# Constraints on distributions imposed by properties of linear forms

ESAIM: Probability and Statistics (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topBelomestny, Denis. "Constraints on distributions imposed by properties of linear forms." ESAIM: Probability and Statistics 7 (2010): 313-328. <http://eudml.org/doc/104310>.

@article{Belomestny2010,

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.
},

author = {Belomestny, Denis},

journal = {ESAIM: Probability and Statistics},

keywords = {Equidistribution; independence; linear
forms; characteristic functions.; equidistribution; linear forms; characteristic functions},

language = {eng},

month = {3},

pages = {313-328},

publisher = {EDP Sciences},

title = {Constraints on distributions imposed by properties of linear forms},

url = {http://eudml.org/doc/104310},

volume = {7},

year = {2010},

}

TY - JOUR

AU - Belomestny, Denis

TI - Constraints on distributions imposed by properties of linear forms

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 7

SP - 313

EP - 328

AB -
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.

LA - eng

KW - Equidistribution; independence; linear
forms; characteristic functions.; equidistribution; linear forms; characteristic functions

UR - http://eudml.org/doc/104310

ER -

## References

top- 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. Zbl0022.35302
- T. Kawata, Fourier analysis in probability theory. Academic Press, New York and London (1972). Zbl0271.60022
- 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. Zbl0052.36701
- I. Marcinkiewicz, 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. Zbl0826.60001
- 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). Zbl0999.60500

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.