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The Bootstrap of Mean for Dependent Heterogeneous Arrays

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(1)Cahier 2001-19. The Bootstrap of Mean for Dependent Heterogeneous Arrays GONÇALVES, Silvia WHITE, Halbert.

(2) Département de sciences économiques Université de Montréal Faculté des arts et des sciences C.P. 6128, succursale Centre-Ville Montréal (Québec) H3C 3J7 Canada http://www.sceco.umontreal.ca SCECO-information@UMontreal.CA Téléphone : (514) 343-6539 Télécopieur : (514) 343-7221. Ce cahier a également été publié par le Centre interuniversitaire de recherche en économie quantitative (CIREQ) sous le numéro 19-2001. This working paper was also published by the Center for Interuniversity Research in Quantitative Economics (CIREQ), under number 19-2001.. ISSN 0709-9231.

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(38) @ 4  q q lv wkh Srolwlv dqg Urpdqr +4<<7d, zhljkw/ zlwk vprrwklqj sdudphwhu s? @ s  c3 1 Dv lv hylghqw iurp +514, dqg +517,/ wkh PEE dqg wkh VE fryduldqfh pdwul{ hvwlpdwruv iru wkh vdpsoh phdq duh forvho| uhodwhg wr d odj zlqgrz hvwlpdwru ri wkh vshfwudo ghqvlw| pdwul{ dw iuhtxhqf| }hur1 Lq sduwlfxodu/ dv uhpdunhg lq wkh xqlyduldwh frqwh{w e| Ilw}hqehujhu +4<<:, dqg e| Srolwlv dqg Urpdqr a ?c lv dssur{lpdwho| htxlydohqw wr wkh Eduwohww nhuqho yduldqfh +4<<7d,/ wkh PEE yduldqfh hvwlpdwru hvwlpdwru frqvlghuhg e| Qhzh| dqg Zhvw +4<;:,1 Srolwlv dqg Urpdqr +4<<7d, dovr glvfxvv wkh uhodwlrq a ?c dqg a ?c2 1 Wkh| rhu dq lqwhusuhwdwlrq iru wkh VE yduldqfh hvwlpdwru dv d zhljkwhg dyhudjh ehwzhhq a ?c2 vkrxog eh ohvv vhqvlwlyh ryhu c ri PEE yduldqfh hvwlpdwruv zlwk {hg ohqjwk c/ zklfk vxjjhvwv wkdw a ?c lv wr wkh fkrlfh ri c= Vhh Kdoo hw do1 +4<<8,/ Srolwlv hw do1 +4<<:,/ Ilw}hqehujhu wr wkh fkrlfh ri s wkdq +4<<:,/ Krurzlw} +4<<<, dqg Srolwlv dqg Zklwh +5334, iru glvfxvvlrq ri wkh lpsruwdqw lvvxh ri eorfnvl}h fkrlfh lq wkh frqwh{w ri pl{lqj revhuydwlrqv1 Lq d uhfhqw wkhruhwlfdo vwxg|/ Odklul +4<<<, frpsduhv 9.

(39) vhyhudo eorfn errwvwuds yduldqfh hvwlpdwruv/ lqfoxglqj wkh PEE dqg wkh VE1 Kh frqfoxghv wkdw zkloh a ?c dqg a ?c2 kdyh wkh vdph dv|pswrwlf eldv/ wkh yduldqfh ri a ?c2 lv odujhu +wkh xqlyduldwh dqdorjv ri, a ?c / vxjjhvwlqj wkdw wkh VE phwkrg lv dv|pswrwlfdoo| ohvv h!flhqw wkdq wkh PEE1 wkdq wkdw ri Dvvxpswlrq 514 lv xvhg wr hvwdeolvk rxu pdlq frqvlvwhqf| wkhruhp1 Assumption 2.1 2.1.a) Iru vrph u A 5/ n[?| n

(40) o   ? 4 iru doo q> w @ 4> 5> = = = 1 2.1.b) i[?| j lv qhdu hsrfk ghshqghqw +QHG, rq iY| j zlwk QHG frh!flhqwv y& ri vl}h  2Eo3 Eo32 > iY| j 2o lv dq 0pl{lqj vhtxhqfh zlwk & ri vl}h  o32 1.   Theorem 2.1. Dvvxph i[?| j vdwlvhv Dvvxpswlrq 5141 Wkhq/ li c? $ 4 dqg c? @ r q*2 / iru   S  WE WE a ?c  + ? . X?c , $ dqg ?| lv wkh uhvdpsohg 3> zkhuh X?c  yduW q3*2 ?|' ?| m @ 4> 5> yhuvlrq ri ?| 1 a ?c / m @ 4> 5> duh Xqghu duelwudu| khwhurjhqhlw| lq i[?| j wkh eorfn errwvwuds fryduldqfh hvwlpdwruv qrw frqvlvwhqw iru ? / exw iru ? . X?c 1 Wkh eldv whup X?c lv uhodwhg wr wkh khwhurjhqhlw| lq wkh phdqv S s WE WE i?| j dqg fdq eh lqwhusuhwhg dv wkh eorfn errwvwuds fryduldqfh pdwul{ ri q ? @ q3*2 ?|' ?| wkdw zrxog uhvxow li zh frxog uhvdpsoh wkh yhfwru wlph vhulhv i?| j1 Wkhruhp 514 pdnhv fohdu wkdw d a ?c iru ? lv wkdw X?c $ 3 dv q $ 41 D vx!flhqw frqglwlrq qhfhvvdu| frqglwlrq iru wkh frqvlvwhqf| ri iru X?c wr ydqlvk lv uvw rughu vwdwlrqdulw| ri i[?| j = li ?| @  iru doo q> w / wkhq X?c @ 31 Zh kdyh wkh iroorzlqj ohppd1 Lemma 2.1. Li zh ohw c? @ X?c @. ? [. |'  ?| ?| /. wkhq.     ?c|cf ?|  c? ?|  c?. |' 3  [.

(41) ' ? [ 3. dqg X?c2 @ q. |' ?3 [. +?|  ? , +?|  ? , 3. e?

(42) q.

(43) '. k      l [  ?c|c

(44) ?|  c? ?c|n

(45)  c? . ?c|n

(46)  c? ?|  c? > c |' ?3

(47). 4. .. .. S?. ?3

(48) [k |'. l     +?|  ? , ?c|n

(49)  ? . ?c|n

(50)  ? +?|  ? , = :.

