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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. The Phenomenon of Statistical Stability and Its Properties

Here we examine the main manifestations of the phenomenon of statistical stability: the statistical stability of the relative frequency and sample average. Attention is drawn to an emergent property of the phenomenon of statistical stability. We discuss the hypothesis of perfect (absolute or ideal) statistical stability, which assumes the convergence of relative frequencies and averages. Examples of statistically unstable processes are presented. We discuss the terms “identical statistical conditions” and “unpredictable statistical conditions”. Hilbert’s sixth problem concerning the axiomatization of physics is then described. The universally recognized mathematical principles of axiomatization of probability theory and mechanics are considered. We propose a new approach for solution of the sixth problem, supplementing the mathematical axioms by physical adequacy hypotheses which establish a connection between the existing axiomatized mathematical theories and the real world. The basic concepts of probability theory and the theory of hyper-random phenomena are considered, and adequacy hypotheses are formalized for the two theories. Attention is drawn to the key point that the concept of probability has no physical interpretation in the real world.
Igor I. Gorban

### Chapter 2. Determinism and Uncertainty

Various conceptual views of the structure of the world are examined from the standpoint of determinism and uncertainty. A classification of uncertainties is presented. To present the different types of models, a uniform method using the distribution function is described. A classification of mathematical models is proposed. We examine random variables and stochastic processes that are statistically unstable with respect to different statistics. Various types of non-stationary processes are analyzed from the standpoint of statistical stability.
Igor I. Gorban

### Chapter 3. Formalization of the Statistical Stability Concept

The notion of statistical stability is formalized and the parameters of statistical instability are introduced. Measurement units are proposed for the statistical instability parameters. We introduce the concepts of statistical stability/instability of processes in both narrow and broad senses and study the statistical stability of several models for these processes.
Igor I. Gorban

### Chapter 4. Dependence of the Statistical Stability of a Stochastic Process on Its Spectrum-Correlation Characteristics

The Wiener–Khinchin transformation is examined. It is noted that there are stochastic processes which do not simultaneously have a correlation function that is typical for a stationary process, and a power spectral density. We determine the dependence of the statistical stability on the power spectral density of the process and investigate the statistical stability of a process for which the power spectral density is described by a power function. Results are obtained for continuous and discrete processes. We then present simulation results which confirm the correctness of the formulas describing the dependence of the statistical instability parameters on the power spectral density of the process. The dependence of the statistical stability of a process on its correlation characteristics is analyzed. The statistical stability of low frequency and narrowband stochastic processes is investigated.
Igor I. Gorban

### Chapter 5. Experimental Investigation of the Statistical Stability of Physical Processes Over Long Observation Intervals

Here we discuss experimental studies of the statistical stability of various physical processes. These include the city mains voltage, the height and period of sea waves, variations in the Earth’s magnetic field, currency fluctuations, and variations in the temperature and speed of sound in the Pacific Ocean. Attention is drawn to the fact that, in all cases involving small observation intervals, statistical stability violations are not visible, whereas for those involving broad observation intervals, they become explicit.
Igor I. Gorban

### Chapter 6. Experimental Investigation of the Statistical Stability of Meteorological Data

We present experimental studies of the statistical stability of air temperature and precipitation in the Moscow and Kiev areas, and also the wind speed in Chernobyl. It is shown that all these processes are statistically unstable, but that the degree of instability is different in each case. For example, the temperature fluctuations are much more unstable than the precipitation oscillations.
Igor I. Gorban

### Chapter 7. Experimental Studies of the Statistical Stability of Radiation from Astrophysical Objects

Here we discuss experimental studies over long observation intervals (13 years) of the statistical stability of X-ray radiation from three astrophysical objects. It is found that all the studied radiation intensities are statistically unstable. The most stable oscillations are from the pulsar PSRJ 1012 + 5307. It is found that, over the whole observation interval, its oscillations are statistically stable with respect to the average, but unstable with respect to the standard deviation.
Igor I. Gorban

### Chapter 8. Statistical Stability of Different Types of Noise and Process

Different types of noise are studied, in particular, the color noises, flicker noise, and self-similar (fractal) noise. The results of studies of statistical stability of various noises and processes are generalized and the causes of statistical stability violation are investigated. It is found that statistically unstable processes can arise in different ways: because of inflows from the outside in an open system of matter, energy, and (or) information, as a result of nonlinear and even linear transformations, and due to wave damping.
Igor I. Gorban

### Chapter 9. Hyper-random Events and Variables

The notion of a hyper-random event is introduced. To describe such events, conditional probabilities and probability bounds are used. The properties of these parameters are presented. The concept of a scalar hyper-random variable is introduced. Here we use conditional distribution functions (providing an exhaustive description), bounds of the distribution function, moments of the distribution function, and bounds of these moments. The properties of these characteristics and parameters are presented. The notion of a hyper-random vector variable is introduced. Methods used to describe hyper-random scalar variables are extended to the case of hyper-random vector variables. Properties of the characteristics and parameters of hyper-random vector variables are given.
Igor I. Gorban

