Geodesic
Estimating the conditional expectations for continuous time stationary processes
Kybernetika, Tome 56 (2020) no. 3, pp. 410-431
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.14736/kyb-2020-3-0410
Classification : 60G10, 60G25, 62G05
Keywords: nonparametric estimation; continuous time stationary processes 
@article{10_14736_kyb_2020_3_0410,
     author = {Morvai, Guszt\'av and Weiss, Benjamin},
     title = {Estimating the conditional expectations for continuous time stationary processes},
     journal = {Kybernetika},
     pages = {410--431},
     year = {2020},
     volume = {56},
     number = {3},
     doi = {10.14736/kyb-2020-3-0410},
     mrnumber = {4131737},
     zbl = {07250731},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2020-3-0410/}
}
TY  - JOUR
AU  - Morvai, Gusztáv
AU  - Weiss, Benjamin
TI  - Estimating the conditional expectations for continuous time stationary processes
JO  - Kybernetika
PY  - 2020
SP  - 410
EP  - 431
VL  - 56
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2020-3-0410/
DO  - 10.14736/kyb-2020-3-0410
LA  - en
ID  - 10_14736_kyb_2020_3_0410
ER  - 
%0 Journal Article
%A Morvai, Gusztáv
%A Weiss, Benjamin
%T Estimating the conditional expectations for continuous time stationary processes
%J Kybernetika
%D 2020
%P 410-431
%V 56
%N 3
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2020-3-0410/
%R 10.14736/kyb-2020-3-0410
%G en
%F 10_14736_kyb_2020_3_0410
Morvai, Gusztáv; Weiss, Benjamin. Estimating the conditional expectations for continuous time stationary processes. Kybernetika, Tome 56 (2020) no. 3, pp. 410-431. doi: 10.14736/kyb-2020-3-0410

[1] Algoet, P.: Universal schemes for prediction, gambling and portfolio selection. Ann. Probab. 20 (1992), 901-941. | DOI | MR

[2] Algoet, P.: The strong law of large numbers for sequential decisions under uncertainty. IEEE Trans. Inform. Theory 40 (1994), 609-633. | DOI | MR

[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 | MR | Zbl

[4] Bailey, D.: Sequential Schemes for Classifying and Predicting Ergodic Processes. Ph.D. Thesis, Stanford University 1976. | MR

[5] Breiman, L.: The individual ergodic theorem of information theory. Ann. Math. Statist. 28 (1957), 809-811. | DOI | MR

[6] Cover, T.: Open problems in information theory. In: 1975 IEEE-USSR Joint Workshop on Information Theory 1975, pp. 35-36. | MR

[7] Chow, Y. S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales. Second edition. Springer-Verlag, New York 1978. | MR

[8] Doob, J. L.: Stochastic Processes. Wiley, 1990 | MR

[9] Elton, J.: A law of large numbers for identically distributed martingale differences. Ann. Probab. 9 (1981), 405-412. | DOI | MR

[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 | MR

[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 | MR | Zbl

[12] Györfi, L., Ottucsák, Gy.: Sequential prediction of unbounded stationary time series. IEEE Trans. Inform. Theory 53 (2007), 1866-1872. | DOI | MR

[13] Györfi, L., Ottucsák, Gy., Walk, H.: Machine Learning for Financial Engineering. Imperial College Press, London 2012. | DOI

[14] Hall, P., Heyde, C. C.: Martingale Limit Theory and Its Application. Academic Prress, 1975. | MR

[15] Maker, Ph. T.: The ergodic theorem for a sequence of functions. Duke Math. J. 6 (1940), 27-30. | DOI | MR

[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 | MR

[18] Morvai, G., Weiss, B.: Nonparametric sequential prediction for stationary processes. Ann. Prob. 39 (2011), 1137-1160. | DOI | MR

[19] Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, 1965. | MR

[20] Ornstein, D.: Guessing the next output of a stationary process. Israel J. of Math. 30 (1978), 292-296. | DOI | MR

[21] Ryabko, B.: Prediction of random sequences and universal coding. Probl. Inform. Trans. 24 (1988), 87-96. | MR | Zbl

[22] Scarpellini, B.: Predicting the future of functions on flows. Math. Systems Theory 12 (1979), 281-296. | DOI | MR

[23] Scarpellini, B.: Entropy and nonlinear prediction. Probab. Theory Related Fields 50 (1079, 2, 165-178. | DOI | MR

[24] Scarpellini, B.: Conditional expectations of stationary processes. Z. Wahrsch. Verw. Gebiete 56 (1981), 4, 427-441. | DOI | MR

[25] Shields, P. C.: Cutting and stacking: a method for constructing stationary processes. IEEE Trans. Inform. Theory 37 (1991), 1605-1614. | DOI | MR

[26] Shiryayev, A. N.: Probability. Springer-Verlag, New York 1984. | MR

[27] Weiss, B.: Single Orbit Dynamics. American Mathematical Society, 2000. | MR

Cité par Sources :