Conditional Stein’s method and maximal spanning forests

Terlov, Grigory

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https://hdl.handle.net/2142/120316 Copy

Description

Title

Conditional Stein’s method and maximal spanning forests

Author(s)

Terlov, Grigory

Issue Date

2023-04-27

Director of Research (if dissertation) or Advisor (if thesis)

• Dey, Partha S
• Tserunyan, Anush

Doctoral Committee Chair(s)

Song, Renming

Committee Member(s)

• Baryshnikov, Yuliy
• Bernshteyn, Anton

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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.

Abstract

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.

Graduation Semester

2023-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2023 Grigory Terlov

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