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Constraints on distributions imposed by properties of linear forms

Published online by Cambridge University Press:  15 May 2003

Denis Belomestny*
Affiliation:
Institute fur Angewandte Mathematik, Universität Bonn, Interdisziplinares Zentrum für Komplexe Systeme, Meckenheimer Allee 176, 53115 Bonn, Germany; db@izks.uni-bonn.de.
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Abstract

Let (X1,Y1),...,(Xm,Ym) be m independent identically distributed bivariate vectors and L1 = β1X1 + ... + βmXm, L2 = β1X1 + ... + βmXm are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of L1 and L2 imply the same property for X1 and Y1, and under what conditions does the independence of L1 and L2 entail independence of X1 and Y1? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

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References

D.B. Belomestny, To the problem of reconstructing the distribution of summands by the distribution of their sum. Theory Probab. Appl. 46 (2001).
M. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues. (French) C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940) 17-22.
T. Kawata, Fourier analysis in probability theory. Academic Press, New York and London (1972).
B.Ja. Levin, Distribution of zeros of entire functions. American Mathematical Society, Providence, R.I. (1964) viii+493 pp.
Yu.V. Linnik, Linear forms and statistical criteria. I, II. (Russian) Ukrain. Mat. Zurnal 5 (1953) 207-243, 247-290.
Marcinkiewicz, I., Sur une propriété de la loi de Gauss. Mat. Z. 44 (1938) 622-638.
V.V. Petrov, Limit theorems of probability theory. Sequences of independent random variables. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, Oxford Stud. Probab. 4 (1995) xii+292 pp.
A.V. Prohorov and N.G. Ushakov, On the problem of reconstructing the distribution of summands by the distribution of their sum. Theory Probab. Appl. 46 (2001).
N.G. Ushakov, Selected topics in Characteristic functions. VSP, Utrecht and Tokyo (1999).