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Full-Text Articles in Harmonic Analysis and Representation
Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan
Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan
Honors Theses
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 …
Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su
Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su
All HMC Faculty Publications and Research
We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin2(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.