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JOURNALS // Problemy Peredachi Informatsii // Archive

Probl. Peredachi Inf., 2015 Volume 51, Issue 2, Pages 114–121 (Mi ppi2174)

This article is cited in 5 papers

Large Systems

Coupling of probability distributions and an extremal problem for the divergence

V. V. Prelov

Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

Abstract: Let $X$ and $Y$ be discrete random variables having probability distributions $P_X$ and $P_Y$, respectively. A necessary and sufficient condition is obtained for the existence of an $\alpha$-coupling of these random variables, i.e., for the existence of their joint distribution such that $\operatorname{Pr}\{X=Y\}=\alpha$, where $\alpha$, $0\le\alpha\le1$, is a given constant. This problem is closely related with the problem of determining the minima of the divergences $D(P_Z\,\|\,P_X)$ and $D(P_X\,\|\,P_Z)$ over all probability distributions $P_Z$ of a random variable $Z$ given $P_X$ and under the condition that $\operatorname{Pr}\{Z=X\}=\alpha$. An explicit solution for this problem is also obtained.

UDC: 621.391.1+519.2

Received: 13.01.2015
Revised: 14.05.2015


 English version:
Problems of Information Transmission, 2015, 51:2, 192–199

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© Steklov Math. Inst. of RAS, 2025