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A technique for establishing L1 asymptotic unbiasedness of a kernel density estimator in Rd that does not depend on the form of the kernel function will be demonstrated. We will introduce the concept of a region sequence of a sequence of kernel functions and show how this can be used to give necessary and sufficient conditions for L1 asymptotic unbiasedness. These results are then applied to kernel density estimators whose form is given and a number of known and novel results are obtained.

The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions. We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8. We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution.

A technique for establishing L1 asymptotic unbiasedness of a kernel density estimator in Rd that does not depend on the form of the kernel function will be demonstrated. We will introduce the concept of a region sequence of a sequence of kernel functions and show how this can be used to give necessary and sufficient conditions for L1 asymptotic unbiasedness. These results are then applied to kernel density estimators whose form is given and a number of known and novel results are obtained.

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Thesis (Ph. D.)--UCLA, 2007. / Vita. Includes bibliographical references (leaves 93-96).

<p> This thesis gives new finite and asymptotic estimates of the Ramsey numbers using certain number-theoretical considerations; it also contains a brief historical survey on determination of Ramsey numbers and related number-theoretical problems.</p> / Thesis / Master of Science (MSc)