Aperiodic Markov Chain

Definition

Markov Chain whose period is one.

Facts

If there is a probability of returning to oneself after 1 period or there exist non-zero diagonal elements of the Transition Matrix, then the Markov always aperiodic

Every state $x\in E$ of finite IRR HMC has the same period.

If HMC is IRR and recurrent, then every state $x\in E$ has the same period.

Examples

Periodic Markov Chain with 3-period

flowchart LR  
  0 -->|1| 1  
  1 -->|1| 3  
  2 -->|1| 1  
  3 -->|1/3| 0   
  3 -->|2/3| 2  

Aperiodic Markov Chain with 1-period

flowchart LR
  0 -->|1| 1
  1 -->|1/2| 0
  1 -->|1/2| 2
  2 -->|1| 3
  3 -->|1| 1