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Stochastic processes are considered that admit a time-varying spectral representation, and hence are only locally stationary. The statistical problem under consideration is the nonparametric maximum Whittle-likelihood estimation of characteristics of locally stationary time series. An example is the estimation of the time varying variance of an AR-time series which, for instance, has interesting applications in seismology. From a theoretical perspective the time varying empirical spectral process comes into play. The reason is, that the frequency domain is used as a vehicle to tackle the underlying statistical problem. We present some mathematical theory for the time varying empirical spectral process that parallels modern empirical process theory (which is based on independent data) . We then indicate how this theory can be utilized for deriving asymptotic properties of our (function) estimators of the time varying characteristics. We will also show that in special instances algorithmic issues can be tackled by exploiting ideas from isotonic regression. This is joint work with R. Dahlhaus from University of Heidelberg, Germany.

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