Essays on regular variations in classical and free setup

TL;DR

This thesis explores applications of regularly varying functions across three problems: randomly weighted sums, product behavior under conditional extreme value models, and heavy-tailed measures under free convolution, highlighting their theoretical properties.

Contribution

It introduces new insights into the behavior of heavy-tailed distributions and their interactions in classical and free probability frameworks.

Findings

Analysis of randomly weighted sums with heavy tails

Behavior of products under conditional extreme value models

Heavy-tailed measures under free convolution

Abstract

This is a thesis on some applications of regularly varying functions. Three problems are considered. The first problem is about the randomly weighted sums, the second is on the behavior of the product under conditional extreme value model and the final problem studies heavy tailed measures under free convolution. The first chapter gives a brief overview of the heavy tailed distributions.