  We consider the statistical experiment given by a sample of a stationary
Gaussian process with an unknown smooth spectral density f. Asymptotic
equivalence, in the sense of Le Cam's deficiency Delta-distance, to two
Gaussian experiments with simpler structure is established. The first one is
given by independent zero mean Gaussians with variance approximately the value
of f in points of a uniform grid (nonparametric Gaussian scale regression).
This approximation is closely related to well-known asymptotic independence
results for the periodogram and corresponding inference methods. The second
asymptotic equivalence is to a Gaussian white noise model where the drift
function is the log-spectral density. This represents the step from a Gaussian
scale model to a location model, and also has a counterpart in established
inference methods, i.e. log-periodogram regression. The problem of simple
explicit equivalence maps (Markov kernels), allowing to directly carry over
inference, appears in this context but is not solved here.
