Probability Distributions on a Circle

Abstract

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 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.