This monograph investigates violations of statistical stability of physical events, variables, and processes and develops a new physical-mathematical theory taking into consideration such violations ¿ the theory of hyper-random phenomena. There are five parts. The first describes the phenomenon of statistical stability and its features, and develops methods for detecting violations of statistical stability, in particular when data is limited. The second part presents several examples of real processes of different physical nature and demonstrates the violation of statistical stability over broad observation intervals. The third part outlines the mathematical foundations of the theory of hyper-random phenomena, while the fourth develops the foundations of the mathematical analysis of divergent and many-valued functions. The fifth part contains theoretical and experimental studies of statistical laws where there is violation of statistical stability.
The monograph should be of particular interest to engineers and scientists in general who study the phenomenon of statistical stability and use statistical methods for high-precision measurements, prediction, and signal processing over long observation intervals.

Igor I. Gorban graduated from the Kiev Polytechnic Institute, USSR, major-ing in hydroacoustics. At the MorPhysPribor Central Research Institute, Lenin-grad, he received a Ph.D., and at the Institute of Cybernetics of the Academy of Sciences of Ukraine, Kiev, a Dr. Sc. He was awarded the academic rank of Senior Research Associate and then Full Professor.He worked at the Kiev Research Institute for Hydroequipment, participating in a number of developmental and research programmes. He was in charge of algorithms for several sonar systems, was a research adviser for two scientific expeditions to the Pacific to study hydroacoustic signals, was the first deputy to the Chief Designer and Chief Designer of the sonar complexes. Since 1993 he has been working at the Institute of Mathematical Machines and Systems Problems, National academy of Sciences of Ukraine, as Principal Scientist and Deputy Director for Research. Igor I. Gorban is the author of more than 200 scientific publications and several books devoted to: ¿ the theory of space-time processing of hydroacoustic signals under complex dynamic conditions, ¿ the theory of fast multi-channel processing of hydroacoustic signals, and¿ the physical-mathematical theory of hyper-random phenomena that takes into account violations of statistical stability.

Features of the Statistical Stability Phenomenon.- The Phenomenon of Statistical Stability and its Properties.- Determinism and Uncertainty.- Formalization of the Statistical Stability Concept.- Dependence of the Statistical Stability of a Stochastic Process on its Spectrum-Correlation Characteristics.- Experimental Study of the Statistical Stability Phenomenon.- Experimental Investigation of the Statistical Stability of Physical Processes over Large Observation Intervals.- Experimental Investigation of the Statistical Stability of Meteorological Data.- Experimental Studies of the Statistical Stability of Radiation from Astrophysical Objects.- Statistical Stability of Different Types of Noise and Process.- The Theory of Hyper-random Phenomena.- Hyper-random Events and Variables.- Hyper-random Functions.- Stationary and Ergodic Hyper-random Functions.- Transformations of Hyper-random Variables and Processes.- Fundamentals of the Statistics of Hyper-random Phenomena.- Principles of the Mathematical Analysis of Divergent and Many-valued Functions.- Divergent Sequences and Functions.- Description of Divergent Sequences and Functions.- Divergent Sequences.- Many-valued Variables, Sequences, and Functions.- Principles of the Mathematical Analysis of Many-valued Functions.- Statistical Laws in Statistical Stability Violation.- The Law of Large Numbers.- The Central Limit Theorem.- Accuracy and Measurement Models.- The Problem of Uncertainty.- Epilogue.- References.