Gaussian Measures in Hilbert Space
Construction and Properties
Alexander Kukush
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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AbbreviationsandNotation
Chapter1.GaussianMeasuresinEuclideanSpace ...........1
1.1.Thechangeofvariablesformula......................1
1.2.InvarianceofLebesguemeasure......................4
1.3.Absenceofinvariantmeasureininfinite-dimensionalHilbertspace..9
1.4.Randomvectorsandtheirdistributions..................10
1.4.1.Randomvariables............................11
1.4.2.Randomvectors.............................12
1.4.3.Distributionsofrandomvectors....................14
1.5.GaussianvectorsandGaussianmeasures.................17
1.5.1.CharacteristicfunctionsofGaussianvectors.............17
1.5.2.ExpansionofGaussianvector.....................20
1.5.3.SupportofGaussianvector.......................22
1.5.4.GaussianmeasuresinEuclideanspace................23
Chapter2.GaussianMeasurein
2.1.Space R
2.1.1.Metricon
2.1.2.Borelandcylindricalsigma-algebrascoincide............30
2.1.3.Weighted l2 space............................31
2.2.Productmeasurein R∞
2.2.1.Kolmogorovextensiontheorem....................34
2.2.2.Constructionofproductmeasureon B (R∞ ) .............36
2.2.3.Propertiesofproductmeasure.....................38
2.3.StandardGaussianmeasurein R∞ .....................42
2.3.1.Alternativeproofofthesecondpartoftheorem2.4.........45
2.4.ConstructionofGaussianmeasurein l2 ..................46
Chapter3.BorelMeasuresinHilbertSpace ................51
3.1.Classesofoperatorsin H ..........................51
3.1.1.Hilbert–Schmidtoperators.......................52
3.1.2.Polardecomposition...........................55
3.1.3.Nuclearoperators............................57
3.1.4. S-operators................................62
3.2.PettisandBochnerintegrals........................68
3.2.1.Weakintegral..............................68
3.2.2.Strongintegral..............................69
3.3.BorelmeasuresinHilbertspace......................75
3.3.1.Weakandstrongmoments.......................75
3.3.2.ExamplesofBorelmeasures......................78
3.3.3.Boundednessofmomentform.....................83
Chapter4.ConstructionofMeasurebyitsCharacteristic Functional ......................................89
4.1.Cylindricalsigma-algebrainnormedspace................89 4.2.Convolutionofmeasures..........................93
4.3.Propertiesofcharacteristicfunctionalsin H ...............96
4.4. S -topologyin H ..............................99
4.5.Minlos–Sazonovtheorem..........................102
Chapter5.GaussianMeasureofGeneralForm
5.1.CharacteristicfunctionalofGaussianmeasure..............111
5.2.DecompositionofGaussianmeasureandGaussianrandomelement..114
5.3.SupportofGaussianmeasureanditsinvariance.............117
5.4.WeakconvergenceofGaussianmeasures.................125
5.5.ExponentialmomentsofGaussianmeasureinnormedspace......129
5.5.1.Gaussianmeasuresinnormedspace..................129
5.5.2.Fernique’stheorem...........................133
Chapter6.EquivalenceandSingularityofGaussianMeasures ...143
6.1.Uniformlyintegrablesequences......................143
6.2.Kakutani’sdichotomyforproductmeasureson R∞ ...........145
6.2.1.Generalpropertiesofabsolutelycontinuousmeasures........145
6.2.2.Kakutani’stheoremforproductmeasures...............148
6.2.3.DichotomyforGaussianproductmeasures..............152
6.3.Feldman–HájekdichotomyforGaussianmeasureson H ........155
6.3.1.ThecasewhereGaussianmeasureshaveequalcorrelation operators.....................................155
6.3.2.NecessaryconditionsforequivalenceofGaussianmeasures....158
6.3.3.CriterionforequivalenceofGaussianmeasures...........165
6.4.Applicationsinstatistics..........................169
6.4.1.EstimationandhypothesistestingformeanofGaussianrandom element.....................................169
6.4.2.Estimationandhypothesistestingforcorrelationoperatorof centeredGaussianrandomelement......................173
Chapter7.Solutions
...............................179
7.1.SolutionsforChapter1...........................179
7.2.SolutionsforChapter2...........................193
7.2.1.GeneralizedKolmogorovextensiontheorem.............196
7.3.SolutionsforChapter3...........................202
7.4.SolutionsforChapter4...........................211
7.5.SolutionsforChapter5...........................217
7.6.SolutionsforChapter6...........................227
SummarizingRemarks ..............................235
Foreword
Thestudyofmoderntheoryofstochasticprocesses,infinite-dimensionalanalysis andMalliavincalculusisimpossiblewithoutasolidknowledgeofGaussian measuresoninfinite-dimensionalspaces.Inspiteoftheimportanceofthistopicand theabundanceofliteratureavailableforexperiencedresearchers,thereisnotextbook suitableforstudentsforafirstreading.
