models of time series-state

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1、ChapterModelsofTimeSeries2EachtimeseriesY1;:::;Yncanbeviewedasaclippingfromasequenceofrandomvariables:::;Y2;Y1;Y0;Y1;Y2;:::InthefollowingwewillintroduceseveralmodelsforsuchastochasticprocessYtwithindexsetZ.2.1LinearFiltersandStochasticProcessesFormat

2、hematicalconveniencewewillconsidercomplexvaluedran-domvariablesY,whoserangeisthesetofcomplexnumbersC=pfu+iv:u;v2Rg,wherei=1.Therefore,wecandecomposeYasY=Y(1)+iY(2),whereY(1)=Re(Y)istherealpartofYandY(2)=Im(Y)isitsimaginarypart.TherandomvariableYiscall

3、edintegrableiftherealvaluedrandomvariablesY(1);Y(2)bothhaveniteexpectations,andinthiscasewedenetheexpectationofYbyE(Y):=E(Y(1))+iE(Y(2))2C:Thisexpectationhas,uptomonotonicity,theusualpropertiessuchasE(aY+bZ)=aE(Y)+bE(Z)ofitsrealcounterpart(seeExercis

4、e2.1).HereaandbarecomplexnumbersandZisafurtherintegrablecom-plexvaluedrandomvariable.InadditionwehaveE(Y)=E(Y),wherea=uivdenotestheconjugatecomplexnumberofa=u+iv.Sincejaj2:=u2+v2=aa=aa,wedenethevarianceofYbyVar(Y):=E((YE(Y))(YE(Y)))0:Thecomple

5、xrandomvariableYiscalledsquareintegrableifthisnumberisnite.TocarrytheequationVar(X)=Cov(X;X)fora48ModelsofTimeSeriesrealrandomvariableXovertocomplexones,wedenethecovarianceofcomplexsquareintegrablerandomvariablesY;ZbyCov(Y;Z):=E((YE(Y))(ZE(Z))):Not

6、ethatthecovarianceCov(Y;Z)isnolongersymmetricwithre-specttoYandZ,asitisforrealvaluedrandomvariables,butitsatisesCov(Y;Z)=Cov(Z;Y).ThefollowinglemmaimpliesthattheCauchy{Schwarzinequalitycar-riesovertocomplexvaluedrandomvariables.Lemma2.1.1.Foranyintegr

7、ablecomplexvaluedrandomvariableY=Y(1)+iY(2)wehavejE(Y)jE(jYj)E(jY(1)j)+E(jY(2)j):Proof.WewriteE(Y)inpolarcoordinatesE(Y)=rei#,wherer=jE(Y)jand#2[0;2).Observethati#Re(eY)=Re(cos(#)isin(#))(Y(1)+iY(2))=cos(#)Y(1)+sin(#)Y(2)221=2221=2(cos(#)+sin(#

8、))(Y+Y)=jYj(1)(2)bytheCauchy{Schwarzinequalityforrealnumbers.Thusweobtaini#jE(Y)j=r=E(eY)i#=ERe(eY)E(jYj):ThesecondinequalityofthelemmafollowsfromjYj=(Y2+Y2)1=2(1)(2)jY(1)j+jY(2)j:Thenextresultisaconsequenceoftheprecedinglemmaandt