Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series

Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=\mu$ and covariance function $\gamma(\tau)=E(X_t-\mu)(X_{t+\tau}-\mu)$. Let $f(\lambda)$ be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density $f(\lambda)$ can be written as the product of a slowly varying function $\tilde{f}(\lambda)$ and the quantity $\lambda ^{-2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, our approach assures posterior consistency both when $f(\cdot)$ and $d$ are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory set-up, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood.