Normal Approximation by Stein’s Method

418 Pages·2011·2.19 MB·English

by  Louis H.Y. Chen

Probability and Its Applications
PublishedinassociationwiththeAppliedProbabilityTrust
Editors:S. Asmussen,J. Gani,P. Jagers,T.G.Kurtz
PhotoofCharlesStein,infront,with,fromlefttorightintherear,Qi-ManShao,LouisChenand
LarryGoldstein,takenataconferenceatStanfordUniversityheldinhonorofCharlesStein’s90th
birthdayonMarch22nd,2010
Forfurthertitlespublishedinthisseries,gotowww.springer.com/series/1560
Louis H.Y. Chen (cid:2) Larry Goldstein (cid:2) Qi-Man Shao
Normal Approximation
by Stein’s Method
LouisH.Y.Chen Qi-ManShao
DepartmentofMathematics DepartmentofMathematics
NationalUniversityofSingapore HongKongUniversityofScienceand
10LowerKentRidgeRoad Technology
Singapore119076 ClearWaterBay,Kowloon
RepublicofSingapore HongKong
[email protected] China
[email protected]
LarryGoldstein
DepartmentofMathematicsKAP108
UniversityofSouthernCalifornia
LosAngeles,CA90089-2532
USA
[email protected]
SeriesEditors:
SørenAsmussen PeterJagers
DepartmentofMathematicalSciences MathematicalStatistics
AarhusUniversity ChalmersUniversityofTechnology
NyMunkegade andUniversityofGothenburg
8000AarhusC 41296Göteborg
Denmark Sweden
[email protected] [email protected]
JoeGani ThomasG.Kurtz
CentreforMathematicsanditsApplications DepartmentofMathematics
MathematicalSciencesInstitute UniversityofWisconsin-Madison
AustralianNationalUniversity 480LincolnDrive
Canberra,ACT0200 Madison,WI53706-1388
Australia USA
[email protected] [email protected]
ISSN1431-7028
ISBN978-3-642-15006-7 e-ISBN978-3-642-15007-4
DOI10.1007/978-3-642-15007-4
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010938379
MathematicsSubjectClassification(2010):60F05,60B12,62E17
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This bookisdedicatedtoCharles Stein.
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Annabelle,Yitian,Yipei
Nancy
JienaandWenqi

