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Wlph Æ Æ Æ Zrunhuv fkrrvh Ilupv r>hu Ilupv dvvljq wkhlu h^ ru h(cid:252) zdjh vfkhgxohv zrunhuv wr wdvnv Iljxuh 4= Wkh Wlph Olqh Vwdjh 7 +Zdjh R>huv,1 Wkh ?upv frpshwh lq wkh oderu pdunhw iru zrunhuv e| vlpxowd0 qhrxvo| dqqrxqflqj zdjh vfkhgxohv1 Wkh zdjh vfkhgxohv pd| ghshqg rq wkh zrunhu*v hgx0 fdwlrqdo dwwdlqphqw/ dv zhoo dv rq wkh whvw vljqdo º= D sxuh vwudwhj| ri ?up l dw wklv vwdjh lv (cid:253) (cid:252) dphdvxudeoh ixqfwlrqz =ik/fjß^3>4‘ $? =Zh zulwh z (cid:236) zß>zQ zkhuh z(cid:230) =^3>4‘ $ (cid:240) n (cid:240) (cid:240) (cid:240) (cid:240) ? 1 ZhzulwhZdvwkhvhwridooihdvleohzdjhr>huv/l1h1/Z(cid:236)izmz =ik>fjß^3>4‘$? j=4: n n Vwdjh 8 +R>hu Dffhswdqfh,1 Wkh zrunhuv revhuyh wkh zdjh vfkhgxohv z dqg z dq0 (cid:129) 2 qrxqfhg lq Vwdjh 7/ dqg ghflgh iru zklfk ?up wr zrun1 D zrunhu*v r>hu dffhswdqfh uxoh lv d sdlu ri pdsv L =Z2ß^3>4‘ßik>fj $^3>4‘>l @4>5= Iru qrwdwlrqdo vlpsolflw|/ zh zulwh (cid:240) L(cid:230) +º, wr uhsuhvhqw L +z >z >º>m,= Iru h{dpsoh/ LQ +º,@ s phdqv wkdw jlyhq (cid:240)(cid:247)Œ(cid:129)cŒ2: (cid:240) (cid:129) 2 (cid:240)’Œ(cid:129)cŒ2(cid:228) z dqg z > d froohjh dwwhqghh zlwk whvw vljqdo º dffhswv ?up l*v r>hu zlwk suredelolw| s= (cid:129) 2 S Zh uhtxluh wkdw 2 L(cid:230) +º, @4 iru doo º 5^3>4‘>m 5ik>fj dqg doo +z >z ,5Z2= (cid:240)’(cid:129) (cid:240)(cid:247)Œ(cid:129)cŒ2: (cid:129) 2 Vwdjh 9 +Wdvn Dvvljqphqw,1 Lq wkh ?qdo vwdjh hdfk ?up doorfdwhv lwv dydlodeoh zrunhuv ehwzhhqwkh frpsoh{dqgvlpsoh wdvnv xvlqjdqdvvljqphqw uxoh zklfklv dphdvxudeoh ixqf0 wlrq w = ^3>4‘ßif/kj $ i3>4j= Wkh lqwhusuhwdwlrq lv wkdw li w +º>m, @ 4 +uhvshfwlyho|/ 3, (cid:240) (cid:240) phdqv wkdw ?up l dvvljqv doo zrunhuv iurp hgxfdwlrq jurxs m zlwk vljqdo º wr wkh frpsoh{ +uhvshfwlyho|/ vlpsoh, wdvn1 Zh vxppdul}h wkh wlplqj ri wkh jdph lq Iljxuh 41 Ilqdoo| zh dvvxph wkdw zrunhuv duh ulvn qhxwudo dqg wkhlu sd|r> ixqfwlrqv duh dgglwlyho| vhsdudeoh lq wkhlu zdjh lqfrph/ wkhlu vnloo lqyhvwphqw frvwv dqg wkhlu xwlolw| frvwv ri dwwhqglqj froohjh1 Zh dvvxph wkdw zrunhuv gr qrw dvvljqdq lqwulqvlf ydoxh wr zrunlqj dw hlwkhu wdvn1 