Abstract
The forecasting problem for a stationary and ergodic binary time series {X n } n=0 ∞ is to estimate the probability that X n+1=1 based on the observations X i , 0≤i≤n without prior knowledge of the distribution of the process {X n }. It is known that this is not possible if one estimates at all values of n. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.
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References
Azuma, K.: Weighted sums of certain dependent random variables, Tohoku Math.J.37(1967), 357–367.
Bailey, D. H.: Sequential schemes for classifying and predicting ergodic processes, PhD thesis, Stanford University, 1976.
Cover, T. M.: Open problems in information theory, In: 1975 IEEE Joint Workshop on Information Theory, IEEE Press, New York, 1975, pp.35–36.
Cover, T. M. and Thomas, J.: Elements of Information Theory, Wiley, New York, 1991.
Csiszár, I. and Shields, P.: The consistency of the BIC Markov order estimator, Ann.Statist.28(2000), 1601–1619.
Gray, R. M.: Probability,Random Processes,and Ergodic Properties, Springer-Verlag, New York, 1988.
Györfi, L., Morvai, G. and Yakowitz, S.: Limits to consistent on-line forecasting for ergodic time series, IEEE Trans.Inform.Theory 44(1998), 886–892.
Kalikow, S.: Random Markov processes and uniform martingales, Israel J.Math.71(1990), 33–54.
Keane, M.: Strongly mixing g-measures, Invent.Math.16(1972), 309–324.
Morvai, G.: Guessing the output of a stationary binary time series, In: Y. Haitovsky, H. R. Levche and Y. Ritov (eds), Foundations of Statistical Inference, Physika Verlag, 2003, pp.205–213.
Morvai, G., Yakowitz, S. and Algoet, P.: Weakly convergent nonparametric forecasting of stationary time series, IEEE Trans.Inform.Theory 43(1997), 483–498.
Morvai, G., Yakowitz, S. and Györfi, L.: Nonparametric inferences for ergodic, stationary time series, Ann.Statist.24(1996), 370–379.
Ornstein, D. S.: Guessing the next output of a stationary process, Israel J.Math.30(1978), 292–296.
Ornstein, D. S.: Ergodic Theory,Randomness,and Dynamical Systems, Yale Univ. Press, 1974.
Ornstein, D. S. and Weiss, B.: Entropy and data compression schemes, IEEE Trans.Inform. Theory 39(1993), 78–83.
Ryabko, B. Ya.: Prediction of random sequences and universal coding, Problems of Inform. Trans.24(1988), 87–96.
Shields, P. C.: Cutting and stacking: a method for constructing stationary processes, IEEE Trans.Inform.Theory 37(1991), 1605–1614.
Weiss, B.: Single Orbit Dynamics, Amer. Math. Soc., Providence, RI, 2000.
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Morvai, G., Weiss, B. Forecasting for Stationary Binary Time Series. Acta Applicandae Mathematicae 79, 25–34 (2003). https://doi.org/10.1023/A:1025862222287
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DOI: https://doi.org/10.1023/A:1025862222287