Abdellaoui, Taoufiq; Heinich, Henri Distance of two laws in case of vector space. (Sur la distance de deux lois dans le cas vectoriel.) (French) Zbl 0808.60008 C. R. Acad. Sci., Paris, Sér. I 319, No. 4, 397-400 (1994). Summary: Let \(P\) and \(Q\) be two probabilities on a separable Banach space \(\mathbb{E}\). We show that, when \(P\) is \(p\)-diffusive and \(Q\) discrete, there is a unique \(p\)-cyclical monotone function \(f\) said to be associated to \((P,Q)\) such that if \(X\) has law \(P\), then \(f(X)\) has law \(Q\) and \(E(\| X - f(X) \|^ p) = \min \{E(\| X - Y\|^ p)\}\), where the minimum is taken over all pairs \(X\), \(Y\) of random variables such that \(P_ X = P\), \(P_ Y = Q\). For a Hilbert space, \(p = 2\) and \(P\) strongly diffuse, without any condition on \(Q\), there is a unique function \(f\) associated to \((P,Q)\) and this is equivalent to the fact that \(f\) is cyclically monotone, weakly continuous \(P\)-a.s. and has law \(Q\). Cited in 7 Documents MSC: 60B05 Probability measures on topological spaces Keywords:distance of two laws; probabilities on a separable Banach space; strongly diffuse; weakly continuous PDFBibTeX XMLCite \textit{T. Abdellaoui} and \textit{H. Heinich}, C. R. Acad. Sci., Paris, Sér. I 319, No. 4, 397--400 (1994; Zbl 0808.60008)