The Functional Central Limit Theorem and Weak. Convergence to Stochastic Integrals II: Fractionally. Integrated Processes1. James Davidson. Cardiff University.
7KH )XQFWLRQDO &HQWUDO /LPLW 7KHRUHP DQG :HDN &RQYHUJHQFH WR 6WRFKDVWLF ,QWHJUDOV ,, )UDFWLRQDOO\ ,QWHJUDWHG 3URFHVVHV -DPHV 'DYLGVRQ &DUGLII 8QLYHUVLW\
5REHUW 0 GH -RQJ 0LFKLJDQ 6WDWH 8QLYHUVLW\ )LQDO 5HYLVLRQ 0D\ $EVWUDFW 7KLV SDSHU GHULYHV D IXQFWLRQDO FHQWUDO OLPLW WKHRUHP IRU WKH SDUWLDO VXPV RI IUDFWLRQDOO\ LQWHJUDWHG SURFHVVHV RWKHUZLVH NQRZQ DV L+g, SURFHVVHV IRU mgm ? 4@5 6XFK SURFHVVHV KDYH ORQJ PHPRU\ DQG WKH OLPLW GLVWULEXWLRQ LV WKH VRFDOOHG IUDFWLRQDO %URZQLDQ PRWLRQ KDYLQJ FRUUHODWHG LQFUHPHQWV HYHQ DV\PSWRWLFDOO\ 7KH XQGHUO\LQJ VKRFN YDULDEOHV PD\ WKHPVHOYHV H[KLELW TXLWH JHQHUDO ZHDN GHSHQGHQFH E\ EHLQJ QHDUHSRFK GHSHQGHQW IXQFWLRQV RI PL[LQJ SURFHVVHV 6HYHUDO ZHDN FRQYHUJHQFH UHVXOWV IRU VWRFKDVWLF LQWHJUDOV KDYLQJ IUDFWLRQDO LQWHJUDQGV DQG ZHDNO\ GHSHQGHQW LQWHJUDWRUV DUH DOVR REWDLQHG 7DNHQ WRJHWKHU WKHVH UHVXOWV SHUPLW L+s . g, LQWHJUDQGV IRU DQ\ LQWHJHU s 4
,QWURGXFWLRQ
,Q 'H -RQJ DQG 'DYLGVRQ ZH REWDLQ IXQFWLRQDO OLPLW UHVXOWV IRU D EURDG FODVV RI VHUL DOO\ GHSHQGHQW DQG KHWHURJHQHRXVO\ GLVWULEXWHG YHFWRU SURFHVVHV ZKLFK KRZHYHU DUH ZHDNO\ GHSHQGHQW RWKHUZLVH FKDUDFWHULVHG DV µVKRUW PHPRU\¶ 7KH GH¿QLQJ IHDWXUH RI VXFK SURFHVVHV LV WKDW WKHLU QRUPDOLVHG SDUWLDO VXPV FRQYHUJH WR SURFHVVHV KDYLQJ LQGHSHQGHQW *DXVVLDQ LQFUH PHQWV VSHFL¿FDOO\ %URZQLDQ PRWLRQ LQ WKH FDVH ZKHUH WKH YDULDQFHV DUH XQLIRUPO\ ERXQGHG DZD\ IURP LQ¿QLW\ DQG ]HUR 7KH SUHVHQW SDSHU H[WHQGV WKHVH UHVXOWV E\ DOORZLQJ WKH SURFHVVHV WR H[KLELW ORQJ PHPRU\ 6SHFL¿FDOO\ ZH FRQVLGHU IUDFWLRQDOO\ LQWHJUDWHG SURFHVVHV RWKHUZLVH NQRZQ DV UE_ SURFHVVHV 7KH FKLHI LQWHUHVW LQ WKLV FODVV RI SURFHVVHV LV WKDW WKH\ GH¿QH D FRQWLQXXP OLQNLQJ WKH VWD WLRQDU\ VKRUWPHPRU\ FDVH _ ' f ZLWK WKH LQWHJUDWHG RU XQLW URRW FDVH _ ' 7KH OLPLW SURFHVVHV IRU WKH SDUWLDO VXPV RI WKHVH YDULDEOHV IRU *2 _ *2 DUH WKH VRFDOOHG IUDF WLRQDO %URZQLDQ PRWLRQV ZKLFK GLIIHU IURP RUGLQDU\ %URZQLDQ PRWLRQ LQ KDYLQJ FRUUHODWHG 4
:H WKDQN %UXFH +DQVHQ 3HWHU 3KLOOLSV DQG WZR DQRQ\PRXV UHIHUHHV IRU WKHLU FRPPHQWV RQ HDUOLHU YHUVLRQV RI WKLV SDSHU $Q\ HUURUV DUH RXUV DORQH
LQFUHPHQWV )RU VLPLODU UHVXOWV VHH DPRQJ RWKHU UHIHUHQFHV 'DY\GRY 7DTTX &KDQ DQG 7HUULQ &V|UJÝR DQG 0LHOQLF]XN &KXQJ 5RELQVRQ DQG 0DULQ XFFL 7KH PDLQ QRYHO IHDWXUH RI WKH SUHVHQW UHVXOWV LV WKDW WKH VKRFN YDULDEOHV IUDFWLRQDO GLIIHUHQFHV DUH SHUPLWWHG WR EH QHDUHSRFK GHSHQGHQW RQ D PL[LQJ SURFHVV D YHU\ JHQHUDO IRUP RI ZHDN GHSHQGHQFH DOORZLQJ YDULRXV IRUPV RI QRQOLQHDU G\QDPLFV VHH 'DYLGVRQ :H DOVR SURYH VWRFKDVWLF LQWHJUDO FRQYHUJHQFH IRU IUDFWLRQDOO\ LQWHJUDWHG SURFHVVHV ZLWK UHVSHFW WR ZHDNO\ GHSHQGHQW LQWHJUDWRU SURFHVVHV 7KHVH UHVXOWV DUH XVHIXO LQ IRU H[DPSOH WKH DQDO\VLV RI FRLQWHJUDWHG UHJUHVVLRQV ZKHUH WKH REVHUYHG YDULDEOHV DUH UE n _ SURFHVVHV ZKHUHDV WKH UHVLGXDOV DUH VKRUW PHPRU\ SURFHVVHV 7KH\ DUH VWUDLJKWIRUZDUGO\ H[WHQGHG WR FDVHV ZKHUH WKH YDULDEOHV DUH UER n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µZHDN GHSHQGHQFH¶ DQG ZLOO DOVR UHIHU WR WKHRUHPV DQG HTXDWLRQV RI WKH SDSHU E\ DWWDFKLQJ WKH SUH¿[ :' WR WKH UHIHUHQFH
)UDFWLRQDOO\ ,QWHJUDWHG 3URFHVVHV
7KH FODVV RI SURFHVVHV ZH FRQVLGHU DUH FXVWRPDULO\ ZULWWHQ LQ WKH IRUP %| ' E u3_ | c
ZKHUH u LV WKH ODJ RSHUDWRU *2 _ *2 DQG | c WKH IUDFWLRQDO GLIIHUHQFH RI %| c LV D VWDWLRQDU\ ZHDNO\ GHSHQGHQW SURFHVV WR EH VSHFL¿HG %\ WKH REYLRXV ELQRPLDO H[SDQVLRQ WKH\ KDYH WKH 0$ UHSUHVHQWDWLRQ " [ %| ' K |3 'f
ZKHUH
KE n _ KE_KE n *UDQJHU DQG -R\HX[ DQG +RVNLQJ DUH VWDQGDUG UHIHUHQFHV RQ WKHVH SURFHVVHV 6WLUOLQJ¶V DSSUR[LPDWLRQ IRUPXOD IRU WKH JDPPD IXQFWLRQ \LHOGV WKH ZHOONQRZQ SURSHUW\ WKDW WKH 0$ FRHI¿FLHQWV GHFOLQH K\SHUEROLFDOO\ WR ]HUR DQG OHWWLQJ % + GHQRWH WKDW % *+ $ DV $ 4 ZH FDQ ZULWH _3 K KE_ 7KH K DUH WKHUHIRUH VTXDUHVXPPDEOH IRU _ *2c ZKLFK LV WKH FRQGLWLRQ QHFHVVDU\ IRU WKH SURFHVVHV WR EH VWDWLRQDU\ ZLWK ¿QLWH YDULDQFH ZKHUHDV _ : *2 LV QHFHVVDU\ IRU WKH SURFHVV WR KDYH DQ S LQYHUWLEOH 0$ UHSUHVHQWDWLRQ²VHH +RVNLQJ IRU GHWDLOV 7KH SDUWLDO VXPV GH¿QHG DV r|' %| IRU r : f UHSUHVHQW D XVHIXO FODVV RI PRGHOV IRU QRQVWDWLRQDU\ VHULHV FRQ WDLQLQJ WKH SRSXODU XQLW URRW FDVH FRUUHVSRQGLQJ WR _ ' f LQ WKLV VHWXS 7KH\ DUH RI SDUWLFXODU LQWHUHVW ZKHQ *2 _ f VLQFH DOWKRXJK QRQVWDWLRQDU\ ZLWK LQ¿QLWH YDULDQFH WKH\ KDYH WKH K '
SURSHUW\ RI HYHQWXDO LQGHSHQGHQFH RI LQLWLDO FRQGLWLRQV ,W LV EHVW WR WKLQN RI WKH FRUUHVSRQGLQJ %| VHULHV DV EHLQJ JHQHUDWHG DV WKH VLPSOH GLIIHUHQFHV RI WKHVH QRQVWDWLRQDU\ VHULHV DQG VR WR H[KLELW D JHQHUDOLVHG S IRUP RI RYHUGLIIHUHQFLQJ 'H¿QLQJ j 2? ' .E ?|' %| 2 c FRQVLGHU WKH VFDOHG SDUWLDO VXP SURFHVV f? E1 '
j 3 ?
