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（连同合著者）打算写了一个非常准确的结果，可以适用于逼近尾部概率的实值和Banach空间值部分和独立变量。根据变量的边际分布定义某些函数，就可以得到精度很高的部分和分位数和p阶矩的近似值。其次，提议者（和合著者）将解决有关自归一化鞅的指数矩和尾概率上界的问题。预计统计应用将因此而出现。概率和统计问题出现在各种各样的理论和应用背景。最常见的问题涉及事件的概率，随机函数的期望值或假设检验。这些结果在真实的世界中的应用非常广泛，从理论到社会科学、制药、金融、经济、工程、物理科学和算法性能的数据分析。该研究员在独立随机变量和领域工作多年。他（与合著者）现在正在追求非常精确的结果。包括在这个列表中的部分和的尾部概率的近似和他们的分位数的位置，期望界，加上尾部概率，指数和矩生成函数的所谓的自归一化鞅界的实质性改进。

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.