Mathematical Sciences: Lp and Tail Probability Approximations for Sums of Dependent Variables

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9626175 de la Pena ABSTRACT The study of the behavior of sums of dependent random variables plays a central role in both the theory and applications of probability and statistics. U-statistics are commonly encountered in problems concerning estimation, while multilinear forms arise in research pertaining to multiple stochastic integration, regression and covariance analysis, and invertibility of matrices. Randomly stopped sums of independent random variables and martingales are found at the core of the studies of sequential analysis, and such diverse areas as queuing theory, inventory theory and reliability theory. The investigator and his colleagues consider fairly general problems involving sums of dependent random variables, including extensions of Wald's equation in the case of martingales, quadratic forms and double stochastic integrals; approximations of the tail probabilities of multilinear forms; generalization of the principle of conditioning to a wider class of problems; and determination of speeds of convergence of related limit theorems. In tackling the proposed problems, the investigator and his colleagues draw from their recent results which include two fundamental contributions to the theories of sequential analysis, U-statistics and empirical processes. The investigator and his colleagues deal with several problems whose solution would have a beneficial impact on several areas of probability and statistics. The broad area consists of the study of the properties of phenomena that exhibit high levels of inter-dependence and, hence, are hard to analyze on their own. The difficulty stems from the strong links present with other components. The approach followed by the investigator and his colleagues when dealing with this type of problem consists in introducing a new set of phenomena that closely resembles the original one but which in addition has desirable independence properties. In probabilistic language, one calls this approach to dealing with dependence a de-coupling of t he dependence of a phenomenon. Up to now there are several results developed in this area which typically produce optimal results. The investigator and his colleagues continue to apply this theory to problems in sequential analysis, U-statistics and stochastic integration. The statistical theory of sequential analysis was introduced during World War II as a means of optimizing resources. In sharp contrast with the typical statistical approach of assigning a prefixed sample and analyzing the data only after the sample size has been achieved, the sequential approach permits the optimization of resources by closely following the development of the process at each stage of the experiment. The sequential approach is particularly useful in cases of destructive sampling, as in equipment and supply lifetime studies in military equipment and industrial applications. In medical studies, its application allows both to optimize resources and to deal with the ethical issues of terminating early a clinical trial when there is strong indication that the drug under study is harmful or proven to be beneficial, without having to wait until a pre-fixed sample size has been attained in accordance with the classical approach.