Ergodic State

Definition

A state is ergodic if it is both:

• Positive recurrent: expected return time is finite
• Aperiodic: period $d=1$

Properties

• In a finite irreducible Markov chain, all states are positive recurrent
• If the chain is also aperiodic, all states are ergodic
• For an irreducible ergodic Markov chain, the limiting distribution always exists and equals the unique stationary distribution
• The limiting distribution, when it exists, is always a stationary distribution

About Positive Recurrent

Positive recurrent: “Closed” communicative state. In Markov chain 0→1→2→0, ${\mu }_{00}=3$ because we expect to return to 0 in 3 steps.

When there are other possible states, we deal with the expectation of time to return.

Null recurrent: Random walks on $\mathbb{Z}$ cause ${\mu }_{ii}=\infty$. We can be anywhere on $\mathbb{Z}=(-\infty ,\infty )$, meaning it’s possible to return to state $i$, but the expected time is infinite.

Related

• Recurrent vs Transient
• Period (Stochastic)
• Stationary Distribution
• Positive Recurrent State