Local Limit Theorems on Finitely Generated Abelian Groups

Author (Your Name)

Yutong Yan, Colby CollegeFollow

Date of Award

2025

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Evan Randles

Second Advisor

Leo Livshits

Abstract

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility to avoid difficulties in analysis. In this thesis, we show that for finite-range random walks on finitely generated abelian groups, many such results can be stated in a form without such assumptions. Using tools from harmonic analysis, we formulate the local limit theorems in terms of a product of an attractor, which is in a uniform or Gaussian measure, and a prefactor (which we will refer to as a ``dance function") that describes the periodic/reducible structure of the random walk. The result generalizes the classical results for random walks on an integer lattice and introduces a formalism of the local limit theorems without the aforementioned hypotheses. The results of this thesis will be presented in a forthcoming publication coauthored by the author and E. Randles.

Keywords

random walk, probability, harmonic analysis

Recommended Citation