Closure properties and heavy tails: random vectors in the presence of dependence

TL;DR

This paper investigates how heavy-tailed distribution classes behave under dependence, focusing on closure properties of products, sums, and mixtures, and introduces new classes to better understand multivariate heavy tails.

Contribution

It establishes closure properties of heavy-tailed classes under dependence, introduces new distribution classes, and explores their applications in risk modeling and multivariate analysis.

Findings

Closure of heavy-tailed classes under product convolution with dependence

Introduction of new distribution classes with closure properties

Applications to risk models and multivariate heavy-tailed vectors

Abstract

This paper is organized in three parts closely related to closure properties of heavy-tailed distributions and heavy-tailed random vectors. In the first part we consider two random variables X and Y with distributions F and G respectively. We assume that these random variables satisfy one type of a weak dependence structure. Under some mild conditions, we examine whether their product convolution distribution H belongs in the same distribution class of the distribution F. Namely we establish the closure property with respect to the product convolution, under this specific weak dependence structure, in the classes ERV, C, D, M, OS, OL, PD and K. Further in the second part we introduce a new distribution class, which satisfies some closure properties such as product and mixture.Further, we provide some applications on randomly weighted sums and on discrete-time risk model with dependent insurance and financial risks. Although the multivariate regular variation is well-established distribution, it does not happen in other heavy tailed random vectors. Therefore in the third part we introduced the class of dominatedly varying vectors and positively decreasing random vectors and we study the closure property of the independent scalar product under. Furthermore we study the closure property of the first class under sum and mixture, and we study the distribution of stopped sums where the summands are random vectors which belongs to this class. Some of these results holds and for positively decreasing random vectors.