Mathematical Sciences: Some Limit Theorems in Probability Theory

摘要研究概率论中的几种极限定理，它们与渐近统计和积分算子谱的实际计算有关。第一组问题涉及退化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/或截尾数据的寿命分布的渐近性中的应用。在第二组相关的问题中，研究者和合作者用某些随机矩阵的谱近似紧积分算子的谱，并研究了这种近似的几个方面。该过程相当于用随机网格代替通常的确定性网格对核进行离散。第三个主要研究领域涉及与学生t统计量相关的独立同分布随机变量的自归一化和，其目的是确定t统计量为渐近标准正态分布或分布收敛或紧的分布集。研究者还研究了一些非标准情况下的自举。这些看似不同的问题集共同使用了极限理论中的许多技术，特别是经验过程和巴拿赫空间中的概率，其中一些先前由研究人员与其他人合作开发。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型统计函数类的行为，因为u统计是它们的构建块。特别是，其中一些研究将有助于更好地评估目前用于研究医学、天文学和其他领域的删减和（或）删减数据的统计程序的准确性。另一个项目涉及算子谱的近似，算子谱是描述诸如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”，这是几乎所有从事数据分析的人都熟悉的一个经典统计对象。这项研究与确定何时可以安全地使用t统计量有关。

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.