Digital Search Trees and Chaos Game Representation
31
0
0
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Digital Search Trees and Chaos Game Representation Peggy Cénac — Brigitte Chauvin — Stéphane Ginouillac — Nicolas Pouyanne. N° 5856 March 2006. ISSN 0249-6399. ISRN INRIA/RR--5856--FR+ENG. Thème BIO. apport de recherche.
(3)
(4)
(5) ! #"$%&(')+* , - . ! /1032424576983:;4< =?>9@BA 2 ADCEC 0F69G?;4H,I A : =KJLC 8NMG?;4:,0PO A :Q4H ASRTR ;4< =?UVA <NQ R ;4WX/1QYH,5Y;4::0 Z.[]\_^F`Va.bcdfehgYikj\_^F`limonqpsrqpstunqvhw]`li xyp3zk`_jxy`_{u|3r }9|3~o~,psykj o`Xy`l[]`_y[]` uu ud|3y[uuuVdf ~L|3t`li ∗. †. †. †. lNou bjk[onqi~L|3~,`_yN1`lps]inqo`_y |.~puiikn¡morq`+y`_~oy`li`_hj|3jknqps¢p3£L|.¤X¥-¦§i`lvhw]`_]l`n¡ |XvhwL|3j`_ykL|3ykg jky`l`un¡¨[onq[ps]`|3F{hnqiwL|3r¡n¡©`.y`_~,`_jkn¡jknqps]i1p3£,ikwomh1psyoi-ª«i`l`_¬|siikwY¨®o`li p3£?ikwom]i`lvhw]`_]l`liB¯Z.[]`-°±}²³jky`l`jkwoyk]i´|Xi`lvµw]`_]l`p3£,rq`_jkj`_yi¶n¡µjpV|¤9n¡tun¡j|3r,e4`|3y[Zy`l` ª«¤·e4Z¯Ehpsmoj|3n¡]`l¨£«yps^¸jk[]` ikwY¨®o`li¶p3£?jk[]`y`_{`_yi`l¬i`lvhw]`_]l`9e4`_{`_y|3r y`likwor¡ji1|3y`¹h]pº lps]l`_ykon¡otVjk[]`1[]`_n¡tu[hjjk[]`.n¡]i`_ykjknqpsFo`_~ojk[»£¼psy¤·e4Z½mowon¡r¡j+£«yps^¾n¡]o`_~,`_]o`_hj1ikw]ll`liikn¡{` y|3]ops^¿i`lvµw]`_]l`li[L|{hn¡otÀjk[]`¨iB|3^F` Ynqijkykn¡mowojknqps7Á-`_y`»jk[]`»ikw]ll`liikn¡{`»n¡]i`_ykj`l 1psyoi |3y`-ikjkypsotur¡gFo`_~`_]o`_µj´ÂÃ` tun¡{` jk[]`9|sikg4^¨~ojpsjknq m`_[L|l{4nqpswoy´p3£?jk[]` n¡]i`_ykjknqpso`_~ojk[P|3] rq`_otujk[§p3£ moy|3][]`li¢£¼psyVjk[]`°±}²³jky`l`¬psmoj|3n¡]`l7£«yps^Äjk[]`¬iwY¨®Y`li¢p3£-|Ày`_{`_yi`l7n³Å n³ÅÆ?Å psy |3yk¹p{4nD|3i`lvhw]`_]l`9¦9i |¢mµgh²³~oyphYw]_j]|sigh^¨~ojpsjknqXy`likwor¡jipsPjk[]`Xrq`_otujk[Àp3£ rqpsot`lij ykwo]i n¡Ç|»|3yk¹pº{hnD|3Çi`lvµw]`_]l`¢|3y`·psmoj|3n¡]`l ÈÉÊËÌ¢Í NÎu }9|3]ops^jky`l`uµ¤9n¡tun¡j|3re4`|3y[Zy`l`uY°±X}¢urq`_otujk[]i¶p3£?jk[]`~L|3jk[]iº[]`_n¡tu[µj n¡]i`_ykjknqpsÏo`_~ojk[KL|sikg4^¨~ojpsjknqtuypjk[K ikjkypsot¬lpsh{`_ykt`_]l`. ∗ †. ÐSë4Ñ ÓÒ õXÐSÓÓÔâÕEöÖ ÷³ØøÖ×BØÚÛáÙNÙÛ÷ÜNòNÛÛáó³ÝLÖ ÕEÞÚ×ß3÷kàáùºÛkú×Bß9ÙNØÚû_â Ûáã ó³äoî¼ãÚØÆå÷³æ_üç ÜNèBÛé-Ý åkÛáçBó³î³ê×ëµØÚÞøÞÚÛ¶Ûîì,ò4í3ÛýSèBî¼Ù3éEæB×kê_ï9ç´ì,ÝLÛkÛáܺó³Ûáî³ð¢×BØÚñqÞÚÞÚòNÛóî ×BÙNàÛô. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.
