TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 11, Pages 4261–4318 S 0002-9947(01)02819-7 Article electronically published on June 20, 2001 LIMIT THEOREMS FOR FUNCTIONALS OF MIXING PROCESSES WITH APPLICATIONS TO U -STATISTICS AND DIMENSION ESTIMATION SVETLANA BOROVKOVA, ROBERT BURTON, AND HEROLD DEHLING 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 reg- ular 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 re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U -statistics Un(h)= 1 ( n 2 )  1≤i<j≤n h(X i ,X j ) with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for ab- solutely regular processes. We also prove a central limit theorem under a differ- ent set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U -processes (Un(h)) h , in- dexed 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. 1. Introduction In this section we provide some motivation for the research presented in this paper, and present some key examples. We will mainly show how functionals of mixing processes occur naturally in the study of dynamical systems, and how U - statistics enter in the context of dimension estimation. 1.1. Examples. Definition 1.1. Let (Ω, F ,P ) be a probability space and let (Z n ) n∈Z be a station- ary stochastic process. (i) We call a sequence (X n ) n∈Z a two-sided functional (or simply functional) of (Z n ) n∈Z if there is a measurable function f defined on R Z such that X n = f ((Z n+k ) k∈Z ). (1.1) Note that (X n ) n∈Z is necessarily a stationary stochastic process. Received by the editors October 28, 1999 and, in revised form, December 14, 2000. 1991 Mathematics Subject Classification. Primary 60F05, 62M10. Research supported by the Netherlands Organization for Scientific Research (NWO) grant NLS 61-277, NSF grant DMS 96-26575 and NATO collaborative research grant CRG 930819. c 2001 American Mathematical Society 4261 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use