Some_limit_theorems_for_the_eigenvalues_of_a_sample_covariance_matrix.pdf

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JOURNALOFMULTIVARIATEANALYSIS12,l-38(1982)SomeLimitTheoremsfortheEigenvaluesofaSampleCovarianceMatrixDAGJONSSONVppsalaUniversity,Vppsala,SwedenCommunicatedbyP.R.KrishnaiahLimittheoremsaregivenfortheeigenvaluesofasamplecovariancematrixwhenthedimensionofthematrixaswellasthesamplesizetendtoinfinity.Thelimitofthecumulativedistributionfunctionoftheeigenvaluesisdeterminedbyuseofamethodofmoments.Theproofismainlycombinatorial.Byavariantofthemethodofmomentsitisshownthatthesumoftheeigenvalues,raisedtok-thpower,k=1,2,...,misasymptoticallynormal.AlimittheoremforthelogsumoftheeigenvaluesiscompletedwithestimatesofexpectedvalueandvarianceandwithboundsofBerry-Esseentype.1.INTR~DWTI~NLetX,,i=1,2,...,p,j=1,2,...,n,beindependentnormalvariableswithzeromeanandunitvariance.V-1)LetYr,=C;=,XrJXs/denoteelementsofapxpmatrix,SF),i.e.,Y=Y.ThenSF’hasacentralWishartdistributionwithndegreesoffriedomr(d.f.).(1.2)Oneaimofthispaperistostudytheasymptoticbehaviouroftheeigen-valuesofSF)whenpandnbothtendtoinfinity,inawayindicatedbyArharov[2].Differentfunctionsoftheeigenvaluesusedastestcriteriainmultivariateanalysisareofspecialinterest.Examplesofsuchfunctionsarethesumoftheeigenvalues=traceSF)andtheproductoftheeigenvalues=1SF)](thedeterminant).Arharov[2]assumedcondition(1.1).WithatechniqueintroducedbyArnold[3a],Arharov’sresultscanbeprovedunderweakerconditionsonthevariablesX,.Theyneednotbenormalandtheirmomentsneednotallbefinite.ReceivedJanuary13,1978;revisedAugust6,1981.AMS1980subjectclassifications:Primary62E20,60F05;Secondary62HlO.Keywordsandphrases:Eigenvaluesofasamplecovariancematrix,cumulativedistributionfunction,methodofmoments,limittheorems,sumsofeigenvalues,generalizedvariance.10047-259X/82/010001-38$02.00/0Copyright01982byAcademicPress,Inc.Allrightsofreproductioninanyformreserved. 2DAGJONSSONFirst,letn+coandthenp-+co.Whenn+co,forfixedp,thematrixWlmSjr’-$J&Iisthepxpidentitymatrix)convergesindistributiontoastochasticmatrixZ,,whoseelementsZ,,arenormallyandforr1.Underthesameconditions,Grenander[8,pp.177-1801hasshownthatW,(x)convergestoW(x)inprobability,whileArnold[3a]hasprovedthattheconvergenceisvalidwithprobability1.Forthepurposeofstandardizationweconsiderthematrix(l/n)SF)ratherthanSF).LetA$)Ofori=kt1,kt2,...,rt1,and-(ak+,tak+2t~“ta,+~ta,t~~~tai)t(r-kti)=(b,+1tr)t(biti)-(bktk)=(biti)-(bk+k)>0fori=1,2,...,k.(7)Ineverycyclethereisexactlyonepossiblesequence,i.e.,thecorrespondingsuitehasallitspartialsumsnon-negative.Supposethisistruefor-a,,1,-a,,I,...,1,--a,+,aswellasfor-uk,1,-ak+l,l,...,1,-a,+,, LIMITTHEOREMSFOREIGENVALUES111,-ur)l)...)1,-uk-r.Fromthefirstsuiteitfollowsthatb,-1+k-2>0orb,_,~-(k-2),whilethesecondonegivesb,+,-b,_,+r-k+1~0orbk-,(-(k-1)(sinceb,+l+r=0),whichmeansacontradiction.