(51) Wkxv/ wkh frqglwlrq olp?<" X?c @ 3/ m @ 4> 5> fdq eh lqwhusuhwhg dv dq krprjhqhlw| frqglwlrq rq wkh phdqv1 Wkh iroorzlqj dvvxpswlrq hqvxuhv olp?<" X?c @ 3> m @ 4> 5= Assumption 2.2 q3. S?. |' +?|.   > zkhuh c? $ 4 dqg c? @ r +q,1  ? , +?|  ? , @ r c3 ?. Wkh iroorzlqj frqvlvwhqf| uhvxow krogv xqghu Dvvxpswlrqv 514 dqg 515 dqg lv dq lpphgldwh frqvh0 txhqfh ri wkh suhylrxv uhpdun1   Corollary 2.1. Dvvxph i[?| j vdwlvhv Dvvxpswlrqv 514 dqg 5151 Wkhq/ li c? $ 4 dqg c? @ r q*2 /  a ?c  ? $. 3> m @ 4> 5=. Wklv uhvxow h{whqgv wkh suhylrxv frqvlvwhqf| uhvxowv e| Nÿqvfk +4<;<, dqg Srolwlv dqg Urpdqr +4<<7d, wr wkh fdvh ri ghshqghqw khwhurjhqhrxv grxeoh duud|v ri udqgrp yhfwruv/ zkhuh wkh vwdwlrqdu| pl{lqj dvvxpswlrq lv uhsodfhg e| wkh pruh jhqhudo dvvxpswlrq ri d +srvvleo| khwhurjhqhrxv, grxeoh duud| qhdu hsrfk ghshqghqw rq d pl{lqj surfhvv1 Lq sduwlfxodu/ iru m @ 5/ Fruroodu| 514 frqwdlqv d yhuvlrq ri Srolwlv dqg Urpdqr*v +4<<7d, Wkhruhp 4 dv d vshfldo fdvh1 Frqvlghu d vwulfwo| vwdwlrqdu| 0pl{lqj vhtxhqfh i[ > = = = > [? j vdwlvi|lqj Dvvxpswlrq 5141 Ehfdxvh d pl{lqj surfhvv lv wulyldoo| qhdu hsrfk ghshqghqw rq lwvhoi/ wkh QHG uhtxluhphqw lv dxwrpdwlfdoo| vdwlvhg1 Fruroodu| 514 dfklhyhv wkh vdph frqfoxvlrq dv Srolwlv dqg Urpdqr*v +4<<7,   Wkhruhp 4 xqghu wkh vdph prphqw frqglwlrqv exw zhdnhu 0pl{lqj vl}h frqglwlrqv +& @ R n3b iru   2o vrph  A o32 dqg u A 5 khuh lq frqwudvw wr & @ R n3b iru vrph  A

(52) ESn0 dqg % A 3 wkhuh,1 0 Zh doorz pruh ghshqghqfh khuh/ zlwk wkh idploldu wudgh0r ehwzhhq prphqw dqg phpru| frqglwlrqv1   Qhyhuwkhohvv/ zh uhtxluh wkh vwurqjhu frqglwlrq wkdw c? @ r q*2 / l1h1 q*2 s? $ 4 +zlwk s? @ c3 ? $ 3,> zkloh Srolwlv dqg Urpdqr +4<<7d, rqo| uhtxluh c? @ r +q,/ l1h1 qs? $ 41 Lpsrvlqj vwdwlrqdulw| lq S rxu iudphzrun zloo hqvxuh wkdw ? $ " dv q $ 4/ zkhuh " @ ydu +[ , . 5 "

(53) ' ydu +[ > [n

(54) ,> a ?c2 $ " lq suredelolw|/ dv Srolwlv dqg Urpdqr +4<<7d, frqfoxgh1 khqfh/ Vlploduo|/ iru m @ 4/ rxu Fruroodu| 514 vshfldol}hv wr Nÿqvfk*v +4<;<, Fruroodu| 614 zkhq i[| j lv d vwdwlrqdu| 0pl{lqj vhtxhqfh/ xqghu wkh vdph prphqw frqglwlrqv dqg zhdnhu 0pl{lqj frqglwlrqv/ exw   xqghu wkh vwurqjhu uhtxluhphqw wkdw c? @ r q*2 lqvwhdg ri c? @ r +q,1 ;.

(55) Wkh qh{w wkhruhp hvwdeolvkhv wkh uvw rughu dv|pswrwlf htxlydohqfh ehwzhhq wkh prylqj eorfnv dqg vwdwlrqdu| errwvwuds glvwulexwlrqv dqg wkh qrupdo glvwulexwlrq iru wkh pxowlyduldwh vdpsoh phdq1 D voljkwo| vwurqjhu ghshqghqfh dvvxpswlrq lv lpsrvhg wr dfklhyh wklv uhvxow1 Vshflfdoo|/ zh uhtxluh i[?| j wr eh O2nB QHG rq d pl{lqj surfhvv +vhh Dqguhzv +4<;;,,1 Zh vwuhqjwkhq Dvvxpswlrq 5141e, voljkwo|= 2.1.b ) Iru vrph vpdoo  A 3/ i[?| j lv O2nB QHG rq iY| j zlwk QHG frh!flhqwv y& ri vl}h  2Eo3 Eo32 > iY| j lv dq 0pl{lqj vhtxhqfh zlwk & ri vl}h  E2nBo o32 1 Theorem 2.2. Dvvxph i[?| j vdwlvhv Dvvxpswlrqv 514 dqg 515 vwuhqjwkhqhg e| 5141e ,1 Ohw ?   s  ? eh srvlwlyh ghqlwh xqlirupo| lq q/ l1h1 ? lv srvlwlyh vhplghqlwh iru doo q dqg ghw ?  ydu q[  A 3 iru doo q vx!flhqwo| odujh1 Wkhq ? @ R +4, dqg    ?  ? , Q +3> L_ , xqghu S> dqg q [. 3*2 s. (i) ?.  ks  l      vxs S q 3*2   [  { . +{,  $ 3/ ? ? ?. %MR_. zkhuh lv wkh vwdqgdug pxowlyduldwh qrupdo fxpxodwlyh glvwulexwlrq ixqfwlrq/ _  % dssolhv wr hdfk frpsrqhqw ri wkh uhohydqw yhfwru  *2 dqg _ , % ghqrwhv frqyhujhqfh lq glvwulexwlrq1 > wkhq iru dq| % A 3> dqg iru m @ 4> 5> Pruhryhu/ li c? $ 4 dqg c? @ r q   3*2 s  ?WE  [  ? , Q +3> L_ , xqghu S W zlwk suredelolw| S dssurdfklqj rqh/ dqg q [ (ii) ? + S. ,  ks   l   s   W WE ?  [ ?  {  S  ?  ?  {  A % $ 3/ q [ q [ vxs S. %MR_. zkhuh S W lv wkh suredelolw| phdvxuh lqgxfhg e| wkh errwvwuds/ frqglwlrqdo rq i[?| j?|' = Iru m @ 5> wklv lv dq h{whqvlrq ri Wkhruhp 6 ri Srolwlv dqg Urpdqr +4<<7d, iru vwdwlrqdu| pl{lqj revhuydwlrqv wr wkh fdvh ri QHG ixqfwlrqv ri d pl{lqj surfhvv1 Iru m @ 4 dqg g @ 4/ Wkhruhp 515 vwdwhv d zhdnhu frqfoxvlrq wkdq grhv Wkhruhp 618 ri Nÿqvfk +4<;<,/ vlqfh zh suryh frqyhujhqfh lq suredelolw|/ exw qrw doprvw vxuh frqyhujhqfh1 Rq wkh rwkhu kdqg/ zh shuplw khwhurjhqhlw| dqg juhdwhu ghshqghqfh1 Lq sduw +l, zh vwdwh wkh xvxdo dv|pswrwlf qrupdolw| uhvxow iru wkh pxowlyduldwh vdpsoh phdq1 Lq sduw +ll, zh suryh wkh xqlirup frqyhujhqfh wr }hur +lq suredelolw|, ri wkh glvfuhsdqf| ehwzhhq wkh dfwxdo  s  glvwulexwlrq ri q [ ?  ? dqg wkh eorfn errwvwuds dssur{lpdwlrq wr lw1 Wklv uhvxow iroorzv iurp wkh <.

