Mathematical Sciences: Inequalities in Multivariate Analysisand Reliability

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The principal investigator will continue his work on probability inequalities in multivariate analysis and reliability theory. Specific dependence properties of random variables are described via conditions such as log.concave or Schure.concave densities, positive dependence, and arrangement increasing functions. The probability inequalities are deduced in terms of properties of the sums of random variables or the order statistics. For example it will be shown that the Schur.concave density of random variables leads to Schur.concave density of the partial sums. The assumption of positive dependence for exchangeable random variables leads to more spread out order statistics and hence bigger variance. The resulting comparisons and order arrangements of random variables are useful in reliability theory for finding the optimal assembly of systems.