Multilinear forms and measures of dependence between random variables
type: Item
instance of: scholarly article
title: Multilinear forms and measures of dependence between random variables
zbMATH Open document ID: 0586.62086
DOI: 10.1016/0047-259X(85)90025-9
author: Włodek Bryc
author: Richard C. Bradley
published in: Journal of Multivariate Analysis
publication date: 1985-01-01T00:00:00Z
review text: Equivalence relations for different dependence measures between two or more given families of real-valued and, respectively, Hilbert-space valued random variables are derived. The authors follow the ideas of \textit{M. Rosenblatt} [in M. L. Puri (ed.), Nonparametric techniques in statistical inference, pp. 199--210 (1970; Zbl 0209.21102)] and apply functional analytical convexity and interpolation methods. The paper is partly expository. Different measures of dependence, e.g. strong mixing, \(\phi\)-mixing and \(\psi\)-mixing conditions as well as the needed Riesz-Thorin and Marcinkiewicz interpolation theorems and their multilinear extensions are reviewed. A typical result is e.g. the following one: For any \(\sigma\)-fields \({\mathcal F}\) and \({\mathcal G}\) and any r,s\(\geq 0\) satisfying \(r+s<1\), one has \(\alpha_{0,0}({\mathcal F},{\mathcal G})\leq \alpha_{r,s}({\mathcal F},{\mathcal G})\leq [\alpha_{0,0}({\mathcal F},{\mathcal G})]^{1-r-s}\) with \[ \alpha_{r,s}({\mathcal F},{\mathcal G})=\sup | P(A\cap B)-P(A)P(B)| /[P(A)]^ r[P(B)]^ s,\quad A\in {\mathcal F},\quad B\in {\mathcal G};\quad P(A),P(B)>0. \]
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 46M35
zbMATH DE Number: 3940471
zbMATH Keywords: multilinear forms
zbMATH Keywords: moment inequalities
zbMATH Keywords: multilinear operators
zbMATH Keywords: Equivalence relations
zbMATH Keywords: dependence measures
zbMATH Keywords: Hilbert-space valued random variables
zbMATH Keywords: mixing conditions
zbMATH Keywords: Riesz-Thorin
zbMATH Keywords: Marcinkiewicz interpolation theorems
MaRDI profile type: Publication
cites work: Q4112938
cites work: On the strong law of large numbers for a class of stochastic processes
cites work: Equivalent measures of dependence
cites work: Q3670282
cites work: Q3268586
cites work: Convergence of Distributions Generated by Stationary Stochastic Processes
cites work: The Invariance Principle for Stationary Processes
cites work: Almost sure invariance principles for weakly dependent vector-valued random variables
cites work: On U-statistics and v. mise? statistics for weakly dependent processes
cites work: A note on empirical processes of strong-mixing sequences
cites work: Q3038274
cites work: Q5671422
cites work: Interpolation of Orlicz spaces
cites work: Q5612941
cites work: Q5759902
cites work: Q3258161
cites work: Some Limit Theorems for Stationary Processes
cites work: A Note on the Central Limit Theorems for Dependent Random Variables
cites work: Q5624460
cites work: Examples of mixing sequences
cites work: Q4136031
cites work: Central Limit Theorem for Stationary Processes in Hilbert Space
cites work: Properties of fourth-order strong mixing rates and its application to random set theory
cites work: Convolution operators and L(p, q) spaces
cites work: A new approach in interpolation spaces
cites work: A note on two measures of dependence and mixing sequences
cites work: Q4183861
cites work: A proof of the Grothendieck inequality
cites work: A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION
cites work: Q5646158
cites work: Q4142391
cites work: Q3254065
cites work: Some Limit Theorems for Random Functions. I
cites work: Central Limit Theorems for dependent variables. I
cites work: Q3957705
cites work: Estimates of the higher-order spectral densities of stationary processes which satisfy the Cramer condition with 'Rosenblatt' mixing
full work available at URL: 0047-259x(85)90025-9
OpenAlex ID: W1974399474
description: scientific article; zbMATH DE number 3940471