# A note on prediction for discrete time series

Gusztáv Morvai; Benjamin Weiss

Kybernetika (2012)

- Volume: 48, Issue: 4, page 809-823
- ISSN: 0023-5954

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topMorvai, Gusztáv, and Weiss, Benjamin. "A note on prediction for discrete time series." Kybernetika 48.4 (2012): 809-823. <http://eudml.org/doc/246722>.

@article{Morvai2012,

abstract = {Let $\lbrace X_n\rbrace $ be a stationary and ergodic time series taking values from a finite or countably infinite set $\{\mathcal \{X\}\}$ 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\rightarrow \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.},

author = {Morvai, Gusztáv, Weiss, Benjamin},

journal = {Kybernetika},

keywords = {nonparametric estimation; stationary processes; stationary processes; nonparametric estimation},

language = {eng},

number = {4},

pages = {809-823},

publisher = {Institute of Information Theory and Automation AS CR},

title = {A note on prediction for discrete time series},

url = {http://eudml.org/doc/246722},

volume = {48},

year = {2012},

}

TY - JOUR

AU - Morvai, Gusztáv

AU - Weiss, Benjamin

TI - A note on prediction for discrete time series

JO - Kybernetika

PY - 2012

PB - Institute of Information Theory and Automation AS CR

VL - 48

IS - 4

SP - 809

EP - 823

AB - Let $\lbrace X_n\rbrace $ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\mathcal {X}}$ 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\rightarrow \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.

LA - eng

KW - nonparametric estimation; stationary processes; stationary processes; nonparametric estimation

UR - http://eudml.org/doc/246722

ER -

## References

top- D. H. Bailey, Sequential Schemes for Classifying and Predicting Ergodic Processes., Ph.D. Thesis, Stanford University 1976. MR2626644
- A. Berlinet, I. Vajda, E. C. van der Meulen, 10.1109/18.669143, IEEE Trans. Inform. Theory 44 (1998), 3, 999-1009. Zbl0952.62029MR1616679DOI10.1109/18.669143
- K. L. Chung, 10.1214/aoms/1177705069, Ann. Math. Statist. 32 (1961), 612-614. Zbl0115.35503MR0131782DOI10.1214/aoms/1177705069
- T. M. Cover, J. Thomas, Elements of Information Theory., Wiley, 1991. Zbl1140.94001MR1122806
- I. Csiszár, P. Shields, 10.1214/aos/1015957472, Ann. Statist. 28 (2000), 1601-1619. Zbl1105.62311MR1835033DOI10.1214/aos/1015957472
- I. Csiszár, 10.1109/TIT.2002.1003842, IEEE Trans. Inform. Theory 48 (2002), 1616-1628. Zbl1060.62092MR1909476DOI10.1109/TIT.2002.1003842
- G. A. Darbellay, I. Vajda, 10.1109/18.761290, {IEEE Trans. Inform. Theory 45 (1999), 4, 1315-1321.} Zbl0957.94006MR1686274DOI10.1109/18.761290
- J. Feistauerová, I. Vajda, 10.1109/21.260666, IEEE Trans. Systems Man Cybernet. 23 (1993), 5 1352-1358. DOI10.1109/21.260666
- L. Györfi, G. Morvai, S. Yakowitz, 10.1109/18.661540, {IEEE Trans. Inform. Theory} 44 (1998), 886-892. Zbl0899.62122MR1607704DOI10.1109/18.661540
- 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.
- W. Hoeffding, 10.1080/01621459.1963.10500830, {J. Amer. Statist. Assoc.} 58 (1963), 13-30. Zbl0127.10602MR0144363DOI10.1080/01621459.1963.10500830
- S. Kalikow, 10.1007/BF02807249, {Israel J. Math.} 71 (1990), 33-54. Zbl0711.60041MR1074503DOI10.1007/BF02807249
- M. Keane, 10.1007/BF01425715, {Invent. Math. } 16 (1972), 309-324. Zbl0241.28014MR0310193DOI10.1007/BF01425715
- H. Luschgy, L. A. Rukhin, I. Vajda, Adaptive tests for stochastic processes in the ergodic case., {Stochastic Process. Appl.} 45 (1993), 1, 45-59. Zbl0770.62071MR1204860
- G. Morvai, I. Vajda, A survay on log-optimum portfolio selection., In: Second European Congress on Systems Science, Afcet, Paris 1993, pp. 936-944.
- G. Morvai, B. Weiss, 10.1007/s00440-004-0386-3, {Probab. Theory Related Fields} 132 (2005), 1-12. MR2136864DOI10.1007/s00440-004-0386-3
- G. Morvai, B. Weiss, 10.1016/j.anihpb.2005.11.001, { Ann. Inst. H. Poincaré Probab. Statist.} 43 (2007), 15-30. Zbl1106.62094MR2288267DOI10.1016/j.anihpb.2005.11.001
- B. Ya. Ryabko, Prediction of random sequences and universal coding., {Problems Inform. Transmission} 24 (1988), 87-96. Zbl0666.94009MR0955983
- B. Ryabko, 10.1109/TIT.2009.2025546, {IEEE Trans. Inform. Theory} 55 (2009), 9, 4309-4315. MR2582884DOI10.1109/TIT.2009.2025546
- I. Vajda, F. Österreicher, Existence, uniqueness and evaluation of log-optimal investment portfolio., {Kybernetika} 29 (1993), 2, 105-120. Zbl0799.90013MR1227745
- I. Vajda, P. Harremoës, 10.1109/TIT.2007.911155, IEEE Trans. Inform. Theory 54 (2008), 321-331. MR2446756DOI10.1109/TIT.2007.911155

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