  Drees and Rootz\'en (2010) have established limit theorems for a general
class of empirical processes of statistics that are useful for the extreme
value analysis of time series, but do not apply to statistics of sliding
blocks, including so-called runs estimators. We generalize these results to
empirical processes which cover both the class considered by Drees and
Rootz\'en (2010) and processes of sliding blocks statistics. Using this
approach, one can analyze different types of statistics in a unified framework.
We show that statistics based on sliding blocks are asymptotically normal with
an asymptotic variance which, under rather mild conditions, is smaller than or
equal to the asymptotic variance of the corresponding estimator based on
disjoint blocks. Finally, the general theory is applied to three well-known
estimators of the extremal index. It turns out that they all have the same
limit distribution, a fact which has so far been overlooked in the literature.
