  In this paper we investigate the problem of testing the assumption of
stationarity in locally stationary processes. The test is based on an estimate
of a Kolmogorov-Smirnov type distance between the true time varying spectral
density and its best approximation through a stationary spectral density.
Convergence of a time varying empirical spectral process indexed by a class of
certain functions is proved, and furthermore the consistency of a bootstrap
procedure is shown which is used to approximate the limiting distribution of
the test statistic. Compared to other methods proposed in the literature for
the problem of testing for stationarity the new approach has at least two
advantages: On one hand, the test can detect local alternatives converging to
the null hypothesis at any rate $g_T\to0$ such that $g_TT^{1/2}\to \infty$,
where $T$ denotes the sample size. On the other hand, the estimator is based on
only one regularization parameter while most alternative procedures require
two. Finite sample properties of the method are investigated by means of a
simulation study, and a comparison with several other tests is provided which
have been proposed in the literature.
