Tauberian Korevaar

TL;DR

This paper reviews Korevaar's contributions to Tauberian theory, especially its connections with probability, covering foundational concepts, key theorems, and applications in areas like large deviations and laws of large numbers.

Contribution

It provides a comprehensive overview of Korevaar's work on Tauberian theorems, Beurling's concepts, and their applications in probability theory.

Findings

Connections between Tauberian theorems and probability applications

Development of Beurling's slow variation and Tauberian theorem

Applications to extremes, laws of large numbers, and large deviations

Abstract

We focus on the Tauberian work for which Jaap Korevaar is best known, together with its connections with probability theory. We begin (Section 1) with a brief sketch of the field up to Beurling's work. We follow with three sections on Beurling aspects: Beurling slow variation (Section 2); the Beurling Tauberian theorem for which it was developed (Section 3); Riesz means and Beurling moving averages (Section 4). We then give three applications from probability theory: extremes (Section 5), laws of large numbers (Section 6), and large deviations (Section 7). We turn briefly to other areas of Korevaar's work in Section 8. We close with a personal postscript (whence our title).