Multilinear forms and measures of dependence between random variables
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 \[ \alpha_{r,s}({\mathcal F},{\mathcal G})=\sup | P(A\cap B)-P(A)P(B)| /[P(A)]^ r[P(B)]^ s,\quad A\in {\mathcal F},\quad B\in {\mathcal G};\quad P(A),P(B)>0. \]
scientific article; zbMATH DE number 3940471
scientific article