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, -mixing and -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 -fields and and any r,s satisfying , one has with