The monograph compares two approaches that describe the statistical stability phenomenon ¿ one proposed by the probability theory that ignores violations of statistical stability and another proposed by the theory of hyper-random phenomena that takes these violations into account. There are five parts. The first describes the phenomenon of statistical stability. The second outlines the mathematical foundations of probability theory. The third develops methods for detecting violations of statistical stability and presents the results of experimental research on actual processes of different physical nature that demonstrate the violations of statistical stability over broad observation intervals. The fourth part outlines the mathematical foundations of the theory of hyper-random phenomena. The fifth part discusses the problem of how to provide an adequate description of the world.
The monograph should be interest to a wide readership: from university students on a first course majoring in physics, engineering, and mathematics to engineers, post-graduate students, and scientists carrying out research on the statistical laws of natural physical phenomena, developing and using statistical methods for high-precision measurement, prediction, and signal processing over broad observation intervals.
To read the book, it is sufficient to be familiar with a standard first university course on mathematics.

Igor I. Gorban graduated from the Kiev Polytechnic Institute, USSR, majoring in hydroacoustics. At the MorPhysPribor Central Research Institute, Leningrad, 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.

Part I The Phenomenon of Statistical Stability.- The Physical Phenomenon of Statistical Stability.- Part II The Probability Theory.- Basis of the Probability Theory.- Stochastic Functions.- Fundamentals of the Mathematical Statistics of the Probability Theory.- Assessment of the Measurement Accuracy on the Basis of Probability Theory.- Part III Experimental Study of the Statistical Stability Phenomenon.- Methodology and Results from Investigation of the Statistical Stability of Processes.- Part IV The Theory of Hyper-Random Phenomena.- Basis of the Theory of Hyper-Random Phenomena.- Hyper-Random Functions.- Fundamentals of the Mathematical Statistics of Hyper-Random Phenomena.- Assessing Measurement Accuracy on the Basis of the Theory of Hyper-Random Phenomena.- Part V The Problem of Adequate Description of the World.- Determinism, Uncertainty, Randomness, And Hyperrandomness.- Epilogue.- References.- Index.