This paper investigates the effect of serial dependence in the data on the efficiency of some robust estimators. When the observations are from a stationary process satisfying certain mixing conditions, linear combinations of order statistics and the Hodges-Lehmann estimator are shown to be asymptotically normally distributed. Gaussian processes are studied in detail and it is shown that when all the serial correlations $(\rho_n)$ are $\geqq 0$, the efficiency of the robust estimators relative to the mean is greater than in the case of independent observations.