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. G L -statistics have only been studied under independence; in this paper, we develop an asymptotic theory for G L -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 G L -statistics. We obtain central limit theorems and laws of the iterated logarithm for U -processes, U -quantile processes and G L -statistics as straightforward corollaries.