zbMATH — the first resource for mathematics

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.
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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.