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：本文针对混合过程泛函，建立了样本分位数与$r$阶绝对中心样本矩的联合（二元）泛函中心极限定理。具体而言，我们考察$L_2$近邻依赖过程，该类过程为$\phi$混合过程或绝对正则过程的泛函。所得一般性结果可应用于两类重要且常用的过程：具有独立同分布新息的增广GARCH($p$,$q$)过程族（涵盖实践中多种GARCH变体），以及具有混合新息的ARMA($p$,$q$)过程族（例如ARMA-GARCH过程）。针对部分典型示例，我们给出了保证联合渐近性成立的矩条件与过程参数的确切条件。