Thepresentmanualisanexcellentget-to-knowcourseinGaussianmeasureson infinite-dimensionalspaces,whichhasbeengivenbytheauthorformanyyearsat theFacultyofMechanics&MathematicsofTarasShevchenkoNationalUniversityof Kyiv,Ukraine.Thepresentationofthematerialiswellthoughtout,andthecourseis self-contained.Afterreadingthebookitmayseemthatthetopicisverysimple.But thatisnottrue!Apparentsimplicityisachievedbycarefulorganizationofthebook. ForexpertsandPhDstudentshavingexperienceininfinite-dimensionalanalysis,I prefertorecommendthemonographV.I.Bogachev, GaussianMeasures (1998).But forfirstacquaintancewiththetopic,Irecommendthisnewmanual.
Prerequisitesforthebookareonlyabasicknowledgeofprobabilitytheory,linear algebra,measuretheoryandfunctionalanalysis.Theexpositionissupplementedwith abulkofexamplesandexerciseswithsolutions,whichareveryusefulforunassisted workandcontrolofstudiedmaterial.
Inthisbook,manydelicateandimportanttopicsofinfinite-dimensionalanalysis areanalyzedindetail,e.g.Borelandcylindricalsigma-algebrasin infinite-dimensionalspaces,BochnerandPettisintegrals,nuclearoperatorsandthe topologyofnuclearconvergence,etc.Wepresentthecontentsofthebook, emphasizingplaceswherefinite-dimensionalresultsneedreconsideration (everywhereexceptChapter1).
–Chapter1. Gaussiandistributionsonafinite-dimensionalspace. Thechapteris preparatorybutnecessary.Lateron,manyanalogieswithfinite-dimensionalspacewill begiven,andtheplaceswillbevisiblewhereanewtechniqueisneeded.
–Chapter2. Space R∞ ,Kolmogorovtheoremabouttheexistenceofprobability measure,productmeasures,Gaussianproductmeasures,Gaussianproductmeasures in l2 space. Afterreadingthechapter,thestudentwillstarttounderstandthaton infinite-dimensionalspacethereareseveralwaystodefineasigma-algebra(luckily, inourcaseBorelandcylindricalsigma-algebrascoincide).Moreover,itwillbecome clearthatinfinite-dimensionalLebesguemeasuredoesnotexist,henceconstruction ofmeasurebymeansofdensityneedsreconsideration.
–Chapter3. BochnerandPettisintegrals,Hilbert–Schmidtoperatorsandnuclear operators,strongandweakmoments. Thechapterisapreparationforthedefinition oftheexpectationandcorrelationoperatorofGaussian(orevenarbitrary)random element.Weseethatitisnotsoeasytointroduceexpectationofarandomelement distributedinHilbertorBanachspace.Asopposedtofinite-dimensionalspace,itis notenoughjusttointegrateoverbasisvectorsandthenaugmenttheresultsinasingle vector.
