Hilbertsche Zerlegungen eingebetteter Prozessräume und ihre Anwendung auf die Vorhersage von Zeitreihen

The Theory of Time Series or Stochastic Processes is partly of functional analytic character. Well-known examples are the Reproducing Kernel Hilbert space associated with a process, and the Karhunen-Loève expansion. More generally, the Spectral Theory of stationary processes, and orthogonal projection as a principle of prediction, are functional analytic aspects of time series. This dissertation aims at enhancing this theory by transfering modern methods of functional analysis to the field of Stochastic Processes, providing new resp. broadened results. The above-mentioned topics are just a few out of Time Series Analysis but they convey the interface between Stochastic Processes and Analysis on which this thesis focuses. We give a more detailed description: To begin with, it is the structure of stationary processes that allows successful application of analytic tools. For instance, the stationary Prediction Theory started by Wiener and Kolmogorov is of abstract and (Fourier) analytic nature. Generalizations leading to similar results without the restriction of stationarity are still of interest. They might require alternative time domain methods formalized only by means of the process' indexset (time). Besides, Representation Theory of Stochastic Processes seems limited to its original setting. As initiated by Karhunen and Loève the theory depends on elementary isometries between the process space and a space of quadratically integrable functions - called Spectral Domain. Continuous processes on compact intervals allow precise deductions and reveal the refineable connection to eigenvector bases of integral operators (Mercer's Theorem). Finally, the Theory of Hilbert Subspaces associated with processes could be modernized, too. Parzen published the relationship by linking a Kernel Hilbert space (in the sense of Aronszajn) to a process. This space of functions on the indexset gives rise to an isometric description of the process space. The relationship is still of relevance, even though it is of rather elementary character due to the discreteness of the time domain. In summary, the open problems and the way the presented thesis approaches them are as follows: (1) Hilbert Subspaces associated with processes are introduced until now assuming a discrete topology on the indexset and this leads solely to Kernel Hilbert spaces in the sense of Aronszajn. This dissertation analyses in how far the actual topology of the time domain could be preserved and what consequences this brings for the construction as well as for the properties of the process space. Time is modelled topologically in form of a (pivotal) Hilbert space and covariance functions are understood as generalized functions. The developed theory of embeddable processes provides associated Hilbert subspaces within this generalized setting. A 'Reproducing Property' of the embedded process space is proved. (2) Issues of bases (for the process spaces) and their constructions follow immediately. So far, they have been formulated as representation problems for the related process by means of a denumerable orthonormal system. Solutions depend on elementary isometries within Hilbert spaces. This thesis shows how modern decomposition techniques of Hilbert Subspaces give rise to new (esp. continuous) bases and describes two construction methods: image and spectral decompositions. Both are independent of conditions of denumerability, incorporate existing methods and allow a representation of the process. (3) The known Karhunen-Loève expansion builts on the usual isometry arguments limited to a formalism of denumerability. However, it is the Spectral Theory of special, positive integral operators which yields the expansion. The dissertation clarifies the abstract formalism by means of unbounded positive operators. The relevance of such operators' Spectral Theory for the resulting spectral decomposition is illustrated, abstracting the rather specific influence of Mercer's Theorem. (4) The use of decompositions for prediction purposes is obvious. So far, preferred Fourier analytic approaches seem to disguise important time domain aspects of forecasting, though. The presented thesis aims at characterizing a general prediction decomposition using time domain terminology. The found prediction methods show an intrinsic 'Gram-Schmidt Principle' and indicate the relationship with Cholesky's Factorization. Predictors in terms of decomposition bases are given as well as their connection to previous results.