# Constraints on distributions imposed by properties of linear forms

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

top

## Abstract

top
Let $\left({X}_{1},{Y}_{1}\right),...,\left({X}_{m},{Y}_{m}\right)$ be $m$ independent identically distributed bivariate vectors and ${L}_{1}={\beta }_{1}{X}_{1}+...+{\beta }_{m}{X}_{m}$, ${L}_{2}={\beta }_{1}{Y}_{1}+...+{\beta }_{m}{Y}_{m}$ 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

top

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

top
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

## NotesEmbed?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.