(6) ¢ , , +, o %& ¢-%&
(7) ¢ *! ¢ ¢ ( ¢
(8) ¢»! !
(9) ¢ |»y`_~o y li`_µj|3jknqps "!]onq`nq_n`likj wo]`Xy`_~oy li`_µj|3jknqps~,puiin¡morq`·o`i lvhw]`_]l` $#Ʀ9¤X¥¸]|3]iwo7|3ykmoy`¨vhwL|3j`_ykL|3n¡y`Fopshj·rD|lps]ikjkykw]_jknqpsÃ~`_yk^F`_jVo` {hnqikwL|3r¡nqi`_yrq`liX y E² ~% _jkn¡jknqps]i.o`XikwY¨®Y`l'i & ~L|3ykjkn¡y$ # wo]`X i lvhw]`_]l`Xo`rq`_jkjky`liYpsÀlps]ikjkykwon¡j.wo|3ykmoy`XYn¡tun¡j|3r o`-y`l[]`_y[]`Fª (*),+-)/.0-1 2%340-546"7985 3 3N¯1ikwoy1r #Æ`_]i`_^Vmorq`Xo`li.ikwY¨®o`li1o`-rD|¢ i lvµw]`_]l`n¡µ{`_y i l` ¤X`li y liwor¡j|3ji·ikwoyrD|[L|3woj`_woy¢`_jrD|~oyp3£¼ps]o`_woyV$ # n¡]i`_ykjknqps§psµ j _ j : _j|3mor¡nqiVrqpsyivhw]`»rq`li ;i lvµw]`_]l`l i <~orD|sl`_y·]|3]ir #Ú|3ykmoy`¨ipsµjn¡] _~`_]]|3µj`li¢rq`liXwo]`li·o`li·|3wojky`li·bá_nrq`li·^Fpsji <n¡] i _y`_yipshjX£«psykj`_^F`_hjV _~`_]]|3µji¢c- opso]`¨o`liX~oyps~oyk=n _ j liV|sikg4^¨~ojpsjknqvhw]`li·ikwoyXrD| ~oyp3£¼ps]o`_woyX$ # n¡]i`_ykjknqps`_j-rq`lirqpsotuw]`_woyiXo`li-moy|3][]`li ~pswoy-wo |3ykmoy`Vpsmoj`_µ>w <F~L|3ykjkn¡y o`li+ikwY¨®o`li+$ # wo]`. i lvµw]`_]l`´n³Å n³ÅÆ?Åspsw ^¬|3yk¹p{4nq`_o]`.y`_jpswoyk l`¤9`1~or¡w]iul`_ykj|3n¡]i+ y likwor¡j|3ji ~`_wo{`_µjX|3w]iiknK?i # n¡hj`_yk~o y _j`_y-lps^¨^F`¢o`li y likwor¡j|3jio`Vlpsµ{`_ykt`_]l` ikwoy rq`lirqpsotuw]`_woyi-o` ~or¡w]i rqpsotuw]`li y _%~ _jkn¡jknqps]i$ # wo]`rq`_jkjky`·]|3]iwo]` i lvhw]`_]l`·|3yk¹pº{hnq`_o]`uÅ @#Í Ë B A ¦ykmoy`À-| |3jpsn¡y`u¶|3ykmoy`Po`y`l[]`_y[]`Yn¡tun¡j|3r³rqpsotuw]`_woyi¨o`li moy|3][]`li [L|3woj`_woy]~oyp3£«ps]o`_woy9$ # n¡]i`_ykjknqpsKL_ypsnqiiB|3]l`·|sikgh^¨~ojpsjknqvhw]`uLlpsh{`_ykt`_]l`£¼psykj`.
