Optimal coupling of multivariate distributions and stochastic processes
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.
scientific article; zbMATH DE number 445133