d?1o [
%| c
|'
ZKHUH %| LV GH¿QHG E\ ZLWK m_m *2 :H VKRZ WKDW XQGHU DSSURSULDWH FRQGLWLRQV _ f? $ fc ZKHUH f LV D IUDFWLRQDO %URZQLDQ PRWLRQ GH¿QHG IRU _ 5 E*2c *2 E\ ] 1 fE1 ' E1 r_ _Er *2 KE_ n T_ f ] f _ _ n E1 r Er _Er 3"
IRU f 1 +HUH LV VWDQGDUG %URZQLDQ PRWLRQ DQG ] " _ _ 2 n T_ ' EE n _ c KE_ n 2 2_ n f
WKLV VFDOH FRQVWDQW EHLQJ FKRVHQ WR PDNH .fE2 ' 6HH 0DQGHOEURW DQG YDQ 1HVV IRU DGGLWLRQDO GHWDLOV 1RWH WKDW f ' ZKHQ _ ' f )RU WKH RWKHU FDVHV WKHVH SURFHVVHV KDYH FRUUHODWHG LQFUHPHQWV SRVLWLYHO\ FRUUHODWHG ZKHQ _ : f DQG QHJDWLYHO\ FRUUHODWHG RWKHUZLVH 7KXV LW LV HDVLO\ YHUL¿HG IURP WKH GH¿QLWLRQ LQ WKDW .EfE1 n B fE12 ' B 2_n
IRU 1 5 dfc DQG f B 1 DQG KHQFH WKDW IRU H[DPSOH E1 n B2_n 1 2_n B 2_n .EfE1EfE1 n B fE1 ' 2 7KHUH DUH D QXPEHU RI H[WHQVLRQV LPSOLFLW LQ RXU UHVXOWV ZKLFK ZH PHQWLRQ KHUH DOWKRXJK WKH GHWDLOV PXVW EH OHIW WR IXWXUH ZRUN 6WDWLRQDULW\ RI WKH IUDFWLRQDO GLIIHUHQFHV HQVXUHV WKDW WKH OLPLW SURFHVV LV DQG ZH ZLOO PDLQWDLQ WKLV DVVXPSWLRQ LQ WKH VHTXHO EXW LW LV QRW D SUHUHT XLVLWH IRU ZHDN FRQYHUJHQFH DV VXFK ,Q :' ZH SUHVHQW UHVXOWV XQGHU ZHDN GHSHQGHQFH ZKLFK DOORZ JOREDO QRQVWDWLRQDULW\ ZLWK WUHQGV LQ WKH YDULDQFHV RI WKH SURFHVV 7KH OLPLW SURFHVVHV LQ VXFK FDVHV DUH QRW %URZQLDQ PRWLRQ EXW WUDQVIRUPDWLRQV RI %URZQLDQ PRWLRQ LQYROYLQJ VRPH VTXHH]LQJ DQG VWUHWFKLQJ RI WKH WLPH GRPDLQ &RPELQLQJ JOREDO QRQVWDWLRQDULW\ DQG VWURQJ GH SHQGHQFH FRXOG \LHOG DQ HQODUJHG FODVV RI DV FRQWLQXRXV *DXVVLDQ OLPLW SURFHVVHV H[WHQGLQJ $Q LPSRUWDQW IHDWXUH RI WKH UHVXOWV LV WKDW IRU WKH FDVH f _ *2c WKH\ JHQHUDOLVH LQ SULQFLSOH EH\RQG WKH IUDFWLRQDO PRGHO 7KH JHQHUDO FODVV RI ORQJ PHPRU\ 0$ SURFHVVHV LQ ZKLFK WKH FRHI¿FLHQWV PHUHO\ VDWLVI\ K uE _3
