Mathematical Sciences: Some Limit Theorems in Probability Theory

数学科学：概率论中的一些极限定理

• 批准号: 9625457
• 负责人: Evarist Gine
• 金额: $ 7.84万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625457&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Limit; Theorems

9625457 Gine ABSTRACT The investigator does research on several types of limit theorems in Probability Theory, with incidence in Asymptotic Statistics and actual computation of spectra of integral operators. The first set of questions relates to the law of the iterated logarithm for degenerate U-statistics, particularly when the kernel is not square integrable, and to applications of U-processes to the asymptotics of lifetime distributions for truncated and/or censored data. In a second related set of questions the investigator and collaborators approximate spectra of compact integral operators by the spectra of certain random matrices and study several aspects of this approximation. The procedure amounts to discretizing the kernel by means of a random grid instead of the usual deterministic grids. The third main area of research concerns selfnormalized sums of independent identically distributed random variables, which are related to the Student t-statistic, and the aim is to determine the set of distributions for which the t-statistic is asymptotically standard normal, or converges in distribution, or is tight. The investigator also studies the bootstrap in some non-standard situations. These seemingly diverse sets of problems have in common the use of many techniques from limit theory, particularly empirical processes and Probability in Banach Spaces, some of them previously developed by the researcher in collaboration with others. One ongoing project of this researcher to study so-called U-statistics and U-processes and to expand the scope of their applicability. Knowing their properties should help us better understand the behavior of large classes of statistical functionals since U-statistics are their building blocks. In particular, some of this research will help to better assess the accuracy of statistical procedures currently used in the study of censored and/or truncated data in Medicine, Astronomy and other fields. Another project involves approximation of spectra of operators, which are mathematical objects that describe the behavior of systems such as chemical reactions. The third main topic of proposed research concerns the celebrated 'Student t-statistic,' a classical statistical object familiar to virtually everyone who does data analysis. This research is related to determining when the t-statistic can be safely used.

9625457 Gine 摘要 研究者研究概率论中的几种类型的极限定理，以及渐近统计中的关联和积分算子谱的实际计算。第一组问题涉及简并 U 统计量的迭代对数定律，特别是当核不可平方可积时，以及 U 过程在截断和/或审查数据的寿命分布渐近中的应用。在第二组相关问题中，研究者和合作者通过某些随机矩阵的谱来近似紧积分算子的谱，并研究该近似的几个方面。该过程相当于通过随机网格而不是通常的确定性网格来离散化内核。第三个主要研究领域涉及独立同分布随机变量的自归一化总和，这些变量与学生 t 统计量相关，目的是确定 t 统计量渐近标准正态分布的分布集，或收敛于分布还是偏紧。研究者还研究了一些非标准情况下的引导程序。这些看似不同的问题组的共同点是使用了极限理论中的许多技术，特别是经验过程和巴纳赫空间中的概率，其中一些是研究人员之前与其他人合作开发的。 该研究人员正在进行的一个项目是研究所谓的 U 统计和 U 过程并扩大其适用范围。了解它们的属性应该有助于我们更好地理解大类统计泛函的行为，因为 U 统计量是它们的构建块。特别是，其中一些研究将有助于更好地评估目前在医学、天文学和其他领域的审查和/或截断数据研究中使用的统计程序的准确性。 另一个项目涉及算子谱的近似，算子是描述化学反应等系统行为的数学对象。拟议研究的第三个主要主题涉及著名的“学生 t 统计量”，这是几乎所有进行数据分析的人都熟悉的经典统计对象。 这项研究涉及确定何时可以安全使用 t 统计量。