Ergodic theorems arising in correlation dimension estimation

The Grassberger-Procaccia (GP) empirical spatial correlation integral, which plays an important role in dimension estimation, is the proportion of pairs of points in a segment of an orbit of length n, of a dynamical system defined on a metric space, which are no more than a distance r apart. It is used as an estimator of the GP spatial correlation integral, which is the probability that two points sampled independently from an invariant measure of the system are no more than a distance r apart. It has recently been proven, for the case of an ergodic dynamical system defined on a separable metric spacey that the GP empirical correlation integral converges as to the GP correlation integral at continuity points of the latter as n left-arrow ∞. It is shown here that for ergodic systems defined on open-quote A close-quote with the open-quotes maxclose quotes metric the convergence is uniform in r. Further, a simplified proof based on weak convergence arguments of the result in separable spaces is given. Finally, the Glivenko-Cantelli theorem is used to obtain ergodic theorems for both the moment estimators and least square estimators of correlation dimension