Geodesic
Limit distributions of the number of vectors satisfying a linear relation
Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 120-135 
V. I. Kruglov. Limit distributions of the number of vectors satisfying a linear relation. Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 120-135. http://geodesic.mathdoc.fr/item/DM_2008_20_4_a10/
@article{DM_2008_20_4_a10,
     author = {V. I. Kruglov},
     title = {Limit distributions of the number of vectors satisfying a~linear relation},
     journal = {Diskretnaya Matematika},
     pages = {120--135},
     year = {2008},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_4_a10/}
}
TY  - JOUR
AU  - V. I. Kruglov
TI  - Limit distributions of the number of vectors satisfying a linear relation
JO  - Diskretnaya Matematika
PY  - 2008
SP  - 120
EP  - 135
VL  - 20
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/DM_2008_20_4_a10/
LA  - ru
ID  - DM_2008_20_4_a10
ER  - 
%0 Journal Article
%A V. I. Kruglov
%T Limit distributions of the number of vectors satisfying a linear relation
%J Diskretnaya Matematika
%D 2008
%P 120-135
%V 20
%N 4
%U http://geodesic.mathdoc.fr/item/DM_2008_20_4_a10/
%G ru
%F DM_2008_20_4_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $X_1,\dots,X_T$ be independent random elements uniformly distributed on a finite Abelian group $G$. In this paper, we give conditions under which the number of ordered sets $(i_1,\dots,i_k)$ of pairwise distinct numbers in $\{1,\dots,T\}$ such that $a_1X_{i_1}+\dots+a_kX_{i_k}=0$ where $a_1,\dots,a_k$ are fixed integers has the Poisson limit distribution as $T\to\infty$ and the group $G$ varies with $T$. We give an example of a sequence of groups $G$ for which the limit distribution of the number of ordered sets is the compound Poisson distribution.

[1] Aldous D. J., Probability approximations via the Poisson clumping heuristics, Springer, New York, 1989 | MR

[2] Barbour A. D., Chen L. H. Y., An introduction to Stein's method, World Scientific, New York, 2005 | Zbl

[3] Glukhov M. M., Elizarov V. P., Nechaev A. A., Algebra, T. 1, Gelios ARV, Moskva, 2003

[4] Klykova N. V., “Predelnoe raspredelenie chisla sovpadayuschikh promezhutkov”, Teoriya veroyatnostei i ee primeneniya, 47:1 (2002), 147–152 | MR | Zbl

[5] Kolchin V. F., Sevastyanov B. A., Chistyakov V. P., Sluchainye razmescheniya, Nauka, Moskva, 1976 | MR | Zbl

[6] Kostrikin A. I., Vvedenie v algebru. I. Osnovy algebry, Fizmatlit, Moskva, 2000 | MR

[7] Leng S., Algebra, Mir, Moskva, 1968

[8] Sevastyanov B. A., “Predelnyi zakon Puassona v skheme summ zavisimykh sluchainykh velichin”, Teoriya veroyatnostei i ee primeneniya, 17:4 (1972), 733–738