Sufficient condition for a Markov chain to be Aperiodic

Question

If I want to prove that a Markov chain is aperiodic, then if I can show that $P(X_{n+1}=i\mid X_n=i)\gt 0$ $ \forall i$. Then can I say that the chain is aperiodic?

Answer 1

It is true that $p_{ii}(1) > 0$ implies that the state $i$ is aperiodic. So your idea can work if you're lucky.

However, it is possible for a state $i$ to be aperiodic when $p_{ii} (1) = 0$. For example, consider the two-state Markov chain with transition matrix $$ \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22}\end{bmatrix} = \begin{bmatrix} 0 & 1 \\ \tfrac 1 2 & \tfrac 1 2 \end{bmatrix}.$$

Then

$$ p_{11}(1) = 0, \ \ \ \ p_{11}(2) = \tfrac 1 2 , \ \ \ \ p_{11}(3) = \tfrac 1 4 ,$$

so the period of state $1$ is

$$ d_1 = {\rm gcd} \left\{ n \in \mathbb N : p_{11}(n) > 0 \right\} = 1.$$

Thus state $1$ is aperiodic, despite $p_{11}(1)$ being zero.

[Notation: $p_{ij}(n) := P(X_n = j \ | \ X_0 = i)$.]