# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 43: Aperiodicity of Markov Chains

A state $s$ in a Markov chain $X$ is aperiodic iff there exists an $n$ such that

$P(X_{n'} = s | X_0 = s) > 0 \quad \forall n' \geq n$.

That is, a state is aperiodic if the chain can always loop back to this state in an arbitrary number of steps.

A Markov chain is aperiodic iff every state in the chain is aperiodic. If the chain is irreducible, then the existence of a single aperiodic state implies that the entire chain is aperiodic.

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