Norms on complex matrices induced by random vectors [r-libre/3283]

Morneau-Guérin, Frédéric (2024). Norms on complex matrices induced by random vectors [compte rendu de l'ouvrage de Chávez, Ángel; Garcia, Stephan Ramon et Hurley, Jackson]. zbMATH Open.

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Catégorie de document :	Comptes rendus d'ouvrages
Évaluation par un comité de lecture :	Oui
Étape de publication :	Publié
Résumé :	This paper introduces a family of norms on the space M_n of n x n complex matrices. These norms arise from a probabilistic framework as they are induced by random vectors whose entries are independent and identically distributed (iid) real-valued random variables with sufficiently many moments. Initially, these norms are defined on complex Hermitian matrices as symmetric functions of their (necessarily real) eigenvalues. This contrasts with Schatten p-norms, which are defined in terms of singular values. To be more specific, these random vector norms do not arise from the machinery of symmetric gauge functions. Rather, they are generalizations of the complete homogeneous symmetric (CHS) polynomial norms introduced in Aguilar, K., Chávez, Á., Garcia, S. R., & Volčič, J. (2022). Norms on complex matrices induced by complete homogeneous symmetric polynomials. \textit{Bulletin of the London Mathematical Society, 54}(6), 2078-2100. The paper is organized as follows: In Section 1, the preliminary concepts and notation are covered. Following this, the main result, which is lengthy and highly technical in nature, is stated. Norms arising from familiar distributions (namely Gamma random variables, Normal random variables, Uniform random variables, Laplace random variables, Bernoulli random variables, Finite discrete random variables, Poisson random variables, and Pareto random variables) are examined in Section 2. Various examples and applications are also provided, including a powerful generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials. The proof of the main result, which involves a wide range of topics, such as cumulants, Bell polynomials, partitions, and Schur convexity, is contained in Section 3. The paper concludes with a list of open questions.
Informations complémentaires :	© FIZ Karlsruhe GmbH 2024. ALL RIGHTS RESERVED.
Adresse de la version officielle :	https://zbmath.org/7741833
Déposant:	Morneau-Guérin, Frédéric
Responsable :	Frédéric Morneau-Guérin
Dépôt :	03 juin 2024 13:08
Dernière modification :	03 juin 2024 13:08

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