–Chapter4. Characteristicfunctionals,Minlos–Sazonovtheorem. Oneofthemost importantmethodstoinvestigateprobabilitymeasuresonfinite-dimensionalspaceis themethodofcharacteristicfunctions.Aswell-knownfromthecourseofprobability theory,thesewillbeallcontinuouspositivedefinitefunctionsequaltooneatzero,and onlythem.Oninfinite-dimensionalspacethisisnottrue.Forthestatement“theyand onlythem”,continuityinthetopologyofnuclearconvergenceisrequired,andthis topologyisexplainedindetail.
–Chapter5. GeneralGaussianmeasures. Basedonresultsofpreviouschapters, weseethenecessaryandsufficientconditionsthathavetobesatisfiedbythe characteristicfunctionalofaGaussianmeasureinHilbertspace.Werealizethatwe haveusedalltheknowledgefromChapters2–4(concerningintegrationofrandom elements,aboutHilbert–Schmidtandnuclearoperators,Minlos–Sazonovtheorem, etc.).Wenoticethatfortheeigenbasisofthecorrelationoperator,aGaussian measureisjustaproductmeasurewhichweconstructedinChapter2.Thisseems natural;butonourwayitwasimpossibletodiscardanysinglestepwithoutlossof mathematicalrigor.Inthischapter,Fernique’stheoremaboutfinitenessofan exponentialmomentofthenormofaGaussianrandomelementisprovedandthe criterionfortheweakconvergenceofGaussianmeasuresisstated.
–Chapter6. Equivalenceandmutualsingularityofmeasures. Here,Kakutani’s theoremisprovenabouttheequivalenceoftheinfiniteproductofmeasures.Aswe sawinthepreviouschapter,GaussianmeasuresonHilbertspacesareproduct measures,inaway.Therefore,asaconsequenceofgeneraltheory,wegetacriterion fortheequivalenceofGaussianmeasures(Feldman–Hájektheorem).Theobtained resultsareappliedtoproblemsofinfinite-dimensionalstatistics.Oneshouldbe
carefulhere,asduetotheabsenceoftheinfinite-dimensionalLebesguemeasure,the Radon–Nikodymdensityshouldbewrittenw.r.t.oneoftheGaussianmeasures.
Theauthorofthisbook,ProfessorA.G.Kukush,hasbeenworkingattheFaculty ofMechanics&MathematicsofTarasShevchenkoNationalUniversityfor40years. Heisanexcellentteacherandafamousexpertinstatisticsandprobabilitytheory.In particular,heusedtogivelecturestostudentsofmathematicsandstatisticson MeasureTheory,FunctionalAnalysis,StatisticsandEconometrics.Asastudent,I wasluckytoattendhisfascinatingcourseoninfinite-dimensionalanalysis.
AndreyP
ILIPENKO LeadingResearcherattheInstituteofMathematics ofUkrainianNationalAcademyofSciences, ProfessorofMathematicsattheNationalTechnicalUniversity ofUkraine,“IgorSikorskyKyivPolytechnicInstitute” August2019
Preface
Thisbookiswrittenforgraduatestudentsofmathematicsandmathematical statisticswhoknowalgebra,measuretheoryandfunctionalanalysis(generalized functionsarenotusedhere);theknowledgeofmathematicalstatisticsisdesirable onlytounderstandsection6.4.Thetopicofthisbookcanbeconsideredas supplementarychaptersofmeasuretheoryandliesbetweenmeasuretheoryandthe theoryofstochasticprocesses;possibleapplicationsareinfunctionalanalysisand statisticsofstochasticprocesses.For20years,theauthorhasbeengivingaspecial course“GaussianMeasures”atTarasShevchenkoNationalUniversityofKyiv, Ukraine,andin2018–2019,preliminaryversionsofthisbookhavebeenusedasa textbookforthiscourse.
