Communication

Definition

Two states $i$ and $j$ communicate if they are mutually accessible:

Notation: $i \leftrightarrow j$

Communication is an equivalence relation:

Reflexive: $i \leftrightarrow i$ (since $P_{ii}^{0} = 1$ )
Symmetric: If $i \leftrightarrow j$ , then $j \leftrightarrow i$
Transitive: If $i \leftrightarrow j$ and $j \leftrightarrow k$ , then $i \leftrightarrow k$

If $P_{ij}^{n} > 0$ and $P_{jk}^{m} > 0$ , then by Chapman-Kolmogorov:

$P_{ik}^{n + m} = \sum_{r} P_{i r}^{n} P_{r k}^{m} \geq P_{ij}^{n} P_{jk}^{m} > 0$