On the maximum entropy principle for uniformly ergodic Markov chains. (English) Zbl 0691.60023

Let \(\{X_ j:\) \(j\in N_ 0\}\), \(N_ 0=N\cup \{0\}\), be a Markov chain on (\(\Omega\),\({\mathcal A},P)\) with Polish state space E, and let \(L_ n=n^{-1}\sum^{n}_{j=1}\delta_{X_ j}\). Let H be some function defined on the set of probability measures on E with values in [- \(\infty,\infty)\) which is nice enough. Transformed laws are defined by \[ \hat P_ n(A)=(\int_{A}\exp \{nH(L_ n)\}dP)(\int_{\Omega}\exp \{nH(L_ n)\}dP)^{-1},\quad A\in {\mathcal A}. \] The possible limit laws of \(\{X_ j:\) \(j\in N_ 0\}\) under \(\hat P_ n\) are described. The main assumption is that \(\{X_ j:\) \(j\in N_ 0\}\) is uniformly ergodic. Roughly speaking, the limit laws are mixtures of Markov chains minimizing a certain free energy. The method of proof strongly relies on large deviation techniques.
Reviewer: B.Kryžienė


60F10 Large deviations
60J05 Discrete-time Markov processes on general state spaces
60F05 Central limit and other weak theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
Full Text: DOI


[1] Bolthausen, E., Markov process large deviations in the τ-topology, Stochastic Process Appl., 25, 97-108 (1987) · Zbl 0625.60026
[2] Choi, B. S.; Cover, Th. M.; Csiszar, I., Conditional limit theorems under Markov conditioning, IEEE Trans. Inform. Theory, IT-33, 788-801 (1987) · Zbl 0628.60037
[3] Csiszar, I., \(I\)-divergence geometry of probability distributions and minimization problems, Ann. Probab., 3, 146-158 (1975) · Zbl 0318.60013
[4] Csiszar, I., Sanov property, generalised \(I\)-projection and a conditional limit theorem, Ann. Probab., 12, 768-793 (1984) · Zbl 0544.60011
[5] Darroch, J. N.; Seneta, E., On quasi-stationary distributions in absorbing discrete time finite Markov chains, J. Appl. Probab., 2, 88-100 (1965) · Zbl 0134.34704
[6] Donsker, M. D.; Varadhan, S. R.S., Asymptotic evaluation of certain Markov process expectations for large time, Comm. Pure Appl. Math., 28, 1-47 (1975), I · Zbl 0323.60069
[7] Donsker, M. D.; Varadhan, S. R.S., Asymptotic evaluation of certain Markov process expectations for large time, Comm. Pure Appl. Math., 29, 389-461 (1976), III · Zbl 0348.60032
[8] Ethier, S. N.; Kurtz, T. G., Markov Processes, Characterization and Convergence (1986), Wiley: Wiley New York · Zbl 0592.60049
[9] Messer, J.; Spohn, H., Statistics of the isothermal Lane-Emden equation, J. Statist. Phys., 29, 561-577 (1982)
[10] Neveu, J., Discrete-Parameter Martingales (1975), North-Holland: North-Holland Amsterdam · Zbl 0345.60026
[11] Parthasarathy, K. R., Probability Measures on Metric Spaces (1967), Academic Press: Academic Press New York · Zbl 0153.19101
[12] Pinsky, R. G., On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes, Ann. Probab., 13, 363-378 (1985) · Zbl 0567.60076
[13] Seneta, E.; Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Probab., 3, 404-434 (1966) · Zbl 0147.36603
[14] Varadhan, S. R.S., Large Deviations and Applications (1984), SIAM: SIAM Philadelphia, PA · Zbl 0549.60023
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.