(56)  s   WE  ? / frqglwlrqdo rq [? > = = = > [?? / idfw wkdw xqghu rxu dvvxpswlrqv wkh glvwulexwlrq ri 3 q [?  [ ? frqyhujhv zhdno| wr wkh vwdqgdug pxowlyduldwh qrupdo glvwulexwlrq iru doo grxeoh duud|v i[?| j lq d vhw zlwk suredelolw| whqglqj wr rqh1 Lq sduwlfxodu/ zh dsso| d fhqwudo olplw wkhruhp iru wuldqjxodu duud|v dqg xvh Dvvxpswlrq 5141e , wr hqvxuh wkdw O|dsrxqry*v frqglwlrq lv vdwlvhg xqghu rxu khwhurjhqhrxv QHG frqwh{w1 Dvvxpswlrq 5141e, pljkw zhoo eh vx!flhqw wr yhuli| wkh zhdnhu Olqghehuj frqglwlrq/ dv lq Nÿqvfk +4<;</ Wkhruhp 618,/ dowkrxjk zh kdyh qrw yhulhg wklv1 Ilw}hqehujhu +4<<:, kdv uhfhqwo| suryhq wkh frqvlvwhqf| ri wkh prylqj eorfnv errwvwuds dssur{lpd0 wlrq wr wkh wuxh vdpsolqj glvwulexwlrq ri wkh vdpsoh phdq iru khwhurjhqhrxv 0pl{lqj surfhvvhv1 Rxu uhvxow h{whqgv klv e| doorzlqj iru qhdu hsrfk ghshqghqfh rq pl{lqj surfhvvhv1 Krzhyhu/ lq wkh sxuho| vwurqj pl{lqj fdvh/ rxu prphqw dqg phpru| frqglwlrqv duh pruh vwulqjhqw wkdq klv1 Lq sduwlfxodu/ klv Wkhruhp 614 rqo| uhtxluhv H m[| m2RnB ? F/ iru vpdoo  A 3 dqg s A 5/ dqg i[| j vwurqj pl{lqj ri R vl}h  R32 1 Srolwlv hw do1 +4<<:/ Wkhruhp 614, kdyh dovr hvwdeolvkhg wkh urexvwqhvv ri wkh vxevdpsolqj. phwkrg iru frqvlvwhqw vdpsolqj glvwulexwlrq hvwlpdwlrq iru khwhurjhqhrxv dqg ghshqghqw gdwd xqghu plog prphqw frqglwlrqv +H m[| m2n20   ? 4 iru vrph % A 3,1 Qhyhuwkhohvv/ wkh| dovr dvvxph d vwurqj pl{lqj surfhvv/ dv|pswrwlf fryduldqfh vwdwlrqdulw|/ dqg voljkwo| vwurqjhu vl}h uhtxluhphqwv wkdq rxuv rq wkh pl{lqj frh!flhqwv +& ri vl}h 

(57) Een0 0 ,1.  s   WE  ? lv wkdw lwv +frqglwlrqdo, h{shfwhg ydoxh q [?  [    ?WE @ S?  ?| [?| / zkhuh wkh zhljkwv lv qrw }hur1 Lqghhg/ xqghu wkh PEE uhvdpsolqj vfkhph H W [ |'      ?WE @ [  ? .R ? +vhh h1j1 Ohppd  ?| duh ghqhg dv lq +515,1 Li c? @ r +q,/ rqh fdq vkrz wkdw H W [ ? s  W   WE    D14 ri Ilw}hqehujhu/ 4<<:,1 Wkxv/ wkh PEE glvwulexwlrq kdv d udqgrp eldv q H [?  [? >     ?  ?WE2 @ [  ? =, Dv srlqwhg zklfk lv ri rughu R ?1*2 1 +Iru wkh VE qr vxfk sureohp h{lvwv vlqfh H W [ D zhoo nqrzq surshuw| ri wkh PEE vwdwlvwlf. rxw e| Odklul +4<<5,/ wklv udqgrp eldv ehfrphv suhgrplqdqw iru vhfrqg rughu dqdo|vlv dqg suhyhqwv wkh PEE iurp surylglqj vhfrqg rughu lpsuryhphqwv ryhu wkh vwdqgdug qrupdo dssur{lpdwlrq1 Wr fruuhfw iru wklv eldv/ kh vxjjhvwv uhfhqwhulqj wkh PEE glvwulexwlrq durxqg wkh errwvwuds phdq/ wkdw lv/ wr   s   WE  ?WE 1 Wkh iroorzlqj uhvxow vkrzv wkdw xqghu frqvlghu wkh errwvwuds glvwulexwlrq ri q [  HW [ ? wkh dvvxpswlrqv ri Wkhruhp 515 uhfhqwhulqj wkh PEE glvwulexwlrq durxqg wkh PEE errwvwuds phdq lv dv|pswrwlfdoo| ydolg +wr uvw rughu, lq wklv khwhurjhqhrxv QHG frqwh{w1. 43.

(58) Corollary 2.2. Xqghu wkh dvvxpswlrqv ri Wkhruhp 515/ iru doo % A 3> + ,   ks    l s      W  WE  ?WE  H W [  ?  ?  {  A % $ 3> S vxs S { S q [ q [ ? %MR_. zkhuh S W lv wkh suredelolw| phdvxuh lqgxfhg e| wkh PEE errwvwuds/ frqglwlrqdo rq i[?| j?|' 1 Wkhruhp 515 dqg Fruroodu| 515 mxvwli| wkh xvh ri wkh PEE dqg VE glvwulexwlrqv wr rewdlq dq dv|pswrwlfdoo| ydolg frqghqfh lqwhuydo iru ? lqvwhdg ri xvlqj d frqvlvwhqw hvwlpdwh ri wkh yduldqfh dorqj zlwk wkh qrupdo dssur{lpdwlrq1 Iru h{dpsoh/ dq htxdo wdlohg +4  , 433( vwdwlrqdu| errwvwuds        ?  tW k `> zkhuh t W k dqg  ?  t?W 4  k > [ frqghqfh lqwhuydo iru ? zkhq g @ 4 zrxog eh ^[ ? 2 ? 2 2    ?WE2  [ ?1 t?W 4  k2 duh wkh k2 dqg 4  k2 txdqwlohv ri wkh VE errwvwuds glvwulexwlrq ri [. 3. Conclusion Lq wklv sdshu zh hvwdeolvk wkh uvw rughu dv|pswrwlf ydolglw| ri eorfn errwvwuds phwkrgv iru wkh vdpsoh phdq ri ghshqghqw khwhurjhqhrxv gdwd1 Rxu uhvxowv dsso| wr wkh prylqj eorfnv errwvwuds ri Nÿqvfk +4<;<, dqg Olx dqg Vlqjk +4<<5, dv zhoo dv wr wkh vwdwlrqdu| errwvwuds ri Srolwlv dqg Urpdqr +4<<7d,1 Lq sduwlfxodu/ zh vkrz wkdw wkh PEE dqg wkh VE fryduldqfh hvwlpdwruv iru wkh pxowlyduldwh vdpsoh phdq duh frqvlvwhqw xqghu d zlgh fodvv ri gdwd jhqhudwlqj surfhvvhv/ wkh surfhvvhv qhdu hsrfk ghshqghqw rq d pl{lqj surfhvv1 Zh dovr suryh wkh uvw rughu dv|pswrwlf htxlydohqfh ehwzhhq wkh eorfn errwvwuds glvwulexwlrqv dqg wkh qrupdo glvwulexwlrq lq wklv khwhurjhqhrxv qhdu hsrfk ghshqghqw frqwh{w1. A. Appendix Wklv dsshqgl{ frqwdlqv deeuhyldwhg yhuvlrqv ri wkh surriv ri rxu uhvxowv wr frqvhuyh vsdfh1 D yhuvlrq ri wklv dsshqgl{ frqwdlqlqj ghwdlohg surriv lv dydlodeoh iurp wkh dxwkruv xsrq uhtxhvw1 Lq wkh surriv iru vlpsolflw| zh zloo frqvlghu i[?| j wr eh uhdo0ydoxhg +l1h1 g @ 4,1 Wkh uhvxowv iru wkh pxowlyduldwh fdvh iroorz e| vkrzlqj wkdw wkh dvvxpswlrqv duh vdwlvhg iru olqhdu frpelqdwlrqv \?|   [?| iru dq| qrq0}hur  5 R_ 1 Wkurxjkrxw wkh Dsshqgl{/ N zloo ghqrwh d jhqhulf frqvwdqw wkdw   pd| fkdqjh iurp rqh xvdjh wr dqrwkhu1 Ixuwkhupruh/ zh vkdoo xvh wkh qrwdwdwlrq Hr| +, @ H mIr| iru w  v/ zkhuh Ir| @  +Y| > = = = > Yr , > zlwk w ru v rplwwhg wr ghqrwh .4 dqg 4/ uhvshfwlyho|1 ^{` zloo ghqrwh wkh lqwhjhu sduw ri {1 Ilqdoo|/ wkh vxevfulsw q lq c? dqg s? zloo eh lpsolflwo| dvvxphg wkurxjkrxw1 Wkh pl{lqjdoh surshuw| ri }hur phdq QHG surfhvvhv rq d pl{lqj surfhvv lv dq lpsruwdqw wrro lq 44.

