# Homogeneous mappings of regularly varying vectors

## TL;DR

This paper extends the theory of regular variation from univariate variables to multivariate vectors, providing conditions for products of such vectors and matrices to maintain regular variation, with applications to stochastic recurrences.

## Contribution

It offers sharp sufficient conditions for regular variation of product-type functions of multivariate regularly varying vectors, generalizing univariate results and applying to matrices and stochastic equations.

## Key findings

- Conditions for regular variation of products of vectors
- Characterization of regular variation in matrix products
- Application to affine stochastic recurrence solutions

## Abstract

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation property of product-type functions of regularly varying random vectors, generalizing and extending the univariate theory in various directions. The main result is then applied to characterize the regular variation property of products of iid regularly varying quadratic random matrices and of solutions to affine stochastic recurrence equations under non-standard conditions.