Preface
Stein’s method has developed considerably since its first appearance in 1972, and
presentlyshowseverysignthatitsrangeintheoryandapplicationswillcontinueto
expand.Nevertheless,theremustbesomepointalongthiscontinuingpathwhenthe
methodreachesacertainlevelofmaturitythatathorough,selfcontainedtreatment,
highlightedwithasamplingofitsmanysuccesses,iswarranted.Theauthorsofthis
bookbelievethatnowisthistime.
IntheyearssinceStein’smethodforthenormalwasintroduced,therecognition
ofitspowerhasonlyslowlybeguntopercolatethroughouttheprobabilitycommu-
nity,helpedalong,nodoubt,bythemainreferencesinthefieldoverthelastmany
years,first,themonographofStein(1986),thecompilationofDiaconisandHolmes
(2004),andtheseriesofBarbourandChen(2005a,2005b).Nevertheless,touseone
barometer,todatethereexistonlyasmallnumberofbooksormonographs,targeted
generallyandaccessibleatthegraduateorundergraduatelevel,thatmakeanymen-
tionofStein’smethodforthenormalatall,inparticular,thetextsofStroock(2000)
andRossandPeköz(2007).Withathoroughbuildingupofthefundamentalsnec-
essarytocoverthemanyformsthatStein’smethodforthenormalcantaketodate,
andtheinclusionofalargenumberofrecentdevelopmentsinboththeoryandap-
plications,wehopethisbookonnormalapproximationwillcontinuetoaccelerate
the appreciation, understanding, and use of Stein’s method. Indeed, as interest in
the method has steadily grown, this book was partly written to add to the list we
cangiveinresponsetothemanyquerieswehavereceivedovertheyears,regarding
sourceswhereonecangotolearnmoreaboutthemethod,and,moreover,togeta
senseofwhetheritcanbeappliedtonewsituations.
Wehavemanytothankforthisbook’sexistence.Thefirstauthorwouldliketo
thankCharlesSteinforhisideaswhichtheformerlearnedfromhimasastudentand
whichhasbeenarichsourceofinspirationtohimovertheyears.Hewouldalsolike
to thank his co-authors, Andrew Barbour, Kwok-Pui Choi, Xiao Fang, Yu-Kiang
Leong, Qi-Man Shao and Aihua Xia, from whom he has benefited substantially
throughmanystimulatingdiscussions.
ThesecondauthorfirstheardaboutStein’smethod,forthePoissoncase,inalec-
turebyPersiDiaconis,andhethankshisfirstteachersinthatarea,RichardArratia
andLouisGordon,forconveyingarealsenseoftheuseoftheSteinequation,and
vii
viii Preface
MichaelWatermanforprovidingafountainofwonderfulapplications.Helearned
themostaboutthenormalapproximationversionofthemethod,andaboutitsap-
plications,fromhisworkwithYosiRinott,towhomheismostgrateful.Hehasalso
benefitedgreatlythroughall his othercollaborationswhereStein’smethodplayed
arole,mostnotablythosewithGesineReinert,aswellaswithAihuaXia,Mathew
Penrose,andHaimengZhang.
ThethirdauthorwouldliketothankLouisChenforintroducinghimtoStein’s
method,andfortheinspirationandinsighthehasprovided.
AlltheauthorswouldliketothanktheInstituteforMathematicalSciences,atthe
National University of Singapore, for their support of the many Singapore confer-
ences,whichservedasanexusforthedisseminationofthemostrecentdiscoveries
bytheparticipants,andfor thecreationof a perfectenvironmentfor theinvention
ofnewideas.
Forcommentsandsuggestionsregardingthepreparationofthisworktheauthors
wouldparticularlyliketothankJasonFulman,IvanNourdinandGiovanniPeccati
for their guidance on the material in Chap. 14. Additionally, we thank Subhankar
Ghosh and Wenxin Zhou for their help in various stages of the preparation of this
book,andproofreading,andXiaoFangforhisassistanceandhelpinwritingparts
of Chap. 7, on Discretized Normal Approximation. The first author was partially
supported by the Tan Chin Tuan Centennial Professorship Grant C-389-000-010-
101 at the National University of Singapore during the time this manuscript was
prepared,thesecondauthoracknowledgesthegrantsupportofNSA-AMS091026,
andthethirdauthoracknowledgesgrantsupportfromHongKongResearchGrants
Council(CERG-602608and603710).
For updates and further information on this book, please visit: http://mizar.usc.
edu/~larry/nabsm.html.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 TheCentralLimitTheorem . . . . . . . . . . . . . . . . . . . . . 1
1.2 ABriefHistoryofStein’sMethod . . . . . . . . . . . . . . . . . . 3
1.3 TheBasicIdeaofStein’sMethod . . . . . . . . . . . . . . . . . . 4
1.4 OutlineandSummary . . . . . . . . . . . . . . . . . . . . . . . . 8
2 FundamentalsofStein’sMethod . . . . . . . . . . . . . . . . . . . . 13
2.1 Stein’sEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 PropertiesoftheSolutions . . . . . . . . . . . . . . . . . . . . . . 15
2.3 ConstructionofSteinIdentities . . . . . . . . . . . . . . . . . . . 18
2.3.1 SumsofIndependentRandomVariables . . . . . . . . . . 19
2.3.2 ExchangeablePairs . . . . . . . . . . . . . . . . . . . . . 21
2.3.3 ZeroBias. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.4 SizeBias . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 A General Framework for Stein Identities and Normal
ApproximationforLipschitzFunctions . . . . . . . . . . . . . . . 36
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Berry–EsseenBoundsforIndependentRandomVariables . . . . . . 45
3.1 NormalApproximationwithLipschitzFunctions . . . . . . . . . . 46
3.2 TheLindebergCentralLimitTheorem . . . . . . . . . . . . . . . 48
3.3 Berry–EsseenInequality:TheBoundedCase . . . . . . . . . . . . 49
3.4 TheBerry–EsseenInequalityforUnboundedVariables. . . . . . . 53
3.4.1 TheConcentrationInequalityApproach . . . . . . . . . . 53
3.4.2 AnInductiveApproach . . . . . . . . . . . . . . . . . . . 57
3.5 ALowerBerry–EsseenBound . . . . . . . . . . . . . . . . . . . 59
4 L1 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 SumsofIndependentVariables . . . . . . . . . . . . . . . . . . . 65
4.1.1 L1 Berry–EsseenBounds . . . . . . . . . . . . . . . . . . 65
4.1.2 ContractionPrinciple . . . . . . . . . . . . . . . . . . . . 69
ix