Refined Approximation of Tail Probabilities, Expectation and Exponential Bounds for Partial Sums and Self-Normalized Martingales

研究人员计划在两个主要领域开展工作，总和和自归一化总和。他(和一位合著者)打算写出一个非常准确的结果，该结果可以应用于独立变量的实值和Banach空间值部分和的尾概率的近似。利用由变量的边缘分布定义的某些函数，可以得到高精度的部分和分位数和p阶矩的近似。其次，提出者(和合著者)将讨论关于自正规鞅的指数矩和尾概率上界的问题。因此，预计将出现统计应用程序。概率和统计问题出现在各种各样的理论和应用背景下。最常见的问题涉及事件的概率、随机函数的期望或对假设的检验。这样的结果在现实世界中的应用是广泛的，从理论到数据分析，在社会科学、制药、金融、经济、工程、物理科学和算法性能方面。这位研究者在独立随机变量和领域工作了很多年。他(与合著者一起)现在正在追求非常精准的结果。在这个列表中包括了在部分和的尾概率及其分位数的位置、期望界、加上尾概率、指数和矩生成函数界的所谓的自归一化鞅的近似方面的实质性改进。

The investigator plans to do work in two principal areas, sums and self-normalized sums. He (together with a co-author) intends to write up a very accurate result which can be applied to the approximation of tail probabilities of both real-valued and Banach space-valued partial sums of independent variates. Armed with certain functions defined from the marginal distributions of the variates, approximations of partial sum quantiles and p-th moments of great precision should follow. Secondly, the proposer (and co-authors) will address questions concerning exponential moment and tail probability upper bounds for self-normalized martingales. It is anticipated that statistical applications will occur as a consequence. Probabilistic and statistical issues arise in a broad variety of theoretical and applied contexts. Most commonly the issues involve the probability of an event, the expectation of a random function, or a test of hypothesis. Real world applications of such results are wide-spread, extending from theory to data analysis in the social sciences, pharmaceuticals, finance, economics, engineering, the physical sciences, and the performance of algorithms. The investigator has worked in the area of sums of independent random variables for many years. He (together with co-authors) now is pursuing results of very refined precision. Included in this list are substantial improvements in the approximation of tail probabilities of partial sums and the location of their quantiles, expectation bounds, plus tail probability, exponential and moment generating function bounds for so-called self-normalized martingales.