### RANDOM COVERS OF FINITE HOMOGENEOUS LATTICES

**A. N. ALEKSEYCHUK**

*Theory of Stochastic Processes*

*Vol. 12 (28), no. 1–2, 2006, pp. 12–19*

We develop and extend some results for the scheme of independent random elements distributed on a finite lattice. In particular, we introduce the concept of the cover of a homogeneous lattice $L_n$ of rank $n$ and derive the exact equations and estimations for the number of covers with a given number of blocks and for the total covers number of the lattice $L_n$. A theorem about the asymptotic normality of the blocks number in a random equiprobable cover of the lattice $L_n$ is proved. The concept of the cover index of the lattice $L_n$, that extend the notion of the cover index of a finite set by its independent random subsets, is introduced. Applying the lattice moments method, the limit distribution as $n\rightarrow\infty$ for the cover index of a subspace lattice of the $n$-dimensional vector space over a finite field is determined.

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