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The positive-divergence and blowing-up properties. (English) Zbl 0797.60044

Summary: A property of ergodic finite-alphabet processes, called the blowing-up property, is shown to imply exponential rates of convergence for frequencies and entropy, which in turn imply a positive-divergence property. Furthermore, processes with the blowing-up property are finitely determined and the finitely determined property plus exponential rates of convergence for frequencies and for entropy implies blowing-up. It is also shown that finitary codings of i.i.d. processes have the blowing-up property.

MSC:

60G99 Stochastic processes
60F99 Limit theorems in probability theory
94A17 Measures of information, entropy
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[1] R. Ahlswede, P. Gács and J. Körner,Bounds on conditional probabilities with applications in multi-user communications, Z. Wahr. Ver. Geb.34 (1976), 157–177. · Zbl 0349.94038
[2] M. Akcoglu, M. Rahe, and A. del Junco,Finitary codes between Markov Processes, Z. Wahr. Ver. Geb.47 (1979), 305–314. · Zbl 0403.28017
[3] I. Csiszár and J. Körner,Information Theory. Coding Theorems for Discrete Memoryless Systems, Akadémiai Kiadó, Budapest, 1981.
[4] K. Marton,A simple proof of the blowing-up lemma, IEEE Trans. Inform. Th.IT-42 (1986), 445–447. · Zbl 0594.94003
[5] D. Ornstein,Ergodic Theory, Randomness, and Dynamical Systems, Yale University Press, New Haven, 1974. · Zbl 0296.28016
[6] D. Ornstein and B. Weiss,How sampling reveals a process, Annals of Probab.18 (1990), 905–930. · Zbl 0709.60036
[7] M. Pinsker,Information and information stability of random variables and processes (In Russian), Vol. 7 of the seriesProblemy Peredači Informacii, AN SSSR, Moscow, 1960. English translation: Holden-Day, San Francisco, 1964. · Zbl 0104.36702
[8] D. Rudolph,If a two-point extension of a Bernoulli shift has an ergodic square, then it is Bernoulli, Israel J. Math.30 (1978), 159–180. · Zbl 0415.28010
[9] P. Shields,Two divergence-rate counterexamples, J. Theoretical Probability6 (1993), 521–545. · Zbl 0799.60046
[10] P. Shields,Stationary coding of Processes, IEEE Trans. Inform. Th.IT-25 (1979), 283–291. · Zbl 0401.94018
[11] M. Smorodinsky and M. Keane,Finitary isomorphism of irreducible Markov shifts, Israel J. Math.34 (1979), 281–286. · Zbl 0431.28015
[12] M. Smorodinsky,Finitary isomorphism of m-dependent processes, Contemporary Math.135 (1992), 373–376. · Zbl 0768.28009
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