Sur une inégalité de Littlewood-Salem. (On a Littlewood-Salem inequality)
type: Item
instance of: scholarly article
title: Sur une inégalité de Littlewood-Salem. (On a Littlewood-Salem inequality)
zbMATH Open document ID: 1019.42004
DOI: 10.1016/S0246-0203(02)00014-6
author: Dominique Schneider
author: Ai-Hua Fan
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2003-04-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77759
review text: The random series \(\sum^\infty_{k= 1} X_k f(n_k x)\) is considered, where \(\{X_k\}\) is a sequence of independent complex random variables with \(EX_k= 0\) and \(E|X_k|^2< \infty\), \(k\geq 1\); \(f\) is a function in the class \(A(\mathbb{T})\), that is, \[ \|f\|_{A(\mathbb{T})}:= \sum_{n\in\mathbb{Z}} |\widehat f(n)|< \infty, \] and \(\{n_k\}\) is a strictly increasing sequence of positive integers. Let \(\sigma^2_N:= \sum^N_{k=1} E|X_k|^2\). Under a certain condition on the sequence \(\{X_k\}\), the following is proved \[ \int_{\Omega} \sup_{N\geq 1} \sup_{x\in\mathbb{T}} \Biggl|\sum^N_{k= 1} X_kf(n_k x)\Biggr|/\{\sigma^2_N\log n_N\}^{1/2} d\mathbb{P}< \infty. \] This inequality is applied to study the local regularities of random series of Rademacher type \(\sum a_k\varepsilon_k f(n_kx)\) and nonstationary Gaussian type \(\sum a_k g_kf(n_k x)\). The case when the \(n_k\) are real numbers is also studied.
Mathematics Subject Classification ID: 42A61
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 1901038
zbMATH Keywords: Littlewood-Salem inequality
zbMATH Keywords: random polynomial
zbMATH Keywords: uniform norm
zbMATH Keywords: spectral measure
zbMATH Keywords: Gaussian process
zbMATH Keywords: local regularities
zbMATH Keywords: random series of Rademacher type
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901038