Theses and Dissertations

Fall 2023

Rahmatidehkordi, Ardalan

this thesis, we give a formula for calculating the Kantrovich distance between mod 1 probability measures. We then use this distance to study the convergence behavior of the (mod 1) empirical distributions associated with real sequences (xn)∞ n=1 for which limn→∞ n(xn−xn−1) exists. We find that for

such sequences, every probability distribution in the limit set of the empirical distributions is a rotated version of a certain exponential distribution. We also describe the speed of convergence to this limit set of distributions.

Distributions of sequences modulo one (mod 1) have been studied over the past century with applications in algebra, number theory, statistics, and computer science. For a given sequence, the weak convergence of the associated empirical distributions has been the usual approach to these studies. In

Fall 2018

Xu,Chuang

Approximation of probability measures, quantization, Kantorovich metric, Levy metric, Kolmogorov metric, Benford's Law, slowly changing sequences, asymptotic distribution, invariance property.","This thesis is based on four papers. The first two papers fall into the field of approximation of one

upper estimate $\lt(N^{-1}\lt(\log N\rt)^{1/2}\rt)$ is obtained for the rate of convergence w.r.t. the Kantorovich metric on the circle. Moreover, a sharp rate of convergence $\lt(N^{-1}\log N\rt)$ w.r.t. the Kantorovich and the discrepancy (or Kolmogorov) metrics on the real line is derived. The last

paper proves a threshold result on the existence of a circularly invariant and uniform probability measure (CIUPM) for non-constant linear transformations on the real line, which shows that there is a constant $c$ depending only on the slope of the linear transformation such that there exists a CIUPM if

Fall 2011

Slevinsky, Richard

develop an algorithm for the G transformation, we derive explicit approximations to incomplete Bessel functions and tail probabilities of five probability distributions from the recursive algorithm for the G transformation, and we present all extant work on the analysis of the convergence properties of

This thesis is concerned with the development of new formulae for higher order derivatives, and the algorithmic, numerical, and analytical development of the G transformation, a method for computing infinite-range integrals. We introduce the Slevinsky-Safouhi formulae I and II with applications, we

Spring 2016

Usman, Iram

simulation studies to investigate right censoring. Different datasets from the exponential, Weibull, log-Normal, and gamma probability distributions have been generated in order to test the robustness of the SSS's. Three differential censoring settings were imposed on the generated datasets to test the

time to events if the SSS uses appropriate distribution. Other authors have proposed the exponential and Weibull distributions for the event times. We have established the log-Weibull distribution as a new and alternative approach for the SSS, and compared and contrasted the three distributions through

The spatial scan statistic (SSS) has been used for the identification of geographical clusters of higher than expected numbers of cases of a condition such as an illness. Disease outbreaks in a geographic area are a typical example. These statistics can also identify geographic areas with longer

Fall 2011

Gong, Jiafen

model from a birth-death process. The calculation of this NTCP model provides an alternative proof to a formula derived by Hanin (Hanin, 2004) to compute the probability distribution of the tumor size from its generating function. My formula is computationally more efficient, compared to Hanin’s

, used for quantifying normal tissue complication. In this thesis, I begin with a simple Poisson TCP based on mean cell population dynamics. Optimal treatment schedules are obtained by maximizing this TCP while constraining the CRE under a given threshold. Some of the optimal results suggest the usage

Cancer is one of the major causes of death in the world. In the field of Oncology, clinical trials form the crux of medical effort to find better treatment schedules. These trials are expensive, time consuming, and carry great risks for the patients involved. Mathematical models provide a

Spring 2012

Rivasplata, Omar D

In this thesis probability estimates on the smallest singular value of random matrices with independent entries are extended to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider

matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix, and help to clarify the role of the variances in the corresponding estimates. We also show that it is enough to require boundedness from

Fall 2010

Pivovarov, Peter

facets) should be sampled in order for such a polytope to capture significant volume. Various criteria for what exactly it means to capture significant volume are discussed. We also study similar problems for random polytopes generated by points on the Euclidean sphere. The second paper is

about volume distribution in convex bodies. The first main result is about convex bodies that are (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on such bodies exhibit super-Gaussian tail-decay

This thesis is based on three papers on selected topics in Asymptotic Geometric Analysis. The first paper is about the volume of high-dimensional random polytopes; in particular, on polytopes generated by Gaussian random vectors. We consider the question of how many random vertices (or

Spring 2014

Susanna, Spektor

for a random variable with hypergeometric distribution, improving previously known estimates. The fourth paper devoted to the quantitative version of a Silverstein's Theorem on the 4-th moment condition for convergence in probability of the norm of a random matrix. More precisely, we show that for a

This thesis is mostly based on six papers on selected topics in Asymptotic Geometric Analysis, Wavelet Analysis and Applied Fourier Analysis. The first two papers are devoted to Ball's integral inequality. We prove this inequality via spline functions. We also provide a method for computing all

under the assumption that the sum of the Rademacher random variables is zero. We also discuss other approaches to the problem. In particular, one may use simple random walks on graph, concentration and the chaining argument. As a special case of Khinchine's type inequality, we provide a tail estimate

Fall 2021

Pak, Andrey

ruin probabilities of such models is shown and supported by diverse examples. The main object of the final part of this thesis is a general regression model in an optional setting – when an observed process is an optional semimartingale depending on an unknown parameter. The cases when the model

stochastic differential equations are stated and proved using a local time approach. Furthermore, these results are applied to the pricing of financial derivatives. Second, the estimates of N. V. Krylov for distributions of stochastic integrals by means of Lebesgue norm of a measurable function are well

This thesis is dedicated to the study of the general class of random processes, called optional processes, and their various applications in Mathematical Finance, Risk Theory, and Statistics. First, different versions of a comparison theorem and a uniqueness theorem for a general class of optional