An extension of the bivariate method of polynomials and a reduction formula for Bonferroni-type inequalities (Q1290932)

Let \(A_1\), \(A_2,\dots,A_N\) and \(B_1\), \(B_2,\dots,B_M\) be two sequences of events, and let \(\nu_N(A)\) and \(\nu_M(B)\) be the number of those \(A_j\) and \(B_i\), respectively, that occur. Let \(S_{k,t}\) be the \((k,t)\)th bivariate binomial moment of the sequences \(A_j\) and \(B_i\). Set \(p(u,v)=P(\nu_N(A)=u\), \(\nu_M(B)=v)\) and \(q(u,v)=P(\nu_N(A)\geq u\), \(\nu_M(B)\geq v)\). In the main result the author establishes that Bonferroni-type bounds on \(q(u,v)\), that is, bounds by means of linear combinations of the moments \(S_{k,t}\), are equivalent to specific bivariate polynomial inequalities. The author then deduces from the main result that several Bonferroni-type inequalities on \(q(u,v)\) are equivalent to Bonferroni-type inequalities on \(p(0,0)\). This brings together the inequalities on \(q(u,v)\) and \(p(u,v)\), since from inequalities on \(p(0,0)\) general inequalities on \(p(u,v)\) can be generated. With this new method, the parallel theory on \(p(u,v)\) and \(q(u,v)\) is unified which is a remarkable result on its own. The method is also used for the establishment of new inequalities. For earlier results on Bonferroni-type inequalities, see the book of the reviewer and the author [``Bonferroni-type inequalities with applications'' (1996; Zbl 0869.60014)].