{"ID":23507304,"CreatedAt":"2026-09-18T02:21:44.056544415Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20469","arxiv_id":"2609.20469","title":"On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes","abstract":"The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many real-world signals exhibiting heavy-tailed or non-Gaussian behavior. In this paper, we extend Whittle's result to multidimensional, real-valued stationary compound Gaussian processes (CGPs) with arbitrary mean, under a unified framework encompassing fully known, parameterized, and completely unknown density generators. Building upon the Slepian-Bangs formula for $n$ consecutive observations and extending it to the semiparametric setting, we leverage the asymptotic properties of block Toeplitz matrices to obtain a closed-form spectral-domain expression for the asymptotic FIM rate. The resulting expression, common to all density generator families, generalizes the standard zero-mean Gaussian formula by incorporating a distribution-dependent contribution associated with the nonzero mean, and an additional covariance term that is invariant to the choice of non-Gaussian distribution. This extension enables efficient and theoretically grounded performance analysis for heavy-tailed signal processing applications.","short_abstract":"The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. Howev...","url_abs":"https://arxiv.org/abs/2609.20469","url_pdf":"https://arxiv.org/pdf/2609.20469v1","authors":"[\"Jean-Pierre Delmas\",\"Habti Abeida\",\"Stefano Fortunati\"]","published":"2026-09-17T14:27:05Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"eess.SP\"]","methods":"[]","has_code":false}