Limit theorems for the number of points of a given set covered by a random linear subspace

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.