Summary
We prove large deviation theorems for empirical measures of independent random fields whose distributions depend measurably on an auxiliary parameter. This dependence respects the action of the shift group, and a large deviation principle holds whenever a certain ergodicity condition is satisfied. We also investigate the entropy functions for these processes, especially in relation to the usual relative entropy.
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Baxter, J.R., Jain, N.C.: Convexity and compactness in large deviation theory. (Preprint 1991)
Baxter, J.R., Jain, N.C., Seppäläinen, T.O.: Large deviations for nonstationary arrays and sequences. Ill. J. Math. (to appear)
Comets, F.: Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Theory Relat. Fields.80, 407–432 (1989)
Deuschel, J.-D., Stroock, D.W.: Large deviations. San Diego: Academic Press 1989
Dunford, N., Schwartz, J.T.: Linear operators, part I: General theory. New York: Wiley 1988
Dynkin, E.B.: The initial and final behaviour of the trajectories of Markov processes. Russ. Math. Surv.26, 165–185 (1971).
Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab.16, 961–977 (1988)
Krengel, U.: Ergodic theorems. Berlin: de Gruyter 1985
Lanford, O.E.: Statistical mechanics and mathematical problems. (Lect. Notes Phys., vol. 20, pp. 1–113) Berlin Heidelberg New York: Springer 1973
Ledrappier, F.: Pressure and variational principle for random Ising model. Commun. Math. Phys.56, 297–302 (1977)
Orey, S.: Large deviations in ergodic theory. In: Çinlar, E., Chung, K.-L., Getoor, R. (eds.) Seminar on stochastic processes, pp. 195–249. Boston: Birkhäuser 1986
Oxtoby, J.C.: Ergodic sets. Bull.58, 116–132 (1952)
Seppäläinen, T.: Large deviations for lattice systems. II. Nonstationary independent fields. Probab. Theory Relat. Fields (to appear)
Varadhan, S.R.S.: Large deviations and applications. Philadelphia: SIAM 1984
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Seppäläinen, T. Large deviations for lattice systems I. Parametrized independent fields. Probab. Th. Rel. Fields 96, 241–260 (1993). https://doi.org/10.1007/BF01192135
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DOI: https://doi.org/10.1007/BF01192135