On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes
Abstract
The classical Whittle formula provides a closedform 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 maximumlikelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many realworld 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.
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BibTeX
@misc{delmas2026parametric,
title = {{On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes}},
author = {Jean-Pierre Delmas and Habti Abeida and Stefano Fortunati},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20469},
eprint = {2609.20469},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05753545v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes
AU - Jean-Pierre Delmas
AU - Habti Abeida
AU - Stefano Fortunati
PY - 2026
DA - 2026/09/17
PB - HAL
DO - 10.48550/arxiv.2609.20469
UR - https://hal.science/hal-05753545v1
AB - The classical Whittle formula provides a closedform 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 maximumlikelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many realworld 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.
ER -