FURTHER-STUDIES OF BIVARIATE BONFERRONI-TYPE INEQUALITIES

Let A1, A2 ,..., A(n) and B1, B2 ,..., B(N) be two sequences of events on the same probability space. Let m(n)(A) and m(N)(B), respectively, be the number of those A(j) and B(j) which occur. Let S(i,j) denote t he joint ith binomial moment of m(n)(A) and jth binomial moment of m(N )(B), 0 less-than-or-equal-to i less-than-or-equal-to n, 0 less-than-o r-equal-to j less-than-or-equal-to N. For fixed non-negative integers a and b, we establish both lower and upper bounds on the distribution P(m(n)(A) = r, m(N)(B) = u) by linear combinations of S(i,j), 0 less-t han-or-equal-to i less-than-or-equal-to a, 0 less-than-or-equal-to j l ess-than-or-equal-to b. When both a and b are even, all mentioned S(i, j) are utilized in both the upper and the lower bound. In a set of rem arks the results are analyzed and their relation to the existing liter ature, including the univariate case, is discussed.