\Pi }_i$ is called limit probability of state $i$. If the limit probabilities exist for all states $i\in \mathcal{X}$,then the row vector

$$\Pi \triangleq \left[{\Pi }_1,{\Pi }_2,{\Pi }_3,\dots \right]$$

Is called limit state probability vector.

Properties of the Limit State Probability Vector:

$i$) Existence

$ii$) Uniqueness

• There exist only one $\Pi$ such that: $\Pi \triangleq {\lim }_{t\to +\infty }\Pi (t)$

$iii$) Consistency

• ${\sum }_i{\Pi }_i=1$

Which are given by this Theorem.

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~Example: