Is whether or not a Markov Chain is periodic dependent on the initial distribution?

Question

I know that whether or not a MC is irreducible does not depend on the initial distribution (it depends only on the transitional matrix). But is the same (independence) true for periodicity?

Answer 1

The period of a state $i$ in a Markov chain is $\gcd\{n > 0: P(X_n=i \mid X_0=i) > 0\}$, and a state is aperiodic if its period is $1$. You can see that the period depends only on the quantities $$P(X_n=i \mid X_0=i) = \sum_{x_{n-1},\ldots,x_1} P(X_n=i \mid X_{n-1}=x_{n-1}) \cdot P(X_{n-1}=x_{n-1} \mid X_{n-2} = x_{n-2}) \cdots P(X_2=x_2 \mid X_1=x_1)\cdot P(X_1=x_1 \mid X_0=i)$$ so it does not depend on the initial distribution.