### Chapter 10. Hyper-random Functions

The notion of a hyper-random scalar function is introduced. Various ways of presenting it are examined, including those based on conditional distribution functions (which provide the most complete characterization of hyper-random functions), the bounds of the distribution function, the probability densities of the bounds, the moments of the bounds, and the bounds of the moments. We then outline the mathematical analysis of random functions, and present the notion of convergence for sequences of random variables and for stochastic functions, and also the derivative and integral of a random function. We introduce the concepts of convergence for sequences of hyper-random variables and for hyper-random functions, and discuss the concepts of continuity, differentiability, and integrability of hyper-random functions.
Igor I. Gorban

### Chapter 11. Stationary and Ergodic Hyper-random Functions

Concepts such as stationarity and ergodicity, well known for stochastic functions, are generalized to hyper-random functions. Spectral methods are discussed for the description of stationary hyper-random functions and the properties of stationary and ergodic hyper-random functions are presented.
Igor I. Gorban

### Chapter 12. Transformations of Hyper-random Variables and Processes

Here we analyze different ways of describing hyper-random variables and processes with respect to appropriateness of their use in different types of transforms. We present relationships between the characteristics and parameters of the original and transformed hyper-random variables and processes, and then develop recommendations for the use of the various ways of describing hyper-random variables in the case of linear and nonlinear transforms, and hyper-random processes in the case of both inertialess and inertial transforms.
Igor I. Gorban

### Chapter 13. Fundamentals of the Statistics of Hyper-random Phenomena

The notion of a hyper-random sample and its properties are formalized. We then describe ways of forming estimators of the characteristics of the hyper-random variables. We discuss in particular the existence of convergence violation of real estimators and the adequate description of these estimators using hyper-random models.
Igor I. Gorban

### Chapter 14. Divergent Sequences and Functions

The notion of limit for convergent numerical sequences is generalized to divergent sequences and functions. In contrast to the fact that conventional limits necessarily possess a single value, the generalized limit has a set of values. For a divergent numerical sequence, we introduce the concept of a spectrum of limit points. A theorem on the sequence of averages is then proven.
Igor I. Gorban

### Chapter 15. Description of Divergent Sequences and Functions

In order to describe divergent sequences and functions, we present an approach based on use of the distribution function. We then prove a theorem on the spectrum of relative frequencies of class values. Examples of divergent functions are then described.
Igor I. Gorban

### Chapter 16. Many-Valued Variables, Sequences, and Functions

Here we analyze different ways to describe many-valued variables and functions. Using the mathematical tools developed in the theory of hyper-random phenomena, the notions of many-valued variable and many-valued function are formalized. A correspondence between many-valuedness and violation of convergence is established. We introduce the notions of spectrum and distribution function for many-valued variables and functions.
Igor I. Gorban

### Chapter 17. Principles of the Mathematical Analysis of Many-Valued Functions

For many-valued functions, the concepts of continuous function, derivative, indefinite and definite integrals, and spectrum of principal values of a definite integral are introduced.
Igor I. Gorban

### Chapter 18. The Law of Large Numbers

It is established that the law of large numbers, known for a sequence of random variables, is valid both with and without convergence of the sample mean. In the absence of convergence, the sample average tends to the average of expectations fluctuating synchronously with it in a certain range. The law of large numbers is generalized to sequences of hyper-random variables. Peculiarities of the generalized law of large numbers are studied.
Igor I. Gorban

### Chapter 19. The Central Limit Theorem

Here we investigate the particularities of the central limit theorem for a sequence of random variables in both the presence and the absence of convergence of the sample mean to a fixed number. The central limit theorem is generalized to a sequence of hyper-random variables. We present experimental results demonstrating the lack of convergence of the sample means of real physical processes to fixed numbers.
Igor I. Gorban

### Chapter 20. Accuracy and Measurement Models

Two concepts for assessing measurement accuracy are analyzed: error and uncertainty. A number of measurement models are considered, including the classical determinate—random measurement model which ignores statistical stability violations of the estimators, and the determinate—hyper-random model which accounts for such violations. We consider the point and interval estimators and their properties. It is found that the limited accuracy of real measurements is caused by the statistical volatility (inconsistency) of real estimators. It is shown that, in the general case, the measurement error cannot be divided into components. In particular cases, the error can be presented as the sum of three components: a random one, whose value decreases with increasing sample size, along with the systematic and interval components, whose values do not depend on sample size.
Igor I. Gorban

### Chapter 21. The Problem of Uncertainty

Different definitions of the entropy concept are analyzed. The concept of Shannon entropy for random variables is extended to uncertain variables that do not have a probability measure. The entropy concept is introduced for hyper-random and interval variables. We investigate different ways that uncertainty can arise. It is found that uncertainty may arise as a result of a certain type of nonlinear transformation and in the process of averaging determinate variables in the absence of convergence. We explain why the interval, multi-interval, and hyper-random models can adequately depict reality, while the random models are mathematical abstractions.
Igor I. Gorban

### Backmatter

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