This paper is concerned with the asymptotic behaviour of the empirical distribution function for a large class of continuous-time weakly dependent stationary processes. Under mild mixing conditions the empirical distribution function is an unbiased consistent estimator of the marginal distribution function of the process. For strongly mixing processes this estimator is asymptotically normal. We propose a consistent estimator of the asymptotic variance, and then study the functional central limit theorem for the empirical distribution function.
"On invariant distribution function estimation for continuous-time stationary processes." Bernoulli 11 (5) 933 - 948, October 2005. https://doi.org/10.3150/bj/1130077600