Geodesic
Limit theorems for the number of points of a given set covered by a random linear subspace
Diskretnaya Matematika, Tome 15 (2003) no. 2, pp. 128-137
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $V^T$ be the $T$-dimensional linear space over a finite field $K$, and let $B_1,\ldots,B_m$ be subsets of $V^T$ not containing the zero-point. Let a subspace $L$ be chosen randomly and equiprobably from the set of all $n$-dimensional linear subspaces of $V^T$. We consider the number $\mu(B_i)$ of points in the intersections $L\cap B_i$, $i=1,\ldots,m$. We study the limit behaviour of the distribution of the vector $(\mu(B_1),\ldots,\mu(B_m))$ as $T,n\to \infty$ and the sets vary in such a way that the means of $\mu(B_i)$ tend to finite limits. The field $K$ is fixed. We prove that this random vector has in limit the compound Poisson distribution. Necessary and sufficient conditions for asymptotic independency of the random variables $\mu(B_1),\ldots,\mu(B_m)$ are derived. This research was supported by the Russian Foundation for Basic Research, grants 02–01–00266 and 00–15–96136. 
@article{DM_2003_15_2_a10,
     author = {V. G. Mikhailov},
     title = {Limit theorems for the number of points of a given set covered by a random linear subspace},
     journal = {Diskretnaya Matematika},
     pages = {128--137},
     year = {2003},
     volume = {15},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2003_15_2_a10/}
}
TY  - JOUR
AU  - V. G. Mikhailov
TI  - Limit theorems for the number of points of a given set covered by a random linear subspace
JO  - Diskretnaya Matematika
PY  - 2003
SP  - 128
EP  - 137
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DM_2003_15_2_a10/
LA  - ru
ID  - DM_2003_15_2_a10
ER  - 
%0 Journal Article
%A V. G. Mikhailov
%T Limit theorems for the number of points of a given set covered by a random linear subspace
%J Diskretnaya Matematika
%D 2003
%P 128-137
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/DM_2003_15_2_a10/
%G ru
%F DM_2003_15_2_a10
V. G. Mikhailov. Limit theorems for the number of points of a given set covered by a random linear subspace. Diskretnaya Matematika, Tome 15 (2003) no. 2, pp. 128-137. http://geodesic.mathdoc.fr/item/DM_2003_15_2_a10/

[1] Kolchin V. F., Sistemy sluchainykh uravnenii, MIEM, Moskva, 1988

[2] Kolchin V. F., Sluchainye grafy, Nauka, Moskva, 2000 | MR

[3] Balakin G. V., “Sistemy sluchainykh uravnenii nad konechnym polem”, Trudy po diskretnoi matematike, 2, 1998, 21–37 | MR | Zbl

[4] Mikhailov V. G., “Predelnye teoremy dlya chisla nenulevykh reshenii odnoi sistemy sluchainykh uravnenii nad polem $\mathit{GF}(2)$”, Teoriya veroyatnostei i ee primeneniya, 43:3 (1998), 598–606

[5] Mikhailov V. G., “Predelnye teoremy dlya chisla nenulevykh reshenii odnoi sistemy sluchainykh uravnenii nad polem $\mathit{GF}(2)$”, Diskretnaya matematika, 12:1 (2000), 70–81 | Zbl

[6] Mikhailov V. G., “Predelnaya teorema Puassona dlya chisla nekollinearnykh reshenii sistemy sluchainykh uravnenii spetsialnogo vida”, Diskretnaya matematika, 13:3 (2001), 81–90 | MR | Zbl

[7] Mikhailov V. G., “Skhodimost k protsessu s nezavisimymi prirascheniyami v skheme narastayuschikh summ zavisimykh sluchainykh velichin”, Matem. sb., 94:2 (1974), 283–299

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

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

[10] Mikhailov V. G., Nekotorye predelnye teoremy dlya summ zavisimykh sluchainykh velichin, Diss. na soisk. stepeni kand. fiz.-matem. nauk, MIAN, Moskva, 1974