Let $S$ be a countable set and $\pi$ a probability distribution on $S$. A discrete-time Markov chain $(X_n)$ with state space $S$ is said to be in detailed balance with respect to $\pi$ (or simply in detailed balance) if for all states $x$ and $y$, $$\pi(x)P(x \to y) = \pi(y)P(y \to x).$$ (a) Show that if $(X_n)$ satisfies detailed balance, then $\pi$ is a stationary distribution for $(X_n)$.
(b) Consider the general two-state chain with $P(0 \to 1) = p$ and $P(1 \to 0) = q$, where $p,q > 0$. Let $\pi$ be the (unique) stationary distribution. Show the two-state chain always satisfies detailed balance with respect to $\pi$.
(c) Find an irreducible 3-state chain that does not satisfy detailed balance.
(d) Show that any irreducible, positive-recurrent birth-death process satisfies detailed balance with respect to its (unique) stationary distribution.