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1、Ratesofconvergenceofextremeforlognormaldistributionunderpowernormalization¤JianwenHuang,ShouquanChenSchoolofMathematicsandComputationalScience,ZunyiNormalCollege,ZunyiGuizhou,563002,ChinaSchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing,400715,China(email:sqchen@swu.
2、edu.cn)Abstract:Byapplyingthetheoryofp-maxstablelaws,westudytheratesofconvergenceofextremesforthelogarithmnormaldistributionunderpowernormalization.Weobtaintheexactuniformconvergencerateofthedistributionofmaximumtoitsextremevaluelimit.MathematicsSubjectClassi¯cation(2010):60F15,6
3、0G70Keywords:P-maxstablelaws;Logarithmnormaldistribution;Maximum;Uni-formconvergencerate.1IntroductionLetfXn;n¸1gbeasequenceofindependentandidenticallydistributed(iid)randomvariableswithcommondistributionfunction(df)F.Supposethatthereexistnormalizingconstantsan>0;bn2Randnon-degen
4、eratedistributionfunctionG(x)suchthatnlimP(Mn·anx+bn)=limF(anx+bn)=G(x);(1.1)n!1n!1forallx2C(G),thesetofallcontinuitypointsofG;whereMn=maxfX1;X2;¢¢¢;Xng.ThenG(x)mustbelongtooneofthefollowingthreeclasses:(0;x<0;©®(x)=¡®expf¡xg;x¸0;(expf¡(¡x)®g;x<0;ª®(x)=1;x¸0;¤ProjectsupportedbyNS
5、FS(grantNo.11071199,No.11171275).1¡x¤(x)=expf¡eg;x2R;where®isapositiveparameter,denoteF2Dl(G):CriteriaforF2Dl(G)andthechoiceofnormalizationconstantsanandbncanbefoundindeHaan(1970),Leadbetteretal.(1983)andResnick(1987).Inordertoobtainamoreaccurateapproximationofdistributionofmaxim
6、umbyitslimitingdf,Pancheva(1985)andWeinstein(1973)introducedanonlinearnormalizationcalledpowernormalization.AccordingtoMohanandRavi(1993),Pancheva(1985),FissaidtobelongtothedomainofattractionofadfHunderpowernormalization,denotedbyF2Dp(H)ifthereexistsomeconstants®n>0and¯n>0,suchth
7、at¯¯1¯Mn¯¯nlimP(¯¯sign(M)·x)=limFn(®jxj¯nsign(x))=H(x);(1.2)¯¯nnn!1®nn!1wheresign(x)=¡1;0or1asx<0;x=0;x>0:AdfHiscalledpower-maxstableorp-maxstableforshortbyMohanandRavi(1993)ifitsatis¯esthestabilityrelationn¯nH(®njxjsign(x))=H(x);x2Randn2N;forsomeconstants®n>0and¯n>0.Pancheva(198
8、5)showedthatHisofthepowertypeofoneofthef
