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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 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, φ-mixing and ψ-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 σ-fields and 𝒢 and any r,s0 satisfying r+s<1, one has α 0,0 (,𝒢)α r,s (,𝒢)[α 0,0 (,𝒢)] 1-r-s with

α r,s (,𝒢)=sup|P(AB)-P(A)P(B)|/[P(A)] r [P(B)] s ,A,B𝒢;P(A),P(B)>0·

Reviewer: H.Niemi

MSC:
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