An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities

LetA1,A2, …, ANandB1, B2, …, BMbe two sequences of events, and letνN(A) andνM(B) be the number of thoseAiandBj, respectively, that occur. We prove that Bonferroni-type inequalities forP(νN(A)⩾u,νM(B)⩾v), whereuandvare positive integers, are valid if and only if they are valid for a two dimensional triangular array of independent eventsAiandBj, withP(Ai)=p1andP(Bj)=p2for alliandj. This result allows to derive a formula from which arbitrary Bonferroni-type inequalities of the above type are reduced to the special case of no events occurring. Such methods for proof and similar reduction formula were so far available only for the case of exactlyuandvevents occurring. Several new inequalities are obtained by using our results.