# “multiple” of markov chain properties

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 $$2)$$.