## Probability inequalities for multivariate distributions with dependence structures.(English)Zbl 0546.62024

Summary: Let $$X_ 1,...,X_ n$$ be a sequence of random variables with a given positive or negative dependence structure. In this article we exploit the assumed dependence structure to construct a sequence of bounds for the $$P(X_ i\in C_ i$$; $$i=1,...,n)$$, where $$C_ i$$ are infinite intervals of the same type. These bounds are superior to the well-known product bounds that are based solely on the marginal probabilities. Moreover, the new bounds can serve as respectable approximations for the $$P(X_ i\in C_ i$$; $$i=1,...,n)$$.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 60E15 Inequalities; stochastic orderings
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