Kybernetika 56 no. 3, 410-431, 2020

Estimating the conditional expectations for continuous time stationary processes

Gusztáv Morvai and Benjamin WeissDOI: 10.14736/kyb-2020-3-0410

Abstract:

One of the basic estimation problems for continuous time stationary processes $X_t$, is that of estimating $E\{X_{t+\beta}| X_s : s \in [0, t]\}$ based on the observation of the single block $\{X_s : s \in [0, t]\}$ when the actual distribution of the process is not known. We will give fairly optimal universal estimates of this type that correspond to the optimal results in the case of discrete time processes.

Keywords:

nonparametric estimation, continuous time stationary processes

Classification:

60G10, 60G25, 62G05

paper.pdf

References:

  1. P. Algoet: Universal schemes for prediction, gambling and portfolio selection. Ann. Probab. 20 (1992), 901-941.   DOI:10.1214/aop/1176989811
  2. P. Algoet: The strong law of large numbers for sequential decisions under uncertainty. IEEE Trans. Inform. Theory 40 (1994), 609-633.   DOI:10.1109/18.335876
  3. P. Algoet: Universal schemes for learning the best nonlinear predictor given the infinite past and side information. IEEE Trans. Inform. Theory 45 (1999), 1165-1185.   DOI:10.1109/18.761258
  4. D. Bailey: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph.D. Thesis, Stanford University 1976.   CrossRef
  5. L. Breiman: The individual ergodic theorem of information theory, Ann. Math. Statist. 28 (1957), 809-811.   CrossRef
  6. T. Cover: Open problems in information theory. In: 1975 IEEE-USSR Joint Workshop on Information Theory 1975, pp. 35-36.   CrossRef
  7. Y. S. Chow and H. Teicher: Probability Theory: Independence, Interchangeability, Martingales. Second edition. Springer-Verlag, New York 1978.   CrossRef
  8. J. L. Doob: Stochastic Processes. Wiley, 1990   CrossRef
  9. J. Elton: A law of large numbers for identically distributed martingale differences. Ann. Probab. 9 (1981), 405-412.   DOI:10.1214/aop/1176994414
  10. L. Györfi, M. Kohler, A. Krzy\.zak and H. Walk: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics, Springer-Verlag, New York 2002.   DOI:10.1007/b97848
  11. L. Györfi, G. Morvai and S. Yakowitz: Limits to consistent on-line forecasting for ergodic time series. IEEE Trans. Inform. Theory 44 (1998), 886-892.   DOI:10.1109/18.661540
  12. L. Györfi and Gy. Ottucsák: Sequential prediction of unbounded stationary time series. IEEE Trans. Inform. Theory 53 (2007), 1866-1872.   DOI:10.1109/tit.2007.894660
  13. L. Györfi, Gy. Ottucsák and H. Walk: Machine Learning for Financial Engineering. Imperial College Press, London 2012.   DOI:10.1142/p818
  14. P. Hall and C. C. Heyde: Martingale Limit Theory and Its Application. Academic Prress, 1975.   CrossRef
  15. Ph. T. Maker: The ergodic theorem for a sequence of functions. Duke Math. J. 6 (1940), 27-30.   DOI:10.1215/s0012-7094-40-00602-0
  16. G. Morvai: Estimation of Conditional Distribution for Stationary Time Series. Ph.D. Thesis, Technical University of Budapest 1994.   CrossRef
  17. G. Morvai, S. Yakowitz and L. Györfi: Nonparametric inferences for ergodic, stationary time series. Ann. Statist. 24 (1996), 370-379.   DOI:10.1214/aos/1033066215
  18. G. Morvai and B. Weiss: Nonparametric sequential prediction for stationary processes. Ann. Prob. 39 (2011), 1137-1160.   DOI:10.1214/10-aop576
  19. J. Neveu: Mathematical Foundations of the Calculus of Probability. Holden-Day, 1965.   CrossRef
  20. D. Ornstein: Guessing the next output of a stationary process. Israel J. of Math. 30 (1978), 292-296.   DOI:10.1007/bf02761077
  21. B. Ryabko: Prediction of random sequences and universal coding. Probl. Inform. Trans. 24 (1988), 87-96.   CrossRef
  22. B. Scarpellini: Predicting the future of functions on flows. Math. Systems Theory 12 (1979), 281-296.   DOI:10.1007/bf01776579
  23. B. Scarpellini: Entropy and nonlinear prediction. Probab. Theory Related Fields 50 (1079, 2, 165-178.   DOI:10.1007/bf00533638
  24. B. Scarpellini: Conditional expectations of stationary processes. Z. Wahrsch. Verw. Gebiete 56 (1981), 4, 427-441.   DOI:10.1007/bf00531426
  25. P. C. Shields: Cutting and stacking: a method for constructing stationary processes. IEEE Trans. Inform. Theory 37 (1991), 1605-1614.   DOI:10.1109/18.104321
  26. A. N. Shiryayev: Probability. Springer-Verlag, New York 1984.   CrossRef
  27. B. Weiss: Single Orbit Dynamics. American Mathematical Society, 2000.   CrossRef