UU-processes, UU-quantile processes and generalized linear statistics of dependent data

Generalized linear statistics are a unifying class that contains UU-statistics, UU-quantiles, LL-statistics as well as trimmed and Winsorized UU-statistics. For example, many commonly used estimators of scale fall into this class. GLGL-statistics have only been studied under independence; in this paper, we develop an asymptotic theory for GLGL-statistics of sequences which are strongly mixing or L1L1 near epoch dependent on an absolutely regular process. For this purpose, we prove an almost sure approximation of the empiricalUU-process by a Gaussian process. With the help of a generalized Bahadur representation, it follows that such a strong invariance principle also holds for the empirical UU-quantile process and consequently for GLGL-statistics. We obtain central limit theorems and laws of the iterated logarithm for UU-processes, UU-quantile processes and GLGL-statistics as straightforward corollaries.