(59) rewdlqlqj rxu uhvxowv1 Vhh iru h{dpsoh Gdylgvrq +4<<7/ Ghqlwlrq 4918, iru d ghqlwlrq ri d pl{lqjdoh1 Zh zloo pdnh xvh ri wkh iroorzlqj ohppdv lq rxu surriv1.

(60)

(61) o32 Lemma A.1. +l, Li i[?| j vdwlvhv Dvvxpswlrq 514/ [?|  ?| > I | lv dq O2 0pl{lqjdoh ri vl}h 

(62) Eo32 > u A 5/ zlwk pl{lqjdoh frqvwdqwv xqlirupo| erxqghg f?| 1 +ll, Li i[?| j vdwlvhv Dvvxpswlrq 514 vwuhqjwk0.

(63) hqhg e| Dvvxpswlrq 5141e , zlwk   7 dqg u A 5/ wkhq [?|  ?| > I | lv dq O2nB 0pl{lqjdoh ri vl}h 

(64) o3E2nB

(65) Eo32 > zlwk pl{lqjdoh frqvwdqwv xqlirupo| erxqghg f?| 1

(66). Lemma A.2. Ohw ]?| > I | eh dq O2 pl{lqjdoh ri vl}h 4@5 zlwk pl{lqjdoh frqvwdqwv f?| 1 Wkhq/  S 2 S? 2   H pd{$? @R ] ?| |' |' f?| =.

(67) ]?| > I | eh dq OR pl{lqjdoh iru vrph s  5 zlwk pl{lqjdoh frqvwdqwv f?| dqg S   S   S     ? 2 *2 = pl{lqjdoh frh!flhqwv #& vdwlvi|lqj " &' # & ? 4= Wkhq/ pd{$?  |' ]?|  @ R |' f?| Lemma A.3. Ohw. R. Ohppd D14 iroorzv iurp Fruroodu| 4:191+l, ri Gdylgvrq +4<<7,/ Ohppd D15 lv Wkhruhp 419 ri PfOhlvk +4<:8,/ dqg Ohppd D16 lv d vwudljkwiruzdug jhqhudol}dwlrq ri Kdqvhq*v +4<<4e, pd{lpdo lqhtxdolw| iru OR 0pl{lqjdohv/ zlwk4 s  5> wr wkh grxeoh duud| vhwwlqj1 Wkh iroorzlqj ohppd jhqhudol}hv Ohppd 91: +d, lq Jdoodqw dqg Zklwh +4<;;/ ss1 <<0433,1 Lemma A.4. Dvvxph [?| lv vxfk wkdw H +[?| , @ 3 dqg n[?| n

(68) o   ? 4 iru vrph u A 5 dqg iru doo q> w= Li i[?| j lv QHG rq iY| j dqg iY| j lv 0pl{lqj/ wkhq iru {hg  A 3 dqg doo w ? v  w .  >  2  1 1  1 1 o32 3 +23o, 2(o31) 2 o mfry +[?| [?c|n

(69) > [?r [?crn

(70) ,m  N   r3| . y^ r3| ` .N 2 y r3| .N

(71) 

(72) . y^

(73) ` > ^4` ^ 4 ` ^ 4 ` 4 4 iru vrph qlwh frqvwdqwv N > N2 dqg N

(74) 1  mH +[?| [?c|n

(75) [?r [ ?crn

(76) ,m *23*o . mH +[?| [?c|n

(77) , H +[?r [?crn

(78) ,m = Qh{w/ qrwh wkdw mH +[?| [?c|n

(79) ,m   8

(80) . 5y^

(81) ` +vhh ^4` 4 Jdoodqw dqg Zklwh/ 4<;;/ ss1 43<0443,/ zklfk lpsolhv wkh odvw erxqg rq wkh fryduldqfh1 Wr erxqg |n

(82) n^ 6 2 ` mH +[?| [?c|n

(83) [?r [?crn

(84) ,m > ghqh p @ v  w A 3 dqg \a?6|

(85)  H +[?| [?r [?c|n

(86) , > dqg qrwh wkdw |3

(87) 3^ 6 2 `      mH +[?| [?c|n

(88) [?r [?crn

(89) ,m  H \a?6|

(90) [?crn

(91)        . H [?crn

(92) [?| [?r [?c|n

(93)  \a?6|

(94)  = +D14, Proof.. Iluvw/. qrwh. wkdw. mfry +[?| [?|n

(95) > [?r [?crn

(96) ,m.

(97) Vlqfh \a?6|

(98) lv d phdvxudeoh ixqfwlrq ri Y|3

(99) 3d6*2o > = = = > Y|n

(100) nd6*2o / lw lv phdvxudeoh0I |n

(101) nd6*2o 1 E| dq dssolfdwlrq ri wkh odz ri lwhudwhg h{shfwdwlrqv dqg uhshdwhg dssolfdwlrqv ri Kùoghu*v lqhtxdo0              lw|/ H \a?6|

(102) [?crn

(103)  @ H H |n

(104) nd6*2o \a?6|

(105) [?crn

(106)   

(107) H |n

(108) nd6*2o [?crn

(109) 2 1 Wr erxqg 4. Kdqvhq +4<<5, jlyhv wkh fruuhfwhg yhuvlrq ri wklv pd{lpdo lqhtxdolw| iru 1 < p < 21 Khuh/ zh zloo rqo| xvh wkh uhvxow zkhq p D 2, dqg zh rplw wkh fdvh 1 < p < 2.. 45.

