Bayesian nonparametric estimation of the spectral density of a long or intermediate memory gaussian process

A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density f can be written as f = 2d g, where 0 < d <1/2 (resp., -1/2 < d <0), and g is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both d and g, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our approach is based on the true likelihood and does not resort to Whittle's approximation.