A characterization of uniform distribution

• Autorzy: Chachulska J.

Identyfikatory

DOI

10.4064/ba53-2-9

• Języki publikacji: EN

Abstrakty

EN

Is the Lebesgue measure on [0,1]² a unique product measure on [0,1] ² which is transformed again into a product measure on [0,1]² by the mapping ψ(x, y) = (x, (x+y) mod 1))? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables X and Y constancy of the conditional expectations of X+Y−I(X+Y>1) and its square given X identifies uniform distribution either absolutely continuous or discrete. No assumptions are imposed on the supports of the distributions of X and Y.

Słowa kluczowe

EN

uniform distribution; constancy of regression; characterization of probability distribution; probability distribution on algebraic structures

PL

rozkład jednostajny; stała regresji; charakteryzacja rozkładu prawdopodobieństwa; rozkład prawdopodobieństwa na strukturach algebraicznych

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 2
• Strony: 207--220
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Chachulska J.

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland,

Bibliografia

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• [5] —, Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, Amer. Math. Soc., Providence, RI, 1993.
• [6] —, A characterization of Gaussian distributions on Abelian groups by constancy of regression, Theory Probab. Appl. 43 (1999), 477–480.
• [7] —, A characterization of the Gaussian distribution on Abelian groups, Probab. Theory Related Fields 126 (2003), 91–102.
• [8] G. M. Feldman and P. Graczyk, On the Skitovitch–Darmois theorem for compact Abelian groups, J. Theoret. Probab. 13 (2000), 859–869.
• [9] —, —, Independent linear statistics on finite Abelian groups, Ukrainian Math. J. 53 (2001), 499–506.
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• [11] J. H. Stapleton, A characterization of the uniform distribution on a compact topological group, Ann. Math. Statist. 34 (1963), 319–326.