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.

翻译：本文研究可逆马尔可夫链产生的自协方差序列估计问题。该问题的研究动机源于马尔可夫链中心极限定理中渐近方差的估计。我们提出了一种新颖的基于形状约束的自协方差序列估计方法，其核心发现是自协方差序列作为矩序列的可表示性会施加特定的形状约束。我们检验了所提估计量的理论性质，并为其提供了强相合性保证。特别地，对于几何遍历可逆马尔可夫链，我们证明了该估计量在$\ell_2$距离下对真实自协方差序列具有强相合性，并由此得到渐近方差的强相合估计。最后，我们通过实证研究展示了所提估计量的理论性质，并将其与当前最先进的马尔可夫链蒙特卡罗方差估计方法（包括批均值法、谱方差估计法和初始凸序列估计法）进行了有效性对比。