Limit distributions of the number of vectors satisfying a linear relation

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.