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摘要研究者研究了概率理论中几种类型的限制定理，并在渐近统计和积分运算符光谱的实际计算中发病。第一组问题与迭代的对数的法律有关，尤其是当内核不是正方形集成的情况下，以及U过程中的应用在截短和/或审查数据的寿命分布的渐近分布中的应用。在第二个相关的问题中，研究者和合作者通过某些随机矩阵的光谱近似紧凑型积分运算符的光谱，并研究了此近似的几个方面。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内核。研究的第三个主要领域涉及与学生t统计相关的独立分布式随机变量的自称总和，其目的是确定t统计量渐近标准正常或分布或分布或紧密的分布集。研究人员还在某些非标准情况下研究了引导程序。这些看似多样化的问题共同使用了极限理论中的许多技术，尤其是Banach空间中的经验过程和概率，其中一些以前是由研究人员与他人合作开发的。 该研究人员正在进行的一个正在进行的项目，以研究所谓的U统计数据和U过程，并扩大其适用性的范围。知道它们的属性应该有助于我们更好地了解大量统计功能的行为，因为U统计数据是它们的基础。特别是，这项研究中的一些将有助于更好地评估当前在研究和/或截断医学，天文学和其他领域截断数据研究中使用的统计程序的准确性。 另一个项目涉及操作器光谱的近似，这些谱是描述化学反应等系统行为的数学对象。拟议的研究的第三个主要主题涉及著名的“学生T统计”，这是一个几乎每个进行数据分析的每个人都熟悉的经典统计对象。 这项研究与确定何时可以安全地使用T统计量有关。