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 $f(\lambda )$ can be written as $f(\lambda )=\mathrm{\mid }\lambda {\mathrm{\mid }}^{-2d}g(\mathrm{\mid }\lambda \mathrm{\mid })$, where $0<d<1\mathrm{/}2$ (resp., $-1\mathrm{/}2<d<0$), 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.