Sums of Banach-valued Random Variables

本研究包含两个项目。首先，首席研究员计划研究巴拿赫值独立随机变量和的尾部分布。首席研究员帕维尔·希琴科（Pawel Hitczenko）已经有了许多关于实值随机变量的结果。但现在，首席研究员认为，巴拿赫价值案例也存在一些根本性的问题，迄今为止，这些问题只有部分的、尽管深刻的结果。这个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Jerry Shen合作，将扩展著名的Burkholder不等式，以及涉及所谓切线序列的类似不等式。单个随机事件本身是很难预测的。但是，当将许多随机事件的影响平均起来时，我们知道的要多得多。这种对许多不同随机变量的总和的研究可以追溯到概率论的基础，包括大数定律和中心极限定理等著名的结果，它们都是统计学研究的基础，可以应用于任何领域，例如医学，在这些领域，单个事件很难预测。尽管这项研究历史悠久，但仍有许多问题尚未得到解答。这项工作将是纯粹的研究，专注于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.