The Statistical Stability Phenomenon

349 Pages·2017·4.925 MB·English

by  Igor I. Gorban

Mathematical Engineering
Igor I. Gorban
The
Statistical
Stability
Phenomenon
Mathematical Engineering
Series editors
Jörg Schröder, Essen, Germany
Bernhard Weigand, Stuttgart, Germany
More information about this series at http://www.springer.com/series/8445
Igor I. Gorban
The Statistical Stability
Phenomenon
123
Igor I.Gorban
Institute of Mathematical Machines
andSystemsProblem
National Academy of Sciences ofUkraine
Kiev
Ukraine
Originally published by Naukova Dumka Publishing House of National Academy of
Sciences ofUkraine, Kiev, 2014.
ISSN 2192-4732 ISSN 2192-4740 (electronic)
Mathematical Engineering
ISBN978-3-319-43584-8 ISBN978-3-319-43585-5 (eBook)
DOI 10.1007/978-3-319-43585-5
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Preface
One of the most remarkable physical phenomena is the statistical stability of mass
phenomena as revealed by the stability of statistics. Modern probability theory
(includingmathematicalstatistics)describesmassphenomenabymeansofrandom
(probabilistic or stochastic) mathematical models characterized by a probability
measure. The basis of such models is the physical hypothesis of perfect statistical
stability,whichassumesconvergenceoftherelativefrequencyofactualeventsand
convergence of average values of physical quantities and processes.
For many years the hypothesis of perfect statistical stability did not raise any
doubts. However, recent experimental research on various physical quantities and
processes over long observation intervals has shown that it is not confirmed
experimentally.
Forrelativelyshorttemporal,spatial,orspatio-temporalobservationintervals,an
increase in data volume usually reduces the level of fluctuation in the statistics.
However,whenthevolumesbecomeverylarge,thistendencyisnolongervisible,
andonceacertainlevelisreached,thefluctuationsremainpracticallyunchangedor
even grow. This indicates a lack of convergence for real statistics (their incon-
sistency). The study of violations of statistical stability of physical phenomena
and the development of an effective way to describe the real world, one which
v
vi Preface
accounts for such violations, has resulted in the construction of the new physical-
mathematical theory of hyper-random phenomena.
In probability theory, the main mathematical entities (models) are random
events, random variables, and stochastic functions; in the theory of hyper-random
phenomena these entities are replaced by hyper-random events, hyper-random
variables, and hyper-random functions, which are, respectively, sets of
non-interconnected random events, random variables, and stochastic functions
considered as a whole.
The theory of hyper-random phenomena has mathematical and physical com-
ponents. The mathematical component is based on A.N. Kolmogorov’s classical
axioms of probability theory, while the physical component is based on certain
physical hypotheses, in particular the hypothesis of imperfect statistical stability of
actualevents,variables,processes,andfieldsandthehypothesisthatthesephysical
phenomena are adequately described by the hyper-random models. For mathe-
maticiansthetheoryofhyper-randomphenomenaisabranchofprobabilitytheory;
for physicists, it is a new theory based on a new view of the world.
Thismonographinvestigatesthephysicalphenomenonofstatisticalstabilityand
presents the basic features of the theory of hyper-random phenomena. Both prob-
lemshavebeenstudiedinfourRussianmonographspublishedin2007,2011,2014,
and 2016 (Gorban 2007a, 2011a, 2014a, 2016). All these books are based on the
author’soriginaltheoreticalandexperimentalresults,publishedinvariousscientific
journals, in particular (Gorban 2005a–c, 2006a–e, 2007b, c, 2008c–f, 2009a–d,
2010a–g,2011b–k,2012a–j,2013a–e,2014b–g,2015a–f).Eachofthesebookshas
its own specifics. The monograph of 2007 is devoted mainly to mathematical
aspects of the theory of hyper-random phenomena and the monograph of 2011 to
bothmathematical and physical considerations. The 2014 monographthen focuses
mainlyontheproblemoftheviolationofconvergenceinphysicsandmathematics,
and the 2016 book compares probability theory with the theory of hyper-random
phenomena. The present book is an English version of the 2014 Russian
monograph.
The basic ideas presented in these books were formulated in the late 1970s and
thereafter, in the context of:
(cid:129) research in the field of applied hydroacoustics and sonar design (Gorban
1998b, c, 2008a, b),
(cid:129) lectures on probability theory and mathematical statistics for cadets at the Kiev
Air Force Institute (Gorban 1998a, 2000, 2003), and, of course,
(cid:129) experimental and theoretical study of statistical stability violations in various
physical processes.
The aim of the current monograph, like those before it, is to generalize results
obtained in the experimental study of violations of the statistical stability of real
physical processes, and to develop, systematize, and improve a number of basic
rules for the theory of hyper-random phenomena.
Thebookconsistsoffiveparts.PartIreviewsfeaturesofthestatistical stability
phenomenonanddevelopsmethodstostudystatisticalstabilityviolation,including
Preface vii
the case where data is limited. Part II describes experimental research on the
violationofstatisticalstabilityinvariousprocessesofdifferentphysicalnature,and
PartIIIpresentsabriefdescriptionofthemathematicalfoundationsofthetheoryof
hyper-random phenomena. Part IV contains a mathematical generalization of the
theory of hyper-random phenomena and lays the foundations for mathematical
analysis of divergent and many-valued functions. Finally, Part V contains the
results of theoretical and experimental study of statistical regularities in the viola-
tion of statistical stability.
Themonographiswrittenforresearchers,engineers,andpost-graduatestudents
studyingthestatisticallawsofphysicalphenomena,aswellasthosedevelopingand
using statistical methods for high-precision measurement, prediction, and signal
processing over long observation intervals. The book may also be useful for
high-level courses given to university students majoring in physics, engineering,
and mathematics. To understand the material of the book, it is sufficient to be
familiar with a standard university course on probability theory and mathematical
statistics.
Kiev, Ukraine Igor I. Gorban
November 2015
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