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

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9504758 Mason Abstract The investigator plans to study problems in three different areas. These are (1) universal results for the behavior of sums of independent random variables, (2) local empirical processes and (3) the fractal nature of empirical increments. Area (1) concerns the study of the law of the iterated logarithm for partial sums and their increments that hold under little or no assumptions. Area (2) involves the investigation of a new mathematical entity called the local empirical process, which is the random mechanism governing density estimators, regression function estimators and tail index estimators. Area (3) deals with the uncovering of the somewhat unexpected fractal nature of the increments of various random processes. The proposed research is concerned with the study of the limit behavior of certain random processes and its applications. Very often a random process will exhibit stable properties as it is observed over a long period. The classical example is the random process of tossing a coin. If the coin is fair, the long time average number of heads will with certainty get arbitrarily close to 1/2. This fact is a special case of the law of large numbers. Statisticians take advantage of this law to construct estimators based on a large sample of data that are close with high probability to the population parameter of interest. The areas that will be investigated in this project should have a number of useful applications to the construction of such estimators.