The class of distortion risk measures coincides with the set of coherent risk measures that are law invariant and comonotonically additive. The class includes the renowned expected shortfall which has many nice features and is of frequent use in practice.

To implement the risk management/regulatory procedure using risk measures, it is necessary to estimate the values of such risk measures.
For a distortion risk measure, its form suggests a simple L-statisitics as a natural nonparametric estimator.

In this talk, we mainly focus on asymptotic properties of L-statistics when the data is weakly dependent; more precisely, when they are considered to be realizations of strong mixing process. Some moment inequalities, nearly linear bounds on the empirical distribution function, and some limit theorems will be discussed.