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Multilinear forms and measures of dependence between random variables. (English) Zbl 0586.62086
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 {\it 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 ${\cal F}$ and ${\cal G}$ and any r,s$\ge 0$ satisfying $r+s<1$, one has $\alpha\sb{0,0}({\cal F},{\cal G})\le \alpha\sb{r,s}({\cal F},{\cal G})\le [\alpha\sb{0,0}({\cal F},{\cal G})]\sp{1-r-s}$ with $$ \alpha\sb{r,s}({\cal F},{\cal G})=\sup \vert P(A\cap B)-P(A)P(B)\vert /[P(A)]\sp r[P(B)]\sp s,\quad A\in {\cal F},\quad B\in {\cal G};\quad P(A),P(B)>0.$$
Reviewer: H.Niemi

62H20Statistical measures of associations
60E15Inequalities in probability theory; stochastic orderings
62H05Characterization and structure theory (Multivariate analysis)
60B05Probability measures on topological spaces
46M35Abstract interpolation of topological linear spaces
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