Communication

Definition

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

• $i\to j$ (accessible)
• $j\to i$ (accessible)

Notation: $i\leftrightarrow j$

Properties

Communication is an equivalence relation:

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

Proof of Transitivity

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

$P_{ik}^{n+m}={\sum }_r{P}_{ir}^n{P}_{rk}^m\ge P_{ij}^n{P}_{jk}^m>0$

Related

• Accessible State
• Communicating Classes
• Irreducible