Journal of Theoretical Probability

, Volume 12, Issue 3, pp 661–673

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

  • Liming Wu

DOI: 10.1023/A:1021671630755

Cite this article as:
Wu, L. Journal of Theoretical Probability (1999) 12: 661. doi:10.1023/A:1021671630755


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 probabilitylarge deviationsBrownian motionrandom walks

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

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