Stochastic Processes: Harmonizable Theory (Multivariate Analysis)

341 Pages·2020·9.432 MB·English

by  M M Rao

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SERIES ON MULTIVARIATE ANALYSIS
Editor: M M Rao ISSN: 1793-1169

Published
Vol. 1: Martingales and Stochastic Analysis
J. Yeh
Vol. 2: Multidimensional Second Order Stochastic Processes
Y. Kakihara
Vol. 3: Mathematical Methods in Sample Surveys
H. G. Tucker
Vol. 4: Abstract Methods in Information Theory
Y. Kakihara
Vol. 5: Topics in Circular Statistics
S. R. Jammalamadaka and A. SenGupta
Vol. 6: Linear Models: An Integrated Approach
D. Sengupta and S. R. Jammalamadaka
Vol. 7: Structural Aspects in the Theory of Probability:
A Primer in Probabilities on Algebraic-Topological Structures
H. Heyer
Vol. 8: Structural Aspects in the Theory of Probability (Second Edition)
H. Heyer
Vol. 9: Random and Vector Measures
M. M. Rao
Vol. 10: Abstract Methods in Information Theory (Second Edition)
Y. Kakihara
Vol. 11: Linear Models and Regression with R: An Integrated Approach
D. Sengupta and S. R. Jammalamadaka
Vol. 12: Stochastic Processes: Harmonizable Theory
M. M. Rao
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World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Rao, M. M. (Malempati Madhusudana), 1929– author.
Title: Stochastic processes : harmonizable theory / M.M. Rao, University of California, Riverside.
Description: 1st edition. | Hackensack, NJ : World Scientific Publishing Co. Pte. Ltd., [2020] |
Series: Series on multivariate analysis,
1793-1169 ; vol. 12 | Includes bibliographical references and index.
Identifiers: LCCN 2020012114 | ISBN 9789811213656 (hardcover) | ISBN 9789811213663 (ebook) |
ISBN 9789811213670 (ebook other)
Subjects: LCSH: Stochastic processes.
Classification: LCC QA274 .R3725 2020 | DDC 519.2/3--dc23
LC record available at https://lccn.loc.gov/2020012114
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TotheMemoriesofFriends
P.R.Krishnaiah
forchangingmyattemptstojointheIndianAirForceAcademyin
DehraDuntoMinnesota’sStochasticAnalysisstudy
and
HerbertHeyer
fortheadvocacyandexampletouseabstractmethodsin
StochasticAnalysis
b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Preface
The following work constitutes the third and final part of the original planned three-
volumetreatmentof‘StochasticAnalysis’.Thefirstoneappearedin1995,asthesec-
ond edition of an earlier publication (the 1979 version), and the second part, termed
‘Inference Theory’ (of stochastic processes) appeared in 2010 (as a revised, and en-
larged work of the original 2000 year publication). These two volumes are pub-
lishedoriginallybyNetherlands’publishers.Thatpublicationhousewasacquiredby
Springer and then reissued under their own name (2nd edition of 2nd volume). The
current volume completes the originally planned coverage of the trilogy. Many new
developments are covered in these volumes. I shall now indicate the contents of this
volumeforabriefviewoftheincludedtopicspresentedinsixchapters.Eachchapter
alsohasitscontentsoutlinedinanearlyparagraph.
Chapter 1 details a characterization of the general concept of (weak) harmoniz-
abilityextendingtheclassicalweakstationarityofprocesses,initiatedbyKhintchine,
extendedbyLoe´veandthegeneralformobtainedbyBochner(aftersomeinitialspe-
cial cases treated separately by L. Schwartz and Yu. A. Rozanov). They showed the
central part of Fourier or harmonic analysis is essential in stochastic theory as well,
and the first two chapters of this volume, using this framework, are devoted to ana-
lyzing the structural aspects based on the first two moments. Most of the traditional
treatmentsconcentrateonthesecondorder(i.e.,covariance)properties,assumingthe
processestobecentered.Butthemeanvaluesarenotgenerallyconstants.First,they
arecharacterizedforharmonizableclassesandthentheprimaryfocusshiftstothepos-
itive definiteness of covariances which are characterized. As a key consequence, the
existence of Le´vy’s Brownian Motion and its properties are established. Thereafter
