Multiple Wiener-Itô Integrals: With Applications to Limit Theorems

141 Pages·2014·3.092 MB·English

by  Péter Major (auth.)

Lecture Notes in Mathematics 849
Péter Major
Multiple
Wiener-Itô
Integrals
With Applications to Limit Theorems
Second Edition
Lecture Notes in Mathematics 849
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
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CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
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Péter Major
Multiple Wiener-Itô Integrals
With Applications to Limit Theorems
Second Edition
123
PéterMajor
HungarianAcademyofSciences
AlfrédRényiMathematicalInstitute
Budapest,Hungary
ISBN978-3-319-02641-1 ISBN978-3-319-02642-8(eBook)
DOI10.1007/978-3-319-02642-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013953865
MathematicsSubjectClassification(2010):Primary:60G18,60H05,60F99,
secondary:60G10,60G15,60G60
Firstedition©Springer-VerlagBerlinHeidelberg1981
Secondedition©SpringerInternationalPublishingSwitzerland2014
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Preface to the Second Edition
This text is a slightly modified version of my Lecture Note Multiple Wiener–Itô
integrals with applications to limit theorems published in the Lecture Notes in
Mathematicsseries(number849)oftheSpringerVerlagin1981.Idecidedtowrite
arevisedversionofthisLectureNoteafteraspecialcourseIheldaboutitssubject
in the first semester of the academic year 2011–2012at the Universityof Szeged.
Preparing for this course I observed how difficult the reading of formulas in this
LectureNotewas.ThesedifficultiesarosebecausethisLectureNotewaswrittenat
the time when the TEX program still did not exist, and the highest technical level
oftypingwaswritingonanIBMmachinethatenabledonetotypebesidetheusual
textalsomathematicalformulas.Butthetextswrittenin sucha wayareveryhard
to read. To make my text more readable, I decided to retype it by means of the
TEXprogram.Butitturnedoutthatarealimprovementofthetextdemandsmuch
more than producing nice, readable formulas. To make a really better version of
this work, I also had to explainbetter the results and definitionstogetherwith the
ideasandmotivationbehindthem.Besides, I hadto makenotonlymorereadable
formulas but also more readable explanations. The reader must see at each point
of the discussion what is just going on and why.In the new version of this work,
I tried to satisfy these demands. Naturally, I also corrected the errors I found. At
some points I had to insert a rather long explanation in the proof, because I met
such a statement which seemed to be trivial at the first sight, but its justification
demandedadetaileddiscussion.Ihopethattheseinsertionsdidnotmakethework
lesstransparent.
ThereappearedmanynewresultsaboutthesubjectofthisLectureNotesinceits
first appearance. The question arose naturally whether I should insert them to the
neweditionofthiswork.FinallyIdecidedtomakenoessentialchangesinthetext,
torestrictmyselftothecorrectionoftheerrorsIfoundandtogiveamoredetailed
explanationoftheproofswhereIfeltthatitisuseful.Inmakingsucha decisionI
ThepreparationofthisrevisedversionwassupportedbytheHungarianNationalFoundationfor
Science(OTKA)undergrantnumber91928.
v
vi PrefacetotheSecondEdition
wasinfluencedbyaRussianproverbwhichsays:“Лучше враг хорошего”.Itried
tofollowtheadviceofthisproverb.(IdonotknowofanEnglishcounterpartofthis
proverb,butithasaFrenchversion:“Lemieuxestl’ennemidubien”.)
I made one exception. I decided to explain those basic notions and results in
thetheoryofgeneralizedfunctionswhichwereappliedintheolderversionofthis
work in an implicit way. In particular, I tried to explain with their help how one
gets those results about the spectral representation of the covariance function of
stationaryrandomfieldsthatIhavepresentedunderthenamesBochner’stheorem
and Bochner–Schwartz theorem. This extension of the text is contained in the
attachmentsto Chaps.1 and3. In the firstversionI onlyreferredto a workwhere
these notions and results can be found. But now I found such an approach not
satisfactory, because these notions and results play an important role in some
argumentsof this work.Hence I felt thatto make a self-containedpresentationof
thesubject,Ihavetoexplaintheminmoredetail.
The first edition of this Lecture Note appeared long time ago, but the main
question discussed in it, the description of the limit behaviour of appropriately
normalizedpartialsumsofstronglydependentrandomvariablesremainedanopen
problem. Also the method applied in this work remained an importanttool in the
study of such problems. Hence a self-contained explanation of the theory which
provides a good foundation for this method is useful. By my hopes this Lecture
Notecontainssuchanexplanation,andthereforeitdidnotbecomeoutofdate.This
wasthemainargumentformyselftowriteanewversionofthisworkwhereItried
topresentabetterandmoreaccessiblediscussion.
Iwouldliketowritesomewordsaboutthelastchapterofthiswork,wheresome
results are discussed that seemed to be important at the time of writing the first
version.Iwouldmentiontwoofthemwhichlaterturnedouttobereallyimportant.
ThefirstoneistheNelson–Grossinequalitywhichlaterplayedanimportantrolein
thetheoryofthe so-calledhypercontractiveandlogarithmicSobolievinequalities.
Thesecondoneisamethodforconstructionofnon-trivialself-similarfieldsworked
