Randomness And Hyper-randomness

Mathematical Engineering Igor I. Gorban Randomness and Hyperrandomness Mathematical Engineering Series editors J€ org Schr€ oder, Essen, Germany Bernhard Weigand, Stuttgart, Germany More information about this series at http://www.springer.com/series/8445 Igor I. Gorban Randomness and Hyper-randomness Igor I. Gorban Institute of Mathematical Machines and Systems Problems National Academy of Sciences of Ukraine Kiev, Ukraine Originally published by Naukova Dumka Publishing House of National Academy of Sciences of Ukraine, Kiev, 2016 ISSN 2192-4732 ISSN 2192-4740 (electronic) Mathematical Engineering ISBN 978-3-319-60779-5 ISBN 978-3-319-60780-1 (eBook) DOI 10.1007/978-3-319-60780-1 Library of Congress Control Number: 2017945377 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface One of the most remarkable physical phenomena is the statistical stability (regularity) of mass phenomena as revealed by the stability of statistics (functions of samples). There are two theories describing this phenomenon. The first is classical probability theory, which has a long history, and the second is the theory of hyper-random phenomena developed in recent decades. Probability theory has established itself as the most powerful tool for solving various statistical tasks. It is even widely believed that any statistical problem can be effectively solved within the paradigm of probability theory. However, it turns out that this is not so. Some conclusions of probability theory do not accord with experimental data. A typical example concerns the potential accuracy. According to probability theory, when we increase the number of measurement results of any physical quantity, the error in the averaged estimator tends to zero. But every engineer or physicist knows that the actual measurement accuracy is always limited and that it is not possible to overcome this limit by statistical averaging of the data. Studies of the causes of discrepancies between theory and practice led to the understanding that the problem is related to an unjustified idealization of the phenomenon of statistical stability. Probability theory is in fact a physical-mathematical discipline. The mathematical component is based on A.N. Kolmogorov’s classical axioms, while the physical component is based on certain physical hypotheses, in particular the hypothesis of perfect statistical stability