The Kolmogorov isomorphism theorem and extensions to some nonstationary processes (Q2734971)

This paper consists of six sections. In Section 2, one-dimensional stationary processes on the real line are considered to obtain their integral representations, where the integral is a stochastic integral w.r.t. an orthogonally scattered vector measure. For such a process the Kolmogorov isomorphism theorem (KIT) is stated, i.e. time and spectral domains are defined and are shown to be isomorphic. In Section 3 some classes of non-stationary processes are introduced, namely harmonizable and Karhunen classes. To this end, bimeasure integration together with Dunford-Schwartz integral w.r.t. a vector measure are introduced. Basic properties of bimeasure integrals are mentioned. An RKHS (reproducing kernel Hilbert space) theory is applied to obtain integral representations for weakly harmonizable processes as well as Karhunen processes. NEWLINENEWLINENEWLINEIn Section 4, the spectral domain for a weakly harmonizable process is defined to be the set of measurable functions that are strictly integrable w.r.t. the spectral bimeasure of the process. Stationary dilations of weakly harmonizable processes are considered, which are used to show the completeness of the spectral domain of a weakly harmonizable process and also the KIT for such a process. Section 5 is devoted to an application of the Kolmogorov isomorphism theorem to the estimation and filtering the problems. In Section 6 a multidimensional extension of the KIT is explored for both stationary and harmonizable processes.NEWLINENEWLINEFor the entire collection see [Zbl 0961.60001].