  This paper studies sparse covariance operator estimation for nonstationary
Gaussian processes with sharply varying marginal variance and small correlation
lengthscale. We introduce a covariance operator estimator that adaptively
thresholds the sample covariance function using an estimate of the variance
components. Building on recent results from empirical process theory, we derive
an operator norm bound on the estimation error in terms of the sparsity level
of the covariance and the expected supremum of the normalized process. Our
theory and numerical simulations demonstrate the advantage of adaptive
threshold estimators over universal threshold and sample covariance estimators
in nonstationary settings.
