  In this paper, we study the problem of estimating the autocovariance sequence
resulting from a reversible Markov chain. A motivating application for studying
this problem is the estimation of the asymptotic variance in central limit
theorems for Markov chains. We propose a novel shape-constrained estimator of
the autocovariance sequence, which is based on the key observation that the
representability of the autocovariance sequence as a moment sequence imposes
certain shape constraints. We examine the theoretical properties of the
proposed estimator and provide strong consistency guarantees for our estimator.
In particular, for geometrically ergodic reversible Markov chains, we show that
our estimator is strongly consistent for the true autocovariance sequence with
respect to an $\ell_2$ distance, and that our estimator leads to strongly
consistent estimates of the asymptotic variance. Finally, we perform empirical
studies to illustrate the theoretical properties of the proposed estimator as
well as to demonstrate the effectiveness of our estimator in comparison with
other current state-of-the-art methods for Markov chain Monte Carlo variance
estimation, including batch means, spectral variance estimators, and the
initial convex sequence estimator.