ZKHUH uE GHQRWHV DQ\ VORZO\ YDU\LQJ FRPSRQHQW KDYH HVVHQWLDOO\ WKH VDPH DV\PSWRWLF SURS
5
)RU D UHFHQW DSSOLFDWLRQ RI WKLV W\SH RI PRGHO VHH %\HUV 'DYLGVRQ DQG 3HHO 2XU IRUPXOD GLIIHUV IURP 0DQGHOEURW DQG YDQ 1HVV HTXDWLRQ VLQFH WKH\ GR QRW QRUPDOLVH WKH YDULDQFH RI [+4, WR XQLW\ :H DOVR LPSOLFLWO\ LPSRVH WKH FRQGLWLRQ [+3, @ 3 DV 7 $ VHTXHQFH LV VDLG WR EH VORZO\ YDU\LQJ DW 4 LI LW VDWLV¿HV O+{m,@O+m, $ 4 DV m $ 4> IRU DQ\ { A 3 6
HUWLHV DV WKH IUDFWLRQDO SURFHVV 7KH PRGL¿FDWLRQV WR WKH SURRIV WR LQFOXGH D VORZO\ YDU\LQJ FRPSRQHQW DUH VWUDLJKWIRUZDUG LQ SULQFLSOH 0RUHRYHU JLYHQ D QRQVWDWLRQDU\ SURFHVV RI WKLV JHQHUDO 0$ IRUP ZLWK *2 _ c LWV VLPSOH GLIIHUHQFH KDV WKH VDPH DV\PSWRWLF SURSHUWLHV DV WKH IUDFWLRQDO PRGHO ZLWK *2 _ f LQFOXGLQJ RI FRXUVH WKDW WKH SDUWLDO VXPV FRQYHUJH WR IUDFWLRQDO %URZQLDQ PRWLRQ +RZHYHU EH HTXDOO\ FDUHIXO WR QRWH WKDW WKH QHJDWLYH IUDFWLRQDO PRGHO EHKDYHV GLIIHUHQWO\ IURP WKH JHQHUDO FDVH RI ZLWK *2 _ f VLQFH LW KDV WKH GLVWLQFWLYH µRYHUGLIIHUHQFLQJ¶ SURSHUW\ WKDW WKH 0$ FRHI¿FLHQWV VXP WR ]HUR 7KH ODWWHU SURS HUW\ LV QRW D IHDWXUH E\ WKH JHQHUDO FODVV RI 0$ SURFHVVHV ZLWK VXPPDEOH FRHI¿FLHQWV DQG LI WKH SDUWLDO VXPV RI WKHVH SURFHVVHV FRQYHUJH LW LV WR RUGLQDU\ %URZQLDQ PRWLRQ /LPLW UHVXOWV RI WKH ODWWHU VRUW PD\ EH REWDLQHG DV FRUROODULHV RI WKH WKHRUHPV RI WKLV SDSHU EXW ZH OLNHZLVH GR QRW SXUVXH WKHVH H[WHQVLRQV KHUH
$Q )&/7 IRU )UDFWLRQDOO\ ,QWHJUDWHG 3URFHVVHV
)ROORZLQJ WKH DSSURDFK RI 'DY\GRY ZH QRWH WKDW DIWHU VXEVWLWXWLQJ %| ' DQG VXPPLQJ WKH WHUPV LQ D GLIIHUHQW RUGHU ZH DUH DEOH WR ZULWH
f? E1 f? E1 '
d?1o [
j 3 ?
