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.

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.