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Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem

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  • Eichelsbacher, Peter
  • Schmock, Uwe

Abstract

This paper is devoted to the well known transformations that preserve a large deviation principle (LDP), namely, the contraction principle with approximately continuous maps and the concepts of exponential equivalence and exponential approximations. We generalize these transformations to completely regular topological state spaces, give some examples and, as an illustration, reprove a generalization of Sanov's theorem, due to de Acosta (J. Appl. Probab. 31 A (1994) 41-47). Using partition-dependent couplings, we then extend this version of Sanov's theorem to triangular arrays and prove a full LDP for the empirical measures of exchangeable sequences with a general measurable state space.

Suggested Citation

  • Eichelsbacher, Peter & Schmock, Uwe, 1998. "Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem," Stochastic Processes and their Applications, Elsevier, vol. 77(2), pages 233-251, September.
    • Handle: RePEc:eee:spapps:v:77:y:1998:i:2:p:233-251
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    References listed on IDEAS

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    1. Dembo, Amir & Zajic, Tim, 1997. "Uniform large and moderate deviations for functional empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 195-211, May.
    2. Daras, Tryfon, 1997. "Large and moderate deviations for the empirical measures of an exchangeable sequence," Statistics & Probability Letters, Elsevier, vol. 36(1), pages 91-100, November.
    3. Bolthausen, Erwin, 1987. "Markov process large deviations in [tau]-topology," Stochastic Processes and their Applications, Elsevier, vol. 25, pages 95-108.
    4. Bolthausen, Erwin & Schmock, Uwe, 1989. "On the maximum entropy principle for uniformly ergodic Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 33(1), pages 1-27, October.
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    Cited by:

    1. J. Garcia, 2004. "An Extension of the Contraction Principle," Journal of Theoretical Probability, Springer, vol. 17(2), pages 403-434, April.
    2. Wang, Ran & Wang, Xinyu & Wu, Liming, 2010. "Sanov's theorem in the Wasserstein distance: A necessary and sufficient condition," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 505-512, March.
    3. Quansheng Liu & Emmanuel Rio & Alain Rouault, 2003. "Limit Theorems for Multiplicative Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 971-1014, October.

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