### 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) |
---|---|

Pages (from-to) | 935-965 |

Number of pages | 31 |

Journal | Journal of Theoretical Probability |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1997 |

### Fingerprint

### 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

}

*Journal of Theoretical Probability*, vol. 10, no. 4, pp. 935-965. https://doi.org/10.1023/A:1022614700678

**Large Deviations from the Almost Everywhere Central Limit Theorem.** / March, Peter; Seppäläinen, Timo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Large Deviations from the Almost Everywhere Central Limit Theorem

AU - March, Peter

AU - Seppäläinen, Timo

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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.

AB - 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.

KW - Almost everywhere central limit theorem

KW - Brownian motion

KW - Large deviations

UR - http://www.scopus.com/inward/record.url?scp=0042323774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042323774&partnerID=8YFLogxK

U2 - 10.1023/A:1022614700678

DO - 10.1023/A:1022614700678

M3 - Article

VL - 10

SP - 935

EP - 965

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 4

ER -