# Spring 2023 Seminar

#### Expository

This is the homepage for the UC Berkeley Student Probability Seminar, a venue for graduate students in the departments of mathematics, statistics, and others to study aspects of modern probability theory. We meet on Wednesdays in Evans 891 from 2:00 PM - 3:00 PM

Organizers: Ella Hiesmayr and Adam Quinn Jaffe.

The topic for this semester's seminar is Stein's method. We will mostly follow this survey article by Ross [R] but we may read more recent papers towards the end of the semester.

• 25 January, Ella Hiesmayr, Introductory talk, R Chapters 1 and 2
• We will define the operator that characterizes the normal distribution and show how this can be used to express the distance of a distribution to the normal.
• 01 February, Adam Quinn Jaffe, Wasserstein bounds for a quantitative CLT, R Chapter 3.1
• We will do some "Gaussian calculus" to show that the Wasserstein distance between an arbitrary random variable and a Gaussian random variable can be explicitly controlled. In particular, we will show that solutions to a certain "Gaussian ODE" have great regularity. This constitutes the basic tool for Stein's method, and we will see how it can be applied to give a quantitative CLT for sums of independent random variables.
• 08 February, Daniel Raban, Wasserstein CLT for sums of locally dependent random variables, R Chapter 3.2
• We will use the machinery we have developed to generalize our Wasserstein bounds to sums of random variables which have limited dependence. We will give an example of an application to analysis of the number of triangles in Erdõs-Renyi random graphs.
• 15 February, Mriganka Basu Roy Chowdhury, Exchangeable pairs, R Chapter 3.3
• Building on results from previous weeks, I will present the exchangeable pairs formalism for Stein's method. This variant of the method has a "dynamical" flavor to it, and is well suited to analyzing stationary distributions of Markov chains. In addition to proving the core lemmas, I will try to discuss some basic examples illustrating the method.
• 22 February, Vilas Weinstein, Size-Bias and Isolated Vertices, R Chapter 3.4
• Today we will discuss another special case of Stein’s method, involving the “size-biased” version of a random variable. As an application, we will give a quantitative convergence of the number of isolated vertices in an Erdős-Rényi graph to the normal distribution.
• 01 March, Karissa Huang, Poisson Approximation and the Chen-Stein Method, R Chapter 4.1
• In today's talk, we will see how the Stein Method can be adapted to other random variables and metrics. In particular, we will consider bounding the total variation distance between a distribution of interest and the Poisson distribution. The general framework is quite similar as in the Gaussian case; we will define a characterizing operator of the Possion, show that it has a unique solution, and prove some properties of that solution, which will give us the main theorem -- an upper bound on the TV distance. We will then see a simple, illustrative example of how the theorem can be applied.
• 08 March, No talk.

This page will be updated periodically to keep up with our progress throughout the semester.