Petrova A. Spectral representation of the vector of nonstationary random processes and sequences and their modelling of the spectrum

Object of the research - nonstationary vector random processes. Purpose: to offer the single approach to structure of correlation theory of new classes of nonstationary vector random functions, based on the spectral representations of nonself-adjoint or nonunitary operators in the Hilbert space. Methods of the research: theory of operator colligatios, spectral theory of nonself-adjoint or nonunitary operators, including triangle models of operators, theory of linear system associated with operator colligatios. The new characteristics (infinitesimal correlation matrix (ICM), matrix of correlation differences (MCR), rank of nonstationarity), which describe the deflection from stationary vector random function. Based on this characteristics of nonstationarity and triangle models of operators, the model representations for ICM, MCR and correlation matrices, as well as nonstationary vector random processes and sequences for different cases of (complex valued, infinite-fold) spectrum, which are the analog of the relevant spectral representations in the stationary case, were obtained. The new method of modeling of nonstationary vector random processes and its characteristics only by spectrum was offered. The scientific results of the dissertation may be used for modeling of statistical characteristics of random medium during the study of advance of waves, when the statistical stationarity or homogeneity are violated, for statistical description of transition processes; for statistical data processing, in particular, astronomical interferometers, as well as in the educational process during training of specialists in the area of mathematical modeling.