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.

翻译：本文建立了混合过程泛函的样本分位数与$r$阶绝对中心样本矩的联合（二元）泛函中心极限定理。更精确地，我们考虑作为$\phi$-混合或绝对正则过程泛函的$L_2$-近邻相依过程。所获得的通用结果可应用于两类在实际应用中重要且流行的过程：具有独立同分布新息的增广GARCH($p$,$q$)过程类（包括实践中使用的多种GARCH变体）和具有混合新息的ARMA($p$,$q$)过程类（例如包含ARMA-GARCH过程）。针对选定的实例，我们给出了该联合渐近性成立所需的过程矩与参数的精确条件。