Approximation Methods in Probability Theory

283 Pages·2016·1.33 MB·English

by  Vydas Čekanavičius

Universitext
Vydas Čekanavičius
Approximation
Methods in
Probability
Theory
Universitext
Universitext
SeriesEditors
SheldonAxler
SanFranciscoStateUniversity
VincenzoCapasso
UniversitàdegliStudidiMilano
CarlesCasacuberta
UniversitatdeBarcelona
AngusMacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
CNRS,Ecolepolytechnique,France
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski
CaseWesternReserveUniversity,Cleveland,OH
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ˇ
Vydas Cekanavicˇius
Approximation Methods in
Probability Theory
123
VydasCˇekanavicˇius
VilniusUniversity
Vilnius,Lithuania
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-319-34071-5 ISBN978-3-319-34072-2 (eBook)
DOI10.1007/978-3-319-34072-2
LibraryofCongressControlNumber:2016941172
MathematicsSubjectClassification(2010):62E20,60E10,60G50,60F99,41A25,41A27
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Preface
The limit theorems of probability theory are at the core of multiple models used
in the broad field of scientific research. Their main function is to replace the
initial complicated stochastic model of a phenomenon by its somewhat simpler
approximatesubstitute.Asarule,suchsubstituteiseasiertousesinceitsproperties
arewellknown.However,itraisesakeyquestion:howgoodistheapproximation?
For example, even in the famous central limit theorem, the rate of convergence
to the normal law can be extremely slow. Therefore, it is important to measure
the magnitude of the difference between models or, in other words, to estimate
the accuracyof approximation.However,there is a notable lack of booksthat are
specificallyfocusedonteachinghowtodoit.Onecouldfindnumerousmonographs
and textbooks devoted to approximations, especially related to the central limit
theorem, but the prime concern of their authors is the impressive results, not the
methods that are used in the process. Thus, such books rarely involve more than
one method, not to mention an actual comparison of the applicability of several
approaches.
This book is loosely based on a course I have been teaching during my visit
atHamburgUniversityin June,2004,combinedwith specific methodsandproofs
accumulatedfromteachingPhD-levelcoursessince then. Itpresentsa wide range
of various well-known and less common methods for estimation of the accuracy
of probabilistic approximations. In other words, it is a book on tools for proving
approximationtheorems.Asarule,wedemonstratethecorrectnessofthe‘tool’by
providing an appropriate proof. In a few cases, when the proofs are very long or
sophisticatedornotdirectlyrelatedtothetopicofthe book,theyareomitted,and
anappropriatereferencetothesourceoftheresultisprovided.
Our ultimate goal is to teach the reader the correct usage of various methods.
Therefore, we intentionally present simple cases, examining in detail all steps
required for the proof. We provide one further simplification on placing the
emphasisontheorderofaccuracyofapproximationratherthanonthemagnitudeof
absoluteconstants.Inordertogainabettercommandofthepresentedtechniques,
various exercises are added at the end of each chapter, with the majority of their
v
vi Preface
solutionsatthe endof the book.Bibliographicalnotesprovideinformationonthe
originsofeachmethodanditsmoreadvancedapplications.
Thespecificmethodsdescribedinthisbookincludeaconvolutionmethod,which
canbeeasilyextendedfromdistributionstothemoregeneralcommutativeobjects.
Inaddition,aconsiderablepartofthebookisrelatedtotheclassicalcharacteristic
function method. As usual, we present the Esseen-type inversion formulas. On
the other hand, we systematically treat the lattice case, since, in our opinion, it is
undeservedlyrarely considered in the literature. Furthermore, one of the chapters
is devoted to the powerful but rarely used triangular function method. Though
we usually deal with independent random variables, Heinrich’s method for m-
dependent variables is also included. Due to the fact that Stein’s method has had
a uniquely comprehensive coverage in the literature (with a specific emphasis on
the method in [7, 50, 127]), only its short version is presented, together with a
discussiononsomespecificmethodologicalaspects,whichmightbeofinteresteven
toexperiencedusersofthemethod.
Naturally, this book does not contain all known methods for estimating the
accuracyof approximation.Forexample,we use onlyfourmetricsand,therefore,
the method of metrics is not included. Moreover, it is already comprehensively
treated in [116]. Meanwhile, methods for random vectors or elements of infinite-
dimensionalspacesdeserveaseparatetextbook.
Astandardintermediatecourseinprobabilityshouldbesufficientforthereaders.
This book is expected to be especially useful for masters and PhD students. In
fact, I wish I had similar book during my PhD studies, because learning various
methodsfrom scientific papers is doing it the hard way for no good reason, since
alotofintermediateresultsareusuallyomittedandsmalltricksandtwistsarenot
adequatelyexplained.Iwasluckytomeetmanymathematicianswhohelpedmein
theprocess,andIwouldliketoextendmyspecialthankstoA.Bikelis,J.Kruopis,
A.Yu.Za˘ıtsev,A.D.BarbourandB.Roos.
Vilnius,Lithuania VydasCˇekanavicˇius
2016
Contents
1 DefinitionsandPreliminaryFacts ....................................... 1
1.1 DistributionsandMeasures.......................................... 1
1.2 MomentInequalities ................................................. 5
1.3 NormsandTheirProperties.......................................... 6
1.4 FourierTransforms................................................... 9
1.5 ConcentrationFunction.............................................. 14
1.6 AlgebraicIdentitiesandInequalities................................ 15
1.7 TheSchemesofSequencesandTriangularArrays................. 19
1.8 Problems.............................................................. 19
2 TheMethodofConvolutions ............................................. 21
2.1 ExpansioninFactorialMoments.................................... 21
2.2 ExpansionintheExponent .......................................... 25
2.3 LeCam’sTrick....................................................... 28
2.4 SmoothingEstimatesfortheTotalVariationNorm ................ 29
2.5 EstimatesinTotalVariationviaSmoothing......................... 32
2.6 SmoothingEstimatesfortheKolmogorovNorm................... 37
2.7 EstimatesintheKolmogorovNormviaSmoothing................ 38
