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.
