# Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH($p$,$q$) processes

## TL;DR

This paper extends the asymptotic theory for augmented GARCH($p$,$q$) processes by establishing a joint functional central limit theorem for sample quantiles and dispersion measures, broadening univariate results to a bivariate context.

## Contribution

It provides the first joint (bivariate) asymptotic results for sample quantiles and moments in augmented GARCH processes, building on and extending univariate asymptotic theory.

## Key findings

- Joint asymptotic normality established under specific moment conditions
- Conditions for univariate convergence suffice for bivariate case
- Explicit examples illustrating the applicability of the results

## Abstract

In this paper, we build upon the asymptotic theory for GARCH processes, considering the general class of augmented GARCH($p$, $q$) processes. Our contribution is to complement the well-known univariate asymptotics by providing a joint (bivariate) functional central limit theorem of the sample quantile and the r-th absolute centred sample moment. This extends existing results in the case of identically and independently distributed random variables. We show that the conditions for the convergence of the estimators in the univariate case suffice even for the joint bivariate asymptotics. We illustrate the general results with various specific examples from the class of augmented GARCH($p$, $q$) processes and show explicitly under which conditions on the moments and parameters of the process the joint asymptotics hold.