# Tail probabilities of random linear functions of regularly varying random vectors

## TL;DR

This paper extends Breiman's Theorem to multivariate cases, providing a comprehensive method to compute tail probabilities of linear transformations of regularly varying vectors, with applications in finance and reinsurance risk assessment.

## Contribution

It offers a complete characterization of multivariate regular variation under random linear transformations, expanding tail probability analysis beyond classical models.

## Key findings

- Derived new tail probability formulas for multivariate regular variation
- Applied results to risk assessment in financial and reinsurance contexts
- Demonstrated effectiveness through bipartite network models

## Abstract

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$ under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.