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 $X_n=f((Z_{n+k})_{k\in \mathbf {Z}})$ of absolutely regular stochastic processes $(Z_n)_{n\in \mathbf {Z}}$. Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic 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 re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to $U$-statistics \[ U_n(h) =\frac {1}{{n\choose 2}}\sum _{1\leq i<j\leq n} h(X_i,X_j) \] with symmetric kernel $h:\mathbf {R}\times \mathbf {R}\rightarrow \mathbf {R}$. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for $U$-processes $(U_n(h))_{h}$, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.