Mathematical Sciences: Some Limit Theorems in Probability Theory

9625457 Gine摘要 研究者研究概率论中几种类型的极限定理，在渐近统计和积分算子谱的实际计算中具有关联性。第一组问题涉及退化U-统计量的重对数律，特别是当内核不是平方可积的，并应用U-过程的寿命分布的渐近截断和/或删失数据。在第二组相关的问题的调查和合作者近似谱紧凑的积分算子的谱的某些随机矩阵和研究几个方面的这种近似。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。研究的第三个主要领域涉及独立同分布随机变量的自归一化和，这与学生t-统计量有关，目的是确定t-统计量是渐近标准正态分布或收敛分布或紧分布的分布集。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有共同的使用许多技术从极限理论，特别是经验过程和概率在Banach空间，其中一些以前开发的研究人员与他人合作。 本研究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。了解它们的性质应该有助于我们更好地理解大量统计泛函的行为，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/或截断数据的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，算子谱是描述化学反应等系统行为的数学对象。建议研究的第三个主要课题涉及著名的“学生t统计量”，这是一个经典的统计对象，几乎每个做数据分析的人都很熟悉。 这项研究涉及确定何时可以安全使用t统计量。

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.