Mathematical Sciences: Some Limit Theorems in Probability Theory

9625457 Gine摘要研究概率论中的几类极限定理，涉及渐近统计和积分算子谱的实际计算。第一组问题涉及退化U统计量的重对数律，特别是当核不可平方可积时，以及U过程在截断和/或删失数据的寿命分布的渐近性中的应用。在第二组相关问题中，研究者和合作者用某些随机矩阵的谱来逼近紧积分算子的谱，并研究了这种逼近的几个方面。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和，它与学生t统计量有关，目的是确定t统计量渐近标准正态、分布收敛或紧分布的分布集合。研究人员还研究了一些非标准情况下的自举问题。这些看似不同的问题集有许多来自极限理论的共同使用的技术，特别是Banach空间中的经验过程和概率，其中一些是研究人员以前与其他研究人员合作开发的。这位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，并扩大它们的适用范围。了解它们的性质应该有助于我们更好地理解大类统计泛函的行为，因为U-统计是它们的基石。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。另一个项目涉及对算符光谱的近似，算符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，这是一个几乎每个进行数据分析的人都熟悉的经典统计对象。这项研究与确定t统计量何时可以安全使用有关。

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.