(110)  |n

(111) nd6*2o  H [?crn

(112) 2 > qrwh wkdw I |n

(113) nd6*2o @ I rn

(114) 3&1  I rn

(115) 3d6*2o zlwk n @ p  ^p@5`  ^p@5` >     lpso|lqj wkdw H |n

(116) nd6*2o [?crn

(117) 2  H rn

(118) 3d6*2o [?crn

(119) 2 e| Wkhruhp 4315: ri Gdylgvrq +4<<7,1   E| d vlplodu dujxphqw dv lq PfOhlvk +4<:8/ Wkhruhp 614,/ zh kdyh wkdw H rn

(120) 3d6*2o [?crn

(121) 2       1 1 3   *23*o yd6*eo . 8d6*eo 1 Wkxv/ H \a?6|

(122) [?crn

(123)   

(124) 8 26 o . y^ 6 ` = Wr erxqg wkh vhfrqg whup ^4` 4     lq +D14,/ e| wkh Fdxfk|0Vfkzdu} lqhtxdolw|/ lw vx!fhv wr vkrz wkdw [?| [?r [?c|n

(125)  \a?6|

(126)   2. o32 2(o31) 6 4. Ny = Iroorzlqj dq dujxphqw vlplodu wr wkdw xvhg e| Gdylgvrq +4<<5/ Ohppd 618, dqg zulwlqj ^ ` |n

(127) na Ha +, iru H|3

(128) 3a +, > [ iru [?| > \ iru [?r > dqg ] iru [?c|n

(129) > zh rewdlq n[\ ]  Ha +[\ ],n2  n[\ ]  Ha +[, Ha +\ , Ha +],n2 e| Wkhruhp 43145 ri Gdylgvrq +4<<7,1 Dovr/ dgglqj dqg vxewudfwlqj     a g X> X a / zkhuh dssursuldwho|/ e| wkh wuldqjoh lqhtxdolw|/ m[\ ]  Ha +[, Ha +\ , Ha +],m  E X> X   a @ +Ha +[, > Ha +\ , > Ha +],, > E X> X a @ +m][m . m]Ha +\ ,m . mHa +[, Ha +\ ,m,/ X @ +[> \> ], > X   a @ +m[  Ha +[,m . m\  Ha +\ ,m . m]  Ha +],m, = Qrz dsso| Ohppd 714 ri Jdoodqw dqg dqg g X> X   a @ Ha +[, Ha +\ , Ha +], > dqg zlwk u A 5/ t @

(130) o dqg Zklwh +4<;;/ s1 7:, zlwk e +X, @ [\ ]/ e X 2  2 Eo32*2Eo3 Eo32*2Eo3 ^

(131) o +9ya , 1 Wkh uhvxow s @ ^3 @

(132) o32 = Wklv |lhogv n[\ ]  Ha +[\ ],n2  N 6  r3|  iroorzv xsrq vhwwlqj M @ ^p@5` @ 2 / jlyhq wkdw yEu lv qrqlqfuhdvlqj1 ¥.   ?c  ? $ Proof of Theorem 2.11 Wkh surri frqvlvwv ri wzr vwhsv= +4, vkrz 3/ dqg +5, vkrz    a ?c   ?c . X?c $  ?c zklfk lv lghqwlfdo wr a ?c. 31 Lq +4,/ zh frqvlghu dq lqihdvleoh hvwlpdwru .  c? dqg [  ? lq +514, dqg +517, zlwk ?| iru m @ 4> 5> uhvshfwlyho|1 h{fhsw wkdw lw uhsodfhv [ Proof of step 11 +m @ 4,= Ghqh ]?|  [?|  ?| dqg U?| + , @ H +]?| ]?c|n

(133) ,1 E| wkh wuldqjoh lqhtxdolw|/. ? 3 ?3

(134)  [    [ [      ?c|cf  q3  mU?| +3,m . 5  ?c|c

(135)  q3  mU?| + ,m  ?c  ?   H |' 3 [.

(136) ' |' ?3 ?3

(137) [ [ 3. ?3

(138)   [   ?c|c

(139)  mU?| + ,m . 5 .5 c |'

(140) '. .  ?3n. . . 2 ?3n. . mU?| + ,m =. +D16,. |'. uhvshfwlyho|/ dqg wkhuhiruh wkh| duh r +4,1 Wkh wzr S

(141) 3 S?3

(142) ? ? whupv lq +D16, frpelqhg duh erxqghg e| ?3n  ? / zkhuh ?3n $ 4> dqg  ?  5 ?3 mU?| + ,m

(143) '  q    |'   3   a ?c  ?  @ 3 jlyhq jlyhq wkh dvvxphg vl}h frqglwlrqv rq & dqg y& 1 Wkxv/ olp?<" H lv R c   wkdw c? $ 4 dqg c @ r q*2 1      ?c $ 3/ ghqh U  ?f + , @ S?3ƒ

(144) ƒ  ?c|c

(145) ]?| ]?c|nƒ

(146) ƒ dqg zulwh ydu  ?c @ Wr vkrz wkdw ydu |'        S3 S3 ƒ

(147) ƒ ƒbƒ    ?f + , 4  4  fry U + , > U +, = Zh vkrz wkdw ydu U lv ?f ?f   

(148) '3n b'3n ? R E?3n zklfk e| wkh Fdxfk|0Vfkzdu} lqhtxdolw| lpsolhv wkh uhvxow vlqfh 2 /      S3 S3 ƒ

(149) ƒ ƒbƒ 4  4  @ c2 dqg c @ r q*2 1 Iroorzlqj Srolwlv dqg Urpdqr +4<<7e,/

(150) '3n b'3n   Wkh wzr whupv lq +D15, duh R. dqg R. q.

(151) '. +D15,. 46.

(152) zulwh ?3ƒ

(153) ƒ ?3ƒ

(154) ƒ  ?3ƒ

(155)  [ƒ [ [      2   ?c|c

(156) ydu ]?| ]?c|nƒ

(157) ƒ . 5  ?c|c

(158)  ?crc

(159) fry ]?| ]?c|nƒ

(160) ƒ > ]?r ]?crnƒ

(161) ƒ  ydu U?f + , @ |'. . |' r'|n. ?3ƒ

(162) ƒ ?3ƒ

(163) ƒ |nƒ

(164) ƒ [ [ [      4 5 fry ]?| ]?c|nƒ

(165) ƒ > ]?r ]?crnƒ

(166) ƒ  ydu ] ] . ?| ?c|nƒ

(167) ƒ 2 2 +q  c . 4, |' +q  c . 4, |' r'|n. . jlyhq wkdw  ?c|c

(168)  i]?| j >.  ?3n. ?3ƒ

(169) ƒ ?3ƒ

(170) ƒ [ [    5 fry ]?| ]?c|nƒ

(171) ƒ > ]?r ]?crnƒ

(172) ƒ  > 2 +q  c . 4, |' r'|nƒ

(173) ƒn. iru doo w dqg  = Iru N vx!flhqwo| odujh/ dqg jlyhq wkh phdq }hur surshuw| ri. + , " " "   o32 [ [ [ 1 1 3  ?f + ,  Nq 2 .  2& o . y^ & ` . y 2(o31) +q  c . 4,2 ydu U ^4` ^ &4 ` 4 &' &' &' $ # 1 1 2+ 12 3 1o , 3o, k l + 2 2 .Nq m m k ƒ

(174) ƒ l . m m yk ƒ

(175) ƒ l . 5 m m k ƒ

(176) ƒ l y ƒ

(177) ƒ > 4. 4. 4. +D17,. 4. zkhuh zh xvhg Ohppd D17 wr erxqg wkh fryduldqfhv zkhq w ? v  w . m m dqg d uhvxow vlplodu wr Ohppd 91: +d, lq Jdoodqw dqg Zklwh +4<;;/ ss1<<0433, zkhq v A w . m m1 Wkh vxpv lq wkh fxuo| eudfnhwv duh qlwh/ zkhuhdv wkh odvw whup lq +D17, whqgv wr 3 dv m m $ 4 e| wkh vl}h dvvxpswlrqv rq & dqg y& / wkhlu    ?f + ,  g? 2 1 prqrwrqlflw|/ dqg wkh idfw wkdw wkh| whqg wr 3 dv n $ 4= Khqfh/ ydu U E?3n   S ƒ

(178) ƒ S?3ƒ

(179) ƒ Proof of step 2. +m @ 4,= Ghqh V?c @ 3  [ [ 4  ?c|c

(180) ?| ?c|nƒ

(181) ƒ > dqg zulwh |'

(182) '3n  a ?c @ V?c ..  ?3ƒ

(183) ƒ 3  [   m m [ 2  c? [?|  [  c? [?c|nƒ

(184) ƒ . [  c? > 4  ?c|c

(185) [ c |'.