outinapaperofKestenandSpitzer.Severalimportantlimittheoremsarebasedon
theideasofthispaper.ItisworthmentioningthatitwasRolandL’vovichDobrushin
whocalledmyattentiontotheseresults,andheemphasizedtheirimportance.SoI
wouldliketofinishthisprefacewithapersonalremarkabouthim.
ThisworkistheresultofsomejointresearchwithRolandL’vovichDobrushin.
Althoughthebookwaswrittenbymealone,Dobrushin’sinfluenceisverystrongin
it.Ihavelearnedverymuchfromhim.Itisratherdifficulttoexplainwhatonecould
learnfromhim,becauseitwasmuchmorethanjustsomeresultsormathematical
arguments. There was something beyond it, some world view which is hard to
explain.If I could give back somethingfrom what I had learned from him in this
LectureNote,thenthiswouldjustifytheworkonitbyitself.
Budapest,Hungary PéterMajor
15August2013
Preface to the First Edition
One of the most important problems in probability theory is the investigation of
thelimitdistributionofpartialsumsofappropriatelynormalizedrandomvariables.
The case where the random variables are independent is fairly well understood.
Many results are known also in the case where independence is replaced by an
appropriatemixingconditionorsomeother“almostindependence”property.Much
lessisknownaboutthelimitbehaviourofpartialsumsofreallydependentrandom
variables.On the otherhand,this case is becomingmoreand moreimportant,not
onlyinprobabilitytheorybutalsoinsomeapplicationsinstatisticalphysics.
The problemaboutthe asymptoticbehaviourofpartialsumsof dependentran-
domvariablesleadstotheinvestigationofsomeverycomplicatedtransformations
of probability measures. The classical methods of probability theory do not seem
to work for this problem. On the other hand, although we are still very far from
a satisfactory solution of this problem, we can already present some non-trivial
results.
Theso-calledmultipleWiener–Itôintegralshaveprovedtobeaveryusefultool
in the investigation of this problem. The proofs of almost all rigorous results in
this field are closely related to this technique. The notion of multiple Wiener–
Itô integrals was worked out for the investigation of non-linear functionals over
Gaussian fields. It is closely related to the so-called Wick polynomialswhich can
beconsideredasthemulti-dimensionalgeneralizationofHermitepolynomials.The
notion of Wick polynomials and multiple Wiener–Itô integrals were worked out
atthesametimeandindependentlyofeachother.Actually,wediscussamodified
versionofthemultipleWiener–Itôintegralsingreatestdetail.Thetechnicalchanges
neededin the definitionof these modifiedintegralsare notessential. On the other
hand, these modified integrals are more appropriate for certain investigations,
since they enable us to describe the action of shift transformations and to apply
some sort of random Fourier analysis. There is also some connection between
multipleWiener–ItôintegralsandtheclassicalstochasticItôintegrals.Themaindif-
ferencebetweenthemisthatinthefirstcasedeterministicfunctionsareintegrated,
andinthesecondcaseso-callednon-anticipatingfunctionals.Theconsequenceof
thisdifferenceisthatnotechnicaldifficultyariseswhenwewanttodefinemultiple
vii
viii PrefacetotheFirstEdition
Wiener–Itôintegralsinthemulti-dimensionaltimecase.Ontheotherhand,alarge
classofnon-linearfunctionalsoverGaussianfieldscanberepresentedbymeansof
multipleWiener–Itôintegrals.
Inthisworkweareinterestedinlimitproblemsforsumsofdependentrandom
variables. It is useful to consider this problem together with its continuous time
version. The natural formulation of the continuous time version of this problem
canbegivenbymeansofgeneralizedrandomfields.Consequentlywealsohaveto
discusssomequestionsaboutthem.
I have not tried to formulate all the results in the most generalform. My main
goal was to work out the most important techniques needed in the investigation
of suchproblems.Thisis the reasonwhythe greatestpartof this workdealswith
multipleWiener–Itôintegrals.Ihavetriedtogiveaself-containedexpositionofthis
subjectandalsotoexplainthemotivationbehindtheresults.
IhadtheopportunitytoparticipateintheDobrushin–SinaiseminarinMoscow.
WhatIlearnedtherewasveryusefulalsoforthepreparationofthisLectureNote.
ThereforeIwouldliketothankthemembersofthisseminarforwhatIcouldlearn
fromthem,especiallyP.M.Bleher,R.L.DobrushinandYa.G.Sinai.
Contents
1 OnaLimitProblem ......................................................... 1
1.1 ABriefOverviewAboutSomeResultsonGeneralizedFunctions... 5
2 WickPolynomials............................................................ 9
3 RandomSpectralMeasures................................................. 15
3.1 On the Spectral Representationof the Covariance
FunctionofStationaryRandomFields................................. 21
4 MultipleWiener–ItôIntegrals.............................................. 27
5 The Proof of Itô’s Formula: The Diagram Formula
andSomeofItsConsequences.............................................. 43
6 SubordinatedRandomFields:ConstructionofSelf-similarFields .... 65
7 OntheOriginalWiener–ItôIntegral ...................................... 81
8 Non-centralLimitTheorems ............................................... 87
9 HistoryoftheProblems:Comments....................................... 113
References......................................................................... 123
Index............................................................................... 125
ix