Probability Distributions on a Circle

• Author / Creator

  Rahmatidehkordi, Ardalan

• 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.

• Subjects / Keywords

  • distribution modulo one
  • slow-growing sequence
  • Kantorovich- Wasserstein distance
  • empirical measure
  • weak convergence

• Graduation date

  Fall 2023

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/r3-85pm-r259

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.