First Borel-Cantelli equalities incorporating known dependency structure (Q1425807)

For an infinite sequence \(E_j\) of events on a given probability space, the classical Borel-Cantelli lemma states that if \(\sum P(E_j)\) is finite, then, with probability one, only a finite number of the \(E_j\) occur. It is known [see \textit{J. Galambos} and \textit{I. Simonelli}, ``Bonferroni-type inequalities with applications'' (1996; Zbl 0869.60014), p. 242] that if \(\sum P(E_j)=+\infty\), then a simple condition on the sum of \(P(E_j\cap E_k)\), \(j< k< n\), implies that, with probability one, infinitely many of the \(E_j\) occur. Both forms of the above statements are related to Bonferroni-type inequalities. The author investigates further conditions when, for some special dependent structures, the probability that infinitely many of the \(E_j\) occur is strictly between zero and one.