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

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.