The asymptotic values of certain coverage probabilities

Abstract

Consider n arbitrary subsets of a set X . Suppose x ∈ χ lies in H ( x ) of these subsets and define H & = min x ∈ χ H ( x ) and H ¯ = max x ∈ χ H ( x ) ⁠ . That is, the least and most covered regiona of X are respectively H - and H -covered, with 0 ≤ H ¯ ≤ n . Alternatively, H and ¯ determine the overall coverage of X by, and the maximal concentration of, the subsets in X , respectively. Each subset is taken to be the uniform random image of a fixed congruent subset of X , the n images being stochastically independent. The asymptotic value as n → ∞ of P ( H ¯ ≤ m ) are determined for m = 0, 1, ….