Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields

Notarnicola, Massimo

Annales Henri Lebesgue, Tome 6 (2023), pp. 975-1030

We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis. We argue that these are well-suited for expressing the Wiener–Itô chaos expansion of functionals of the spectral measure associated with Gaussian matrices. More specifically, we obtain the Wiener chaos expansion of Gaussian determinants of the form $det{\left(X{X}^T\right)}^{1/2}$ and prove that, in the setting where the rows of $X$ are i.i.d. Gaussian vectors, its projection coefficients admit a geometric interpretation in terms of intrinsic volumes of ellipsoids, thus extending the framework of Kabluchko and Zaporozhets (2012). Our proofs rely on a crucial relation between Hermite polynomials and Laguerre polynomials. We introduce the matrix analog of the classical Mehler’s formula for the Ornstein-Uhlenbeck semigroup and prove that matrix Hermite polynomials are eigenfunctions of these operators. We apply our results to the asymptotic study of a total variation associated with vectors of Arithmetic Random Waves on the full three-torus.

Nous étudions les polynômes de Hermite généralisés dont la variable est une matrice rectangulaire. Nous montrons que ceux-ci sont adaptés pour obtenir la décomposition en chaos de Wiener–Itô de variables aléatoires qui dépendent de la mesure spectrale associée avec une matrice de loi normale. Plus précisément, nous obtenons la décomposition en chaos de déterminants gaussiens de la forme $det{\left(X{X}^T\right)}^{1/2}$ et montrons que, dans le cas où les lignes de $X$ sont des vecteurs gaussiens i.i.d, les coefficients de projection associés avec cette décomposition admettent une interprétation géométrique en termes du volume intrinsèque d’elliposoides, permettant ainsi de généraliser un résultat de Kabluchko et Zaporozhets (2012). Notre démonstration repose sur une relation entre les polynômes de Hermite et les polynômes de Laguerre. Dans une deuxième partie, nous introduisons l’analogue matriciel de la formule de Mehler pour l’opérateur d’Ornstein–Uhlenbeck et déduisons que les polynômes de Hermite généralisés sont des fonctions propres de ces opérateurs. Nous appliquons nos résultats à l’étude asymptotique d’une notion de variation totale associée aux ondes aléatoires arithmétiques définies sur le tore à trois dimensions.

Classification : 60G60, 60B10, 60D05, 58J50, 35P20
Keywords: Generalized Hermite polynomials, Gaussian random matrices, Zonal polynomials, Wiener chaos expansions, Intrinsic and mixed volumes, Arithmetic Random Waves, Limit Theorems

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