(10) (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(11) 3 3 5 3 3.0 ./) . . !#"$%& . b jk[]`VrD|sikjXg`|3yi¤X¥9¦$[L|si9m,`l`_ y`_~oy`li`_hj`l mµg^F`|3]ip3£1i`_{`_y|3r^F`_jk[]p4oi9n¡ psyo`_y j p^¬|3¹`P~L|3jkj`_yk {4nqikwL|3r¡n¡©º|3jknqps `|siknq`_y¬|3]§jp o`_j`l_j¨rqph|3r psy»turqpsmL|3r ikn¡^¨n¡rD|3ykn¡jknq`liǪ«i`l` £¼psyPn¡]ikj|3]l`Ã}-pg `_jÀ|3r³Å('Úu*)S¯EÅ Z.[]`+ 7 0 , 0
(12) 3- 3 5 3 3.0 ./) ª³°±X}9¯~oypº{hnqo`li mpsjk[ | tuy|3~o[onq|3rXy`_~oy`li`_µj|3jknqps |3] |7ikjpsy|3t`Ïjp4psr³Å/.oyps^ |Ãi`lvhw]`_]l`Ïn¡ |>!]on¡j` |3r¡~o[L|3m`_ju°±X}7o"` !]]`li|.jky|Nz`l_jpsykgXn¡¢|´mpswo]o`l·ikwom]i`_jp3£ jk[L|3jK¹`l`_~]i|3r¡rµikj|3jknqijknq|3r ~oyps~`_ykjknq`lip3£ jk[]` i`lvµw]`_]l`uÅ ° _L|s,'Ú *)° _L|s `_j|3r³Å0'Ú*)VRikjkw]Yg jk[]`7°±}&n¡jk[ |3 `E®Yj`_]iknqpsÃp3£´1psy4²lpswoµjkn¡otmL|si`l^F`_jk[]p4oi·p3£.|3L|3r¡gYiknqi2Å 1` 3y`_4g '657 )´|si·jk[]` !]yikjjp |3~o~or¡gjk[onqin¡j`_y|3jkn¡{`V^F`_jk[]phÀjp¨¤X¥9¦ i`lvhw]`_]l`liÅb Àjk[onqi lpshj`E®4j]i`lvµw]`_]l`li|3y`·^¬|so` p3£ 4 µw]_rq`lpsjknqo`li-L|3^F`lϦ(ª¼|so`_on¡]`º¯E °$ª«_g4jpuikn¡]`º¯E ± ªStuwL|3on¡]`º¯|3]ZªSjk[µg4^¨n¡]`º¯EÅ Z.[]`X°±X} p3£ |¢i`lvhw]`_]l` p3£rq`_jkj`_yi £«yps^(| !]on¡j`|3r¡~o[L|3m,`_j nqi¶jk[]` i`lvµw]`_]l` (X ) p3£+~psn¡µji n¡UÏ|3.. .|3U~o~oy.ps.~o. yknD|3j`Vlps^¨~LU|s_jikwom]i`_j S p3£ R o"` !]]`lAmµg d. 1. n. n. i i≥0. . d. X0 ∈ S X i+1 = θ X i + `Ui+1 ,. []`_y` θ nqi |¢y`|3r?~L|3y|3^F`_j`_y ª 0 < θ < 1¯Eo`|s[rq`_jkj`_y u ∈ A m,`_n¡ot¬|siikn¡tu]`lPjp¨| tun¡{`_ ~psn¡µj ` ∈ S Å b Ïjk[]`·~L|3ykjknq_worD|3y|si`¢p3£!1`3y`_g#Æi-y`_~oy`li`_µj|3jknqpsK A = {A, C, G, T } nqi jk[]`9i`_j´p3£Kµw]_rq`lpsjknqo`li S = [0, 1] nqi1jk[]` woon¡j.ivhwL|3y`uÅ 8¶|s[Prq`_jkj`_y´nqi¶~orD|sl`lP|3j|{`_ykj`E® |si£«psr¡rqpº :i 9 u. 2. |3]Pjk[]` !]yikj~,psn¡hj nqi.jk[]`l`_hj`_yp3£jk[]`ivµwL|3y`uÅZ.