Mathematical Sciences: General Limit Theorems in Probability and Local Empirical Processes

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9503665 Einmahl Abstract This proposal addresses questions from three different areas: (1) Universal Results on the Behavior of Sums of Independent Random Variables (2) Local Empirical Processes (3) A General Law of the Iterated Logarithm in Banach space. The work in area (1) is motivated by some recent universal results on the almost sure behavior of sums of independent random variables. Contrary to the classical results in this direction, these results do not require moment type conditions. The main difficulty is to find suitable norming and centering sequences which are specific to the particular situations under investigation. Among other things, quantile transformations will be used. The work from area (2) is mainly devoted to the limit behavior of local empirical processes which have useful applications in density and regression function estimation. The third part of this proposal finally addresses a number of questions in connection with a recent general law of the iterated logarithm in Banach space. This proposal addresses a number of questions on limit theorems in probability and empirical processes. Limit theorems in probability such as the central limit theorem, and laws of large numbers provide the basis for many widely used statistical techniques to handle large data sets. Some of these results, however, can only be applied under relatively restrictive conditions. Over the last decade, powerful methods have been developed in probability. It is proposed to use these new techniques to further extend the applicability of the basic limit theorems, and, at the same time, to gain some insight into situations, where the classical techniques have failed. Some applications to statistical estimation theory will be considered as well, where the main emphasis will be on so-called empirical processes.