Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation


Authors: Svetlana Borovkova, Robert Burton and Herold Dehling
Journal: Trans. Amer. Math. Soc. 353 (2001), 4261-4318
MSC (1991): Primary 60F05, 62M10
DOI: https://doi.org/10.1090/S0002-9947-01-02819-7
Published electronically: June 20, 2001
MathSciNet review: 1851171
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we develop a general approach for investigating the asymptotic distribution of functionals $X_n=f((Z_{n+k})_{k\in\mathbf{Z}})$of absolutely regular stochastic processes $(Z_n)_{n\in \mathbf{Z}}$. Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to $U$-statistics

\begin{displaymath}U_n(h) =\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq n} h(X_i,X_j) \end{displaymath}

with symmetric kernel $h:\mathbf{R}\times \mathbf{R}\rightarrow \mathbf{R}$. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for $U$-processes $(U_n(h))_{h}$, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.


References [Enhancements On Off] (What's this?)

  • 1. Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T., Weiss, B. (1996). Strong laws for $L$- and $U$-statistics. Trans. Amer. Math. Soc. 348, 2845-2865. MR 97b:60047
  • 2. Adler, R. and Weiss, B. (1970) Similarity of Automorphisms of the Torus. Memoirs Amer. Math. Soc., No. 98. MR 41:1966
  • 3. Arcones, M.A., Yu,B. (1994). Central limit theorems for empirical processes and $U$-processes of stationary mixing sequences. J. Theor. Probab. 7, 47-71. MR 95a:60024
  • 4. Barnsley, M. (1993) Fractals Everywhere, 2nd ed., Academic Press, Orlando FL. MR 94h:58101
  • 5. Berbee, H.C.P. (1979). Random walks with stationary increments and renewal theory. Mathematical Centre Tracts 112, Mathematisch Centrum, Amsterdam. MR 81e:60093
  • 6. Berk, R.H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37, 51-58. MR 32:6603
  • 7. Berkes, I., Philipp, W. (1977). An almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41, 115-137. MR 57:4276
  • 8. Berkes, I., Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Prob. 7, 29-54. MR 80k:60008
  • 9. Billingsley, P. (1968). Convergence of Probability Measures. John Wiley and Sons, NY. MR 38:1718
  • 10. Borovkova, S. (1995). Weak convergence of the empirical process of $U$-statistics structure for dependent observations. Theor. Stoch. Proc. 2 (18), 115-124.
  • 11. Bowen, R. (1975). Smooth partitions of Anosov diffeomorphisms are weak Bernoulli. Israel J. Math. 21, 95 - 100. MR 52:6786
  • 12. Borovkova, S., Burton, R.M., Dehling, H.G. (1999). Consistency of the Takens estimator for the correlation dimension. The Annals of Applied Probability 9, 376-390. MR 2000g:60035
  • 13. Burton, R. and Faris, W. (1996) A self-organizing cluster process. Ann. Appl. Prob 6, 1232-1247. MR 97k:60268
  • 14. Burton, R., C. Kraaikamp, and Schmidt, T. (2000) Natural Extensions for the Rosen Fractions. Trans. Amer. Math. Soc. 352, 1277-1298. MR 2000j:11123
  • 15. Davydov, Yu.A. (1970). The invariance principle for stationary processes. Th. Prob. Appl. 15, 487-498. (Russian orig., MR 44:1102)
  • 16. Dehling, H., Philipp, W. (1982). Almost sure invariance principles for weakly dependent vector-valued random variables. Ann. Probab. 10, 689-701. MR 83m:60011
  • 17. Dehling, H., Denker, M., Philipp, W. (1987). The almost sure invariance principle for the empirical process of $U$-staqtistic structure. Ann. Inst. Henri Poincaré 23, 121-134. MR 88i:60061
  • 18. Dehling, H. and Taqqu, M.S. (1989). The empirical process of some long-range dependent sequences with an application to U-statistics. The Annals of Statistics 17, 1767-1783. MR 91c:60025
  • 19. Denker, M., Grillenberger, C., Siegmund, K. (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics 527, Springer Verlag, Berlin. MR 56:15879
  • 20. Denker, M., Keller, G. (1983) On $U$-Statistics and v. Mises' Statistics for Weakly Dependent Processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 64, 505-522. MR 85e:60044
  • 21. Denker, M., Keller, G. (1986). Rigorous statistical procedures for data from dynamical systems. J. Stat. Phys. 44, 67-93. MR 87k:58149
  • 22. Deo, C.M. (1973). A note on empirical processes of strong-mixing sequences. Ann. Prob. 1, 870-875. MR 50:8631
  • 23. Donsker, M.D. (1951). An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. 6. MR 12:723a
  • 24. Doob, J. L. (1949) Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statistics 20, 393-403. MR 11:43a
