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Distance of two laws in case of vector space. (Sur la distance de deux lois dans le cas vectoriel.) (French) Zbl 0808.60008

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 \(Q\).

MSC:

60B05 Probability measures on topological spaces
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