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Let $A_n$ be a markov chain with transition matrix $P$ and $B$ be a chain defined as $B_n = A_{mn}$ for some positive integer $m$.

First, I was able to show already that $B_n$ is also a markov chain with transition matrix $P^m.$

Now we are interested in looking at some properties of $A_n$ and seeing if they carry to $B_n$. I have several questions!

1) If $A$ is aperiodic, is $B$? If $A$ is aperiodic, then $gcd(\{n > 0: P(A_n = a \vert A_0 = a\}) = 1$. The corresponding statement for $B$ is: $$gcd(\{n > 0: P(B_n = a \vert B_0 = a\}) \\ = gcd(\{n > 0: P(A_{mn} = a \vert B_0 = a\}) \\ = gcd(\{n > 0: P(A_{mn} = a \vert A_0 = a\})$$ I can't think of why this last expression wouldn't be $1$ (it's still the $gcd$'s over all $n$'s), but I also don't see it intuitively. I also tried and failed to come up with a counterexample for when $A$ would be aperiodic and $B$ as described would be periodic, but I am very unsure.

2) If $A$ is recurrent, is $B$? If $A$ is recurrent, then $P(\text{ever revisit state i} \vert A_0 = i) = 1$. The corresponding statement for $B$ is: $$ P(\text{ever revisit state i} \vert B_0 = i) \\ = P(\text{ever revisit state i} \vert A_0 = i)$$ because $A_0 = B_0$, and this quantity is by assumption $1$. Am I grossly oversimplifying things?

3) If $A$ is transient, is $B$? EDIT: this seems true.

EDIT: I am looking for the most assistance on question $2)$.

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