1 Basic Properties of Stationary Random Functions.- 1. Definition of a Random Function.- 2. Moments of a Random Function. Correlation Theory.- 3. Stationarity.- 4. Properties of Correlation Functions. Derivative and Integral of a Random Process.- 5. Complex Random Functions. Spaces of Random Variables.- 2 Examples of Stationary Random Functions. Spectral Representations.- 6. Examples of Stationary Random Sequences.- 7. Examples of Stationary Random Processes.- 8. Spectral Representation of Stationary Random Processes.- 9. Spectral Representation of the Correlation Function.- 10. Examples of Correlation Functions of Stationary Processes.- 11. Linear Transformations of Stationary Random Processes.- 12. Examples of Linear Transformations of Stationary Processes.- 13. Spectral Representation of Stationary Sequences and Their Correlation Functions.- 14. Discrete Samples of Random Processes and Discrete Linear Transformations.- 15. Examples of Discrete Linear Transformations and Correlation Functions of Stationary Sequences.- 3 Determination of the Statistical Characteristics of a Stationary Random Function from Experimental Data.- 16. Determination of the Mean Value of a Stationary Function X(t).- 17. Determination of the Mean Square and Correlation Function of X(t).- 18. Statistical Spectral Analysis. Determination of the Spectral Density Function.- 19. Some Practical Aspects and Additional Methods of Statistical Spectral Analysis. Determination of the Spectral Distribution Function.- 4 Some Generalizations of the Concepts of a Stationary Random Function and of a Spectral Representation.- 20. Multidimensional Stationary Random Functions.- 21. Homogeneous Random Fields.- 21.1 One-Dimensional and Multidimensional Homogeneous Random Fields.- 21.2 Statistical Inference for Homogeneous Fields.- 22. Isotropic Random Fields.- 22.1 Spectral Representation of Isotropic Correlation Functions and Its Consequences.- 22.2 Examples of Isotropic Correlation Functions.- 22.3 Spectral Representation of Isotropic Random Fields.- 22.4 Multidimensional Isotropic Random Fields.- 22.5 Homogeneous Fields on Spheres and Other Homogeneous Spaces.- 23. Random Processes with Stationary Increments.- 24. Generalized Stationary Processes. Processes with Stationary Increments of Order n.- 24.1 Generalized Random Processes.- 24.2 A Novel Approach to Processes with Stationary Increments.- 24.3 Random Processes with Stationary Increments of Order n.- 25. Generalized Homogeneous Fields. Locally Homogeneous and Locally Isotropic Fields.- 25.1 Generalized Homogeneous Fields.- 25.2 Locally Homogeneous Fields.- 25.3 Locally Isotropic Fields.- 26. Further Examples of Various Spectral Representations.- 26.1 Karhunen-Loève Expansion.- 26.2 Moving Average Representations of Stationary Random Processes.- 26.3 Oscillatory Processes and Evolutionary Spectra.- 26.4 Harmonizable Random Processes.- 26.5 Periodically Correlated Processes.- 26.6 Processes Having Time Average Mean Value and Correlation Function.
The theory of random functions is a very important and advanced part of modem probability theory, which is very interesting from the mathematical point of view and has many practical applications. In applications, one has to deal particularly often with the special case of stationary random functions. Such functions naturally arise when one considers a series of observations x(t) which depend on the real-valued or integer-valued ar gument t ("time") and do not undergo any systematic changes, but only fluctuate in a disordered manner about some constant mean level. Such a time series x(t) must naturally be described statistically, and in that case the stationary random function is the most appropriate statistical model. Stationary time series constantly occur in nearly all the areas of modem technology (in particular, in electrical and radio engineering, electronics, and automatic control) as well as in all the physical and geophysical sciences, in many other ap mechanics, economics, biology and medicine, and also plied fields. One of the important trends in the recent development of science and engineering is the ever-increasing role of the fluctuation phenomena associated with the stationary disordered time series. Moreover, at present, more general classes of random functions related to a class of stationary random functions have also been appearing quite often in various applied studies and hence have acquired great practical importance.
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