Li d zrunhu lv ri w|sh +d>y, dqglv sdlgzdjhz>klv sd|r>zloo eh= z(cid:253)y+m,(cid:253)f+h>d,zkhuhy+f,@y>y+k,@3>f+h >d,@n+d, ^ dqg f+h >d, @31 Zh dvvxph wkdw wkh ?upv duh ulvn qhxwudo dqg pd{lpl}h h{shfwhg sur?wv1 (cid:252) 4:Wrvlpsoli|wkhghvfulswlrq riwkh?upv*vwudwhj|vhw/zhdvvxphgwkdwwkh?upvfdq revhuyhqhlwkhuwkh djjuhjdwhglvwulexwlrq riwhvwvljqdovqruwkhfroohjhhquroophqwudwhzkhq dqqrxqflqjzdjhvfkhgxohv1 Wklv dvvxpswlrq grhv qrw d>hfwdq| ri wkh htxloleulxprxwfrphv1 ; 6 Dqdo|vlv ri wkh Prgho Zh ehjlqrxu dqdo|vlv e| ghulylqj wkh glvwulexwlrqri vnloolqyhvwphqw frvwv flqwkh srs0 xodwlrqiurpwkhglvwulexwlrqridelolwlhvı=Ghqrwh wkhf1g1i1 rifdvJdqglwvfruuhvsrqglqj s1g1i1 j= Vlqfh f+h >d,@n+d,/ zh kdyh e| fkdqjh0ri0yduldeoh= ^ æ æ (cid:253) (cid:252)ægn3(cid:129)+f,æ j+f,@! n3(cid:129)+f, æ æ> iru doo f 5F (cid:236)in+d,= d5Dj> +4, æ æ gf zkhuh F lv wkh vxssruw ri j= Ehfdxvh wkhuh lv d rqh0wr0rqh uhodwlrqvkls ehwzhhq dq djhqw*v delolw| d dqg klv txdol?fdwlrq frvw f> lw lv vrphwlphv xvhixo wr lqgh{ dq djhqw ri w|sh +d>y,5DßY e| +f>y, @+n+d,>y,5FßY= Zh xvh Qdvk Htxloleulxp dv rxu htxloleulxp frqfhsw14; D Qdvk htxloleulxp ri wkh hfrqrp| frqvlvwv ri d olvw ri ghflvlrq sur?ohv ri djhqwv= iM>h>L >L j/ dqg d sdlu ri ghflvlrq (cid:129) 2 uxohve|hdfk?upiz >w jirul@4>5>vxfkwkdw= +4,hyhu|djhqw*vghflvlrquxohvduh rswlpdo (cid:240) (cid:240) jlyhq rwkhu djhqwv* dqg wkh ?upv* ghflvlrq uxohv> +5, hdfk ?up*v ghflvlrq uxohv duh rswlpdo jlyhq wkh rwkhu ?up*v ghflvlrq uxohv dqg djhqwv* ghflvlrq sur?ohv1 Wkh Qdvk htxloleulxp ri wkh hfrqrp| fdq eh dqdo|}hg e| edfnzdug lqgxfwlrq1 Htxloleulxp Wdvn Dvvljqphqwv= Lq Vwdjh 9 +wdvn dvvljqphqw vwdjh,/ vxssrvh wkdw d qrq0hpsw| vhw ri wkh zrunhuv kdv dffhswhg ?up l*v zdjh r>hu1 Ghqrwh ßf +uhvshfwlyho|/ ßß, (cid:240) (cid:240) dv wkh vhw ri whvw vljqdov uhdol}hg e| froohjh +uhvshfwlyho|/ kljk vfkrro, dwwhqghhv zkr dffhsw ?up l*v r>hu1 Gh?qh ß (cid:236) ßQ ^ßß= Vlqfh lq wklv vwdjh/ wkh wdvn dvvljqphqw uxoh ri d ?up (cid:240) (cid:240) (cid:240) lv qr orqjhu vwudwhjlf/ zh fdq vlpso| dqdo|}h wkh rswlpdo wdvn dvvljqphqw uxoh iru ?up l idflqj ß = Lw lv suryhg dv Ohppd 5 lq wkh dsshqgl{ wkdw wkh rswlpdo wdvn dvvljqphqw uxoh (cid:240) iru ?up l lv ri wkh irup= + 4 li º Ł(cid:129)º(cid:230) w +º>m, @ (cid:240) >m @k>f= +5, (cid:240) 3 li º ?