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

The Fréchet transform

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

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

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 ℝN by G and μ be the probability measure induced on [0,1]N by C. We construct a certain transformation Φ of subsets of ℝN 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.