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

This Note presents an estimator of the spectral density of a fractional Gau ssian process, f(x) = \1 - e(ix)\(-2d) f*(x), where -1/2 < d < 1/2 and f* i s positive. The rate of convergence of an estimator of f is shown not to de pend on d but only on the smoothness of f*, and thus is the same for a long range and a short range dependent process. When the Fourier coefficients o f f* decrease exponentially fast, an exact constant is obtained. The log-pe riodogram estimator is shown to achieve the best possible rate of convergen ce when the smoothness of f* is known, and to have adaptivity property when this smoothness is unknown. (C) 2000 Academie des sciences/Editions scient ifiques et medicales Elsevier SAS.