Joint FCLT for Sample Quantile and Measures of Dispersion for Functionals of Mixing Processes

TL;DR

This paper proves a joint functional central limit theorem for sample quantiles and measures of dispersion in mixing processes, applicable to GARCH and ARMA models, with specific conditions for asymptotic validity.

Contribution

It establishes the first joint CLT for sample quantiles and dispersion measures in mixing processes, covering GARCH and ARMA models with explicit moment conditions.

Findings

Joint CLT for quantiles and moments in mixing processes

Applicable to GARCH and ARMA models with explicit conditions

Provides exact moment and parameter conditions for asymptotics

Abstract

In this paper, we establish a joint (bivariate) functional central limit theorem of the sample quantile and the $r$-th absolute centred sample moment for functionals of mixing processes. More precisely, we consider $L_2$-near epoch dependent processes that are functionals of either $\phi$-mixing or absolutely regular processes. The general results we obtain can be used for two classes of popular and important processes in applications: The class of augmented GARCH($p$,$q$) processes with independent and identically distributed innovations (including many GARCH variations used in practice) and the class of ARMA($p$,$q$) processes with mixing innovations (including, e.g., ARMA-GARCH processes). For selected examples, we provide exact conditions on the moments and parameters of the process for the joint asymptotics to hold.