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Ratesofconvergenceofextremeforlognormaldistributionunderpowernormalization¤JianwenHuang,ShouquanChenSchoolofMathematicsandComputationalScience,ZunyiNormalCollege,ZunyiGuizhou,563002,ChinaSchoolofMathematicsandStatistics,SouthwestUniversity,Chongqing,400715,China(email:sqchen@swu.edu.cn)Abstract:Byapplyingthetheoryofp-maxstablelaws,westudytheratesofconvergenceofextremesforthelogarithmnormaldistributionunderpowernormalization.Weobtaintheexactuniformconvergencerateofthedistributionofmaximumtoitsextremevaluelimit.MathematicsSubjectClassi¯cation(2010):60F15,60G70Keywords:P-maxstablelaws;Logarithmnormaldistribution;Maximum;Uni-formconvergencerate.1IntroductionLetfXn;n¸1gbeasequenceofindependentandidenticallydistributed(iid)randomvariableswithcommondistributionfunction(df)F.Supposethatthereexistnormalizingconstantsan>0;bn2Randnon-degeneratedistributionfunctionG(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¡x¡®g;x¸0;(expf¡(¡x)®g;x<0;ª®(x)=1;x¸0;¤ProjectsupportedbyNSFS(grantNo.11071199,No.11171275).1¤(x)=expf¡e¡xg;x2R;where®isapositiveparameter,denoteF2Dl(G):CriteriaforF2Dl(G)andthechoiceofnormalizationconstantsanandbncanbefoundindeHaan(1970),Leadbetteretal.(1983)andResnick(1987).Inordertoobtainamoreaccurateapproximationofdistributionofmaximumbyitslimitingdf,Pancheva(1985)andWeinstein(1973)introducedanonlinearnormalizationcalledpowernormalization.AccordingtoMohanandRavi(1993),Pancheva(1985),FissaidtobelongtothedomainofattractionofadfHunderpowernormalization,denotedbyF2Dp(H)ifthereexistsomeconstants®n>0and¯n>0,suchthat¯¯1¯M¯¯n¯n¯n¯nlimP(¯¯sign(Mn)·x)=limF(®njxjsign(x))=H(x);(1.2)n!1®nn!1wheresign(x)=¡1;0or1asx<0;x=0;x>0:AdfHiscalledpower-maxstableorp-maxstableforshortbyMohanandRavi(1993)ifitsatis¯esthestabilityrelationn¯nH(®njxjsign(x))=H(x);x2Randn2N;forsomeconstan
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