Abstract
Let Y t be a homogeneous nonexplosive Markov process with generator R defined on a denumerable state space E (not necessarily ergodic). We introduce the empirical generator G t of Y t and prove the Ruelle–Lanford property, which implies the weak LDP. In a fairly broad setting, we show how to perform almost all classical operations (e.g., contraction) on the weak LDP under suitable assumptions, whence Sanov's theorem follows.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Deuschel, J.-D. and Stroock, D.W., Large Deviations, Boston: Academic, 1989.
Dembo, A. and Zeitouni, O., Large Deviations Techniques and Applications, New York: Springer, 1998.
Bolthausen, E., Markov Process Large Deviations in τ-Topology, Stock. Process. Appl., 1987, vol. 25,no. 1, pp. 95-108.
de Acosta, A., Upper Bounds for Large Deviations of Dependent Random Vectors, Z. Wahrsch. Verw. Geb., 1985, vol. 69,no. 4. pp. 551-565.
de Acosta, A. and Ney, P., Large Deviation Lower Bounds for Arbitrary Additive Functionals of a Markov Chain, Ann. Probab., 1998, vol. 26,no. 4, pp. 1660-1682.
Donsker, M.D. and Varadhan, S.R.S., On Some Problems of Large Deviations for Markov Processes, Proc. 40th Session Int. Stat. Inst. (Warsaw, 1975), vol. 1, Invited Papers, Bull. Inst. Int. Stat. 1975, vol. 46, pp. 409-418.
Iscoe, I., Ney, P., and Nummelin, E., Large Deviations of Uniformly Recurrent Markov Additive Processes, Adv. Appl. Math., 1985, vol. 6,no. 4, pp. 373-412.
Jain, N.C., Large Deviation Lower Bounds for Additive Functionals of Markov Processes, Ann. Probab., 1990, vol. 18,no. 3, pp. 1071-1098.
Ney, P. and Nummelin, E., Markov Additive Processes: Large Deviations for the Continuous Time Case, Probability Theory and Mathematical Statistics, vol. 2 (Vilnius, 1985), Utrecht: VNU Sci. Press, 1987, pp. 377-389.
Sanov, I.N., On the Probability of Large Deviations of Random Variables, Select. Transl. Math. Stat. Probab., vol. 1, Providence: Inst. Math. Stat. and AMS, 1961, pp. 213-244.
Ellis, R.S., Entropy, Large Deviations, and Statistical Mechanics, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Science), vol. 271, New York: Springer, 1985.
Kesidis, G. and Walrand, J., Relative Entropy between Markov Transition Rate Matrices, IEEE Trans. Inf. Theory, 1993, vol. 39,no. 3, pp. 1056-1057.
Baldi, P. and Piccioni, M., A Representation Formula for the Large Deviation Rate Function for the Empirical Law of a Continuous Time Markov Chain, Stat. Probab. Lett., 1999, vol. 41,no. 2, vol. 107-115.
Hornik, K., On Some Large Deviation Probabilities for Finite Markov Processes, Studia Sci. Math. Hungar., 1991, vol. 26,nos. 2–3, pp. 355-362.
Hornik, K., Asymptotically Optimal Tests of Hypotheses on the Generator of a Finite Markov Process, J. Stat. Plan. Inference, 1989, vol. 23.,no. 3, pp. 345-352.
Hornik, K., Exponential Rate Optimal Estimation of the Generator of a Finite Markov Process, Stat. Decisions, 1990, vol. 8,no. 2, pp. 101-113.
Bucklew, J.A., Ney, P., and Sadowsky, J.S., Monte Carlo Simulation and Large Deviations Theory for Uniformly Recurrent Markov Chains. J. Appl. Probab., 1990, vol. 27,no. 1, pp. 44-59.
Lewis, J. and Pfister, C., Thermodynamic Probability Theory: Some Aspects of Large Deviations, Usp. Mat. Nauk, 1995, vol. 50,no. 2, (302), pp. 47-88.
de La Fortelle, A. and Fayolle, G., Large Deviation Principle for Markov Chains in Discrete Time, Rapport de recherche de l'INRIA, Rocquencourt, France, 1999, no. 3791.
Feller, W., On the Integro-Differential Equations of Purely Discontinuous Markoff Processes, Trans. AMS, 1940, vol. 48, pp. 488-515.
Dinwoodie, I.H. and Ney, P., Occupation Measures for Markov Chains, J. Theor. Probab., 1995, vol. 8,no. 3, pp. 679-691.
de Acosta, A., Large Deviations for Empirical Measures of Markov Chains, J. Theor. Probab., 1990, vol. 3,no. 3, pp. 395-431.
Dinwoodie, I.H., Identifying a Large Deviation Rate Function, Ann. Probab., 1993, vol. 21,no. 1, pp. 216-231.
Nummelin, E., General Irreducible Markov Chains and Nonnegative Operators, Cambridge: Cambridge Univ. Press, 1984.
Rights and permissions
About this article
Cite this article
de La Fortelle, A. Large Deviation Principle for Markov Chains in Continuous Time. Problems of Information Transmission 37, 120–139 (2001). https://doi.org/10.1023/A:1010470024888
Issue Date:
DOI: https://doi.org/10.1023/A:1010470024888