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

研究者计划在两个主要领域做工作，和和自规格化和。他（与另一位合著者）打算写出一个非常精确的结果，该结果可以应用于自变量的实值和巴拿赫空间值部分和的尾概率逼近。有了由变量的边际分布定义的某些函数，部分和分位数和p阶矩的近似值就可以得到很高的精度。其次，提案人（和合著者）将解决有关指数矩和尾概率上界的问题。预计统计应用将因此出现。概率和统计问题出现在各种各样的理论和应用环境中。最常见的问题涉及事件的概率、随机函数的期望或假设的检验。这些结果在现实世界中的应用非常广泛，从理论扩展到社会科学、制药、金融、经济、工程、物理科学和算法性能的数据分析。这位研究者在独立随机变量和领域工作多年。他（和他的合作者）现在正在追求非常精确的结果。其中包括部分和的尾部概率近似值及其分位数的位置、期望边界、尾部概率、所谓的自归一化鞅的指数和矩生成函数边界等方面的实质性改进。

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.