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 lengthn, of a dynamical system defined on a metric space, which are no more than a distancer 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 distancer apart. It has recently been proven, for the case of an ergodic dynamical system defined on a separable metric spaceythat the GP empirical correlation integral converges a.s. to the GP correlation integral at continuity points of the latter asn. It is shown here that for ergodic systems defined on ^d with the max metric the convergence is uniform inr. 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.