Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation

Abstract

In this paper we develop a general approach for investigating the asymptotic distribution of functionals Xn = f ((Z n+k) k∈Z) of absolutely regular stochastic processes (Zn) n∈Z. Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study proba-bilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by reproving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Un(h) = 1 n 2 1≤i