Absorbing State
- Definition: A state $i$ is absorbing if $P_{ii}=1$.
- Consequence: Once entered, the chain remains at $i$ forever, so $P_{ij}=0$ for all $j\neq i$.
- Deduction: Since $P_{ii}=1>0$, every absorbing state is automatically aperiodic.
- Structural Property: An absorbing state forms a closed communicating class by itself.