  A new concept of (asymptotic) qualitative robustness for plug-in estimators
based on identically distributed possibly dependent observations is introduced,
and it is shown that Hampel's theorem for general metrics $d$ still holds.
Since Hampel's theorem assumes the UGC property w.r.t. $d$, that is,
convergence in probability of the empirical probability measure to the true
marginal distribution w.r.t. $d$ uniformly in the class of all admissible laws
on the sample path space, this property is shown for a large class of strongly
mixing laws for three different metrics $d$. For real-valued observations, the
UGC property is established for both the Kolomogorov $\phi$-metric and the
L\'{e}vy $\psi$-metric, and for observations in a general locally compact and
second countable Hausdorff space the UGC property is established for a certain
metric generating the $\psi$-weak topology. The key is a new uniform weak LLN
for strongly mixing random variables. The latter is of independent interest and
relies on Rio's maximal inequality.