S"
'f K |3
LQ
@?| E1c 1 |
K
|'3"
IRU 1 : 1 c ZKHUH
d?1o3|
[
@?| E1c 1 '
'4@ tfcd?1 o3|n
:KHQ K KDV WKH IRUP LQ WKLV GHFRPSRVLWLRQ KDV WKH IROORZLQJ SURSHUWLHV /HPPD D
,I 1 % 1 WKHQ @?cd?%o E1c 1
Ed?1o d?%o_ KE_ n
DQG LI 4 % 1 WKHQ @?cd?%o E1c 1 E KHQFH
d?1o [
Ed?1o d?%o_ Ed?1 o d?%o_ KE_ n @?| E1c 1 2 T_ E?E1 1 2_n
|'3"
6HH $SSHQGL[ % IRU SURRIV IRU WKLV VHFWLRQ 7KH IROORZLQJ DVVXPSWLRQ IRUPV WKH EDVLV RI RXU IXQFWLRQDO OLPLW UHVXOWV $VVXPSWLRQ 7KH VHTXHQFH i| c 4 | 4j D KDV ]HUR PHDQ E LV XQLIRUPO\ uo ERXQGHG IRU o : 2 F LV u2 1(' RI VL]H *2 RQ T| ZLWK _| ' ZKHUH T| LV HLWKHU DQ kPL[LQJ VHTXHQFH RI VL]H
o*Eo 2 RU D PL[LQJ VHTXHQFH RI VL]H o*E2Eo G LV FRYDULDQFH VWDWLRQDU\ DQG f j 2 4 ZKHUH ? ? [ [ 3 2 j ' *4 ? .E| r ? DV LQ 7KHRUHP S S 9 )RU D YHFWRU d ndns KHUH GHQRWHV l ndl ns $OVR IRU D PDWUL[ D nDns GHQRWHV l m nDlm ns
ZKHUH T| r LV D LLG VXEYHFWRU RI T| DQG WKH X ^ r DUH PDWULFHV RI FRHI¿FLHQWV |n6 3 *23_30 /HWWLQJ IRU 0 : f 6LQFH | .|36 | ' S" w ' 4@ c& mEX & mc VXSSRVH WKDW w ' E '6n X Tc|3 LQ WKLV FDVH ZH REWDLQ # " $*2 [ | . |n6 | w2 |36
2
'6n 33_30
' E6
IRU ' ^r 4@ $&$r n|& n2 c VR WKDW | LV u2 1(' RI VL]H _ RQ iT| j DQG WKHUHIRUH D u RI VL]H _c E\ 7KHRUHP RI 'DYLGVRQ 0RUHRYHU VLQFH .| |n ' S2 PL[LQJDOH " 'f Xn Tc|3 ZH KDYH $*2 # " [ .| |n . |n6 .| |n w2 |36
2
'6nn
' EE6 n 33_30 2EVHUYH WKDW
" [ 'f
#
" [
$*2 w 2
' E63_30 c
'6nn
VR WKDW KROGV ZLWK l E6 JLYHQ E\ $ SRLQW RI LQWHUHVW DERXW 7KHRUHP LV WKDW WKH SURRI LV QRW YDOLG IRU QHJDWLYH _ 7KLV LV DSSDUHQW RQ FRPSDULQJ H[SUHVVLRQV & DQG & RI WKH SURRI %RWK WKLV UHVXOW DQG 7KHRUHP DUH QHHGHG WR FRYHU WKH FRPSOHWH VSHFWUXP RI FDVHV 7KHUH LV QR DSSDUHQW REVWDFOH WR VKRZLQJ MRLQW FRQYHUJHQFH IRU D YHFWRU FRQWDLQLQJ ERWK SRVLWLYH DQG QHJDWLYH IUDFWLRQDOV SURYLGHG WKH SURFHVV E| c | VDWLV¿HV WKH FRQGLWLRQV RI ERWK UHVXOWV +RZHYHU ZH IRUHJR WKH GHWDLOV KHUH $ SRVVLEO\ XQGHVLUDEOH IHDWXUH RI 7KHRUHP LV WKH DGDSWDWLRQ UHTXLUHPHQW ZKLFK LV LP SRVHG XQGHU WKH WKHRUHP RI .XUW] DQG 3URWWHU RQ ZKLFK LW LV EDVHG 7KLV LPSOLHV WKDW WKH SURFHVVHV | DQG | PXVW QRW GHSHQG RQ IXWXUH YDOXHV RI WKH XQGHUO\LQJ SURFHVV T| :KLOH WKLV LV TXLWH D QRUPDO IHDWXUH RI G\QDPLF HFRQRPHWULF PRGHOV VR WKDW WKH DVVXPSWLRQ QHHG QRW EH UHVWULFWLYH LQ SUDFWLFH LW VKRXOG EH QRWHG WKDW LW LV QRW HVVHQWLDO ,Q RWKHU ZRUGV ZKLOH WKH OLPLWLQJ SURFHVV ` PXVW EH D PDUWLQJDOH ZLWK UHVSHFW WR D ¿OWUDWLRQ WR ZKLFK f PXVW DOVR EH DGDSWHG WKLV QHHG QRW LPSO\ WKDW f?