### 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.

Original language | English (US) |
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Pages (from-to) | 935-965 |

Number of pages | 31 |

Journal | Journal of Theoretical Probability |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 1997 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

### Keywords

- Almost everywhere central limit theorem
- Brownian motion
- Large deviations

### Cite this

*Journal of Theoretical Probability*,

*10*(4), 935-965. https://doi.org/10.1023/A:1022614700678