(8)Thus,thenumberofpossiblesequencesis(l/(r+1)).(‘:)(k;‘>=ak,ryandtheproofofthelemmaiscompleted.Thereisadirectconnectionwithanurnmodel:Consideranurnwithkballs,ofwhichraremarkedwithl’s,k-r-saremarkedwithO’sandtherestwith-i,,-i,,...,-is,respectively.Takeoutballswithequalprobabilitiesoneatatime,withoutreplacement.LetY,,V,,...,Vkbethesuccessiveobtainednumbers.ThenP(V,+V,+...+VU>0,u=1,2,...,k)equalstheprobabilitythatarandomlychosensuite(2.13)hasonlynon-negativepartialsums=l/(rtl),by(4)-(7)(cf.Karlin[11,pp.244-249,2681).Tosumup,fromLemma2.3itfollowsthatEM,=f‘2’ak,r+nkpr.p’+termsinnandpwithtotaldegree=2n(ll-y){2x-2rry}(since~~zeimtdt=Oform#O)=1.(ii)Thecasey=1.Thelimitdistributionisageneralp(f,$)-distributionovertheinterval(0,4)byRemark2.1. 14DAGJONSSON(iii)Thecase10isthesameasthedistributionofpP(‘).Therefore,itmightbenaturaltouseasymmetricstandardizationinpandn,forexample,define1:’as$“/fiinsteadof$‘/n.Thedensityofthelimitdistributionfor00.Proof.AtruncationmethodusedintheproofofArnold’stheorem[3a]isappliedwithslightmodifications.X,correspondstoaijinArnold’spaper.Thefollowinglemmaswillbeneeded.LEMMA3.1.SetM=M(N)=max(p,n),wherepandnsatisfy(1.4),y>0.ThenforeverydistributionfunctionH(x)JxlkdH(x)<00=s-i”,p(r-k-s”2.n-“*.JX”dH(x)=0I+lXIk+122.Proof.ByuseofxrdH(x)=1xrdH(x)+jxrdH(x)i1x1<‘WI*aIx1<‘WV4W/4<1x10,itfollowsthatllm,,,(M/p)<1+l/yMV 20DAGJONSSON.$12..IXlk’j-+’,,,!,,dH(x)11x1>M’/I<($)~“-”.(Lg2./,,,,M,,*,xl”dH(x).(3.3)Butp/Mandn/MarebothM”*.LetItisseenthatlimEM;=li+liEM,=a,(y),k=1,2,....N-X.3NotethatMkinSection2wasexpressedintermsofYrs)sinsteadofX,jsjustinordertosimplifytheassortingofthedifferentindexcombinations.Analogousmodificationsof(c),(d)and(e)of[3a]completetheproofofTheorem3.1.Forexample,p,of(e)heretakestheformpn=22P(Xij#xij>i=lj=l=.$jyP(IX,(&w’*)i=l]=IQW{l-H@P-0)+H(--M”‘)}w/*iNotethattheexistenceofthe4thmomentisnecessaryhere.Itisnotrequestedintheformerpartoftheproof(onlythe2ndmoment).ThecorrespondencetoTheorem2.2isTHEOREM3.2.Withcondition(3.1)insteadof(1.1)andwiththeadditionalconditionExb,<00,thestatementsofTheorem2.2arestilltrue,providedy>0.Thedetailsoftheproofareomitted.Remark.Theresultsofthissectionarestilltrueiftheelementsix;=1xrj-xsjofSF)arereplacedwithC;=i(Xr,-X,)(X,i-x,),wherexr=(l/n)CLxr,. 22DAGJONSSON4.LIMITTHEOREMSFORSUMSOFEIGENVALUESLetA,=traceSk=nkCT=,n:=pnkMk,k=1,2,...(see(2.8)).Arharov[2](Theorem2)hassetupacentrallimittheoremforA,,AZ,...,A,,,.Thetheoremwillberestatedheretogetherwithadetailedproof.Arharov’sversioniscorrectedonseveralpoints.THEOREM4.1.SetThen,underconditions(1.1)and(1.4),thestochasticvector(q,,?I*,...,q,)convergesindistributiontoacertainm-variatenormaldistribution;m=1,2,....Proof:Hereweshalluseamultidimensionalanalogueofthemomentmethod,exploitedintheproofofTheorem2.1.ConsiderthemixedmomentsBS,.S2,...,S,=E(A,-EA,)“(A2-EAJS’+e.