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Mass transport problem and derivation

Nacereddine BeliliHenri Heinich — 1999

Applicationes Mathematicae

A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure λ f * : = λ ( f * ) - 1 , where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

Median for metric spaces

Nacereddine BeliliHenri Heinich — 2001

Applicationes Mathematicae

We consider a Köthe space ( , | | · | | ) of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that | | d ( X , Y ) | | | | d ( X , Z ) | | for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give some applications....

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