The accuracy of a Markov chain Monte Carlo (MCMC) estimator is dictated by the spectral gap of the Markov operator defined by the chain’s transition density. When the Markov operator is non-negative and trace-class, its second largest eigenvalue (and hence its spectral gap) can be bounded above and below by functions of the power sums of its eigenvalues. For a certain class of MCMC algorithms, the power sums have simple integral representations. A classical Monte Carlo method is proposed to estimate the integrals, and a simple sufficient condition for finite variance is provided. This leads to asymptotically valid confidence intervals for the second largest eigenvalue (and spectral gap) of the Markov operator. For illustration, the method is applied to Albert and Chib’s data augmentation (DA) algorithm for Bayesian probit regression, and also to a DA algorithm for Bayesian linear regression with non-Gaussian errors.