On U-statistics and v. mise' statistics for weakly dependent processes

Some probabilistic limit theorems for Hoeffding's U-statistic [13] and v. Mises' functional are established when the underlying processes are not necessarily independent. We consider absolutely regular processes [24] and processes (X_n)_n≧1 which are uniformly mixing [14] as well as their time reversal (X_-n)_n≦-1, called uniformly mixing in both directions of time. Many authors have weakened the hypothesis of independence in statistical limit theorems and considered m-dependent, Markov or weakly dependent processes; in particular for U statistics under weak dependence Sen [22] has considered *-mixing processes and derived a central limit theorem and a law of the iterated logarithm, while Yoshihara [26] proved central limit theorems and a.s. results in the absolutely regular and uniformly mixing case. Here we extend these results considerably and prove central limit theorems and their rate of convergence (in the Prohorov metric and a Berry Esseén type theorem), functional central limit theorems and a.s. approximation by a Brownian motion. Extensions to multisample versions and other extensions are briefly discussed.