Abstract
Two series of binary observations x 1,x 1,… and y 1,y 2,… are presented: x n and y n are given at each time n∈ℕ. It is assumed that the sequences are generated independently of each other by two B-processes. The question of interest is whether the sequences represent a typical realization of two different processes or of the same one. It is demonstrated that this is impossible to decide, in the sense that every discrimination procedure is bound to err with non-negligible frequency when presented with sequences from some B-processes. This contrasts with earlier positive results on B-processes, in particular, those showing that there are consistent \(\bar{d}\) -distance estimates for this class of processes, and on ergodic processes, in particular, those establishing consistent change point estimates.
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References
Adams, T.M., Nobel, A.B.: On density estimation from ergodic processes. Ann. Probab. 26(2), 794–804 (1998)
Gyorfi, L., Morvai, G., Yakowitz, S.: Limits to consistent on-line forecasting for ergodic time series. IEEE Trans. Inf. Theory 44(2), 886–892 (1998)
Gray, R.: Probability, Random Processes, and Ergodic Properties. Springer, Berlin (1988)
Lehmann, E.L.: Testing Statistical Hypotheses. Springer, Berlin (1986)
Morvai, G., Weiss, B.: On classifying processes. Bernoulli 11(3), 523–532 (2005)
Nobel, A.B.: Hypothesis testing for families of ergodic processes. Bernoulli 12(2), 251–269 (2006)
Ornstein, D.S.: Ergodic Theory, Randomness, and Dynamical Systems. Yale Mathematical Monographs, vol. 5. Yale Univ. Press, New Haven (1974)
Ornstein, D.S., Shields, P.: The \(\bar{d}\) -recognition of processes. Adv. Math. 104, 182–224 (1994)
Ornstein, D.S., Weiss, B.: How sampling reveals a process. Ann. Probab. 18(3), 905–930 (1990)
Ryabko, B.: Prediction of random sequences and universal coding. Probl. Inf. Transm. 24, 87–96 (1988)
Ryabko, B., Astola, J., Gammerman, A.: Application of Kolmogorov complexity and universal codes to identity testing and nonparametric testing of serial independence for time series. Theor. Comput. Sci. 359, 440–448 (2006)
Ryabko, D., Ryabko, B.: Nonparametric statistical inference for ergodic processes. IEEE Trans. Inf. Theory (to appear)
Ryabko, D., Ryabko, B.: On hypotheses testing for ergodic processes. In: Proceedings of IEEE Information Theory Workshop (ITW’08), Porto, Portugal, pp. 281–283 (2008)
Shields, P.: Two divergence-rate counterexamples. J. Theor. Probab. 6, 521–545 (1993)
Shields, P.: The interactions between ergodic theory and information theory. IEEE Trans. Inf. Theory 44(6), 2079–2093 (1998)
Shields, P.: The Ergodic Theory of Discrete Sample Paths. AMS Bookstore. AMS, Providence (1996)
Shiryaev, A.: Probability, 2nd edn. Springer, Berlin (1996)
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Ryabko, D. Discrimination Between B-Processes is Impossible. J Theor Probab 23, 565–575 (2010). https://doi.org/10.1007/s10959-009-0263-1
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DOI: https://doi.org/10.1007/s10959-009-0263-1