## 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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