# On the tail behavior of a class of multivariate conditionally heteroskedastic processes

## TL;DR

This paper investigates the tail behavior of multivariate BEKK-ARCH processes, showing their invariant distributions are regularly varying and analyzing implications for sample covariance matrices.

## Contribution

It establishes conditions for geometric ergodicity and characterizes the tail behavior of BEKK-ARCH processes using vector scaling regular variation.

## Key findings

- Invariant distribution is regularly varying.
- Tail indices of marginals can differ.
- Asymptotic properties of sample covariance matrices derived.

## Abstract

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation, in the spirit of Perfekt (1997, Advances in Applied Probability, 29, pp. 138-164). The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.