Limiting_spectral_distribution_for_a_class_of_random_matrices.pdf

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1、JOURNALOFMULTIVARIATEANALYSIS20,5O68(1986)LimitingSpectralDistributionforaClassofRandomMatricesY.Q.YIN*CenterforMultivariateAnalysis,UniversityofPittsburghCommunicatedbytheEditorsLetX={X,:i,j=1,2,...}beaninfinitedimensionalrandommatrix,rpbeapxpnonnegativedefiniterandommatrixindependentofX,fo

2、rp=1,2,....Suppose(l/p)trTi+H,a.s.asp-+ccfork=1,2,...,andxH2;‘/2k0,underthemildconditionsthatX,‘sarei.i.d.andEx;?,

3、ONThespectraforrandommatricesoftheform(l/n)X,X;T,areimportantinmanyfields.ManyresultsareavailableforthespecialcasewhereT,=I,=identity(e.g.,seeGrenanderandSilverstein[4],Wachter[7],andYinandKrishnaiah[9]).InYinandKrishnaiah[S],thecasewhen7’,isanarbitrarypositivedefinitematrixwasinvestigatedfo

4、rthefirsttime.Inthatpaper,itwasassumedthattheentriesX,=[xv:i=l,...,p;j=l,...,n]arei.i.d.andnor-mal.Anewcombinatorialtechniquewasdevelopedinthatpapertoprovetheexistenceofthelimitingspectraldistribution.Theaboveworkcanbegeneralizedintwodirections.FirstwecangeneralizetothecasewhenX,hasisotropic

5、columns.ThisworkwasdoneinYinandKrishnaiah[9]andBai,Yin,andKrishnaiah[l].Intheseconddirection,wewouldprovetheresultbyassumingthatX,hasi.i.d.entrieswithmomentrequirementsasweakaspossible.Thepresentworkisdevotedtothisgoal.Inthispaper,wehavesucceededtoprovetheexistenceoflimitingspectraldistribut

6、ionbyassumingonlythatthesecondmomentexists.ThekeystoreachthisgoalareReceivedJuly15,1985;revisedDecember4,1985.AMS1980SubjectClassifications:Primary62HlO.*Presentaddress:DepartmentofMathematics,UniversityofArizona,Tucson,Arizona85721.500047-259X/86$3.00Copyright01986byAcademicPress.Inc.LIMITI

7、NGSPECTRALDISTRIBUTION51(1)truncationtechniqueand(2)sophisticatedcombinatorialtechniques.Thetwo-stagetruncationmethodworksinprovingthemainresult.Toprovethemainresult,wehavetogeneralizethenotionofQ-graphsintroducedinYinandKrishnaiah[S]toanewkindofgr