Weak Convergence (IIA) - Functional and Random Aspects of the Univariate Extreme Value Theory

TL;DR

This paper provides a comprehensive mathematical and statistical analysis of weak convergence in univariate extreme value theory, focusing on foundational aspects and estimation problems.

Contribution

It offers an extensive synthesis of the mathematical foundations and statistical estimation methods in univariate extreme value theory, integrating new contributions and existing literature.

Findings

Mathematical foundation based on regular variation and weak convergence

Development of statistical estimation techniques for extreme values

Survey of research questions and unpublished results in the field

Abstract

The univariate extreme value theory deals with the convergence in type of powers of elements of sequences of cumulative distribution functions on the real line when the power index gets infinite. In terms of convergence of random variables, this amounts to the the weak convergence, in the sense of probability measures weak convergence, of the partial maximas of a sequence of independent and identically distributed random variables. In this monograph, this theory is comprehensively studied in the broad frame of weak convergence of random vectors as exposed in Lo et al.(2016). It has two main parts. The first is devoted to its nice mathematical foundation. Most of the materials of this part is taken from the most essential Lo\`eve(1936,177) and Haan (1970), based on the stunning theory of regular, pi or gamma variation. To prepare the statistical applications, a number contributions I made in my PhD and my Doctorate of Sciences are added in the last chapter of the last chapter of that part. Our real concern is to put these materials together with others, among them those of the authors from his PhD dissertations and Science doctorate thesis, in a way to have an almost full coverage of the theory on the real line that may serve as a master course of one semester in our universities. As well, it will help the second part of the monograph. This second part will deal with statistical estimations problems related to extreme values. It addresses various estimation questions and should be considered as the beginning of a survey study to be updated progressively. Research questions are tackled therein. Many results of the author, either unpublished or not sufficiently known, are stated and/or updated therein.