U-Processes, U-Quantile Processes and Generalized Linear Statistics of Dependent Data

Generalized linear statistics are a unifying class that contains U-statistics, U-quantiles, L-statistics as well as trimmed and winsorized U-statistics. For example, many commonly used estimators of scale fall into this class. GL-statistics only have been studied under independence; in this paper, we develop an asymptotic theory for GL-statistics of sequences which are strongly mixing or L^1 near epoch dependent on an absolutely regular process. For this purpose, we prove an almost sure approximation of the empirical U-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 U-quantile process and consequently for GL-statistics. We obtain central limit theorems and laws of the iterated logarithm for U-processes, U-quantile processes and GL-statistics as straightforward corollaries.