Estimating the second largest eigenvalue of a Markov transition matrix

The number of iterations required to estimate accurately the stationary distribution of a Markov chain is determined by a preliminary sample to estimate the convergence rate, which is related to the second largest eigenvalue of the transition operator. The estimator of the second largest eigenvalue, along with those of two nuisance parameters, can be shown to converge to their true values in probability, and a form of the central limit theorem is proved. Explicit expressions for the bias and variance of the asymptotic distribution of this estimator are derived. A theoretical standard is derived against which other estimators of the second largest eigenvalue may be judged. An application is given involving the use of the Gibbs sampler to calculate a posterior distribution.