| DQG `?| DUH K| PHDVXUDEOH IRU ¿QLWH ? 2QH FDQ DYRLG WKH ODWWHU FRQGLWLRQ E\ REWDLQLQJ WKH OLPLWLQJ PDUWLQJDOH WKURXJK D EORFNLQJ DUJXPHQW DV XVHG IROORZLQJ &KDQ DQG :HL LQ 7KHRUHPV :' DQG +RZHYHU WR FRPELQH WKH &KDQ:HL DSSURDFK ZLWK WKH +DQVHQW\SH DUJXPHQW UHTXLUHV DQ DV ERXQGHGQHVV UHVWULFWLRQ RQ | ZKLFK IRU SUDFWLFDO DSSOLFDWLRQV PD\ EH OHVV DWWUDFWLYH WKDQ DGDSWDWLRQ :H WKHUHIRUH JLYH WKLV UHVXOW DV DQ DOWHUQDWLYH WR 7KHRUHP 7KHRUHP /HW WKH DVVXPSWLRQV RI 7KHRUHP KROG H[FHSW WKDW | DQG | QHHG QRW EH DGDSWHG WR K| c EXW m| m 4 DQG WKH VHTXHQFH i.| |n c j LV VXPPDEOH ZLWK SUREDELOLW\ 7KHQ KROGV )LQDOO\ ZH SRLQW RXW WKDW WKHVH UHVXOWV FDQ EH H[WHQGHG WR FDVHV ZKHUH WKH LQWHJUDQG LV S S| _ U1 UE n R n _ IRU DOO SRVLWLYH LQWHJHUV R 1RWLQJ WKDW ?3 d?1o r' f?r $ f fEr_r E\ WKH |' FRQWLQXRXV PDSSLQJ WKHRUHP ZH KDYH WKH IROORZLQJ UHVXOW ZKLFK IROORZV E\ 7KHRUHP L
RI &KDQ DQG :HL 6ROHO\ IRU HDVH RI H[SRVLWLRQ ZH JLYH WKH VFDODU FDVH RI WKH UHVXOW DV IROORZV 7KHRUHP /HW WKH DVVXPSWLRQV RI 7KHRUHP KROG LQ WKH FDVH _ fc DQG WKRVH RI HLWKHU 7KHRUHP RU 7KHRUHP LQ WKH FDVH _ : f /HW `?| ' ?3*2 | DQG OHW " r [ | [ [ f?| ' Rn_n*2 K, rf 3, ? r ' r ' R
f
,'f
IRU R 5 Q 7KHQ LI f? `? DQG C? DUH GH¿QHG DV EHIRUH ] _ f` f? c `? c C? \? $ fc `c f _` f
ZKHUH
] f E1 R '
1R f
]
1 f
~E1 f _1 f _1 R3
LV WKH RIROG LQWHJUDO RI D IUDFWLRQDO %URZQLDQ PRWLRQ ~ DQG ` \f ?
'
?Rn_n
| ?3 [ [
|' rR '
" r [ [
K, .Erf 3, |n
rf ' ,'f
&RQFOXVLRQ
$SSO\LQJ WHFKQLTXHV GHYHORSHG LQ 'H -RQJ DQG 'DYLGVRQ LQ WKLV SDSHU ZH REWDLQ ZHDN FRQYHUJHQFH UHVXOWV IRU WKH IUDFWLRQDOO\ LQWHJUDWHG FODVV RI ORQJ PHPRU\ SURFHVVHV LQ ZKLFK WKH OLPLW SURFHVVHV DUH IUDFWLRQDO %URZQLDQ PRWLRQ 7KH IXQFWLRQDO FHQWUDO OLPLW WKHRUHP LV SURYHG DVVXPLQJ WKDW WKH IUDFWLRQDO GLIIHUHQFHV RI WKH SURFHVV DUH QHDUHSRFK GHSHQGHQW RQ D PL[LQJ SURFHVV :H DOVR SURYH WKH ZHDN FRQYHUJHQFH RI VWRFKDVWLF LQWHJUDOV KDYLQJ WKHVH SURFHVVHV DV LQWHJUDQGV XQGHU D UDQJH RI FROOHFWLYHO\ PLOG GHSHQGHQFH UHVWULFWLRQV %\ FRPELQLQJ RXU UH VXOWV ZLWK WKH FRQWLQXRXV PDSSLQJ WKHRUHP DSSOLHG WR SDUWLDO VXPV LW LV SRVVLEOH WR FKDUDFWHULVH WKHVH ZHDN OLPLWV IRU QRUPDOLVHG UE n _ SURFHVVHV IRU DQ\ _ : *2
