A stationary Gaussian process is said to be long-range dependent(resp., anti-persistent)if its spectral density f(λ)can be written as f(λ)= |λ|~2dg(|λ|), 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.
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