Date

2013

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mathematics

First Adviser

Dobric, Vladimir

Other advisers/committee members

Stanley, Lee; Napier, Terrence; Fisher, Evan

Abstract

The principal result of Chapter 1 is a new, direct and elementary proof of the general Central Limit Theorem (CLT). Two important stepping-stones are, first, a new, similarly direct and elementary proof of the CLT for Bernoulli random variables defined on [0,1]; this was initially proved by Bernoulli in the 1700's. The second important stepping-stone is a new result for Bernstein polynomials of continuous functions. Bernstein polynomials are a fundamental object of mathematical analysis. It is well known that Bernstein polynomials of a continuous function on intervals $[0,b_{n}]$ when $n$ tends to infinity return the value of the function for an appropriate rate of $b_{n}$, but uniform convergence is sacrificed. Nothing was known for the symmetric interval $[-b_{n},b_{n}]$. We have proven that for these intervals the limit does not recover the function but rather its integral with respect to Gaussian measure. The extension to our direct proof of the of the general CLT involves a new and surprising connection between the CLT and the Haar basis on [0, 1]: the i.i.d. sequence of random variable is transformed to a sequence defined on [0,1] and the random variables in the transformed sequence are then expanded with respect to the Haar basis.Our work on the estimation of the concentration of measure for fractional Brownian motion requires finding the intersections of ellipsoidal and spherical shells for Gaussian measure in $\mathbb{R}^{N}.$ Gaussian measure is concentrated on a small shell of a sphere of radius the square root of N. We want to determine how large this shell must be to include the majority of the Gaussian measure. This result determines the rate of convergence of averages of squares for fractional Brownian increments. It requires understanding the spectrum of the covariance operator as a function of dimension N and the Hurst index. To help understand the spectrum, we compute the exact rate of the largest eigenvalue of this operator.

Included in

Mathematics Commons

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