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1、Probab.TheoryRelat.Fields(2007)139:605635DOI10.1007/s00440-007-0055-4Exactinequalitiesforsumsofasymmetricrandomvariables,withapplicationsIosifPinelisReceived:25May2006/Revised:26December2006/Publishedonline:7February2007©Springer-Verlag2007AbstractLetB
2、S1,...,BSnbeindependentidenticallydistributedrandomvariableseachhavingthestandardizedBernoullidistributionwithparame-terp∈(0,1).Letm2221∗(p):=(1+p+2p)/(2p−p+4p)if0
3、ex.Thenitisprovedthatforallnonnegativenumbersc1,...,cnonehastheinequalityBS+···+c(m)Ef(c11nBSn)Efs(BS1+···+BSn),1wheres(m):=1nc2m2m.Thelowerboundmni=1i∗(p)onmisexactforeachp∈(0,1).Moreover,Ef(c1BS1+···+cnBSn)isSchur-concavein(c2m,...,c2nm).1Anumb
4、erofcorollariesareobtained,includingupperboundsongener-alizedmomentsandtailprobabilitiesof(super)martingaleswithdifferencesofboundedasymmetry,andalsoupperboundsonthemaximalfunctionofsuch(super)martingales.Applicationstogeneralizedself-normalizedsumsand
5、t-statisticsaregiven.Keywords(super)martingales·Probabilityinequalities·Generalizedmoments·Self-normalizedsums·t-statisticMathematicsSubjectClassification(2000)Primary:60E15·60G50·60G42·60G48·62F03·62F25·62G10·60G15;Secondary:60E05·62E10·62G35I.Pinelis(
6、B)DepartmentofMathematicalSciences,MichiganTechnologicalUniversity,Houghton,MI49931,USAe-mail:ipinelis@mtu.edu606I.Pinelis1IntroductionExponentialupperbounds,sayoftheforme−x2/(2B2),onthetailsprobabilitiesofsumsofindependentrandomvariables(r.v.s)or,more
7、generally,martingaleshavehadmanyapplicationsinprobabilityandstatistics.Intheapplications,theratioB/xisusuallysmall.Suchexponentialboundsarebasedonupperboundsonthecorrespond-ingexponentialmoments.However,incomparisonwiththeideal,normalestimate1−Φ(x/B)∼√
8、Be−x2/(2B2)ofthetailprobability(whereΦ(u):=x2πP(Zu),withZ∼N(0,1)),theexponentialupperbounde−x2/(2B2)missestheusuallysmallfactorB/x.Theapparentcauseofthisdeficiencyisthattheclassofexponentialmomentfunctionsistoosmall(andsoistheclassofth