[]`_KYn¡j`_y|3jkn¡{`_r¡gLjk[]`X~,psn¡hj nqijk[]`^¨nqoYrq`·p3£+jk[]`·Xi`_tu^F`_hjm`_j1`l`_ X |3]Àjk[]`·ivµwL|3y` #Æi{`_ykj`E® ` 9 `A = (0, 0),. θ = 21 X i+1. psyo`lvhwon¡{u|3rq`_µjkr¡g. `C = (0, 1),. `G = (1, 1),. `T = (1, 0),. 0. i. X i+1 =. Xi =. Ui+1. X i + `Ui+1 , 2. i X ` Uk X0 + i . i−k+1 2 2. n¡tuwoy` 5Xy`_~oy`li`_µji-jk[]`lps]ikjkykw]_jknqpsÇp3£+jk[]`1psyϦZ±·°±X¦±XZ±XZÅ Â n¡jk[7`|s[½o`_j`_yk^¨n¡onqikjknq¬1psy w = u . . . u 1`|siip4_nD|3j`¨jk[]`¨[L|3r £S²ps~,`_]`l½ikwomY² ivµwL|3y` Sw o`"!]]`lÀmµgPjk[]`9£¼psyk^VworD| .. k=1. 1. Sw ; =<>=. n X k=1. Ò Ò Ù?]çBéEçA@. ` uk n−k+1 2. n. +. 1 [0, 1[2 ; 2n.
(13) 3 +?+ 0B6 . 5"),+-) . . 3 7 0
(14) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. ¡n jÀ[L|siÏl`_µj`_y P ` /2 + X /2 |3]iknqo` 1/2 Å .]psyÇ|§tun¡{`_y|3]ops^ psy o`_j`_yk^¨n¡onqijknqÃi`lvhw]`_]l` £«psyÏ|3hg ´psy |3] |3hg lpswoµjkn¡ot jk[]`ϵwo^¢m,`_yÀp3£·~psn¡µji (XU ). . . U . jk.[L. |3jm`_rqpsot½jp7jk[]`Ïw ikwom]ivµwL|3y` nSw≥ nqi¬|w|j|3hj|3^Fpswoµj jpÀlpswohjkn¡otÇjk[]`»hwo^Vm`_y p3£p4l_woy`_]l`liVp3£ w |siV|Pikwomh1psy½p3£ U . . . U ÅFb ]o`l`l? Sw lpshj|3n¡]i¶|3r¡rYjk[]`ikw]ll`liikn¡{`´´psyoi+£Syps^¾jk[]`.i`lvhw]`_]l`´[L|l{4n¡ot w |si|-iwY¨®Åe4`l` .n¡tuwoy`05 £¼psy|3`E®]|3^¨~orq`n¡jk[jk[oy`l`E²³rq`_jkj`_y-iwomµ1psyoiºÅ¶Z.[onqi~oypº{hnqo`lij|3morq`lip3£+1psyP£«y`lvµw]`_]_nq`li ª«i`l`ϱXpsrqY^¬|3 '65:*)S¯EÅ c-]`À|3 t`_]`_y|3r¡n¡©`Àn¡j¬jpÃ|3µg ikwom,Yn¡{4nqiknqps p3£Xjk[]`Pwoon¡j¬ivhwL|3y` []`_½jk[]`Fhwo^Vm`_y p3£ ikwom]ivhwL|3y`li·nqiV]psj |P~p´`_y p3£ 4 jk[]`Fj|3morq`¬p3£´psy £«y`lvµw]`_]_nq`li o"` !]]`li9|»lpswoµjkn¡otp3£´psyoi n¡jk[]pson¡hj`_t`_y-rq`_otujk[½ª«i`l`¦r¡^F`_nq]|¨`_j9|3r³Å 'Ú7 )S¯EÅ Z.[]` £«psr¡rqpºn¡ot~oyps~,`_ykj g p3£1jk[]`¬°±X}$nqi·n¡^¨~,psykj|3h:j 9 . 7 3
