  In nonparametric classification and regression problems, regularized kernel
methods, in particular support vector machines, attract much attention in
theoretical and in applied statistics. In an abstract sense, regularized kernel
methods (simply called SVMs here) can be seen as regularized M-estimators for a
parameter in a (typically infinite dimensional) reproducing kernel Hilbert
space. For smooth loss functions, it is shown that the difference between the
estimator, i.e.\ the empirical SVM, and the theoretical SVM is asymptotically
normal with rate $\sqrt{n}$. That is, the standardized difference converges
weakly to a Gaussian process in the reproducing kernel Hilbert space. As common
in real applications, the choice of the regularization parameter may depend on
the data. The proof is done by an application of the functional delta-method
and by showing that the SVM-functional is suitably Hadamard-differentiable.
