9504614 Rinott Abstract The proposed research centers on stochastic inequalities and dependence structures which arise in several contexts: a) Multiple tests for positively dependent test statistics, with the goal of replacing known error probability rates for independent tests by useful bounds for suitably dependent tests. b) Prophet inequalities for dependent random variables, which essentially provide measures on the value of future information in sequentially observed data. c) Positive and negative dependence inequalities: some conjectures on correlation inequalities and negative dependence inequalities for random variables obtained from certain distributions on random graphs are shown to be equivalent to interesting determinental inequalities. Special cases have been proved directly, and by numerical calculations. Another conjecture concerns integral inequalities which extend the well-known van Den Berg-Kesten Conjecture (rumored to have been proved recently) and suggest interesting new applications. Special cases of the new aonjecture have already been established using FKG type and rearrangement inequalities; numerical calculations seem to support these extensions. d) Limit theorems for dependent random variables, with the goal of obtaining convergence rates in terms of meaningful characteristics of the dependence structures, using Stein's method and classical approaches. The assumption of statistical independence of sampled measurements is basic to much of classical data analysis. It is relevant when one can assume that the data arises from repeated controlled experiments in which there is no carryover of errors between repetitions. However, situations with more complex data are ubiquitous. The proposed research centers on the study of certain aspects of samples which consist of observations (random variables) which are not independent, and various models for dependence structures. Such structures are also relevant in various models in physics, and certain issues arising in percolation theory, for example, are also studied. Another aspect of this research is relevant to statistical large sample theory. A good portion of statistical theory is based on approximations which are valid only for large samples. Methods for such approximations for dependent data are investigated, in order to determine the validity of the theory in complex situation. The quality of the approximations is important in deciding how large samples must be in order for large sample approximations to be valid.
