For a Markov chain, I define a reversible distribution to be a distribution wrt which the MC is reversible to. A stationary distribution is defined as a distribution that once reached will stay. A reversible distribution is a stationary distribution. But not vice versa.
I was wondering if a MC that has a reversible distribution can have a stationary distribution which is not a reversible distribution?
My question comes from Wikipedia:
Let $X$ be a finite set and let $K(x,y)$ be the transition probability for a reversible Markov chain on $X$. Assume this chain has stationary distribution $\pi$.
$\pi$ seems to be used as a reversible distribution in the article. I was wondering why it doesn't say $\pi$ is a distribution wrt which the MC is reversible.
Thanks and regards!