International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1993 / Article

Open Access

Volume 16 |Article ID 291508 | https://doi.org/10.1155/S0161171293000183

Piotor Mikusiński, Morgan Phillips, Howard Sherwood, Michael D. Taylor, "The Fréchet transform", International Journal of Mathematics and Mathematical Sciences, vol. 16, Article ID 291508, 10 pages, 1993. https://doi.org/10.1155/S0161171293000183

The Fréchet transform

Received19 Apr 1991
Revised07 Jul 1992

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.

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.


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