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() = ()-2dg(()), where 0 < d < 1/2(resp. -1/2 < d < 0), and g is continuous. We propose a novel Bayesian nonparametricapproach for the estimation of the spectral density of such processes. Within this approach,we prove posterior consistency for both d and g, under appropriate conditions on the priordistribution. We establish the rate of convergence for a general class of priors, and applyour results to the family of fractionally exponential priors. Our approach is based on thetrue likelihood function, a