Gebelein inequality in a Hilbert space
type: Item
instance of: scholarly article
title: Gebelein inequality in a Hilbert space
zbMATH Open document ID: 1543.60005
author: Marek Beśka
published in: Applicationes Mathematicae
publication date: 2024-07-09T00:00:00Z
review text: Letting \(\mu\) be the standard Gaussian measure and \(f\in L^2(\mu)\) with \(\int_\mathbb{R}f(x)\text{d}\mu(x)=0\), the Gebelein inequality states that \(\lVert P_\rho f\rVert_2\leq|\rho|\lVert f\rVert_2\) for \(-1\leq\rho\leq1\), where \(P_\rho\) is the Ornstein--Uhlenbeck operator defined by\N\[\N(P_\rho f)(y)=\int_\mathbb{R}f\left(\rho y+\sqrt{1-\rho^2}z\right)\,\text{d}\mu(z)\N\]\Nfor \(y\in\mathbb{R}\) and \(f\in L^2(\mu)\). The author generalises this inequality to a real, separable Hilbert space, and discusses applications to establishing strong laws of large numbers for Gaussian functionals.
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60B11
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 7878419
zbMATH Keywords: Gaussian random vector
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Cameron-Martin space
zbMATH Keywords: white noise mapping
zbMATH Keywords: Ornstein-Uhlenbeck operator
MaRDI profile type: Publication
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DOI: 10.4064/AM2478-1-2024
description: scientific article; zbMATH DE number 7878419