The Fréchet transform

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.

Keywords

• Fréchet transform
• probability distribution function
• doubly stochastic measure
• copula
• distributions with fixed marginals.