Thereareexcellenttextbooksandmonographsonrelatedtopics,suchas Gaussian MeasuresinBanachSpaces [KUO75], GaussianMeasures [BOG98]and Probability DistributionsonBanachSpaces [VAK87].WhydidIwritemyowntextbook?
Inthe1970s,IstudiedattheFacultyofMechanicsandMathematicsofTaras ShevchenkoNationalUniversityofKyiv,atthattimecalledKievStateUniversity. ThereIattendedunforgettablelecturesgivenbyProfessorsAnatoliyYa. Dorogovtsev(calculusandmeasuretheory),LevA.Kaluzhnin(algebra),MykhailoI. Yadrenko(probabilitytheory),MyroslavL.Gorbachuk(functionalanalysis)and YuriyM.Berezansky(spectraltheoryoflinearoperators).MyPhDthesiswas supervisedbyfamousstatisticianA.Ya.Dorogovtsevanddealtwiththeweak convergenceofmeasuresoninfinite-dimensionalspaces.Forlongtime,Iwasa memberoftheresearchseminar“Stochasticprocessesanddistributionsinfunctional spaces”headedbyclassicsofprobabilitytheoryAnatoliyV.SkorokhodandYuriyL. Daletskii.Myseconddoctoralthesiswasaboutasymptoticpropertiesofestimators forparametersofstochasticprocesses.Thus,Iamsomewhattiedupwithmeasures oninfinite-dimensionalspaces.
In1979,Kuo’sfascinatingtextbookwastranslatedintoRussian.Inspiredbythis book,IstartedtogivemylecturesonGaussianmeasuresforgraduatestudents.The subjectseemedhighlytechnicalandextremelydifficult.Idecidedtocreate somethinglikeacomicbookonthistopic,inparticulartodividelengthyproofsinto smallunderstandablestepsandexplaintheideasbehindcomputations.
Itisimpossibletostudymathematicalcourseswithoutsolvingproblems.Each sectionendswithseveralproblems,someofwhichareoriginalandsomearetaken fromdifferentsources.Aseparatechaptercontainsdetailedsolutionstoallthe problems.
Acknowledgments
IwouldliketothankmycolleaguesatTarasShevchenkoNationalUniversityof Kyivwhosupportedmyproject,especiallyYuliyaMishura,OleksiyNesterenkoand IvanFeshchenko.AlsoIwishtothankmystudentsofdifferentgenerationswho followedupontheideasofthematerialandhelpedmetoimprovethepresentation.I amgratefultoFedorNazarov(KentStateUniversity,USA)whocommunicatedthe proofoftheorem3.9.Inparticular,IamgratefultoOksanaChernovaandAndrey Frolkinforpreparingthemanuscriptforpublication.IthankSergiyShklyarforhis valuablecomments.
MywifeMariyadeservesthemostthanksforherencouragementandpatience.
AlexanderK UKUSH Kyiv,Ukraine September2019
ThetheoryofGaussianmeasuresliesonthejunctionoftheoryofstochastic processes,functionalanalysisandmathematicalphysics.Possibleapplicationsarein quantummechanics,statisticalphysics,financialmathematicsandotherbranchesof science.Inthisfield,theideasandmethodsofprobabilitytheory,nonlinearanalysis, geometryandtheoryoflinearoperatorsinteractinanelegantandintriguingway.
TheaimofthisbookistoexplaintheconstructionofGaussianmeasureinHilbert space,presentitsmainpropertiesandalsooutlinepossibleapplicationsinstatistics.
Chapter1dealswithEuclideanspace,wheretheinvarianceofLebesguemeasure isexplainedandGaussianvectorsandGaussianmeasuresareintroduced.Their propertiesarestatedinsuchaformthat(lateron)theycanbeextendedtothe infinite-dimensionalcase.Furthermore,itisshownthatonaninfinite-dimensional Hilbertspacethereisnonon-trivialmeasure,whichisinvariantunderalltranslations (thesameconcerninginvarianceunderallunitaryoperators);henceonsuchaspace thereisnomeasureanalogoustotheLebesgueone.
