Statistical Inequality英文学习材料

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1、LectureNotes21ProbabilityInequalitiesInequalitiesareusefulforboundingquantitiesthatmightotherwisebehardtocompute.Theywillalsobeusedinthetheoryofconvergence.Theorem1(TheGaussianTailInequality)LetXN(0;1).Then2e2=2P(jXj>):IfX1;:::;XnN(0;1)then1n2=2P(jXnj>)pe:nProof

2、.ThedensityofXis(x)=(2)1=2ex2=2.Hence,Z1Z11P(X>)=(s)dss(s)dsZ12=210()e=(s)ds=:Bysymmetry,2e2=2P(jXj>):NowletX;:::;XN(0;1).ThenX=n1PnXN(0;1=n).Thus,X=dn1=2Z1nni=1inwhereZN(0;1)andp121=2pn=2P(jXnj>)=P(njZj>)=P(jZj>n)e:n1Theorem2(Markov

3、'sinequality)LetXbeanon-negativerandomvariableandsupposethatE(X)exists.Foranyt>0,E(X)P(X>t):(1)tProof.SinceX>0,Z1ZtZ1E(X)=xp(x)dx=xp(x)dx+xp(x)dxZ00Zt11xp(x)dxtp(x)dx=tP(X>t):ttTheorem3(Chebyshev'sinequality)Let=E(X)and2=Var(X).Then,21P(jXjt)andP(jZjk)(2)t2k2wh

4、ereZ=(X)=.Inparticular,P(jZj>2)1=4andP(jZj>3)1=9.Proof.WeuseMarkov'sinequalitytoconcludethatE(X)2222P(jXjt)=P(jXjt)=:t2t2Thesecondpartfollowsbysettingt=k.1PnIfX1;:::;XnBernoulli(p)thenandXn=ni=1XiThen,Var(Xn)=Var(X1)=n=p(1p)=nandVar(Xn)p(1p)1P(jXnpj>)

5、=2n24n2sincep(1p)1forallp.42Hoeding'sInequalityHoeding'sinequalityissimilarinspirittoMarkov'sinequalitybutitisasharperinequality.Webeginwiththefollowingimportantresult.Lemma4SuppposethatE(X)=0andthataXb.ThenE(etX)et2(ba)2=8:2Recallthatafunctiongisconvexifforeac

6、hx;yandeach2[0;1],g(x+(1)y)g(x)+(1)g(y):Proof.SinceaXb,wecanwriteXasaconvexcombinationofaandb,namely,X=b+(1)awhere=(Xa)=(ba).Bytheconvexityofthefunctiony!etywehavetXtbtaXatbbXtaee+(1)e=e+e:babaTakeexpectationsofbothsidesandusethefactthatE(X)=0togettXatbbta

7、g(u)Eee+e=e(3)babawhereu=t(ba),g(u)= u+log(1+ eu)and=a=(ba).Notethat000g(0)=g(0)=0.Also,g(u)1=4forallu>0.ByTaylor'stheorem,thereisa2(0;u)suchthatu2u2u2t2(ba)200000g(u)=g(0)+ug(0)+g()=g()=:2288Hence,EetXeg(u)et2(ba)2=8:Next,weneedtouseCherno'smethod.Lemma

8、5LetXbearandomvariable.ThenttXP(X>)infeE(e):t0Proof.Foranyt>0,XtXtttXP(X>