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International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 155-164
http://dx.doi.org/10.1155/S0161171293000183

The Fréchet transform

Department of Mathematics, University of Central Florida, Orlando 32816, FL, USA

Received 19 April 1991; Revised 7 July 1992

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let F1,,FN be 1-dimensional probability distribution functions and C be an N-copula. Define an N-dimensional probability distribution function G by G(x1,,xN)=C(F1(x1),,FN(xN)). Let ν, be the probability measure induced on N by G and μ be the probability measure induced on [0,1]N by C. We construct a certain transformation Φ of subsets of N to subsets of [0,1]N which we call the Fréchet transform and prove that it is measure-preserving. It is intended that this transform be used as a tool to study the types of dependence which can exist between pairs or N-tuples of random variables, but no applications are presented in this paper.