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Let P(c) = P(X1 ≦ c1, · ··, Xp ≦ cp) for a random vector (X1, · ··, 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.
