Geodesic
A note on prediction for discrete time series
Kybernetika, Tome 48 (2012) no. 4, pp. 809-823 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$ and that $f(X)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional expectation $E(f(X_{\lambda_n+1})|X_0,\dots,X_{\lambda_n} )$ from the observations $(X_0,\dots,X_{\lambda_n})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} \frac{n}{\lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upper bounded by a polynomial, eventually almost surely.
Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$ and that $f(X)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional expectation $E(f(X_{\lambda_n+1})|X_0,\dots,X_{\lambda_n} )$ from the observations $(X_0,\dots,X_{\lambda_n})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} \frac{n}{\lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upper bounded by a polynomial, eventually almost surely.
Classification : 60G10, 60G25, 62G05
Keywords: nonparametric estimation; stationary processes 
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Morvai, Gusztáv; Weiss, Benjamin. A note on prediction for discrete time series. Kybernetika, Tome 48 (2012) no. 4, pp. 809-823. http://geodesic.mathdoc.fr/item/KYB_2012_48_4_a9/

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

[2] A. Berlinet, I. Vajda, E. C. van der Meulen: About the asymptotic accuracy of Barron density estimates. IEEE Trans. Inform. Theory 44 (1998), 3, 999-1009. | DOI | MR | Zbl

[3] K. L. Chung: A note on the ergodic theorem of information theory. Ann. Math. Statist. 32 (1961), 612-614. | DOI | MR | Zbl

[4] T. M. Cover, J. Thomas: Elements of Information Theory. Wiley, 1991. | MR | Zbl

[5] I. Csiszár, P. Shields: The consistency of the BIC Markov order estimator. Ann. Statist. 28 (2000), 1601-1619. | DOI | MR | Zbl

[6] I. Csiszár: Large-scale typicality of Markov sample paths and consistency of MDL order estimators. IEEE Trans. Inform. Theory 48 (2002), 1616-1628. | DOI | MR | Zbl

[7] G. A. Darbellay, I. Vajda: Estimation of the information by an adaptive partitioning of the observation space. {IEEE Trans. Inform. Theory 45 (1999), 4, 1315-1321.} | DOI | MR | Zbl

[8] J. Feistauerová, I. Vajda: Testing system entropy and prediction error probability. IEEE Trans. Systems Man Cybernet. 23 (1993), 5 1352-1358. | DOI

[9] L. Györfi, G. Morvai, S. Yakowitz: Limits to consistent on-line forecasting for ergodic time series. {IEEE Trans. Inform. Theory} 44 (1998), 886-892. | DOI | MR | Zbl

[10] L. Györfi, G. Morvai, I. Vajda: Information-theoretic methods in testing the goodness of fit. {In: Proc. 2000 IEEE Internat. Symposium on Information Theory}, ISIT 2000, New York and Sorrento, p. 28.

[11] W. Hoeffding: Probability inequalities for sums of bounded random variables. {J. Amer. Statist. Assoc.} 58 (1963), 13-30. | DOI | MR | Zbl

[12] S. Kalikow: Random Markov processes and uniform martingales. {Israel J. Math.} 71 (1990), 33-54. | DOI | MR | Zbl

[13] M. Keane: Strongly mixing g-measures. {Invent. Math. } 16 (1972), 309-324. | DOI | MR | Zbl

[14] H. Luschgy, L. A. Rukhin, I. Vajda: Adaptive tests for stochastic processes in the ergodic case. {Stochastic Process. Appl.} 45 (1993), 1, 45-59. | MR | Zbl

[15] G. Morvai, I. Vajda: A survay on log-optimum portfolio selection. In: Second European Congress on Systems Science, Afcet, Paris 1993, pp. 936-944.

[16] G. Morvai, B. Weiss: Prediction for discrete time series. {Probab. Theory Related Fields} 132 (2005), 1-12. | DOI | MR

[17] G. Morvai, B. Weiss: Estimating the memory for finitarily Markovian processes. { Ann. Inst. H. Poincaré Probab. Statist.} 43 (2007), 15-30. | DOI | MR | Zbl

[18] B. Ya. Ryabko: Prediction of random sequences and universal coding. {Problems Inform. Transmission} 24 (1988), 87-96. | MR | Zbl

[19] B. Ryabko: Compression-based methods for nonparametric prediction and estimation of some characteristics of time series. {IEEE Trans. Inform. Theory} 55 (2009), 9, 4309-4315. | DOI | MR

[20] I. Vajda, F. Österreicher: Existence, uniqueness and evaluation of log-optimal investment portfolio. {Kybernetika} 29 (1993), 2, 105-120. | MR | Zbl

[21] I. Vajda, P. Harremoës: On the Bahadur-efficient testing of uniformity by means of entropy. IEEE Trans. Inform. Theory 54 (2008), 321-331. | DOI | MR