Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T10:17:31.552Z Has data issue: false hasContentIssue false
  • English
  • Français

Constraints on distributions imposedby properties of linear forms

Published online by Cambridge University Press:  15 May 2003

Denis Belomestny
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.
Get access

Abstract

Let (X1,Y1),...,(Xm,Ym ) be m independent identicallydistributed bivariate vectorsandL1 = β1X1 + ... + βmXm , L2 = β1X1 + ... + βmXm are two linear forms with positive coefficients.We study two problems:under what conditions does the equidistribution of L 1 and L 2imply the same property forX 1 and Y 1, and under what conditions does the independence of L 1and L 2 entail independenceof X 1 and Y 1?Some analytical sufficient conditions are obtained and it is shownthat in general they can not be weakened.

Keywords

Equidistributionindependencelinearformscharacteristic functions.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 7 , March 2003 , pp. 313 - 328
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).