Exponential Convergence in Probability for Empirical Means of Levy Processes

Let (X(t))(t >= 0) be a Levy process taking values in R(d) with absolutely continuous marginal distributions. Given a real measurable function f on R(d) in Kato's class, we show that the empirical mean 1/t integral(t)(0)f(X(s))ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Levy processes.