2026-09-22T02:55:50Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/120316 2023-09-05 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Dey, Partha S Tserunyan, Anush Song, Renming Baryshnikov, Yuliy Bernshteyn, Anton 2023-05 application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2023-09-01 without embargo terms The student, Grigory Terlov, accepted the attached license on 2023-04-21 at 22:43. The student, Grigory Terlov, submitted this Dissertation for approval on 2023-04-21 at 22:50. This Dissertation was approved for publication on 2023-04-27 at 11:19. DSpace SAF Submission Ingestion Package generated from Vireo submission #19110 on 2023-09-01 at 17:09:24 Conditional Stein’s method and maximal spanning forests Terlov, Grigory 2023-04-27 Stein’s Method Central Limit Theorem Rate Of Convergence Conditional Law Multivariate Normal Approximation Borel Graphs Amenable Countable Borel Equivalence Relations Quasi-pmp Nonsingular Group Actions Radon–nikodym Cocycle Spanning Forest Random Forest Percolation. This dissertation consists of two independent parts. Part 1. In the seventies, Charles Stein revolutionized the way of proving the Central Limit Theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last fifty years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. We develop a novel approach using Stein's method for exchangeable pairs to find a rate of convergence in the Conditional Central Limit Theorem of the form $(X_n\mid Y_n=k)$, where $(X_n, Y_n)$ are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph counts in Erd\H{o}s-R\'enyi random graph. This chapter is joint work with Partha S.~Dey and has appeared in Annals of Probability, 51(2), 723-773, (March 2023). Part 2. We prove the almost everywhere nonamenability of quasi-pmp (measure-class preserving) locally finite Borel graphs whose every component admits at least three nonvanishing ends with respect to the underlying Radon--Nikodym cocycle. We witness their nonamenability by constructing Borel subforests with at least three nonvanishing ends per component, and then applying Tserunyan and Tucker-Drob's recent characterization of amenability for acyclic quasi-pmp Borel graphs. Our main technique is a weighted cycle-cutting algorithm, which yields a weight-maximal spanning forest. We also introduce a random version of this forest, which generalizes the Free Minimal Spanning Forest, to capture nonunimodularity in the context of percolation theory. This chapter is joint work with Ruiyuan Chen and Anush Tserunyan. Thesis eng https://hdl.handle.net/2142/120316 Copyright 2023 Grigory Terlov Ph.D. Dissertation Mathematics University of Illinois at Urbana-Champaign Mathematics