  • 25. Dudley, R. M., Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62, 509-5523. MR 84g:60012
  • 26. Grassberger, P., Procaccia, I. (1983). Characterization of strange attractors. Phys. Rev. Lett. 50, 346-349. MR 84k:58141
  • 27. Halmos, P. A. (1946). The theory of unbiased estimation. Ann. Math. Statist. 17, 34-54. MR 7:463g
  • 28. Helmers, R., Janssen, P., Serfling, R. (1988). Glivenko-Cantelli properties of some generalized empirical df's and strong convergence of generalized $L$-statistics. Probab. Th. Rel. Fields 79, 75-93. MR 89h:60046
  • 29. Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293-325. MR 10:134g
  • 30. Hoeffding, W. (1961). The strong law of large numbers for $U$-statistics. University of North Carolina Mimeo Report No. 302.
  • 31. Hofbauer, F., Keller, G. (1982). Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119-140. MR 83h:28028
  • 32. Ibragimov, I.A., Linnik, Yu.V. (1971). Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen. MR 48:1287
  • 33. Loéve, M. (1977). Probability Theory. I, 4th edition, Springer Verlag, Berlin. MR 58:31324a
  • 34. Nakada, H. (1981). Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4, 399-426. MR 83k:10095
  • 35. Nakada, H., S. Ito, and Tanaka, S. (1977). On the invariant measure for the transformations associated with some real continued franctions. Keio Engineering Reports 30, 159-175. MR 58:16574
  • 36. Nolan, D., Pollard, D. (1988). Functional limit theorems for $U$-processes. Ann. Prob. 16, 1291-1298. MR 89g:60123
  • 37. Petersen, K. (1983). Ergodic Theory, Cambridge University Press, Cambridge. MR 87i:28002
  • 38. Philipp, W. (1977). A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Prob. 5, 319-350. MR 56:1397
  • 39. Philipp, W. (1986). Invariance principles for independent and weakly dependent random variables. in: Dependence in probability and statistics. (Eberlein, E., Taqqu, M., eds) Progress in Probability and Statistics 11, 225-269. Birkhäuser Boston. MR 89a:60093
  • 40. Ritter, H., T. Martinez, and K. Schulten (1992). Neural Computation and Self-Organizing Maps. Addison-Wesley Reading, MA.
  • 41. Serfling, R. (1984). Generalized $L$-, $M$- and $R$-statistics. Ann. Statist. 12, 76-86. MR 85i:62018
  • 42. Silverman, B. (1983). Convergence of a class of empirical distribution functions of dependent random variables. Ann. Prob. 11, 745-751 MR 84m:60033
  • 43. Silverman, B., Brown, T. (1978). Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815-825. MR 80c:60042
  • 44. Skorohod, A.V. (1956). Limit theorems for stochastic processes. Theor. Probab. Appl. 21, 628-632.
  • 45. Strassen, V., Dudley, R.M. (1969). The central limit theorem and $\varepsilon$-entropy. In: Lecture Notes in Mathematics 89, 224-231, Springer-Verlag, Berlin.
  • 46. Takens, F. (1981). Detecting strange attractors in turbulence. In: Dynamical systems and turbulence. Lecture Notes in Mathematics 898, 336-381. Springer-Verlag. MR 83i:58065
  • 47. Takens, F. (1985). On the numerical determination of the dimension of the attractor. In: Dynamical Systems and Bifurcations. Lecture Notes in Mathematics 1125, 99-106. Springer-Verlag. MR 86f:58043
  • 48. Withers, C.S. (1975). Convergence of empirical processes of mixing random variables. Ann. Statist. 3, 1101-1108. MR 52:15593
  • 49. Yoshihara K. (1976). Limiting behaviour of $U$-statistics for stationary, absolutely regular processes. Z. Wahr. Verw. Geb. 35, 237-252. MR 54:6221

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 60F05, 62M10

Retrieve articles in all journals with MSC (1991): 60F05, 62M10


Additional Information

Svetlana Borovkova
Affiliation: ITS-SSOR, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
Email: S.A.Borovkova@its.tudelft.nl

Robert Burton
Affiliation: Department of Mathematics, Oregon State University, Kidder Hall 368, Corvallis Oregon 97331
Email: burton@math.orst.edu

Herold Dehling
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany
Email: herold.dehling@ruhr-uni-bochum.de

DOI: https://doi.org/10.1090/S0002-9947-01-02819-7
Received by editor(s): October 28, 1999
Received by editor(s) in revised form: December 14, 2000
Published electronically: June 20, 2001
Additional Notes: Research supported by the Netherlands Organization for Scientific Research (NWO) grant NLS 61-277, NSF grant DMS 96-26575 and NATO collaborative research grant CRG 930819
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society