Optimal second-order product probability bounds

Let P(c) = P(X 1 ≦ c 1, · ··, Xp ≦ cp ) for a random vector (X 1, · ··, Xp ). Bounds are considered of the form where T is a spanning tree corresponding to the bivariate probability structure and di is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.