Constraints on distributions imposed by properties of linear forms
ESAIM: Probability and Statistics, Tome 7 (2003), pp. 313-328.

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.

DOI : 10.1051/ps:2003014
Classification : 62E10, 60E10
Mots-clés : equidistribution, independence, linear forms, characteristic functions 
@article{PS_2003__7__313_0,
     author = {Belomestny, Denis},
     title = {Constraints on distributions imposed by properties of linear forms},
     journal = {ESAIM: Probability and Statistics},
     pages = {313--328},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2003},
     doi = {10.1051/ps:2003014},
     mrnumber = {1987791},
     zbl = {1013.62059},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2003014/}
}
TY  - JOUR
AU  - Belomestny, Denis
TI  - Constraints on distributions imposed by properties of linear forms
JO  - ESAIM: Probability and Statistics
PY  - 2003
SP  - 313
EP  - 328
VL  - 7
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2003014/
DO  - 10.1051/ps:2003014
LA  - en
ID  - PS_2003__7__313_0
ER  - 
%0 Journal Article
%A Belomestny, Denis
%T Constraints on distributions imposed by properties of linear forms
%J ESAIM: Probability and Statistics
%D 2003
%P 313-328
%V 7
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps:2003014/
%R 10.1051/ps:2003014
%G en
%F PS_2003__7__313_0
Belomestny, Denis. Constraints on distributions imposed by properties of linear forms. ESAIM: Probability and Statistics, Tome 7 (2003), pp. 313-328. doi : 10.1051/ps:2003014. http://archive.numdam.org/articles/10.1051/ps:2003014/

[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] 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. | MR | Zbl

[3] T. Kawata, Fourier analysis in probability theory. Academic Press, New York and London (1972). | MR | Zbl

[4] B.Ja. Levin, Distribution of zeros of entire functions. American Mathematical Society, Providence, R.I. (1964) viii+493 pp. | MR | Zbl

[5] Yu.V. Linnik, Linear forms and statistical criteria. I, II. (Russian) Ukrain. Mat. Žurnal 5 (1953) 207-243, 247-290. | MR | Zbl

[6] I. Marcinkiewicz, Sur une propriété de la loi de Gauss. Mat. Z. 44 (1938) 622-638. | JFM | Zbl

[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. | MR | Zbl

[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). | Zbl

[9] N.G. Ushakov, Selected topics in Characteristic functions. VSP, Utrecht and Tokyo (1999). | MR | Zbl

Cité par Sources :