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On a problem of Dudley. (English. Russian original) Zbl 0587.60014

Sov. Math., Dokl. 29, 162-164 (1984); translation from Dokl. Akad. Nauk SSSR 275, 28-31 (1984).
Let (U,d) denote a separable metric space and let \(D(u_ 1,u_ 2)\) be a real measurable function defined in \(U\times U\). Define \[ P^ 2(P_ 1,P_ 2)=\{L(X_ 1,X_ 2):\quad X_ 1,X_ 2\in {\mathcal X}(U),\quad L(X_ 1)=P_ 1,\quad L(X_ 2)=P_ 2\} \] to be the set of joint distributions of all possible pairs of random variables \(X_ 1\), \(X_ 2\) defined on some probability space \({\mathcal X}(U)\) and taking values in \(U\times U\) with the properties \(L(X_ 1)=P_ 1\) and \(L(X_ 2)=P_ 2.\)
The paper investigates properties of extremal functionals of the type \[ E_ 1D(P_ 1,P_ 2)=\inf \{ED(X_ 1,X_ 2):\quad L(X_ 1,X_ 2)\in P^ 2(P_ 1,P_ 2)\} \]
\[ E_ 2D(P_ 1,P_ 2)=\sup \{ED(X_ 1,X_ 2):\quad L(X_ 1,X_ 2)\in P^ 2(P_ 1,P_ 2)\} \] when D is of the form \(H(d(u_ 1,u_ 2))\) with H(t) as a positive function that is convex on the half-line \(t\geq 0\) and such that \(H(0)=0\) and sup\(\{\) H(2t)/H(t): \(t>0\}<\infty.\)
In this context Dudley’s problem of finding an explicit expression for the functional \([Ed^ p(X_ 1,X_ 2)]^{1/p}\), \(1<p<\infty\), is treated and a solution is provided in the form of a corollary to a more general result.
Reviewer: E.Xekalaki

MSC:

60E05 Probability distributions: general theory
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