Abstract
Given a Brownian motion (B t) t≥0 in R d and a measurable real function f on R d belonging to the Kato class, we show that 1/t ∫ t0 f(B s ) 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.
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Wu, L. Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks. Journal of Theoretical Probability 12, 661–673 (1999). https://doi.org/10.1023/A:1021671630755
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DOI: https://doi.org/10.1023/A:1021671630755