# Bounding the Rates of Convergence Towards the Extreme Value Distributions

5-13-2019

Thesis

## Abstract

Extreme value theory is a branch of probability which examines the extreme outliers of probability distributions. Three extreme value distributions arise as the limits of the maxima of sequences of random variables with certain properties. In this paper, we will first give information about these three distributions and prove that they are the only limit distributions of maxima. After that, we switch to a discussion about Stein's method. Stein's method is commonly used to prove central limit theorems. Stein's method also develops bounds on the distance between probability distributions with regards to a probability metric. There are three essential steps to Stein's method: finding a characterizing operator, solving the Stein equation, and then using the solution to that equation to generate bounds on the distance to the target distribution. We will give a general overview of the method with some basic examples, and then go over various ways to find operators for any distribution. We outline the generalized density method, a recent technique for finding operators, and apply it to the extreme value distributions to end a particularly good operator. Next, we work with two operators simultaneously in an attempt to bound the distance between maxima and the extreme value distributions. Specifically, we apply this idea to the convergence of the exponential distribution to the Gumbel.

Mathematics

## Thesis Comittee

Dr. John Pike, Thesis Advisor

Dr. Stephen Flood, Committee Member

Dr. Laura Gross, Committee Member

## Copyright and Permissions

Original document was submitted as an Honors Program requirement. Copyright is held by the author.

COinS