Sums of Banach-valued Random Variables

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This research contains two projects. First, the principal investigator plans to investigate tail distributions of sums of Banach-valued independent random variables. The principal investigator, with Pawel Hitczenko, already has many results concerning the case of real-valued random variables. But now the principal investigator feels that there are also fundamental questions concerning the Banach-valued case, for which up until now there are only partial, albeit deep, results. The second part of this proposal is to find new martingale inequalities. This work, joint with his Ph.D. student Jerry Shen, will extend the celebrated inequalities of Burkholder, and also similar inequalities involving so called tangent sequences.A single random event in and of itself is hard to predict. But far more is known when the effect of many random events are averaged together. This study of the sum of many different random variables goes back to the foundations of probability theory, including such celebrated results as the laws of large numbers and the central limit theorem, both foundational in the study of statistics, finding applications to any field, for example, medicine, in which single events are hard to predict. Despite the long history of this study, there remain many questions still unanswered. This work will be pure research, focusing on the fundamental aspects of this averaging behavior.