(A,-EA,JSm;sl,s2,...,s,arenon-negativeintegers.Sets=s,+s,+***+s,,a=s,+2s,+asa+ms,andGC~~~:::~k)=Xi,j,.Xiti,.XiJz.Xi~il....*XidK.Xi,jkvk=1,2,....Then.....)!I..... LIMITTHEOREMSFOREIGENVALUES23-EG~p-ms,+ly-m,+2**-*-‘p-lies,-1)(’Ja-ms,+~Ja-ms,+2*****J=--m(s,--l)iamm+,ia-m+2*....i,(4.1)Itistobeshownthatthesemixedmomentsdividedby(n+p)”convergetothecorrespondingmomentsinacertainm-variatenormaldistribution.Thefollowinglemmaswillthenbeneeded.LEMMA4.1.BS,.S*,...,s,isapolynomialinnandpoftotaldegree=aifsiseven,and+(m,+m,)+...+(m,_,+m,)=(1+1+***+1)+(2+2+**‘+2)+***+(m+m+***+m)(s,l’s,s,2’s)...,s,m’s)=s,+2s,+em*+ms,=a.Thecoefficientsarethenobtainedbysummingoverthepossiblepartitionsintos/2pairs.Ifsisodd,itisnotpossibletoformevenpairs.Morethantwofactorswillthenhaveindicesincommonorsomepairoffactorswillhaveindicesincommonwithotherfactors.ThismeansthatthedegreeofBS,,S2,...,S,mustbesxcy;+li*y:.....Yf-l.Yf+l....‘Y:s1i=l+‘*’7i.e.,wegetallpossiblecombinationsoffirstandsecondderivativesofy(t)inrespecttot,,,t,,,...,trs,t,S+I.COROLLARY4.1.LetX,,X2,...,X,,,bejointlynormallydistributed 26DAGJONSSONrandomvariables,allwithexpectationzero,andCov(X,,Xj)denotedbyoij.ThenEX,,-Xr,..a-.Xr,ifsiseven=r,cm,m2*om,m4.*..*om5-,m,=oifsisodd;rl,r2,...,rsareselectedamongthefirstmpositiveintegers(theyneednotallbedifferent).Thesummationistakenoverallpossiblepartitionsofr,,r2,...,rsintopairs(m,,m,),(m3,m,),...,(m,-,,m,)(seeRemark4.1).Proof.Thecharacteristicfunctionof(X,,X,,...,X,)isofthetypeg(t)withcij=-jaij,i.e.,g(t)=exp(-+$$oije’i’tj)yI-1J-1y;=0fort=0andJ$‘~=-uk,;k,I=1,2,...,m.Wehavethefollowingrelationshipbetweenthemomentsandthecharac-teristicfunction:&r(t)=(i)”.at,,*at,******atrst=(JY-”.y"ifsiseven=LYm,m,m3m4*‘*’*Yli-,m,*(9”=oifsisodd=x0mm*om,m,“”~~m_,m,Iifsiseven=oifsisodd.ThestatementofCorollary4.1canbewritteninthefollowingform(thisisLemma3byArharov[2]).COROLLARY4.1’.UndertheconditionsofCorollary4.1EX;l.$2.....X2=Camlml.omsm,.....amSelm,ifs=s,+s,+...+s,iseven=oifsisodd.Thesummationistakenoverallpartitionsofs,I’s,s,2’s,...,s,m’sintopairs(m,,m,),(mJ,mq),...,(m,-,,m,)(seeRemark4.1). LIMITTHEOREMSFOREIGENVALUES27Aformulafordeterminingthecoefficientofthetermum,ml.u,....g(i.e.,thenumberofpartitionswhichgivesthisterm)isgi?:ibythe%i%inglemma.Itisstatedwithoutproof.LEMMA4.3.Thecoeficientofurn,,,,,.IS,,,~,,,~.....o,,,-,,,,,inEX:l.$2.....Xf.,y,wheres=s,+s2+e+a+s,iseven,iss,!s,!***a*s,!2”.j7.....jyvisthenumberofpairswithequalelements,i.e.,withm?i-1=m,i,uisthenumberofdIrerentpairsamong(m,,m,>,(m,,m,),...,(m,-,,m,>andf,,fi,...,f,arethefrequencesfortheupairs.Thetotalsumofcoeflcientsis1.3.....