(186) '3n.  ?c @ V?c ..  ?3ƒ

(187) ƒ 3    [ m m [  ?c|c

(188) ?c|nƒ

(189) ƒ [?|  ?| [?c|nƒ

(190) ƒ . ?| ?c|nƒ

(191) ƒ = 4 c |'.

(192) '3n.  ?c @ D? . D?2 . D?

(193) . D?e > zkhuh a ?c  Wkhq/ D?.  ?3ƒ

(194) ƒ 3   [    m m [  4 @  [c?  c?  ?c|c

(195) ]?| . ]?c|nƒ

(196) ƒ > c |'.

(197) '3n. D?2 @.  ?3ƒ

(198) ƒ 3  [   m m [  ?c|c

(199) ?|  c? ]?c|nƒ

(200) ƒ > 4 c |'.

(201) '3n. D?

(202) @. dqg.  ?3ƒ

(203) ƒ 3    [ m m [  ?c|c

(204) ?c|nƒ

(205) ƒ  c? ]?| > 4 c |'.

(206) '3n. 47.

(207)  ?3ƒ

(208) ƒ 3      [ m m [ 2  c?  c? . ?| ?c|nƒ

(209) ƒ > D?e @  ?c|c

(210) [  ?| . ?c|nƒ

(211) ƒ [ 4 c |'

(212) '3n S? S zlwk c? @ |'  ?| ?| 1 Li ?| @  iru doo w/ c? @  vlqfh ?|'  ?| @ 4/ zklfk lpsolhv D?2 @ D?

(213) @ 3    kc?  kc? 2 c> ehfdxvh S?3ƒ

(214) ƒ  ?c|c

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(216) ƒ 3     [  m m [   [c?  c?  ?c|c

(217) ?| . ?c|nƒ

(218) ƒ > dqg 4 c |'.

(219) '3n.  ?3ƒ

(220) ƒ 3   [   m m [ X?c @  ?c|c

(221) ?|  c? ?c|nƒ

(222) ƒ  c? = 4 c |'

(223) '3n   S WE E| Wkhruhpv 614 dqg 617 ri Nÿqvfk +4<;<,/ X?c @ yduW q3*2 ?|' ?| = Wkxv/ lw vx!fhv wr vkrz wkdw D? > D?2 > D?

(224) dqg D?e duh r +4,1  c?  c? Zh qrz vkrz wkdw [.   @ r c3 1. Ghqh !?| +{,. @ $ ?| {> zkhuh $ ?|. .  c?  plq iw@c> 4> +q  w . 4, @cj/ dqg qrwh wkdw !?| +, lv xqlirupo| Olsvfklw} frqwlqxrxv1 Qh{w/ zulwh [ S c? @ +q  c . 4,3 ?|' \?| > zkhuh \?|  !?| +]?| , lv d phdq }hur QHG duud| rq iY| j ri wkh vdph vl}h dv ]?| e| Wkhruhp 4:145 ri Gdylgvrq +4<<7,/ vdwlvi|lqj wkh vdph prphqw frqglwlrqv1 Khqfh/ e| Ohppd.

(225)

(226) o32 D14 \?| > I | lv dq O2 0pl{lqjdoh ri vl}h 

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(228) ƒ ƒ

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(230) '3n 4   |'  ?c|c

(231) ]?| . ]?c|nƒ

(232) ƒ @ R +c, =.   Wr suryh wkdw D?

(233) @ r +4, > ghqh \?|

(234)  $?|

(235) ?c|nƒ

(236) ƒ  c? ]?| @ !?|

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(238) @ q r   ?3|3ƒ

(239) ƒn | plq 3ƒ

(240) > dqg !?|

(241) +{, @ $ ?|

(242) ?c|nƒ

(243) ƒ  c? { lv xqlirupo| Olsvfklw} frqwlqxrxv1 Du0 ƒ > 4> 3ƒ

(244) ƒ.

(245) jxlqj dv deryh/ iru hdfk  / \?|

(246) > I | lv dq O2 0pl{lqjdoh ri vl}h 4@5 e| Ohppd D14/ zlwk pl{lqjdoh frqvwdqwv fm ?|

(247)  N pd{ in\?|

(248) n

(249) o > 4j  N pd{ in]?| n

(250) o > 4j zklfk duh erxqghg +xqlirupo| lq q> w dqg  ,1 Wkxv/  6  5 5 6    [    ?3ƒ

(251) ?3ƒ

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(253)   %8  \?|

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(255) '3n  |'

(256) '3n. 48.

(257) 5 . 4 9 7 +q  c . 4, %. 3 [.

(258) '3n. 5 . 4 9 7 +q  c . 4, %. 42 4*2 6 3 3   ?3ƒ

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(260) D D 8 H c |'. 4*2 6   ?3ƒ

(261) ƒ cq*2 m m C [  m 2 D : f?|

(262) 4 N $ 3> 8N c qc.4 3. 3 [.

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(264) j iru {hg  / dqg wkh odvw lqhtxdolw| krogv e| wkh xqlirup erxqghgqhvv ri fm ?|

(265) = Wkh surri ri D?2 @ r +4, iroorzv vlploduo|1 Wkh surri ri wkh wkhruhp iru wkh VE iroorzv forvho| wkdw iru wkh PEE/ dqg zh rqo| suhvhqw wkh uho0 a ?f +3, . 5 S?3 e?

(266) U a ?f + , @ q3 S?3

(267) ]?| ]?c|n

(268) / a ?f + , > zkhuh U  ?c2 @ U hydqw ghwdlov1 Lq vwhs 4/ ohw

(269) '  |'  ?c2  ? $ 31 dqg iroorz Jdoodqw dqg Zklwh +4<;;/ s1 444> vhh dovr Qhzh| dqg Zhvw/ 4<;:, wr vkrz H   1 1 3 Ohw i? + ,  4t

(270) $?3 me?

(271)  4m  2

(272) o . y^

(273) ` dqg  eh wkh frxqwlqj phdvxuh rq wkh srvlwlyh lqwhjhuv1 ^4` 4       ?c2  ?  E| wkh grplqdwhg frqyhujhqfh wkhruhp/ olp?<" H   1 1 U" S?3 3 2 o  olp?<" 5N

(274) ' me?

(275)  4m 

(276) . y^

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(278) ' 

(279) . y^

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(281) S?3 2 . 5 3 a Lq vwhs 5/ ghqh V?c2 @ q3 ?|' [?|

(282) ' e?

(283) q |' [?| [?c|n

(284) dqg ohw F? + , S  ?2 ghqrwh wkh circular dxwrfryduldqfh/ zklfk lv edvhg rq wkh extended wlph @ q3 ?