$SSHQGL[ $ 6RPH 7HFKQLFDO /HPPDV /HPPD $ ,I @ K fc WKHQ
; ? E @nKn @ : [ @ K & E & ' E K *L} @ ' = E K @ &' 3
3URRI )RU WKH FDVH @ : WKLV IROORZV E\ DQ LQWHJUDO DSSUR[LPDWLRQ XVLQJ WKH %HWD IXQFWLRQ DQG IRU WKH RWKHU FDVHV E\ HOHPHQWDU\ VXPPDELOLW\ DUJXPHQWV 7KH IROORZLQJ OHPPDV H[WHQG /HPPDV :'$ DQG :'$ /HPPD $ LV VWDWHG IRU DUELWUDU\ FRHI¿FLHQWV K& c DQG KDV DQ DSSOLFDWLRQ WR WKH IUDFWLRQDO PRGHO LQ WKH FDVH _ f 1RWH WKDW IRU _ : f WKH VXP LQ $ GLYHUJHV
/HPPD $ /HW if?| c J?| j DQG it?| c J?| j EH WULDQJXODU u2 PL[LQJDOH DUUD\V RI VL]H *2 t ZLWK PL[LQJDOH PDJQLWXGH LQGLFHV @f ?| DQG @?| UHVSHFWLYHO\ 7KHQ D # ? $*2 # ? $*2 ? ? [ [ [ [ f 2 t 2 m.Ef?| t?r m E@?| E@?| |'3" r'3"
|'3"
|'3"
IRU : f S S? 2 t 2 E ,I ?|'3" E@f ?| ' E DQG |'3" E@?| ' Ec DQG ? LV DQ LQFUHDVLQJ LQWHJHU YDOXHG IXQFWLRQ RI ? ZLWK ? $ 4 DV ? $ 4 WKHQ ? ? [ [ m.Ef?| t?r mUEm| rm : ? ' f *4 ? SHUPLVVDEOH XQGHU RXU VWDWLRQDULW\ DVVXPSWLRQ DQG s @ u= ,Q WKLV FDVH VHW s @ @ 5 LQ +DQVHQ¶V QRWDWLRQ
FRQVWDQWV c 2 : f DQG 0 : fc n.|36 %| 5| .%| 5| n
?3| [ K& .|36 E|3& .| |n .|3& |n ' &'f " ?3| [ [ n2 K& |3& n.| |n n2 ' &'6n 2 # " $*2 " 6 [ [ [ 2 K& E6 & n 2 K& lE 6 [
&'f
' E6 n 6_3*2 c
&'6n
'
30
&
ZKHUH lE DQG E& DUH WKH PL[LQJDOH QXPEHUV IRU i| c K| j DQG WKH SURFHVV LQ & UH VSHFWLYHO\ 7KH ¿UVW RI WKH LQHTXDOLWLHV LQ & XVHV WKH 0LQNRZVNL -HQVHQ DQG &DXFK\ 6FKZDU] LQHTXDOLWLHV 7KH VHFRQG LQHTXDOLW\ DSSOLHV WKH PL[LQJDOH DVVXPSWLRQV DQG /HPPD $D SXWWLQJ t?r ' f?r ' K|3r r 7KH ¿UVW RUGHURIPDJQLWXGH WHUP LV HYDOXDWHG XVLQJ /HPPD $ DQG WKH VHFRQG IROORZV E\ DVVXPSWLRQ 7KH VHTXHQFH i%| 5| c K| j LV WKHUHIRUH D VWDWLRQDU\ u PL[LQJDOH DQG DFFRUGLQJO\ XQLIRUPO\ LQWHJUDEOH ,W IROORZV E\ WKH ZHDN ODZ RI ODUJH QXPEHUV RI 'DYLGVRQ RU VHH 'DYLGVRQ 7KHRUHP WKDW ?3 [ R 3 ? E%| 5| .%| 5| $ f & |'
` FRPSOHWHV WKH SURRI 1RWLQJ WKDW .E2? ' \f ?
