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.
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March, P., Seppäläinen, T. Large Deviations from the Almost Everywhere Central Limit Theorem. Journal of Theoretical Probability 10, 935–965 (1997). https://doi.org/10.1023/A:1022614700678
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DOI: https://doi.org/10.1023/A:1022614700678