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

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.