The Fréchet Transform

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 v 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,l]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. © 1993, Hindawi Publishing Corporation. All rights reserved