InChapter2,aproductmeasureisconstructedonthesequencespace R∞ based onKolmogorovextensiontheorem.ForstandardGaussianmeasure μ on R∞ , Kolmogorov–Khinchincriterionisestablished.Inparticular,itisshownthat μ is concentratedoncertainweightedsequencespaces l2,a ,andbasedonisometry between l2,a and l2 ,aGaussianproductmeasureisconstructedonthelattersequence space.
Chapter3introducesimportantclassesofoperatorsinaseparable infinite-dimensionalHilbertspace H ,inparticular S -operators,i.e.self-adjoint, positiveandnuclearones.Theorem3.9showsthattheconvergenceof S -operatorsis equivalenttocertainconvergenceofcorrespondingquadraticforms.Alsotheweak (Pettis)andstrong(Bochner)integralsaredefinedforafunctionvaluedinaBanach space.
Borelprobabilitymeasureson H andanormedspace X arestudiedwithexamples. Theboundednessofmomentformsofsuchmeasuresisshown,withsimpleproof basedontheclassicalBanach–Steinhaustheorem.Corollary3.3andremark3.8give mildconditionsfortheexistenceofmeanvalueofaprobabilitymeasure μ asPettis integral,andiftheunderlyingspaceisaseparableBanachspace B and μ hasastrong firstmoment,thenitsmeanvalueexistsasBochnerintegral.
InChapter4,propertiesofcharacteristicfunctionalsofBorelprobabilitymeasures on H arestudied.Aspeciallineartopology, S -topology,isintroducedin H witha neighborhoodsystemconsistingofellipsoids.ClassicalMinlos–Sazonovtheoremis provenandproperlyextendsBochner’stheoremfrom Rn to H .AccordingtoMinlos–Sazonovtheorem,thecharacteristicfunctionalofaBorelprobabilitymeasureson H shouldbecontinuousin S -topology.Apartofproofofthistheorem(seelemma4.9) suggeststhewaytoconstructaprobabilitymeasurebyitscharacteristicfunctional.
InChapter5,theorem5.1usestheMinlos–Sazonovtheoremtodescribea Gaussianmeasureon H ofgeneralform.Itturnsoutthatthecorrelationoperatorof suchameasureisalwaysan S -operator.ItisshownthateachGaussianmeasureon H isjustaproductofone-dimensionalGaussianmeasuresw.r.t.theeigenbasisofthe correlationoperator.Thus,everyGaussianmeasureon H canbeconstructedalong theway,asdemonstratedinChapter2.
ThesupportofGaussianmeasureisstudied.ItisshownthatacenteredGaussian measureisinvariantunderquitearichgroupoflineartransforms(seetheorem5.5). Hence,aGaussianmeasureinHilbertspacecanbeconsideredasanaturalinfinitedimensionalanalogueof(invariant)Lebesguemeasure.
AcriterionfortheweakconvergenceofGaussianmeasuresisstated,where(dueto theorem3.9)werecognizetheconvergenceofcorrelationoperatorsinnuclearnorm.
Insection5.5,westudyGaussianmeasuresonaseparablenormedspace X Importantexample5.3showsthataGaussianstochasticprocessgeneratesameasure onthepathspace Lp [0,T ],henceincase p =2,weobtainaGaussianmeasureon Hilbertspace.Lemma5.9presentsacharacterizationofGaussianrandomelementin X
ThefamoustheoremofFerniqueisproven,whichstatesthatcertainexponential momentsofaGaussianmeasureon X arefinite.Inparticular,everyGaussianmeasure onaseparableBanachspace B hasmeanvalueasBochnerintegralanditscorrelation operatoriswell-defined.Theorem5.10derivestheconvergenceofmomentsofweakly convergentGaussianmeasures.
