arXiv:2611.13037v1 Announce Type: new Abstract: This paper develops an information-geometric framework for stationary covariance structures. We begin with the infinite-dimensional setting, where stationary covariance structures can be characterized without finite-window boundary effects. By formulating the covariance structure as a bilinear form on a suitable class of test functions, we establish its spectral representation and show how finite-dimensional Toeplitz covariance matrices arise as restrictions to nested observation subspaces. For zero-mean proper complex Gaussian wide-sense stationary (WSS) processes, we show that the normalized finite-dimensional Fisher information converges to a spectral-domain metric. In logarithmic spectral coordinates, this metric becomes the standard $L^2$ inner product, yielding an explicit Fisher--Rao distance that is also recovered as the asymptotic normalized affine-invariant Riemannian distance between Toeplitz covariance matrices. We then extend the framework to elliptical observation models by separating global covariance scale from shape. The shape Fisher information is orthogonal to radial nuisance directions and is characterized by a dimension- and radial-model-dependent coefficient. We derive a condition under which this coefficient converges to its Gaussian value as the observation dimension tends to infinity. The resulting limiting spectral-shape geometry is the centered $L^2$ geometry of log-spectra and is invariant to positive rescaling. Numerical experiments illustrate the theoretical convergence results and the recovery of the Gaussian shape coefficient.