Ergodic theorems arising in correlation dimension estimation

Journal of Statistical Physics

Volumn 85, Issue 1-2, 1996, Pages 25-40

  Serinko, Regis J ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

Almost sure convergence; Dynamical systems: Chaos; Fractal; Glivenko Cantelli theorem; Least square estimators; Moment estimators

Indexed keywords

EID: 0030502375     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02175554     Document Type: Article

References (20)

1. J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hili, and B. Weiss, Law large numbers for U-statistics of dependent observation. Preprint (1993).
2. P. Billingsley, Ergodic Theory and Information (Wiley, New York, 1965).
3. P. Billingsley, Convergence of Probability Measure (Wiley, New York, 1968).
4. P. Billingsley, Probability and Measure, 2nd ed. (Wiley, New York, 1986).
5. C. Cutler, Some results on the behavior and estimation of fractal dimensions of distributions on attractors, J. Stat. Phys. 62:651-708 (1991).
6. C. Cutler, A review of the theory and estimation of fractal dimension, in Nonlinear Time Series and Chaos, Vol 1: Dimension Estimation and Models, H. Tong, ed. (World Scientific, Singapore, 1993), pp. 1-107.
7. C. Cutler, A theory of correlation dimension for stationary time series, Technical Report STAT-94-17, Department of Statistics, University of Waterloo (1994).
8. M. Denker, and G. Keller, Rigorous statistical procedures for data from dynamical systems, J. Stat. Phys. 44:67-93 (1986).
9. J. -P. Eckmann, and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57:617-656 (1985).
10. J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, and S. Ciliberto, Liapunov exponents from time series, Phys. Rev. A 34:4971-4979 (1986).
11. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York, 1971).
12. P. B. Gaenssler, Around the Glivenko-Cantelli theorem, Coll. Math. Soc. Janos Bolyai 11:93-103 (1974).
13. P. Grassberger, and I. Procaccia, Measuring the strangeness of strange attractors, Physica D 9:189-208 (1983).
14. K. Krickeberg, An alternative approach to Glivenko-Cantelli theorems, in Lecture Notes in Mathematics No. 566, (Springer-Verlag, New York, 1976), pp. 57-67.
15. A. R. Osborne, and A. Provenzale, Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35:357-381 (1989).
16. K. R. Parthasarathy, Probability measures on Metric Spaces (Academic Press, New York, (1967).
17. Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimension, J. Stat. Phys. 71:529-547 (1993).
18. R. Serinko, A consistent approach to least squares estimation of correlation dimension in weak Bernoulli dynamical system, Ann. Appl. Prob. 4:1234-1254 (1994).
19. F. Takens, On the numerical determination of the dimension of an attractor, in Lecture Notes in Mathematics, No. 1125 (Springer-Verlag, Berlin, 1984), pp. 99-106.
20. R. Wells, H. N. Shire, C. Fosmire, and J. A. Doran, Improved algorithms for estimating the correlation dimension and the associated probable errors, Penn State Department of Mathematics Technical Report, # AM114 (1993).