(s-1)=thecoeflcientofaj(’inEz.ThelaststepoftheproofofTheorem4.1istoestablishtheconvergenceofthemomentsB,,,S*,...,Sm.ByLemma4.1AhPIifsiseven+(n+PI”Polynomialofdegree0.However,theasymptoticcovariancesoijarenotthesameasinTheorem4.1if,u,#3,sincethecoeflcientsofthepolynomialBmli-,qm2idependon,uu,.Proof.ThetruncationmethodbyArnold[3a]isusedinthesamewayasintheproofofTheorem3.1.Denoteby(vi,vi,...,q&)thecorrespondenceof;;15;’Y***?q,,JwhenX,isreplacedwithXijdefinedin(3.4).ThenthevectorI,i,...,~6)convergesindistributiontotheactualm-variatenormaldistribution.Itremainstoshowthat(q’-9,)vi-q2,...,r7;-r,,,)convergesinprobabilitytozero.Butwhichtendstozero,N-rco,sincethe4thmomentofXiiisfinite(cf.(3.5)).5.THEGENERALIZEDVARIANCELetXl,X2,...,Xn+,bei.i.d.normalvectorswithpcomponents,withmeanvaluevectorpPandcovariancematrixEP.Considerthematrixcot+I)=;Z’(xj-;ri)(xj-x)‘.PI-1Supposepdx-2nyx1*lOg(l+y-2hCost).l-e2”+e-zitdt=2nI-z1+y-2Jycost217C2log(l-y)+y++loid-Y)=27r(l-y)1-n[I()+(1-y)(e-‘lt+e”‘)+negligibleterms1Ie21t+e-Zitdt2Thespecialcasey=1isconsideredinthenexttheorem.683/12/l-3 32DAGJONSSONTHEOREM5.la.(i)Underconditions(1.1)and(5.2),butwithy=1,1lqr’l-2log(1-p/n)log(n-l&isasymptoticallyEN(0;1),N+co.(ii)Supposep=nforeveryN.Then,underconditions(1.1)1INI?Jzlog(n-l>!asymptoticallyEN(0;1),N--fco.Proof.(i)Thec.f.of1IS:‘1d-2log(1-p/n)log(n-l)Ptrp-2log(1-p/n)it=exp-+1-2log(l-p/n)-j5n-P+j(it)*+62-4lOg(l-p/n)j51n-p+jItl+-?R,,r*-2log(1-p/n)11’Theremainderterm+0,asN+03,since-2log(1-p/n)-+00.Frominequality(5.4)andthefactthat-log(I-*)>-log(l-+)-log2wefindthatcpt)-+exp/%I.(d-2log(1-p,n)(ii)Thec.f.of1logISj7YJ2logn(n-1Y LIMITTHEOREMSFOREIGENVALUES33whichevidently-+exp{(it)*/2}asZV-+00.COROLLARY5.2.Undertheconditionsof(5.2)convergesindistributiontoalognormaldistributionwithparametersp=0andu2=-2log(l-y),asN+00.ThelimitdistributionofISp’I/(n-l&inTheorem5.1hastheexpectedvalueer+(“2)u2=l/(1-y)andthevariance2Y-Y2e2@.eu2(eu2-I)=(1-y)4’whileIS:)1(4,1p2,E(n-l),=(n-l),=-1-y*’nandIs:)/*n+ln+2-1var(n-l&=(n”p)’In-p+l’n-p+2I~2r*-y**(1-y*)4AWehavethefollowingestimatesforlog(lSF’(/(n-l),).THEOREM5.2.If16p1v(z)=logz-&-&,o1ww=$+(&+$)e,0<8<1. LIMITTHEOREMSFOREIGENVALUES35Itfollowsthatvarlog(s;)I11=2.-++28f-yyn-p+jg(n-P+II’+!!+1(5.7)3,T,(n-p+.g3<2f&~+21”+dx++l’---$x“n-pxn-PXnP322y*-y**=-2log(1-y*>+f*y*1-y*+3*n2(1-y*)*2-y*<-2log(l-y”)+A.y*.<+for1-~.