Journal of Statistical Physics

, Volume 85, Issue 1, pp 25–40

Ergodic theorems arising in correlation dimension estimation

  • Regis J. Serinko

DOI: 10.1007/BF02175554

Cite this article as:
Serinko, R.J. J Stat Phys (1996) 85: 25. doi:10.1007/BF02175554


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 ℜd 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.

Key Words

Glivenko-Cantelli theorem fractal almost sure convergence moment estimators least square estimators dynamical systems chaos 
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Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Regis J. Serinko
    • 1
  1. 1.Department of StatisticsPurdue UniversityWest Lafayette

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