2.8 Kerstan’sMethod..................................................... 44
2.9 Problems.............................................................. 47
3 LocalLatticeEstimates ................................................... 51
3.1 TheInversionFormula............................................... 51
3.2 TheLocalPoissonBinomialTheorem.............................. 53
3.3 ApplyingMomentExpansions ...................................... 54
3.4 ALocalFranken-TypeEstimate..................................... 57
3.5 InvolvingtheConcentrationFunction............................... 59
3.6 SwitchingtoOtherMetrics.......................................... 62
3.7 LocalSmoothingEstimates.......................................... 64
3.8 TheMethodofConvolutionsforaLocalMetric ................... 65
3.9 Problems.............................................................. 67
vii
viii Contents
4 UniformLatticeEstimates ................................................ 69
4.1 TheTsaregradskiiInequality ........................................ 69
4.2 TheSecondOrderPoissonApproximation......................... 71
4.3 TakingintoAccountSymmetry ..................................... 74
4.4 Problems.............................................................. 75
5 TotalVariationofLatticeMeasures ..................................... 77
5.1 InversionInequalities ................................................ 77
5.2 ExamplesofApplications............................................ 79
5.3 SmoothingEstimatesforSymmetricDistributions................. 85
5.4 TheBarbour-XiaInequality ......................................... 86
5.5 ApplicationtotheWassersteinNorm ............................... 89
5.6 Problems.............................................................. 91
6 Non-uniformEstimatesforLatticeMeasures .......................... 93
6.1 Non-uniformLocalEstimates ....................................... 93
6.2 Non-uniformEstimatesforDistributionFunctions................. 95
6.3 ApplyingTaylorSeries............................................... 98
6.4 Problems.............................................................. 100
7 DiscreteNon-latticeApproximations..................................... 101
7.1 Arak’sLemma........................................................ 101
7.2 ApplicationtoSymmetricDistributions ............................ 103
7.3 Problems.............................................................. 105
8 AbsolutelyContinuousApproximations................................. 107
8.1 InversionFormula.................................................... 107
8.2 LocalEstimatesforBoundedDensities............................. 109
8.3 ApproximatingProbabilitybyDensity.............................. 110
8.4 EstimatesintheKolmogorovNorm................................. 112
8.5 EstimatesinTotalVariation.......................................... 113
8.6 Non-uniformEstimates .............................................. 117
8.7 Problems.............................................................. 119
9 TheEsseenTypeEstimates................................................ 121
9.1 GeneralInversionInequalities....................................... 121
9.2 TheBerry-EsseenTheorem.......................................... 125
9.3 Distributionswith1CıMoment ................................... 127
9.4 EstimatingCenteredDistributions................................... 130
9.5 DiscontinuousDistributionFunctions............................... 135
9.6 Problems.............................................................. 138
10 LowerEstimates............................................................ 141
10.1 EstimatingTotalVariationviatheFourierTransform.............. 141
10.2 LowerEstimatesfortheTotalVariation............................. 143
10.3 LowerEstimatesforDensities....................................... 146
10.4 LowerEstimatesforProbabilities................................... 147
Contents ix
10.5 LowerEstimatesfortheKolmogorovNorm........................ 149
10.6 Problems.............................................................. 151
11 TheSteinMethod........................................................... 153
11.1 TheBasicIdeaforNormalApproximation......................... 153
11.2 TheLatticeCase...................................................... 155
11.3 EstablishingStein’sOperator........................................ 157
11.4 TheBigThreeDiscreteApproximations............................ 159
11.5 ThePoissonBinomialTheorem..................................... 161
11.6 ThePerturbationApproach ......................................... 163
11.7 EstimatingtheFirstPseudomoment................................. 167
11.8 LowerBoundsforPoissonApproximation......................... 173
11.9 Problems.............................................................. 175
12 TheTriangleFunctionMethod ........................................... 179
12.1 TheMainLemmas ................................................... 179
12.2 AuxiliaryTools....................................................... 184
12.3 FirstExample......................................................... 186
12.4 SecondExample...................................................... 195
12.5 Problems.............................................................. 206
13 Heinrich’sMethodform-DependentVariables......................... 207
13.1 Heinrich’sLemma.................................................... 207
13.2 PoissonApproximation.............................................. 211
13.3 Two-WayRuns ....................................................... 214
13.4 Problems.............................................................. 220
14 OtherMethods.............................................................. 223
14.1 MethodofCompositions............................................. 223
14.2 CouplingofVariables................................................ 229
14.3 TheBentkusApproach............................................... 230
14.4 TheLindebergMethod............................................... 232
14.5 TheTikhomirovMethod............................................. 234
14.6 IntegralsOvertheConcentrationFunction.......................... 236
14.7 AsymptoticallySharpConstants .................................... 237
SolutionstoSelectedProblems ................................................. 241
Bibliography...................................................................... 266
Index............................................................................... 273