Journal of Theoretical Probability

, Volume 12, Issue 3, pp 661–673 | Cite as

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

  • Liming Wu


Given a Brownian motion (Bt)t≥0 in Rd and a measurable real function f on Rd belonging to the Kato class, we show that 1/t ∫0tf(Bs) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.

Exponential convergence in probability large deviations Brownian motion random walks 


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Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • Liming Wu
    • 1
  1. 1.Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620UniversitéAubierreFrance

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