The Fréchet transform

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

International Journal of Mathematics and Mathematical Sciences, 1993, vol. 16, 1-10

Abstract:

Let F 1 , … , F N be 1 -dimensional probability distribution functions and C be an N -copula. Define an N -dimensional probability distribution function G by G ( x 1 , … , x N ) = C ( F 1 ( x 1 ) , … , F N ( x N ) ) . Let ν , be the probability measure induced on ℝ 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 ℝ 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.

Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:291508

DOI: 10.1155/S0161171293000183

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