Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

TL;DR

This paper introduces multi-normex distributions to accurately approximate the sum of iid heavy-tailed random vectors, extending univariate methods to multivariate cases and providing convergence rates.

Contribution

It develops two new multivariate normex models, $d$-Normex and MRV-Normex, with theoretical convergence rates and practical numerical comparisons.

Findings

Both models achieve specified convergence rates under regular variation assumptions.

Numerical experiments demonstrate the models' effectiveness across different dependence structures.

The approach effectively combines Gaussian approximations with extreme value theory for heavy-tailed vectors.

Abstract

We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called 'normex' approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named $d$-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, via the CLT, while the difference between the two versions comes from using the exact distribution or the EV theorem for the maximum. The main theorems provide the rate of convergence for each version of the multi-normex distributions towards the distribution of the sum, assuming second order regular variation property for the norm of the parent random vector when considering the MRV-normex case. Numerical illustrations and comparisons are proposed with various dependence structures on the parent random vector, using QQ-plots based on geometrical quantiles.