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The Fréchet transform. (English) Zbl 0798.60002

The authors point out that to study dependency relations between random variables \(X\) and \(Y\) with distribution functions \(F\) and \(G\), it is helpful to consider the map \(\Phi(x,y) = (F(x), G(y))\). Moreover, the authors note that while this map is useful, it can introduce some severe complications. Consequently, they propose a new map which is a set transform rather than a point transform and which they hope can be used to simplify the analysis of types of dependence. This new map, called the Fréchet transform of a distribution function \(F\) is a map from the subsets of \(F\) to the subsets of \(I\) given by \[ \Phi(A) =\{ y \in [0,1] \mid\text{there is an } x \in A\text{ such that } F(x) \leq y \leq F(x^ +)\}. \] The authors develop several fundamental properties of this map. A particularly interesting aspect is the development of geometric intuition for the Fréchet transform.

MSC:

60A10 Probabilistic measure theory
28A35 Measures and integrals in product spaces
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