We consider the general question of estimating decay of correlations for non-uniformly expanding maps, for classes of observables which are much larger than the usual class of Holder continuous functions. Our results give new estimates for many non-uniformly expanding systems, including Manneville-Pomeau maps, many one-dimensional systems with critical points, and Viana maps. In many situations, we also obtain a Central Limit Theorem for a much larger class of observables than usual. Our main tool is an extension of the coupling method introduced by L.-S. Young for estimating rates of mixing on certain non-uniformly expanding tower maps.
Let us now mention some other results concerning estimates on decay of correlations for non-H "older observables. Most of these are stated in the context of one-sided finite alphabet shift maps, or subshifts of finite type. (For a comprehensive discussion of shift maps and equilibrium measures, we suggest the book of Baladi, @cite_6 .) Shift maps are relatively simple dynamical systems, but are often used to more complicated systems via a semi-conjugacy, in much the same way that each of the examples we consider can be represented by a suitable (see ). Where a system @math being coded has an invariant measure @math which is absolutely continuous with respect to Lebesgue measure, @math is an equilibrium measure for the potential @math , where @math is the Jacobian with respect to Lebesgue measure. Most results for shift maps work with an equilibruim measure given by a potential @math which is H "older continuous (in terms of the usual metric on shift spaces - two sequences are said to be distance @math apart if they agree for exactly the first @math symbols). This assumption corresponds to assuming good distortion for @math .
@cite_6
1582592707
We study the density estimation problem with observations generated by certain dynamical systems that admit a unique underlying invariant Lebesgue density. Observations drawn from dynamical systems are not independent and moreover, usual mixing concepts may not be appropriate for measuring the dependence among these observations. By employing the @math -mixing concept to measure the dependence, we conduct statistical analysis on the consistency and convergence of the kernel density estimator. Our main results are as follows: First, we show that with properly chosen bandwidth, the kernel density estimator is universally consistent under @math -norm; Second, we establish convergence rates for the estimator with respect to several classes of dynamical systems under @math -norm. In the analysis, the density function @math is only assumed to be H " o lder continuous which is a weak assumption in the literature of nonparametric density estimation and also more realistic in the dynamical system context. Last but not least, we prove that the same convergence rates of the estimator under @math -norm and @math -norm can be achieved when the density function is H " o lder continuous, compactly supported and bounded. The bandwidth selection problem of the kernel density estimator for dynamical system is also discussed in our study via numerical simulations.