Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks
- Cite this article as:
- Wu, L. Journal of Theoretical Probability (1999) 12: 661. doi:10.1023/A:1021671630755
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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.