The HyperLogLog (HLL) algorithm is used to estimate the cardinality of large sets. This thesis gives a novel analysis of the HyperLogLog algorithm by using techniques from statistics and probability. Initially, closed form bounds for the mean and variance of the max of n independent and identically distributed geometric random...
This dissertation examines properties and representations of several isotropic Gaussian random fields in the unit ball in d-dimensional Euclidean space. First we consider Lévy's Brownian motion. We use an integral representation for the covariance function to find a new expansion for Lévy's Brownian motion as an infinite linear combination of...
The extreme value index (EVI) links the generalized extreme value (GEV) distribution and the generalized Pareto (GP) distribution. These two distributions are fundamental in extreme value theory (EVT), with the GEV distribution being the only possible non-degenerate limiting distribution of properly normalized maxima of iid random variables, and the GP...
A stochastic process is given by a family of random variables indexed by elements of a set. We have considered stochastic processes of three different types, each involving an associated martingale structure. Martingale is a sequence of random variables for which the conditional expectation at a certain time point given...
Pardoxes in voting has been an interest of voting theorists since the 1800's when Condorcet demonstrated the key example of a voting paradox: voters with individually transitive rankings produce an election outcome which is not transitive. With Arrow's Impossibility Theorem, the hope of finding a fair voting method which accurately...
Integral representations provide a useful framework of study and simulation of fractional Browian motion, which has been used in modeling of many natural situations. In this thesis we extend an integral representation of fractional Brownian motion that is supported on a bounded interval of ℝ to integral representation that is...
This thesis is on the existence and uniqueness of weak solutions to the Navier-Stokes equations in R3 which govern the velocity of incompressible fluid with viscosity ν. The solution is obtained in the space of tempered distributions on R3 given an initial condition and forcing data which are dominated by...
Random fields are frequently used in computational simulations of real-life processes. In particular, in this work they are used in modeling of flow and transport in porous media. Porous media as they arise in geological formations are intrinsically deterministic but there is significant uncertainty involved in determination of their properties...