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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