(285) ' [?| [?c|n

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(287) +4  s,

(288) > a ?c2 @ V?c2 [ q3 ?3

(289) [?c|n

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(294).  4 +4  s,

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(299) 4 

(300) +4  s,

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(302) ' ?        ?   2 @ R q3 e| d FOW iru wkh vdpsoh phdq/ dqg jlyhq wkdw zklfk lv R ? > jlyhq wkdw [ 49.

(303)   3  S?3

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(320) '. ?3

(321) [ |'.   +?|  ? , ?c|n

(322)  ? =.     duh R q3 dqg R ?? uhvshfwlyho|/ dqg wkdw wkh uhpdlqlqj whupv duh. r +4, e| dq dujxphqw vlplodu wr wkh rqh xvhg iru wkh PEE wr vkrz wkdw D? @ r +4, = E| Ohppd 4 lq s  WE2 1 ¥ q ? Srolwlv dqg Urpdqr +4<<7d,/ X?c2  yduW Proof of Lemma 2.11 Lpphgldwh iurp wkh surri ri Wkhruhp 5141 ¥ Proof of Corollary 2.11 Lpphgldwh iurp Wkhruhp 514 dqg wkh uhpdun wkdw iroorzv lw1 ¥ Proof of Theorem 2.21 +l, iroorzv e| Wkhruhp 816 lq Jdoodqw dqg Zklwh +4<;;, dqg Sro|d*v wkhruhp +h1j1 Vhu lqj/ 4<;3,1 Wr suryh +ll,/ uvw qrwh wkdw     s   WE s   WE s  W  WE   ? @ 3*2 3*2 q [?  [ q ]?  H W ]?WE . 3*2 q H ]?  ]? ? ? ?  s  WE . 3*2 q    ? ? ? E E  DE ? . E? . F? > WE. WE. WE. zkhuh ]?|  [?|  ?| dqg ]?|  [?|  ?| 1.     WE E Proof for m @ 4. E| Ohppd D14 ri Ilw}hqehujhu +4<<:,/ H W ]? @ ]? . R ? 1 Wkxv/ E? @       E R c@q*2 @ r +4,/ jlyhq wkdw ? A  A 3 dqg c? @ r q*2 1 Dovr/ H W F? @ R +c@q, $ 3 dqg   W  E 3*2 E E 2 $ 3/ zklfk lpsolhv F yduW F? @ ? X?c ? $ 3= Khqfh/ lw vx!fhv wr suryh wkdw D? , Q +3> 4,/ S WE jlyhq ]? > = = = > ]?? > zlwk suredelolw| dssurdfklqj rqh1 Zulwh ]? @ n3 &' X? > zkhuh iX? j duh     ~ nn~ WE  l1l1g1 zlwk S W X? @ ?c+1  ?c+ @ ?3n @ H W +X? , dqg > m @ 3> = = = > q  c= Wkxv/ H W ]?   S S E 3*2 s 3*2 D? @ ? q n3 &' ^X?  H W +X? ,`  &' ]? > zkhuh ]? @ ? q3*2 c ^X?  H W +X? ,`/     q r. P = E| Ndw}*v jlyhq n @ ? 1 Lq sduwlfxodu/ ]? duh l1l1g1 zlwk H W ]? @ 3 dqg yduW ]? @ n3 P?c1 ? +4<96, Ehuu|0Hvvhhq Erxqg/ iru vrph vpdoo  A 3>. 4:.

(323)   3 4   $33B*2 #   S&   2nB a ?c F   WE ]. ? W  ' F  +{,  N u vxs S E nH =  { ]   ? S   C D ? %MR   & W  ydu   ' ]? Vlqfh. ?c1  P P? $. 4 e| Fruroodu| 514/ lw vx!fhv wr vkrz wkdw.  2nB   W     H nH ]?   . . 2nB.   nH W ]? .  %  ?3 [  [ 4 q   ]   ?c|n B nB*2 n   c +q  c . 4, q 2 ? |' 'f. . $ 31 Exw &2nB . ncH W +X?c ,n2nB. /. +D18,. 2nB. zkhuh wkh lqhtxdolw| iroorzv e| wkh Plqnrzvnl lqhtxdolw|1 Xqghu rxu dvvxpswlrqv/    3 4*2    [   n n  [ [      ]?c|n   pd{  ]?|  NC f2?| D  Nc*2 >      $$     |' |'n |'n 2nB 2nB. e| Ohppdv D16 dqg D17/ jlyhq wkdw wkh f?| duh xqlirupo| erxqghg1 Vlploduo|/ ncH W +X?c ,n2nB @ 2nB   B*2   W     *2     / zklfk iurp +D18,/ lpsolhv H nH ]?   @ R ? $ 3/ jlyhq c? @ r q*2 1 R c   E2 WE W  Proof for m @ 5. Iluvw/ qrwh wkdw E? @ 3 vlqfh iru wkh VE zh kdyh H ]? @ ]? 1 Vhfrqg/    S E2  W WE2 @ 3> e| wkh vwdwlrqdulw| ri wkh VE uhvdpsolqj F? $ 3> jlyhq wkdw H W q3*2 ?|' ?|  ?|    S WE2 ? vfkhph/ dqg jlyhq wkdw yduW q3*2 |' ?|  ?|  X?c2 $ 3 e| Dvvxpswlrq 5151 Wr suryh wkdw E2. D? , Q +3> 4,/ zh yhuli| wkh frqglwlrqv +F4,/ +F5, dqg +F6, lq Srolwlv dqg Urpdqr*v +4<<7, surri ri wkhlu Wkhruhp 5 dqg uhihu wr wkhlu surri iru pruh ghwdlov1 Lq rxu pruh jhqhudo fdvh/ zkhuh ? lv qrw dvvxphg wr kdyh d olplw ydoxh 2" / wkh dssursuldwh yhuvlrq ri wkhvh frqglwlrqv lv dv iroorzv= (C1). 2  7? ?~ ?R $. 3>. S 

(324) a (C2) Fa? +3, . 5 "

(325) ' +4  s, F? + ,  ? $ 3>  2nB   R S" S? ~7?   +4  s,K3 s $ 3> (C3) 1+ V

(326) cK B K'

(327) ' R  ?. 2. zkhuh lq +F6, V

(328) cK lv ghqhg dv wkh vxp ri revhuydwlrqv lq eorfn E

(329) cK 1   Proof of (C1)= Wklv iroorzv iurp ]? @ R q3*2 +e| wkh FOW, dqg qs? $ 41 a ?c" @ Fa? +3,.5 S" +4  s,

(330) Fa? + , dqg a ? @ Fa? +3,.5 S?3 +4  s,

(331) Fa? + ,1 Proof of (C2)= Iluvw/ ghqh

(332)

(333) ' Qh{w/ qrwh wkdw Fa? +l, @ Fa? +l . qm, iru doo l @ 3> 4> = = = > q  4 dqg m @ 4> 5> = = =/ e| wkh flufxodulw| surs0 a ?c" @ a ? . 5 S" S?3 +4  s,?n Fa? +qm . l, @ huwlhv ri wkh h{whqghg wlph vhulhv1 Lw iroorzv wkdw ' 'f  S " ? a a a ? . ? . F? +3, ' +4  s, = E| dq dujxphqw vlplodu wr wkh surri ri Wkhruhp 514 iru m @ 5/ zh   a ?  ? $ a ?c"  ? $ a ? dqg Fa? +3, duh R +4, dqg S" +4  s,? fdq vkrz wkdw 31 Khqfh/ 3> vlqfh ' lv r +4,/ jlyhq qs? $ 4 dqg s? $ 3= 4;.

