  We present a novel weighted $\ell_2$ projection method for estimating
autocovariance sequences and spectral density functions from reversible Markov
chains. Berg and Song (2023) introduced a least-squares shape-constrained
estimation approach for the autocovariance function by projecting an initial
estimate onto a shape-constrained space using an $\ell_2$ projection. While the
least-squares objective is commonly used in shape-constrained regression, it
can be suboptimal due to correlation and unequal variances in the input
function. To address this, we propose a weighted least-squares method that
defines a weighted norm on transformed data. Specifically, we transform an
input autocovariance sequence into the Fourier domain and apply weights based
on the asymptotic variance of the sample periodogram, leveraging the asymptotic
independence of periodogram ordinates. Our proposal can equivalently be viewed
as estimating a spectral density function by applying shape constraints to its
Fourier series. We demonstrate that our weighted approach yields strongly
consistent estimates for both the spectral density and the autocovariance
sequence. Empirical studies show its effectiveness in uncertainty
quantification for Markov chain Monte Carlo estimation, outperforming the
unweighted moment LS estimator and other state-of-the-art methods.
