Adaptive estimation of the spectral density of a weakly or strongly dependent Gaussian process

This paper deals with estimation of the spectral density f(x) = |1− eix|−2df∗(x), of a stationary fractional Gaussian process, where −1/2 < d < 1/2 and f∗ is positive. The optimal rate of convergence of an estimate of f is shown not to depend on d but only on the smoothness of f∗, and thus is the same for a long range (d > 0) and a short range dependent (d = 0) process. When the Fourier coefficients of f∗ decrease exponentially fast, the exact asymptotic behaviour of the minimax risk is obtained. The log-periodogram regression estimate is shown to achieve the best possible rate of convergence when the smoothness of f∗ is known, and to have adaptivity property when this smoothness is unknown.