Springer Statistics

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-Loeve 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.