The question:
Q. Let $\pi$ and $\pi^{\prime}$ be two distinct stationary distributions of a Markov chain with state space $S$. We say $\pi^{\prime}$ is absolutely continuous with respect to $\pi$ if, for every $x \in S, \pi(x)=0$ implies $\pi^{\prime}(x)=0$
Assume $\pi^{\prime}$ is absolutely continuous with respect to $\pi$. Prove that if the Markov chain with initial distribution $\pi$ is reversible, then the Markov chain with initial distribution $\pi^{\prime}$ is also reversible.
Is the above still true if we remove the absolute continuity assumption? Explain why.
My attempt at this question:
$\pi$ and $\pi'$ are two distinct stationary distributions. First, we prove that $\pi$ is reversible. Assume that the two distributions have the transitions probabilities $p_{i j}$.
(Side note: Reversibility condition is as follows:
A Markov chain with invariant measure $\pi$ is reversible if and only if $$ \pi_{i} P_{i j}=\pi_{j} P_{j i} $$ for all states $i$ and $j$.)
i.e. \begin{aligned} &P_{ji} = P\left(X_{k}=i \mid X_{k+1}=j, X_{k+2}=i_{k+2}, \ldots, X_{n}=i_{n}\right) \\ &=\frac{P\left(X_{k}=i, X_{k+1}=j, X_{k+2}=i_{k+2}, \ldots, X_{n}=i_{n}\right)}{\left.X_{k+1}=j, X_{k+2}=i_{k+2}, \ldots, X_{n}=i_{n}\right)} \\ &=\frac{\pi_{i} P_{i j} P_{j i_{k+2}} \cdots P_{i_{n-1} i_{n}}}{\pi_{j} P_{j i_{k+2}} \cdots P_{i_{n-1} i_{n}}} \\ &=\frac{\pi_{i} P_{i j}}{\pi_{j}} \end{aligned}
Thus, $ P_{ij} \pi_{i} = P_{ji} \pi_{j} $
Now, we prove the other way, i.e. $P\left(X_{k+1}=i \mid X_{k}=j\right)=P_{j i}$
i.e. \begin{aligned} &P_{ji} = P\left(X_{k+1}=i \mid X_{k}=j, X_{k-1}=i_{k-1}, \ldots, X_{0}=i_{0}\right) \\ &=\frac{P\left(X_{k+1}=i, X_{k}=j, X_{k-1}=i_{k-1}, \ldots, X_{0}=i_{0}\right)}{\left.X_{k}=j, X_{k-1}=i_{k-1}, \ldots, X_{0}=i_{0}\right)} \\ &=\frac{\pi_{j} P_{j i} P_{i j_{k-1}} \cdots P_{i_{1} i_{0}}}{\pi_{i} P_{i j_{k-1}} \cdots P_{i_{1} i_{0}}} \\ &=\frac{\pi_{j} P_{ j i}}{\pi_{i}} \end{aligned}
Thus, $ P_{ij} \pi_{i} = P_{ji} \pi_{j} $
Hence, we can say that $\pi$ is reversible and since we know that $\pi$ and $\pi'$ are absolutely continuous, $\pi'$ is also reversible by the same logic.
I'm not sure how to explain the answer if we remove the absolute continuity condition.
Does this answer make sense? Please let me know how to correct it if it's wrong. Thank you!!