Randomness and Hyper-Randomness

240 Pages·2018·3.177 MB·English

by  Gorban, Igor I

Mathematical Engineering
Igor I. Gorban
Randomness
and Hyper-
randomness
Mathematical Engineering
Serieseditors
J€orgSchr€oder,Essen,Germany
BernhardWeigand,Stuttgart,Germany
Moreinformationaboutthisseriesathttp://www.springer.com/series/8445
Igor I. Gorban
Randomness and
Hyper-randomness
IgorI.Gorban
InstituteofMathematicalMachinesandSystemsProblems
NationalAcademyofSciencesofUkraine
Kiev,Ukraine
Originally published by Naukova Dumka Publishing House of National Academy of
Sciences ofUkraine, Kiev,2016
ISSN2192-4732 ISSN2192-4740 (electronic)
MathematicalEngineering
ISBN978-3-319-60779-5 ISBN978-3-319-60780-1 (eBook)
DOI10.1007/978-3-319-60780-1
LibraryofCongressControlNumber:2017945377
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Preface
One of the most remarkable physical phenomena is the statistical stability
(regularity)ofmassphenomenaasrevealedbythestabilityofstatistics(functions
of samples). There are two theories describing this phenomenon. The first is
classicalprobabilitytheory,whichhasalonghistory,andthesecondisthetheory
ofhyper-randomphenomenadevelopedinrecentdecades.
Probability theory has established itself as the most powerful tool for solving
variousstatisticaltasks.Itisevenwidelybelievedthatanystatisticalproblemcan
beeffectivelysolvedwithintheparadigmofprobabilitytheory.However,itturns
outthatthisisnotso.
Someconclusionsofprobabilitytheorydonotaccordwithexperimentaldata.A
typical example concerns the potential accuracy. According to probability theory,
whenweincreasethenumberofmeasurementresultsofanyphysicalquantity,the
errorintheaveragedestimatortendstozero.Buteveryengineerorphysicistknows
thattheactualmeasurementaccuracyisalwayslimitedandthatitisnotpossibleto
overcomethislimitbystatisticalaveragingofthedata.
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
phenomenonofstatisticalstability.
Probabilitytheoryisinfactaphysical-mathematicaldiscipline.Themathemat-
icalcomponentisbasedonA.N.Kolmogorov’sclassicalaxioms,whilethephysical
componentisbasedoncertainphysicalhypotheses,inparticularthehypothesisof
perfectstatisticalstabilityofactualevents,variables,processes,andfieldsassum-
ingtheconvergenceofstatisticswhenthesamplesizegoestoinfinity.
Experimentalinvestigationsofvariousprocessesofdifferentphysicalkindsover
broad observation intervals have shown that the hypothesis of perfect statistical
stability is not confirmed experimentally. For relatively short temporal, spatial, or
spatio-temporal observation intervals, an increase in data volume usually reduces
the level of fluctuation inthe statistics.However,whenthe volumes become very
large, this tendency is no longer visible, and once a certain level is reached, the
v
vi Preface
fluctuations remain practically unchanged or even grow. This indicates a lack of
convergenceforrealstatistics(theirinconsistency).
If the volume of processing data is small, the violation of convergence has
practically no influence on the results, but if this volume is large, the influence is
verysignificant.Thestudyofviolationsofstatisticalstabilityinphysicalphenom-
ena and the development of an effective way to describe the actual world, one
which accounts for such violations, has resulted in the construction of the new
physical-mathematicaltheoryofhyper-randomphenomena.
Thetheoryofhyper-randomphenomenaisalsoaphysical-mathematicaltheory.
Its mathematical component is based on the axioms and statements of the mathe-
matical component of the probability theory, and its physical component is based
on hypotheses that differ essentially from the physical hypotheses of probability
theory, in particular the hypothesis of limited statistical stability assuming the
absence of convergence in the actual statistics. Therefore, for mathematicians the
theory of hyper-random phenomena is a branch of probability theory, and for
physicists,itisanewphysicaltheorybasedonanewviewoftheworld.
There is much literature describing probability theory from various points of
view (Kolmogorov 1929, 1956; Mises 1964; Bernshtein 1946, etc.) and oriented
toward readerships with different mathematical knowledge (Feller 1968; Loe´ve
1977; Gnedenko 1988; Angot 1957; Devor 2012; Gorban 2003; Pugachev 1979;
Peebles 1987; Tutubalin 1972; Rozhkov 1996; Ventsel 1962, etc.). Quite a few
studieshavealsobeenpublishedintheareaofstatisticalstabilityviolationandthe
theory of hyper-random phenomena. Among the latter, we can mention three
monographs in Russian (Gorban 2007, 2011, 2014), two monographs in
Ukrainian(UvarovandZinkovskiy2011a,b),andamonographinEnglish(Gorban
2017)devotedtovariousmathematical,physical,andpracticalquestions.
Probability theory and the theory of hyper-random phenomena give different
descriptionsofthephenomenonofstatisticalstability.Untilrecently,therewereno
