= Solution
\b[Neither <Markov chain irreducibility> nor fast convergence follows from <detailed balance>.] On two states, the identity <transition matrix> satisfies <detailed balance> for the uniform <probability distribution>, but neither state can reach the other.
Even after imposing <Markov chain irreducibility> and <aperiodicity>, consider
$$
P_\varepsilon=\begin{pmatrix}1-\varepsilon&\varepsilon\\\varepsilon&1-\varepsilon\end{pmatrix},\qquad 0<\varepsilon<\tfrac12.
$$
It satisfies <detailed balance> for $\pi=(1/2,1/2)$. Its nontrivial <eigenvalue> is $1-2\varepsilon$, and, from state zero, its <total variation distance> from $\pi$ after $n$ steps is $\tfrac12(1-2\varepsilon)^n$. This approaches zero arbitrarily slowly as $\varepsilon$ decreases. <Detailed balance without rapid convergence> separates correctness of a <stationary distribution> from reachability and speed.
Back to article page
