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Optimal second-order product probability bounds. (English) Zbl 0782.62057

Summary: Let \(P(c)= P(X_ 1\leq c_ 1,\dots,X_ p\leq c_ p)\) for a random vector \((X_ 1,\dots,X_ p)\). Bounds are considered of the form \[ \beta^ T_ 2(c)= \prod_{(i,j)\in T} P(X_ i\leq c_ i,\;X_ j\leq c_ j)\Bigl/\prod^ p_{i=1} P(X_ i\leq c_ i)^{d_ i-1}, \] where \(T\) is a spanning tree corresponding to the bivariate probability structure and \(d_ i\) is the degree of the vertex \(i\) in \(T\). An optimized version of this inequality is obtained. The main result is that \(\beta^ T_ 2(c)\) always dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a \(N(0,\Sigma)\) distribution are given so that this bound applies, and various applications are given.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E15 Inequalities; stochastic orderings
05C90 Applications of graph theory
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