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A strong limit theorem for the average of ternary functions of Markov chains in bi-infinite random environments. (English) Zbl 1316.60039
Summary: We consider strong limit theorems for Markov chains in bi-infinite random environments. We first give a new proof of a strong limit theorem of W. Liu and W. Yang [Stat. Probab. Lett. 22, No. 4, 295–301 (1995; Zbl 0833.60034)] on the average of functions of non-homogeneous Markov chains by constructing a nonnegative martingale. As corollaries, for a Markov chain in a bi-infinite random environment, we obtain strong limit theorems for the conditional relative entropy and for the number of times that the Markov chain and related processes reach a given point.
##### MSC:
 60F15 Strong limit theorems 60K37 Processes in random environments 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F05 Central limit and other weak theorems
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##### References:
 [1] Breiman, L., The individual ergodic theorem of information theory, Ann. Math. Statist., 28, 809-811, (1957) · Zbl 0078.31801 [2] Cogburn, R., Markov chains in random environments: the case of Markovian environments, Ann. Probab., 8, 908-916, (1980) · Zbl 0444.60053 [3] Cogburn, R., The ergodic theory of Markov chains in random environments, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 66, 109-128, (1984) · Zbl 0525.60074 [4] Cogburn, R., On direct convergence and periodicity for transition probabilities of Markov chains in random environments, Ann. Probab., 18, 642-654, (1990) · Zbl 0707.60057 [5] Cogburn, R., On the central limit theorem for Markov chains in random environments, Ann. Probab., 19, 587-604, (1991) · Zbl 0733.60038 [6] Durrett, R., Probability: theory and examples, (2010), Cambridge University Press · Zbl 1202.60001 [7] Li, Y. Q., Recurrence and invariant measure of Markov chains in bi-infinite random environments, Sci. China Ser. A, 44, 1294-1299, (2001) · Zbl 1007.60103 [8] Liu, W., Relative entropy densities and a class of limit theorems of the sequence of m-valued random variables, Ann. Probab., 18, 829-839, (1990) · Zbl 0711.60026 [9] Liu, G. X.; Liu, W., Some strong limit theorems relative to the geometric average of random transition probabilities of arbitrary finite nonhomogeneous Markov chains, Statist. Probab. Lett., 21, 77-83, (1994) · Zbl 0814.60027 [10] Liu, W.; Liu, G. X., An extension of Shannon-mcmillan theorem and some limit properties for nonhomogeneous Markov chains, Stochastic Process. Appl., 61, 129-145, (1996) · Zbl 0861.60042 [11] Liu, W.; Yang, W. G., A limit theorem for the entropy desity of nonhomogeneous Markov information source, Statist. Probab. Lett., 22, 295-301, (1995) · Zbl 0833.60034 [12] McMillan, B., The basic theorems of information theory, Ann. Math. Statist., 24, 196-219, (1953) · Zbl 0050.35501 [13] Meyn, S. P.; Tweedie, R. L., Markov chains and stochastic stability, (1993), Springer-Verlag London · Zbl 0925.60001 [14] Orey, S., Markov chains with stochastically stationary transition probabilities, Ann. Probab., 19, 907-928, (1991) · Zbl 0735.60040 [15] Revuz, D., Markov chains, (1984), North-Holland Mathematical Library · Zbl 0539.60073 [16] Seppalainen, T., Large deviations for Markov chains with random transitions, Ann. Probab., 22, 713-748, (1994) · Zbl 0809.60032 [17] Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 379-423, (1948), 623-656 · Zbl 1154.94303
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