  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.
