Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heavy-tailed random vectros: theory and applications

Dimitrios G. Konstantinides, Charalampos D. Passalidis

TL;DR

The work addresses understanding multivariate heavy-tailed behavior beyond standard MRV by developing a unified framework around multivariate positively decreasing distributions and their intersections with key tail classes. It establishes convolution-closure results, scale-mixing stability, and Breiman-type extensions in the multivariate setting, and demonstrates a linear single-big-jump principle for scale mixture sums. The paper then applies these tools to time-dependent multivariate risk models to derive finite-time ruin probabilities and provides lower bounds for precise large deviations in multivariate setups. Together, these results offer a comprehensive toolkit for analyzing dependent heavy-tailed vectors in risk, queuing, and related areas. The findings advance both theory (closure properties, convolution roots, MRV connections) and practice (risk metrics under dependence and scale modulation).

Abstract

In this paper we introduce and study several multivariate, heavy-tailed distribution classes, and we explore their closure properties and their applications. We consider the class of multivariate, positively decreasing distributions, and its intersection with other multivariate distribution classes.

Heavy-tailed random vectros: theory and applications

TL;DR

The work addresses understanding multivariate heavy-tailed behavior beyond standard MRV by developing a unified framework around multivariate positively decreasing distributions and their intersections with key tail classes. It establishes convolution-closure results, scale-mixing stability, and Breiman-type extensions in the multivariate setting, and demonstrates a linear single-big-jump principle for scale mixture sums. The paper then applies these tools to time-dependent multivariate risk models to derive finite-time ruin probabilities and provides lower bounds for precise large deviations in multivariate setups. Together, these results offer a comprehensive toolkit for analyzing dependent heavy-tailed vectors in risk, queuing, and related areas. The findings advance both theory (closure properties, convolution roots, MRV connections) and practice (risk metrics under dependence and scale modulation).

Abstract

In this paper we introduce and study several multivariate, heavy-tailed distribution classes, and we explore their closure properties and their applications. We consider the class of multivariate, positively decreasing distributions, and its intersection with other multivariate distribution classes.

Paper Structure

This paper contains 10 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Heavy tails and preliminary results
  3. Heavy-tailed distributions
  4. Heavy-tailed random vectors
  5. Convolution closure and related properties
  6. Dependence scale mixtures
  7. Closure properties with repsect to scale mixtures.
  8. Scale mixture sums.
  9. Finite-time ruin probability in time-dependent risk model
  10. Lower bounds of precise large deviations in mutlivariate set up

Key Result

Proposition 3.1

It holds $MRV(\alpha,\,V,\,\mu) \subsetneq (\mathcal{D}\cap \mathcal{A})_{\mathscr{R}}$, for any $\alpha \in (0,\,\infty)$.

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.1
  • Proposition 3.1
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.2
  • Theorem 3.1
  • Lemma 3.1
  • ...and 18 more