Large Deviations from the Almost Everywhere Central Limit Theorem

Peter March, Timo Seppäläinen

Research output: Contribution to journalArticle

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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 languageEnglish (US)
Pages (from-to)935-965
Number of pages31
JournalJournal of Theoretical Probability
Issue number4
Publication statusPublished - Jan 1 1997
Externally publishedYes


All Science Journal Classification (ASJC) codes

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


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

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