Further studies of bivariate Bonferroni-type inequalities

Abstract

Let A 1, A 2, …, An and B 1, B 2,. ., BN be two sequences of events on the same probability space. Let mn (A) and mN (B), respectively, be the number of those Aj and Bj which occur. Let Si,j denote the joint ith binomial moment of mn (A) and jth binomial moment of mN (B), 0 ≤ i ≤ n, 0 ≤ j ≤ N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(mn (A) = r, mN (B) = u) by linear combinations of Si,j , 0 ≤ i ≤ a, 0 ≤ j ≤ b. When both a and b are even, all mentioned S¡,j are utilized in both the upper and the lower bound. In a set of remarks the results are analyzed and their relation to the existing literature, including the univariate case, is discussed.