(334) S 2nB B  nK3  Proof of (C3)= Wklv iroorzv surylghg H 

(335) |'

(336) ]?|   Nen 2 > zkhuh wkh frqvwdqw N rqo| ghshqgv rq wkh pl{lqjdoh frh!flhqwv ri i]?| j1 Dsso| Ohppd D171 ¥ Proof of Corollary 2.21 Lpphgldwh iurp wkh surri ri Wkhruhp 515 iru m @ 4= ¥. References ^4` Dqguhzv/ G1Z1N1 +4<;7, Qrq0vwurqj pl{lqj dxwruhjuhvvlyh surfhvvhv1 Journal of Applied Probability 54/ <630<671 ^5` Dqguhzv/ G1Z1N1 +4<;;, Odzv ri odujh qxpehuv iru ghshqghqw qrqlghqwlfdoo| glvwulexwhg udqgrp yduldeohv1 Econometric Theory 7/ 78;079:1 ^6` Elfnho/ S1M1 ) G1D1 Iuhhgpdq +4<;4, Vrph dv|pswrwlf wkhru| iru wkh errwvwuds1 Annals of Statistics </ 44<90454:1 ^7` Eloolqjvoh|/ S1 +4<9;, Convergence of probability measures1 Qhz \run= Zloh|1 ^8` Eroohuvohy/ W1 +4<;9, Jhqhudol}hg dxwruhjuhvvlyh frqglwlrqdo khwhurvnhgdvwlflw|1 Journal of Econometrics 64/ 63:065:1 ^9` Fduudvfr/ P1 ) [1 Fkhq +4<<<, Ehwd0pl{lqj dqg prphqw surshuwlhv ri ydulrxv JDUFK/ vwrfkdvwlf yrodwlolw| dqg DFG prghov1 Orqgrq Vfkrro ri Hfrqrplfv/ pdqxvfulsw1 ^:` Gdylgvrq/ M1 +4<<7, Stochastic limit theory1 R{irug= R{irug Xqlyhuvlw| Suhvv1 ^;` Hqjoh/ U1I1 +4<;5, Dxwruhjuhvvlyh frqglwlrqdo khwhurvnhgdvwlflw| zlwk hvwlpdwhv ri wkh yduldqfh ri Xqlwhg Nlqjgrp lq dwlrq1 Econometrica 83/ <;:043391 ^<` Ilw}hqehujhu/ E1 +4<<:, Wkh prylqj eorfnv errwvwuds dqg urexvw lqihuhqfh iru olqhdu ohdvw vtxduhv dqg txdqwloh uhjuhvvlrqv1 Journal of Econometrics ;5/ 56805;:1 ^43` Jdoodqw/ D1U1 ) K1 Zklwh +4<;;, A unified theory of estimation and inference for nonlinear dynamic models1 Qhz \run= Edvlo Eodfnzhoo1 ^44` Jrqêdoyhv/ V1 ) K1 Zklwh +5333, Pd{lpxp olnholkrrg dqg wkh errwvwuds iru qrqolqhdu g|qdplf prghov1 XFVG Zrunlqj Sdshu 53330651 ^45` Kdoo/ S/ M1 Krurzlw}/ ) E1 Mlqj +4<<8, Rq eorfnlqj uxohv iru wkh errwvwuds zlwk ghshqghqw gdwd1 Biometrika ;5/ 8940:71. 4<.

(337) ^46` Kdqvhq/ E1H1 +4<<4d, JDUFK+4/4, surfhvvhv duh qhdu hsrfk ghshqghqw1 Economics Letters 69/ 4;404;91 ^47` Kdqvhq/ E1H1 +4<<4e, Vwurqj odzv iru ghshqghqw khwhurjhqhrxv surfhvvhv1 Econometric Theory :/ 54605541 ^48` Kdqvhq/ E1H1 +4<<5, Huudwxp1 Econometric Theory ;/ 75407551 ^49` Krurzlw}/ M1 +4<<<, Wkh errwvwuds1 Iruwkfrplqj lq Handbook of Econometrics/ Yro1 81 ^4:` Ndw}/ P1O1 +4<96, Qrwh rq wkh Ehuu|0Hvvhhq wkhruhp1 Annals of Mathematical Statistics 67/ 443:0 443;1 ^4;` Nÿqvfk/ K1U1 +4<;<, Wkh mdfnnqlih dqg wkh errwvwuds iru jhqhudo vwdwlrqdu| revhuydwlrqv1 Annals of Statistics 4:/ 454:045741 ^4<` Odklul/ V1Q1 +4<<5, Hgjhzruwk fruuhfwlrq e| cprylqj eorfn* errwvwuds iru vwdwlrqdu| dqg qrqvwd0 wlrqdu| gdwd1 Lq OhSdjh dqg Elooldug +hgv,/ Exploring the limits of the bootstrap/ ss1 4;605471 Qhz \run= Zloh|1 ^53` Odklul/ V1Q1 +4<<<, Wkhruhwlfdo frpsdulvrqv ri eorfn errwvwuds phwkrgv1 Annals of Statistics 5:/ 6;907371 ^54` Olx/ U1\1 ) N1 Vlqjk +4<<5, Prylqj eorfnv mdfnnqlih dqg errwvwuds fdswxuh zhdn ghshqghqfh Lq OhSdjh dqg Elooldug +hgv,/ Exploring the limits of the bootstrap/ ss1 557057;1 Qhz \run= Zloh|1 ^55` PfOhlvk/ G1O1 +4<:8, D pd{lpdo lqhtxdolw| dqg ghshqghqw vwurqj odzv1 Annals of Probability 8/ 94909541 ^56` Qhzh|/ Z1N1 ) N1G1 Zhvw +4<;:, D vlpsoh srvlwlyh vhpl0ghqlwh/ khwhurvnhgdvwlf dqg dxwrfruuh0 odwlrq frqvlvwhqw fryduldqfh pdwul{1 Econometrica 88/ :360:3;1 ^57` Srolwlv/ G1 ) K1 Zklwh +5334, Dxwrpdwlf eorfn0ohqjwk vhohfwlrq iru wkh ghshqghqw errwvwuds1 XFVG pdqxvfulsw1 ^58` Srolwlv/ G1 ) M1 Urpdqr +4<<7d, Wkh vwdwlrqdu| errwvwuds1 Journal of American Statistics Association ;</ 4636046461 ^59` Srolwlv/ G1 ) M1 Urpdqr +4<<7e, Olplw wkhruhpv iru zhdno| ghshqghqw kloehuw vsdfh ydoxhg udqgrp yduldeohv zlwk dssolfdwlrqv wr wkh vwdwlrqdu| errwvwuds1 Statistica Sinica 7/ 79407:91 53.

(338) ^5:` Srolwlv/ G1/ Urpdqr/ M1 ) P1 Zroi +4<<:, Vxevdpsolqj iru khwhurvnhgdvwlf wlph vhulhv1 Journal of Econometrics ;4/ 5;4064:1 ^5;` Vhu lqj/ U1M1 +4<;3, Approximation theorems of mathematical statistics1 Qhz \run= Zloh|1 ^5<` Vlq/ F10\1 ) K1 Zklwh +4<<9, Lqirupdwlrq fulwhuld iru vhohfwlqj srvvleo| plvvshflhg sdudphwulf prghov1 Journal of Econometrics :4/ 53:05581 ^63` Vlqjk/ N1 +4<;4, Rq wkh dv|pswrwlf dffxudf| ri Hiurq*v errwvwuds1 Annals of Statistics </ 44;:0 44<81. 54.

(339)

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