Abstract
Three limit theorems describing asymptotic distribution of vacancy in general multivariate coverage problems are proved, in which n k-dimensional spheres are distributed within a k-dimensional unit cube according to a density f. The first result (a central limit theorem) describes the case where the proportion of vacancy converges to a fixed constant lying between 0 and 1. The last two results treat the case where the proportion of vacancy tends to 1 as n → ∞. Results of this nature have hitherto been available only for restricted k and/or for f equal to the uniform density.
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