Conference Agenda

We are interested in the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and globally. Our approach achieves faster rates through a two-stage process: we apply first the Laplace mechanism to the truncated value and then use the former privatized sample to gain knowledge on the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter α, since the pointwise rates depend on nα² instead of nα⁴. Moreover, we show that the rate 1/(nα⁴) is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the L2 rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona-fide, locally differentially private estimator of the full covariance matrix.