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Large Deviations from the Almost Everywhere Central Limit Theorem

Author

Listed:
  • Peter March

    (Ohio State University)

  • Timo Seppäläinen

    (Iowa State University)

Abstract

We prove large deviation principles for the almost everywhere central limit theorem, assuming that the i.i.d. summands have finite moments of all orders. The level 3 rate function is a specific entropy relative to Wiener measure and the level 2 rate the Donsker-Varadhan entropy of the Ornstein-Uhlenbeck process. In particular, the rate functions are independent of the particular distribution of the i.i.d. process under study. We deduce these results from a large deviation theory for Brownian motion via Skorokhod's representation of random walk as Brownian motion evaluated at random times. The results for Brownian motion come from the well-known large deviation theory of the Ornstein-Uhlenbeck process, by a mapping between the two processes.

Suggested Citation

  • Peter March & Timo Seppäläinen, 1997. "Large Deviations from the Almost Everywhere Central Limit Theorem," Journal of Theoretical Probability, Springer, vol. 10(4), pages 935-965, October.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:4:d:10.1023_a:1022614700678
    DOI: 10.1023/A:1022614700678
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    Cited by:

    1. Alain Rouault & Marc Yor & Marguerite Zani, 2002. "A Large Deviations Principle Related to the Strong Arc-Sine Law," Journal of Theoretical Probability, Springer, vol. 15(3), pages 793-815, July.

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