thegeneralharmonizabilityaspectbecomesthekeycomponentofourstudy.
From this point on, the property of V-boundedness of process, becomes primary
and Bochner’s characterization of it, and consequences take on a central place in the
ensuing analysis. Chapters 2 and 3 consider extensions of weakly harmonizable pro-
cesses, as well as a detailed study in much of the (new) classes by Karhunen and
Crame´r,andtheirdilationstolarger(orsuper)Hilbertspaces.Theyhaveseveralcon-
sequences. Also integral representations of (Gel’fand’s) local functionals given for
vii
August20,2020 9:16 svmono-9x6 11648-main pageviii
viii Preface
the first time, as well as a new and notable probabilistic proof of the long open and
interestingRiemannhypothesis,withadetaileddiscussion,areincludedhere.
Inthesechapterssomeapplicationsoftheworktotheharmonizableprocessesand
other extensions including the Crame´r, Karhunen classes using generalized Fourier
analysis methods as well as some summability analysis of Kampe´ de Feriet and
Frankiel(andindependentlyofParzen’s)onnonstationaryprocessesarestudied.They
areofinterestinapplicationsinvolvingworksonsignalextractionproblems.
When the indexing set is higher (2 or more) dimensional, then one has to study
randomfieldssothatnewmethodsandanalysesareneeded.Thisisconsideredfrom
Chapter 4 on. Here the concept of isotropy arises and we need to use properties of
Bessel and related special functions. In these cases, a weakening of Lebesgue’s inte-
gral is needed and fortunately it is found in a work of M. Morse and W. Transue. It
isrefinedbyD.K.Changandtheauthor(1986)toobtainasuitableformtousehere.
This is employed and with it we consider the isotropic random fields analysis which
isdetailedinChapter4.Theyincludethecorresponding(integral)representationsof
randomfieldsforbothharmonizable,KarhunenandCrame´rclassesthatextendthefor-
mer.Chapter5isdevotedtosomeextensionsoftheprecedinganalysisiftheindexing
isanLCAgroupaswellasanextensionwhenitisahypergroup.Theworkindicates
manypossibilitiesofthestudywhentheindexingisconsideredonthesestructuresthat
havebothpracticalandtheoreticalconsequences.Thereisalsosomeanalysisgivenin
thischapteriftheoptimalitycriterionisnottheusualsquaredlossbutageneralnon-
negativeconvexfunctionthatmaybespecializedformanyapplications.
Thelastchaptercontainsananalysisofbistochasticoperators,theircharacteriza-
tionsaswellastheirconvergentproperties.Alsosomeaccountsofisotropicanalysis
ofrandomfieldsareincluded,andtheareaisactive.Itisalsousefulforreaderstolearn
more about random measures with applications here. They are detailed in my earlier
workinthevolumeentitledRandomandVectorMeasures(2012),especiallyinitslast
threechapters.Thisgivesagoodmotivationaswellassomenewideasonextending
the present work in other directions, motivated by the work in (the long) Chapter 7
of the above book. An extended presentation of Crame´r and Karhunen classes as a
far-reachinggeneralizationsof(weak)harmonizability(andweakstationarityinmany
respects) is also detailed in last two chapters that exhibit the extent of growth of the
subjectswithpossiblefurtherapplicationsandexplorationforsecondorderprocesses.
In the preparation of this monograph, I have been awarded a UCR Senate’s
EdwardA.Dickson Professorship which has really assisted me in completing this
volume. The material is composed using the LATEX version with the great help of
Ms.AmbikaVanchinathan for a period of over a year, from Chennai, India, and I
am very grateful to her for this assistance. I am also much obliged to my friends
Drs.Y.KakiharaandJ.H.Park,fortheirgreathelpinproofreadingandeffectivecor-
rectionsoftheTeXversion.AlsofromourDepartmentoffice,Ihadbeenassistedby
JamesMarberry,aswellasCrissyReisingandJoyceSphabmisaiinsomepreparations
August20,2020 9:16 svmono-9x6 11648-main pageix
Preface ix
ofthiswork.Ihopethematerialinthisvolume,alongwithitspredecessors,willen-
courage new as well as seasoned researchers in the subject, to significantly advance
thisworkintothefuture.
Riverside,CA
August18,2020 M.M.Rao