# Article

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Keywords:
nonparametric estimation; continuous time stationary processes
Summary:
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.
References:
[1] Algoet, P.: Universal schemes for prediction, gambling and portfolio selection. Ann. Probab. 20 (1992), 901-941. DOI 10.1214/aop/1176989811 | MR 1159579
[2] Algoet, P.: The strong law of large numbers for sequential decisions under uncertainty. IEEE Trans. Inform. Theory 40 (1994), 609-633. DOI 10.1109/18.335876 | MR 1295308
[3] Algoet, P.: 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 | MR 1686250 | Zbl 0959.62078
[4] Bailey, D.: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph.D. Thesis, Stanford University 1976. MR 2626644
[5] Breiman, L.: The individual ergodic theorem of information theory. Ann. Math. Statist. 28 (1957), 809-811. DOI 10.1214/aoms/1177706899 | MR 0092710
[6] Cover, T.: Open problems in information theory. In: 1975 IEEE-USSR Joint Workshop on Information Theory 1975, pp. 35-36. MR 0469507
[7] Chow, Y. S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales. Second edition. Springer-Verlag, New York 1978. MR 0513230
[8] Doob, J. L.: Stochastic Processes. Wiley, 1990 MR 1038526
[9] Elton, J.: A law of large numbers for identically distributed martingale differences. Ann. Probab. 9 (1981), 405-412. DOI 10.1214/aop/1176994414 | MR 0614626
[10] Györfi, L., Kohler, M., Krzyzak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics, Springer-Verlag, New York 2002. DOI 10.1007/b97848 | MR 1920390
[11] Györfi, L., Morvai, G., Yakowitz, S.: Limits to consistent on-line forecasting for ergodic time series. IEEE Trans. Inform. Theory 44 (1998), 886-892. DOI 10.1109/18.661540 | MR 1607704 | Zbl 0899.62122
[12] Györfi, L., Ottucsák, Gy.: Sequential prediction of unbounded stationary time series. IEEE Trans. Inform. Theory 53 (2007), 1866-1872. DOI 10.1109/tit.2007.894660 | MR 2317147
[13] Györfi, L., Ottucsák, Gy., Walk, H.: Machine Learning for Financial Engineering. Imperial College Press, London 2012. DOI 10.1142/p818
[14] Hall, P., Heyde, C. C.: Martingale Limit Theory and Its Application. Academic Prress, 1975. MR 0624435
[15] Maker, Ph. T.: The ergodic theorem for a sequence of functions. Duke Math. J. 6 (1940), 27-30. DOI 10.1215/s0012-7094-40-00602-0 | MR 0002028
[16] Morvai, G.: Estimation of Conditional Distribution for Stationary Time Series. Ph.D. Thesis, Technical University of Budapest 1994.
[17] Morvai, G., Yakowitz, S., Györfi, L.: Nonparametric inferences for ergodic, stationary time series. Ann. Statist. 24 (1996), 370-379. DOI 10.1214/aos/1033066215 | MR 1389896
[18] Morvai, G., Weiss, B.: Nonparametric sequential prediction for stationary processes. Ann. Prob. 39 (2011), 1137-1160. DOI 10.1214/10-aop576 | MR 2789586
[19] Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, 1965. MR 0198505
[20] Ornstein, D.: Guessing the next output of a stationary process. Israel J. of Math. 30 (1978), 292-296. DOI 10.1007/bf02761077 | MR 0508271
[21] Ryabko, B.: Prediction of random sequences and universal coding. Probl. Inform. Trans. 24 (1988), 87-96. MR 0955983 | Zbl 0666.94009
[22] Scarpellini, B.: Predicting the future of functions on flows. Math. Systems Theory 12 (1979), 281-296. DOI 10.1007/bf01776579 | MR 0529563
[23] Scarpellini, B.: Entropy and nonlinear prediction. Probab. Theory Related Fields 50 (1079, 2, 165-178. DOI 10.1007/bf00533638 | MR 0551610
[24] Scarpellini, B.: Conditional expectations of stationary processes. Z. Wahrsch. Verw. Gebiete 56 (1981), 4, 427-441. DOI 10.1007/bf00531426 | MR 0621658
[25] Shields, P. C.: Cutting and stacking: a method for constructing stationary processes. IEEE Trans. Inform. Theory 37 (1991), 1605-1614. DOI 10.1109/18.104321 | MR 1134300
[26] Shiryayev, A. N.: Probability. Springer-Verlag, New York 1984. MR 0737192
[27] Weiss, B.: Single Orbit Dynamics. American Mathematical Society, 2000. MR 1727510

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