with [n\omega₁] denoting the integral part of n\omega₁, and that this inequality is an equality for some collection of events A₁, A₂,...,A_n whatever \omega₁ and n.
Here the authors consider the closely related more general problem of the determination, for any natural n and a in (0,1), of the constant \epsilon_n (\alpha) defined as the least real number \epsilon such that for any collection of events A₁,A₂,...,A_n subject to the only condition (2) Pr(A_i \cap A_j) \leq \alpha² for i \ne j we have the inequality sum Pr(A_i) \leq n\alpha+\epsilon. With \nu denoting the largest integer such that \nu(\nu-1) \leq n(n-1)\alpha² the constant sought for is found to be given by

The second term in this formula vanishes if n\alpha or (n-1)\alpha is an integer; otherwise for n ––> oo it is of the order of 1/n. An explicit extremal collection of events A₁,A₂,...,A_n is constructed in the case of \alpha = ¹/₂ and n \equiv 3 (mod 4) by the use of the method of quadratic residues.
Reviewer:  S.Zubrzycki
Classif.:  * 60C05 Combinatorial probability
                   60E05 General theory of probability distributions
Index Words:  probability theory

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