Sharp minimax rates for spectral density estimation under sequential local differential privacy

We determine the minimax integrated squared error for estimating the spectral density of a centered stationary Gaussian time series when each observation is released under local differential privacy and the release may depend on all earlier private outputs. On a variance-bounded class whose normalized covariances have Sobolev smoothness $s>1/2$, the risk is, up to constants, $1\wedge(n\alpha^4)^{-2s/(2s+1)}$, uniformly for $n\ge2$ and $0<\alpha\le1$. Thus sequential interaction preserves the sharp non-interactive rate, without a logarithmic loss. The lower bound follows by representing a covariance perturbation through independent Gaussian innovations. An innovation that enters only one past raw observation remains protected by the same privacy inequality after the entire past output has been released. Gaussian smoothing and the current private channel then give a divergence bound of order $n\alpha^4$. The matching upper bound is the non-interactive estimator of Issartel and Roueff.