InChapter6,Kakutani’sremarkabledichotomyforproductmeasureson R∞ is proven.Inparticular,twosuchproductmeasureswithabsolutelycontinuous
componentsareeitherabsolutelycontinuousormutuallysingular.Thisimpliesthe dichotomyforGaussianmeasureson R∞ :twosuchmeasuresareeitherequivalentor mutuallysingular.Section6.3provesthefamousFeldman–Hájekdichotomyfor Gaussianmeasureson H ,andincaseofequivalentmeasures,expressionsfor Radon–Nikodymderivativesareprovided.
Insection6.4,theresultsofChapter6areappliedinstatistics.Basedonasingle observationofGaussianrandomelementin H ,weconstructunbiasedestimatorsfor itsmeanandforparametersofitscorrelationoperator;alsowecheckahypothesis aboutthemeanandthecorrelationoperator(thelatterhypothesisisinthecasewhere theGaussianelementiscentered).Inviewofexample5.3with p =2,thesestatistical procedurescanbeusedforasingleobservationofaGaussianprocessonfinitetime interval.
Thebookisaimedforadvancedundergraduatestudentsandgraduatestudentsin mathematicsandstatistics,andalsofortheoreticallyinterestedstudentsfromother disciplines,sayphysics.
Prerequisites forthebookarecalculus,algebra,measuretheory,basicprobability theoryandfunctionalanalysis(wedonotusegeneralizedfunctions).Insection6.4, theknowledgeofbasicmathematicalstatisticsisrequired.
Somewordsabout thestructureofthebook :wepresenttheresultsinlemmas, theorems,corollariesandremarks.Allstatementsareproven.Importantand illustrativeexamplesaregiven.Furthermore,eachsectionendswithalistof problems.DetailedsolutionstotheproblemsareprovidedinChapter7.
Theabbreviationsandnotationusedinthebookaredefinedinthecorresponding chapters;anoverviewofthemisgiveninthefollowinglist.
AbbreviationsandNotation
a.e.almosteverywherew.r.t.Lebesguemeasure a.s.almostsurely cdfcumulativedistributionfunction pdfprobabilitydensityfunction i.i.d.independentandidenticallydistributed(randomvariablesor vectors)
r.v.randomvariable
LHSleft-handside
RHSright-handside
MLEmaximumlikelihoodestimator
|A| numberofpointsinset A
Ac complementofset A
A closureofset A
TB imageofset B undertransformation T
T 1 A preimageofset A undertransformation T
x , A transposedvectorandtransposedmatrix,respectively R extendedrealline,i.e. R = R ∪{−∞, +∞}
Rn×m spaceofreal n × m matrices
B (x,r ), B (x,r ) openandclosedball,respectively,centeredat x withradius r> 0 inametricspace
f+ positivepartoffunction f , f+ =max(f, 0) f negativepartoffunction f , f = min(f, 0)
δij Kroneckerdelta, δij =1 if i = j ,and δij =0, otherwise
an ∼ bn {an } isequivalentto {bn } as n →∞,i.e. an /bn → 1 as n →∞
C (X ) spaceofallrealcontinuousfunctionson X
R∞ spaceofallrealsequences
B (X ) Borelsigma-algebraonmetric(ortopological)space X
λm Lebesguemeasureon Rm
Sm sigma-algebraofLebesguemeasurablesetson Rm
IA indicatorfunction,i.e. IA (x)=1 if x ∈ A,else IA (x)=0 μT 1 measureinducedbymeasurabletransformation T basedon measure μ,i.e. (μT 1 )(A)= μ(T 1 A),foreachmeasurable set A
