  The asymptotic variance is an important criterion to evaluate the performance
of Markov chains, especially for the central limit theorems. We give the
variational formulas for the asymptotic variance of discrete-time
(non-reversible) Markov chains on general state space. The variational formulas
provide many applications, extending the classical Peskun's comparison theorem
to non-reversible Markov chains, and obtaining several comparison theorems
between Markov chains with various perturbations.
