Geodesic
Limit distributions of the number of vectors satisfying a~linear relation
Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 120-135

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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. 
@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},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_4_a10/}
}
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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/