(cid:129)º(cid:230) (cid:240) Wklv fxw0r> surshuw| lv d frqvhtxhqfh ri wkh vwulfw POUS ri wkh whvw whfkqrorj|1 Htxloleulxp R>hu Dffhswdqfh Ghflvlrqv= Lq Vwdjh 8/ zkhq d zrunhu iurp hgxfdwlrq q r jurxs m zlwk whvw vljqdo º lv r>huhg d sdlu ri zdjh vfkhgxohv/ z(cid:230)+º,>z(cid:230)+º, / klv rswlpdo (cid:129) 2 r>hu dffhswdqfh ghflvlrq/ dvvxplqj zlwkrxw orvv ri jhqhudolw| wkdw kh lsv d frlq zkhq kh lv lqgl>huhqw/ lv= ; A? 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(cid:252) ª(cid:230)H(cid:230)^dmh+d>y,@h ‘> i +º, ^ i +º, (cid:252) Z(cid:230) Z(cid:230) (cid:226) (cid:228)(cid:230)i +º, ^ ª(cid:230)H(cid:230)^dmh+d>y, @h ‘{ = +7, i +º, ^ ^ Z(cid:230) Qrwhwkdw(cid:228)(cid:230) vhuyhvdvwkh?upv*sulrueholhiderxwdudqgrpo|gudzqzrunhuiurphgxfdwlrq jurxs m ehlqj txdol?hg/ dqg (cid:228)(cid:230)i +º,@i +º, lv wkhlu srvwhulru eholhi wkdw wkh zrunhu lv ^ Z(cid:230) txdol?hg zkhq d whvw vljqdo º lv uhdol}hg1 Lq wkh hyhqw wkdw wkh zrunhu lv txdol?hg/ klv h{shfwhg surgxfwlylw| rq wkh vlpsoh wdvn lv ª(cid:230) wlphv wkh h{shfwhglqqdwh delolw| frqglwlrqdo rq klv fkrrvlqj h > vlploduo| lq wkh hyhqw wkdw kh lv xqtxdol?hg/ klv h{shfwhg surgxfwlylw| ^ rq wkh vlpsoh wdvn lv ª(cid:230) wlphv wkh h{shfwhg lqqdwh delolw| frqglwlrqdo rq klv fkrrvlqj h = (cid:252) Klv h{shfwhg surgxfwlylw| rq wkh frpsoh{ wdvn lv wkh suredelolw| wkdw kh lv txdol?hg wlphv wkh h{shfwhg h!flhqf| xqlwv frqglwlrqdo rq klv ehlqj txdol?hg wlphv { = Wkxv irupxod +7, ^ hvvhqwldoo| vwdwhv wkdw hdfk zrunhu lv sdlg klv h{shfwhg surgxfwlylw| lq klv pruh surgxfwlyh wdvn1 Zh zloo khqfhiruwk zulwh H(cid:230)+dmh , dqg H(cid:230)+dmh , dv vkruw0kdqg iru H(cid:230)^dmh+d>y,@h ‘ ^ (cid:252) ^ dqg H(cid:230)^dmh+d>y,@h ‘/ uhvshfwlyho|1 (cid:252) Sursrvlwlrq 4 VxssrvhlqdQdvkhtxloleulxpdjhqwv*hgxfdwlrqfkrlfhdqgvnloolqyhvwphqw ghflvlrq sur?ohv duh uhvshfwlyho| M dqg h= Wkhq wkh ?upv* zdjh r>huv pxvw eh jlyhq e| +7, 43
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