I'm working on the following question:
Consider an irreducible homogeneous Markov chain over a discrete state space $S$, all of whose states are positive recurrent. Prove that its stationary distribution $\pi$ satisfies the detailed balance conditions (i.e., is reversible) if and only if $$p(x_0,x_1) p(x_1,x_2)\cdots p(x_{n-1},x_n)p(x_n,x_0) = p(x_0,x_n)p(x_n,x_{n-1})\cdots p(x_2,x_1)p(x_1,x_0)$$ for all finite sequences $x_0,x_1,\ldots,x_n \in S$.
I'm having some trouble even concocting ideas for this. Does anyone have any specific hints or advice?