# Coupling of probability distributions and an extremal problem for the divergence

Coupling of probability distributions and an extremal problem for the divergence 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 α-coupling of these random variables, i.e., for the existence of their joint distribution such that Pr{X = Y} = α, where α, 0 ≤ α ≤ 1, 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 Pr{Z = X} = α. An explicit solution for this problem is also obtained. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Coupling of probability distributions and an extremal problem for the divergence

, Volume 51 (2) – Jul 7, 2015
8 pages

/lp/springer_journal/coupling-of-probability-distributions-and-an-extremal-problem-for-the-YXPIx2wXms
Publisher
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S003294601502009X
Publisher site
See Article on Publisher Site

### 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 α-coupling of these random variables, i.e., for the existence of their joint distribution such that Pr{X = Y} = α, where α, 0 ≤ α ≤ 1, 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 Pr{Z = X} = α. An explicit solution for this problem is also obtained.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 7, 2015

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