Sums of Banach-valued Random Variables

本研究包括两个项目。首先，主要研究者计划研究Banach值独立随机变量和的尾部分布。主要研究人员与Pawel Hitczenko一起，已经有了许多关于实值随机变量的结果。但现在首席调查员认为，关于Banach-Value案件也存在一些基本问题，到目前为止，对该案件只有部分结果，尽管是深刻的结果。这一建议的第二部分是寻找新的鞅不等式。这项工作与他的博士生曾傑瑞·沈一起，将推广著名的Burkholder不等式，以及涉及所谓正切序列的类似不等式。单个随机事件本身很难预测。但将许多随机事件的影响平均在一起，就能知道更多的信息。这种对许多不同随机变量之和的研究回到了概率论的基础上，包括著名的结果，如大数定律和中心极限定理，这两个结果都是统计学研究的基础，可以在任何领域找到应用，例如医学，在这些领域中，单一事件很难预测。尽管这项研究有很长的历史，但仍然有许多问题没有得到回答。这项工作将是纯粹的研究，重点是这种平均行为的基本方面。

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.