&
3URRI RI 7KHRUHP
$V EHIRUH DVVXPH %| DQG | DUH VFDODUV 7KH ¿UVW SDUW RI WKH DUJXPHQW IROORZV WKH VDPH PRGL¿ R FDWLRQ RI 7KHRUHP :' DV XVHG LQ 7KHRUHP ,W UHPDLQV WR VKRZ WKDW mC? CW? \L? ` m $ fc VLPLODUO\ WR H[SUHVVLRQ :' 8VLQJ WKH GHFRPSRVLWLRQ LQ & DQG QRWLQJ WKDW ? 3
[
? 3
| [
%r E5| 5|n '
|'? 3 n r'? 3 n
ZH FDQ ZULWH ZKHUH
? 3
[
%| 5|
|'? 3 n
C? CW? ' ? n 2? ?
[
% | 5 ? c
&
|'? 3
&
|3?3 3
? ' ?
33_
2? ' ?33_
[
%|36 "|n c
6'f ? 3 &? [ [
&
%| 5|
&
% | 5 ?
&
' |'?3 n
DQG ? ' ?
33_
3 &? ?[ [
' |'? 3
1RZ E\ WKH DVVXPSWLRQ RQ | m"| m 4 DV DQG E\ +|OGHU¶V LQHTXDOLW\ #|3?3 3 $2 #|3?3 3 $2 [ [ ."2|n %|36 n"|n n2" . %|36 6'f
&
6'f
7KHUHIRUH LQ YLHZ RI WKH PDUWLQJDOH GLIIHUHQFH SURSHUW\ RI "| c .E2?
?
3
? 3
&? [
[
2 . f? E|*? f? E1 3
' |'?3 n
?
3
'
&? [
? E1 1 3 2_n
'
4@
$$&?
2_n
E1 1 3
' JE
&
IRU FRQVWDQW : f 7KH SURRI LV FRPSOHWHG E\ DUJXPHQWV VLPLODU WR WKRVH XVHG IRU 7KHRUHP QRWLQJ WKDW WKH WHUP 2? .E2? FRQYHUJHV LQ SUREDELOLW\ WR f WKDW .m ? m ?3_ DQG WKDW m.E2? \f` m $ f ?
5HIHUHQFHV %\HUV ' - 'DYLGVRQ DQG ' 3HHO 0RGHOOLQJ SROLWLFDO SRSXODULW\ DQ DQDO\VLV RI ORQJ UDQJH GHSHQGHQFH LQ RSLQLRQ SROO GDWD -RXUQDO RI WKH 5R\DO 6WDWLVWLFDO 6RFLHW\ 6HULHV $ &KDQ 1 + DQG 1 7HUULQ ,QIHUHQFH IRU XQVWDEOH ORQJ PHPRU\ SURFHVVHV ZLWK DSSOLFD WLRQV WR IUDFWLRQDO XQLW URRW DXWRUHJUHVVLRQV $QQDOV RI 6WDWLVWLFV &KDQ 1+ DQG &= :HL /LPLWLQJ GLVWULEXWLRQV RI OHDVWVTXDUHV HVWLPDWHV RI XQVWDEOH DXWRUHJUHVVLYH SURFHVVHV $QQDOV RI 6WDWLVWLFV &V|UJÝR 6 DQG - 0LHOQLF]XN 'LVWDQW ORQJUDQJH GHSHQGHQW VXPV DQG UHJUHVVLRQ HVWLPD WLRQ 6WRFKDVWLF 3URFHVVHV DQG WKHLU $SSOLFDWLRQV &KXQJ &) 6DPSOH YDULDQFHV VDPSOH FRYDULDQFHV DQG OLQHDU UHJUHVVLRQ RI VWDWLRQDU\ PXOWLYDULDWH ORQJ PHPRU\ SURFHVVHV :RUNLQJ SDSHU 1DWLRQDO 7DLZDQ 8QLYHUVLW\ 'DYLGVRQ - $Q u FRQYHUJHQFH WKHRUHP IRU KHWHURJHQHRXV PL[LQJDOH DUUD\V ZLWK WUHQG LQJ PRPHQWV 6WDWLVWLFV DQG 3UREDELOLW\ /HWWHUV 'DYLGVRQ - 6WRFKDVWLF /LPLW 7KHRU\ 2[IRUG 2[IRUG 8QLYHUVLW\ 3UHVV 'DYLGVRQ - :KHQ LV D WLPH VHULHV , " (YDOXDWLQJ WKH PHPRU\ SURSHUWLHV RI QRQOLQHDU G\QDPLF PRGHOV :RUNLQJ 3DSHU UHYLVHG 'DY\GRY