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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Svetlana Borovkova, Robert Burton and Herold Dehling
Journal: Trans. Amer. Math. Soc. 353 (2001), 4261-4318
MSC (1991): Primary 60F05, 62M10
Published electronically: June 20, 2001
MathSciNet review: 1851171
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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

\begin{displaymath}U_n(h) =\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq n} h(X_i,X_j) \end{displaymath}

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.

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Additional Information

Svetlana Borovkova
Affiliation: ITS-SSOR, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

Robert Burton
Affiliation: Department of Mathematics, Oregon State University, Kidder Hall 368, Corvallis Oregon 97331

Herold Dehling
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany

Received by editor(s): October 28, 1999
Received by editor(s) in revised form: December 14, 2000
Published electronically: June 20, 2001
Additional Notes: 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
Article copyright: © Copyright 2001 American Mathematical Society