L(X,μ) spaceofLebesgueintegrablefunctionson X w.r.t.measure μ f = g (mod μ)functions f and g areequalalmosteverywherew.r.t.measure μ
δx Diracmeasureatpoint x, δx (B )= IB (x)
ν μ signedmeasure ν isabsolutelycontinuousw.r.t.measure μ dν dμ theRadon–Nikodymderivativeof ν w.r.t. μ
ν ∼ μ measures ν and μ areequivalent
ν ⊥ μ signedmeasure ν andmeasure μ aremutuallysingular (x,y ) innerproductofvectors x and y inEuclideanorHilbertspace x Euclideannormofvector x A Euclideannormofmatrix A, A =sup x=0 Ax x Im theidentitymatrixofsize m rk (S ) rankofmatrix S Pn projectiveoperator, Pn x =(x1 ,...,xn ) , x ∈ R∞ √A squarerootofpositivesemidefinitematrix A,itispositive semidefiniteaswellwith (√A)2 = A
x,x∗ or x∗ ,x valueoffunctional x∗ atvector x I theidentityoperator
L(X ) spaceoflinearboundedoperatorsonnormedspace X R(A) rangeofoperator A, R(A)= {y : ∃x,y = Ax}
L⊥ orthogonalcomplementtoset L
L2 [a,b] Hilbertspaceofsquareintegrablerealfunctionswithinner product (x,y )= b a x(t)y (t)dt,thelatterisLebesgueintegral lp spaceofrealsequences x =(xn )∞ 1 withnorm x p = ( ∞ n=1 |xn |p )1/p if 1 ≤ p< ∞,and x ∞ =supn≥1 |xn | if p = ∞.For p =2, l2 isHilbertspacewithinner product (x,y )= ∞ 1 xn yn
l2,a weighted l2 space span(M ) spanofset M, i.e.,setofallfinitelinearcombinationsofvectors from M
ˆ An cylinderin R∞ withbase An ∈B (Rn )
A = A L(X ) operatornormoflinearboundedoperator A, A =sup x=0 Ax x
A∗ adjointoperator
√B = B 1/2 squarerootofself-adjointpositiveoperator B
|A| modulusofcompactoperator A, |A| =(A∗ A)1/2
AbbreviationsandNotationxxi
A 1 nuclearnormofoperator A
A 2 Hilbert–Schmidtnormofoperator A
An ⇒ A operators An uniformlyconvergetooperator A
S0 (H ) classoffinite-dimensionaloperatorson H
S1 (H ) classofnuclearoperatorson H
S2 (H ) classofHilbert–Schmidtoperatorson H
S∞ (H ) classofcompactoperatorson H
LS (H ) classof S -operatorson H
A ≥ 0 operator A ispositive,i.e. (Ax,x) ≥ 0 forall x
A ≥ B comparisoninLoewnerorderofself-adjointoperators: A B is positiveoperator
n k =1 Ak Cartesianproductofsets A1 ,...,An
n
k =1 μk productofmeasures μ1 ,...,μn
∞
k =1 μk productmeasureon R∞ oronHilbertspace
mμ meanvalueofmeasure μ
Cov (μ) variance-covariancematrixofmeasure μ on Rn
ϕμ or ˆ μ characteristicfunction(orfunctional)ofmeasure μ
Aμ operatorofsecondmomentofmeasure μ
Sμ correlationoperatorofmeasure μ
σn (z1 ,...,zn ) weakmomentsoforder n ofBorelprobabilitymeasureon H
μX distributionofrandomvector X orrandomelement X , μX (B )= P(X ∈ B ) forallBorelsets B
ϕX characteristicfunction(functional)ofrandomvector (element) X
E X expectationofrandomvector(element) X
D X varianceofrandomvariable X
Cov (X ) variance-covariancematrixofrandomvector X
X d = Y randomvectors(elements) X and Y areidenticallydistributed
N (m,σ 2 ) Gaussiandistributionwithmean m andvariance σ 2 , σ ≥ 0
N (m,S ) Gaussiandistributionon Rn (oron H )withmeanvalue m and variance-covariancematrix(orcorrelationoperator) S d −→ convergenceindistributionofrandomelements
