Mean Time Spent in Transient States

Definition

For a finite Markov chain, let $T=\{1,2,\dots ,t\}$ be the set of transient states.

The transition matrix restricted to transient states is $\mathbf{Q}$ (or ${\mathbf{P}}_T$).

The matrix $\mathbf{\mu }$ contains the expected number of time periods spent in transient state $j$ given starting in transient state $i$:

$\mathbf{\mu }=(\mathbf{I}-\mathbf{Q}{)}^{-1}$

Properties

• ${\mu }_{ij}$ (or ${\mu }_i(j)$): Expected number of visits to state $j$ starting from $i$
• $\mathbf{I}$: Identity matrix of size $t$

Relation to Absorption Time

The mean time matrix $\mathbf{\mu }$ relates to expected time to absorption ${\mu }_i$:

${\mu }_i={\sum }_j{\mathbf{\mu }}_{ij}$

Where:

• ${\mu }_i$: single value representing total time until any absorption
• $\mathbf{\mu }$: matrix containing detailed information about visits to each transient state $j$ before absorption

Related

• Hitting Probability (Transient)
• First Step Analysis