Constraints on distributions imposed by properties of linear forms

Denis Belomestny

ESAIM: Probability and Statistics (2003)

  • Volume: 7, page 313-328
  • ISSN: 1292-8100

Abstract

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Let ( X 1 , Y 1 ) , ... , ( X m , Y m ) be m independent identically distributed bivariate vectors and L 1 = β 1 X 1 + ... + β m X m , L 2 = β 1 Y 1 + ... + β m Y m are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of L 1 and L 2 imply the same property for X 1 and Y 1 , and under what conditions does the independence of L 1 and L 2 entail independence of X 1 and Y 1 ? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

How to cite

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Belomestny, Denis. "Constraints on distributions imposed by properties of linear forms." ESAIM: Probability and Statistics 7 (2003): 313-328. <http://eudml.org/doc/245879>.

@article{Belomestny2003,
abstract = {Let $(X_1,Y_1),\ldots ,(X_m,Y_m)$ be $m$ independent identically distributed bivariate vectors and $L_1=\beta _1X_1+\ldots +\beta _mX_m$, $L_2=\beta _1Y_1+\ldots +\beta _mY_m$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_1$ and $L_2$ imply the same property for $X_1$ and $Y_1$, and under what conditions does the independence of $L_1$ and $L_2$ entail independence of $X_1$ and $Y_1$? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.},
author = {Belomestny, Denis},
journal = {ESAIM: Probability and Statistics},
keywords = {equidistribution; independence; linear forms; characteristic functions},
language = {eng},
pages = {313-328},
publisher = {EDP-Sciences},
title = {Constraints on distributions imposed by properties of linear forms},
url = {http://eudml.org/doc/245879},
volume = {7},
year = {2003},
}

TY - JOUR
AU - Belomestny, Denis
TI - Constraints on distributions imposed by properties of linear forms
JO - ESAIM: Probability and Statistics
PY - 2003
PB - EDP-Sciences
VL - 7
SP - 313
EP - 328
AB - Let $(X_1,Y_1),\ldots ,(X_m,Y_m)$ be $m$ independent identically distributed bivariate vectors and $L_1=\beta _1X_1+\ldots +\beta _mX_m$, $L_2=\beta _1Y_1+\ldots +\beta _mY_m$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_1$ and $L_2$ imply the same property for $X_1$ and $Y_1$, and under what conditions does the independence of $L_1$ and $L_2$ entail independence of $X_1$ and $Y_1$? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.
LA - eng
KW - equidistribution; independence; linear forms; characteristic functions
UR - http://eudml.org/doc/245879
ER -

References

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  1. [1] D.B. Belomestny, To the problem of reconstructing the distribution of summands by the distribution of their sum. Theory Probab. Appl. 46 (2001). 
  2. [2] 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. Zbl0022.35302MR4333
  3. [3] T. Kawata, Fourier analysis in probability theory. Academic Press, New York and London (1972). Zbl0271.60022MR464353
  4. [4] B.Ja. Levin, Distribution of zeros of entire functions. American Mathematical Society, Providence, R.I. (1964) viii+493 pp. Zbl0152.06703MR156975
  5. [5] Yu.V. Linnik, Linear forms and statistical criteria. I, II. (Russian) Ukrain. Mat. Žurnal 5 (1953) 207-243, 247-290. Zbl0052.36701MR60767
  6. [6] I. Marcinkiewicz, Sur une propriété de la loi de Gauss. Mat. Z. 44 (1938) 622-638. Zbl0019.31705JFM64.0521.02
  7. [7] 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. Zbl0826.60001MR1353441
  8. [8] 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). Zbl1032.60010
  9. [9] N.G. Ushakov, Selected topics in Characteristic functions. VSP, Utrecht and Tokyo (1999). Zbl0999.60500MR1745554

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