Cuesta-Albertos, J. A.; Rüschendorf, L.; Tuero-Diaz, A. Optimal coupling of multivariate distributions and stochastic processes. (English) Zbl 0788.60025 J. Multivariate Anal. 46, No. 2, 335-361 (1993). Let \(({\mathcal U},{\mathcal B})\) be a measure space and \((x,y)\in M(P,Q)\) be \({\mathcal U}\times {\mathcal U}\)-valued random variables in a common probability space \((\Omega,{\mathcal F},\lambda)\) with marginal distributions \(P\) and \(Q\), respectively. A pair \((x,y)\) is optimal coupling (o.c.) if the Kantorovich functional \(\sigma(P,Q)\) induced by \(\sigma: {\mathcal U}\times {\mathcal U} \to \mathbb{R}_ +\) equals to \(\int \sigma(x,y)d\lambda\). Chosen some \(\sigma\), i.e. some metric, the problem is to characterize such pairs. The authors consider minimal metrics of \(\ell_ p\)-type, spaces \(\mathbb{R}^ N\) and \(\mathbb{R}^ T\) as \(\mathcal U\) and discuss some transformations \(\varphi\) (in particular, radial transformations, positive transformations and monotone transformations of the components) for which \((x,\varphi(x))\) are o.c. pairs. Reviewer: N.Kalinauskaitė (Vilnius) Cited in 22 Documents MSC: 60E05 Probability distributions: general theory 60G99 Stochastic processes 62H20 Measures of association (correlation, canonical correlation, etc.) Keywords:optimal coupling; marginal distributions; Kantorovich functional PDFBibTeX XMLCite \textit{J. A. Cuesta-Albertos} et al., J. Multivariate Anal. 46, No. 2, 335--361 (1993; Zbl 0788.60025) Full Text: DOI