International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 155-164

The Fréchet transform

Piotor Mikusiński, Morgan Phillips, Howard Sherwood, and Michael D. Taylor

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

Received 19 April 1991; Revised 7 July 1992

Copyright © 1993 Piotor Mikusiński et al. 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.


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.