books comparing these theories. However, this gap was closed in 2016 with the
monographinRussian(Gorban2016).ThisbookisanEnglishversionofthat.
Theaimsofthecurrentmonograph,likethosebeforeit,are:
• Toacquaintthereaderwiththephenomenonofstatisticalstability
• Todescribeprobabilitytheoryandthetheoryofhyper-randomphenomenafrom
asinglestandpoint
• Tocomparethesetheories
• Todescribetheirphysicalandmathematicalessenceattheconceptuallevel
This monograph consists of five parts. The first entitled The Phenomenon of
Statistical Stability consists of the introductory chapter. Manifestations of this
phenomenon and different approaches for its description are described in it. The
second part entitled Probability Theory contains four chapters (Chaps. 2–5) and
describesthefoundationsofprobabilitytheory.ThethirdpartentitledExperimen-
talStudyoftheStatisticalStabilityPhenomenoncontainsonlyChap.6,dedicatedto
a description of the techniques developed to assess statistical stability violations
andalsotheresultsofexperimentalinvestigationsofstatisticalstabilityviolations
Preface vii
inactualphysicalprocessesofvariouskinds.ThetitleofthefourthpartisTheoryof
Hyper-randomPhenomena.Itincludesfourchapters(Chaps.7–10)presentingthe
foundationsofthetheoryofhyper-randomphenomena.ThefifthpartentitledThe
Problem of an Adequate Description of the World includes only Chap. 11, which
discussestheconceptofworldbuilding.
The book aims at a wide readership: from university students of a first course
majoring in physics, engineering, and mathematics to engineers, postgraduate
students,andscientistsresearchingthestatisticallawsofnaturalphysicalphenom-
enaanddevelopingandusingstatisticalmethodsforhigh-precisionmeasurement,
prediction, and signal processing over broad observation intervals. To understand
thematerialinthebook,itissufficienttobefamiliarwithastandardfirstuniversity
courseonmathematics.
Kiev,Ukraine IgorI.Gorban
25June2016
References
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Moskow–Leningrad(1934),(1946)
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Cole,CengageLearning,Boston(2012)
Feller,W.:AnIntroductiontoProbabilityTheoryandItsAplications.JohnWiley
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nomena).IMMSP,NASofUkraine,Kiev(2007)
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Kolmogorov,A.N.:FoundationsoftheTheoryofProbability.ChelseaPub.Comp.,
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Loe´ve,M.:ProbabilityTheory(part1,2).Springer-Verlag(1977)
Mises, R.: Mathematical Theory of Probability and Statistics. Acad. Press., N.
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Pugachev,V.S.:TeoriyaVeroyatnosteyiMatematicheskayaStatistica(Probability
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Rozhkov, V.A.:TeoriyaVeroyatnosteySluchainikhSobytiy,VelichiniFunkziys
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Acknowledgments
The issues discussed in this monograph lie at the intersection of physics, mathe-
matics,andengineering.Forthisreason,forofficialandunofficialreviewingofthe
current book and its Russian version, scientists of different specialties have been
involved. The author is grateful to all those who read the books, made critical
remarks,andparticipatedinconstructivediscussion.
Theauthorwouldliketoexpressspecialappreciationtotheanonymousrefereesof
the present monograph,English monograph (Gorban 2017),and also to the official
reviewers of the previous Russian monographs (Gorban 2007, 2011, 2014, 2016),
dedicatedtoresearchonthesameproblem,inparticulartoProfessorI.N.Kovalenko,
academician of the National Academy of Sciences of Ukraine (NASU), Professor
V.F.Gubarev,correspondentmemberofNASU,ProfessorP.S.Knopov,correspon-
dent member of NASU, Professor N. Yu. Kuznetsov, correspondent member of
NASU, Professor G.P. Butsan, Professor A.S. Mazmanishvili, Professor
M.A. Popov, Dr. Sc. A.M. Reznick, Professor S. Ya. Zhuk, and Professor
Yu.F.Zinkovskiy.
The author is grateful to Professor P.M. Tomchuk, correspondent member of
NASU, K.V. Gaindrik, correspondent member of Sciences of Moldova, Professor
O.G. Sarbey, Professor V.I. Ivanenko, Professor V.A. Kasyanov, and Professor
M.I. Schlesinger for the opportunity to present the material of these books at
seminars they organized, as well as to all the participants of these seminars for
usefuldiscussions.
TheauthorisgratefultoProfessorV.T.Grinchenko,ProfessorV.MKuntsevich,
bothacademiciansofNASU,ProfessorV.A.Akulichev,ProfessorR.I.Nigmatulin,
andProfessorYu.I.Shokin,allacademiciansoftheRussianAcademyofSciences
(RAS), Professor V.S. Lysenko, correspondent member of NASU, Professor V.
N. Tutubalin, Professor A.V. Kharchenko, Professor S.P. Shary, Professor
V.Kreinovich,Dr.Sc.I.O.Yaroshchuk,andmanyothers,whohaveshownastrong
interestintheauthor’sresearchonthestatisticalstabilityofrealphysicalprocesses
andthetheoryofhyper-randomphenomena.
ix