+2log(l-Y*)~Jn-p-1xby(5.6).EXAMPLE5.1.Letp=5,n=20,i.e.,y*=0,25.Thecorrectmeanvalueis0.003.Theboundsare-0.006and0.017.Thecorrectvarianceis0.592.Theapproximatingvalueis-2log0.75=0.575.Theboundsare0.542and0.625.EXAMPLE5.2.Letp=10,n=40,i.e.,y*=0.25.Thecorrectmeanvalueis0.0014.Theboundsare-0.0028and0.0083.Thecorrectvarianceis0.568.Theapproximatingvalueis0.575,whiletheboundsare0.559and0.600.Thenexttheoremconcernstheconvergencerateoflog/Sr’(.THEOREM5.3.LetG:‘(x)bethedistributionfunctionof,ogIs~‘I-E1ogls;‘l=,&(‘%Uj-ElogUJ/Varlog)SF’I~~~=lVarloguj’ 36DAGJONSSONwherey*=p/nandCisanabsoluteconstant,11givesw’(z)=++e*$o’ZW”‘(Z)=$+3;,0<8’<1,i.e.,v/“‘(Z)+3(w’(z))2<3forsomeconstantc,.Thus24c,&‘>andPV’-2logl-j=l(5)2Y*>-2log(l-y*>--p;n1-y*iEIlogUj-ElogUj13j=lY*-logl-ny*>(n+l12-Mn+l))Y*‘2-(n/(n+1))y”whichgivestheupperboundc16/Cl-Y*)Wl/(l-Y*)>-y*/n’Theboundisprimarlyoforderl/rnexceptwheny*isnear0ornear1,i.e.,whenpisverysmallornearn. 38DAGJONSSONACKNOWLEDGMENTMyteacher,ProfessorCarl-GustavEsseen.introducedmetothetopicofthispaper.Iwishtothankhimforhisencouragingsupport,valuableadviceandconstructivecriticism.REFERENCES[I]ANDERSON,T.W.(1958).AnIntroductiontoMultivariateStatisticalAna@sis.Wiley,NewYork.121ARHAROV,L.V.(1971).Limittheoremsforthecharacteristicalrootsofasamplecovariancematrix.SovietMath.Dokl.12.1206-1209.[31[a]ARNOLD.L.(1967).Ontheasymptoticdistributionoftheeigenvaluesofrandommatrices.J.Math.Anal.Appl.20,262-268.[b]ARNOLD,L.(1971).OnWigner’sSemicircleLawfortheEigenvaluesofRandomMatrices.Z.Wahrsch.Verw.Gebiete19,191-198.14]CARMELI.M.(1974).Statisticaltheoryofenergylevelsandrandommatricesinphysics.J.Statist.Phys.10.259-297.15]CHUNG.K.L.(1968).ACourseinProbabilityTheory.Harcourt,Brace&World.NewYork.[G]ERDI?LYI,A.,MAGNUS,W..OBERHETTINGER,F.,ANDTRICOMI,F.(1953).HigherTranscendentalFunctions,I.McGraw-Hill,NewYork.(71[a]FELLER,W.(1968).AnIntroductiontoProbabilityTheoryandItsApplications,1.Wiley,NewYork.(blFELLER,W.(1971).AnIntroductiontoProbabilityTheoryandItsApplications,II.Wiley,NewYork.[81GRENANDER.~.(I963).ProbabilitiesonAlgebraicStructures.Almqvist&Wiksell.Stockholm.191GRENANDER.U.,ANDSILVERSTEIN.J.(1977).Spectralanalysisofnetworkswithrandomtopologies.SIAMJ.Appl.Math.32(2),499-519.IlO]JONSSON.D.(1976).Somelimittheoremsfortheeigenvaluesofasamplecovariancematrix.TechnicalReportNo.6,DepartmentofMathematics,UppsalaUniversity,Uppsala.II1IKARLIN,~.(1969).AFirstCourseinStochasticProcesses.AcademicPress,NewYork.[121KRISHNAIAH,P.R.(1978).Somerecentdevelopmentsonrealmultivariatedistributions.Develop.Statist.1.135-169.[131MAREENKO.V.A.,ANDPASTUR,L.A.(1967).Distributionsofeigenvalusofsomesetsofrandommatrices.Math.USSR-Sb.1,507-536.[141RAO,C.R.(1973).LinearStatisticalInferenceandItsApplications.Wiley,NewYork.[151WACHTER.K.W.(1978).Thestronglimitsofrandommatrixspectraforsamplematricesofindependentelements.Ann.Probab.6,1-18.[161[a]WIGNER,E.P.(1955).Characteristicvectorsofborderedmatriceswithinfinitedimensions.Ann.ofMath.62,548-564.[b]WIGNER,E.P.(1958).Onthedistributionsoftherootsofcertainsymmetricmatrices.Ann.ofMath.67,325-327.