(15) 0-1 3 0 X 6 .0 ) b ]o`l`l?4]psjknql` !]yikj´jk[L|3j . 7 3 7B) . -5") 6 0-1%) -5
(16) 0 ./) . 7 3 7$ 1,3 3 3 6 3 X ,...X mhg lps]ikjkykw]_jknqpsK n¡jk[ U = u jk[]`À[]psrq`i`lvhw]`_]l`Ànqi¨]pº tun¡{`_ mhg§jk[]` Å n¡]Yw]_jkn¡{`·£«psyk^¢worD| XX ∈ Su = 2X − ` ÂÃ`o"` !]]`|9y`_~oy`li`_µj|3jknqpsp3£?|9y|3]ops^¤9¥9¦ i`lvµw]`_]l` |si1|9y|3]ops^ vhwL|3j`_ykL|3ykgjky`l`uKjk[]` ! ./543 3ln¡ [onq[7ps]`¨|3Ã{4nqikwL|3r¡n¡©`»Uy=`_~,`_(Ujkn¡jknq)ps]i¢p3£´ikwomh1psyoiÅ ÂÃ` |sops~oj-jk[]`¢_rD|siiknq|3rpsyo`_y (A, C, G, T ) psÇrq`_jkj`_yiÅ `_j T m`jk[]`¢lps^¨~orq`_j`¢n¡ !]on¡j` ² |3ykg jky`l ` `|s[½]p4o`p3£ [L|siV£¼pswoyVmoy|3][]`li»lpsyky`lik~ps]Yn¡otjpÏrq`_jkj`_yi 4 jk[L|3j¬|3y`Àpsyo`_y`l n¡ jk[]`ÀiBT|3^F`P´|lgÅ Z.[]`Ï°±X} ²³jky`l`Àp3£ U nqiF|3 n¡]_y`|sin¡o(A,tÃiC,`lvµw]G,`_]Tl`) p3£ !]on¡j`·ikwomY²³jky`l`lip3£ L`|s[ [L|l{4n¡ot n ]p4o`liÅZ.[]` T #Æi T ⊂ T ... ⊂ T ⊂ ... |3y`Xmowon¡r¡jmµgÀikw]ll`liikn¡{`_r¡gn¡]i`_ykjkn¡otFjk[]` 543
(17) -3"5 3 T 5 3#"$ 3 T ª 5º¯ W (n) = U . . . U |si£«psr¡rqpº iVn¡ jk[]`lps^¨~orq`_j`Fn¡ !]on¡j`Fjky`l`uÅZ.[]`F¤X¥-¦ i`lvhw]`_]l` nqiVy`|s £«yps^¿rq`E£SjVjp ykn¡tu[hjÅ . n¡yijrq`_jkj`_y W (1) = U nqi n¡]i`_ykj`lÏn¡jk[]`Vlps^¨~orq`_j`¢n¡ !]oUn¡j`·jky`l` |3j rq`_{`_r 1 ) 3 zw]ikj-wo]o`_y-jk[]`Vyphpsj|3j-jk[]`V]pho`¢jk[L|3jXlpsyky`lik~,ps]oi-jpjk[]`·rq`_jkj`_y U Å b ]Yw]_jkn¡{`_r¡g,jk[]` n¡]i`_ykjknqpsÃp3£¶jk[]`»1psy W (n) = U . . . U nqiX^¬|so`F|si9£«psr¡rqpº :i 9-jkykgÏjpn¡]i`_ykjn¡j·|3jXrq`_{`_r |3jjk[]`·]p4o` N jk[L|3j9lpsyky`lik~ps]oijp¬jk[]`·rq`_jkj`_y U Åb³£+jk[onqi]p4o` N nqi{s|s|3hjLn¡]i`_ykj 1 |3j N ´n £ N nqiF]psj¨{s|s|3hjjkykg§jpÇn¡]i`_ykj W (n) n¡ jk[]`Àikwomojky`l`P[L|l{4n¡ot N |si W (n) yp4psj?|3j-jk[]` ]p4o`Vjk[L|3jlpsyky`lik~ps]oiXjpjk[]`¢rq`_jkj`_y U ?|3] ippsKÅVc-]`¢y`_~`|3jiXjk[onqi ps~`_y|3jknqpswoµjkn¡r,jk[]`]p4o`-|3j1rq`_{`_r k jk[L|3j.lpsyky`lik~ps]oi´jp·rq`_jkj`_y U nqi¶{u|s|3µ%j Y´psy nqijk[]`_n¡]i`_ykj`lÇ|3jjk[L|3j ]pho`uÅ W (n) ÂÃ`lps^¨~orq`_j`pswoy lps]ikjkykw]_jknqpsmµgrD|3m`_r¡r¡n¡ot¨jk[]` n²³jk[Pn¡]i`_ykj`lP]p4o`Xn¡jk[Àjk[]`X1psy ÅFc-]`¨y`|sYn¡r¡gpsmoj|3n¡]iVjk[onqi´|g jk[]`»~oyphl`liiVp3£|PYn¡tun¡j|3r¶i`|3y[Ãjky`l`ª«¤·e4Z¯E|si W (n) ij|3j`lPn¡jk[]`X£¼psr¡rqpn¡otF~oyps~puikn¡jknqpsKÅ n k=1 uk. n−k+1. 1. n. 0. n. n. i 1≤i≤n. 1. n. i. 1. i. i. i. i−1. i. Ui. n n≥1. 1. 2. n. n. n. n. 1. 1. 1. n. 1. n. n−1. n−k+1. &('*),+)- .0/.),1324524 8 7 3 ! ./5 3 3 0 5 0
(18) 3 3 6 3 ) , 0 -),+-)/.0-1 = U U ... 3 40-546"7 ./543;36 7 .0-) 3 87 ) 3"5 ./) ) 09 0 . 3"5 0-5 ./5 3 3 U9 . 7 3 1 26;6;3 ):
(19) -3 543
(20) -3"5 3 543#" B $ 3 . 7 3 3< 3. 6;3 U1 U2 U1 U3 U2 U1 ; . ÐSÑÒЫÓ.
(21) . (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(22) 3 3 5 3 3.0 ./). C(0, 1). G(1, 1) S. C(0, 1). S. S. S A(0, 0). T (1, 0). G(1, 1). S . S . S . S . S . S . S . S . S . S
(23) . S . S . S . S . S . S . A(0, 0). T (1, 0). C [0, 1]. G [1, 1]. CCC GCC CGC GGC CCG GCG 5 CGG GGG ACC TCC AGC TGC ACG TCG AGG TGG 4 7 9 CAC GAC CTC GTC CAG GAG CTG GTG 3. AAC TAC ATC TTC AAG TAG ATG TTG 6 GGA CCT GCT CGT GGT CCA GCA CGA 8. ACA TCA AGA TGA ACT TCT AGT TGT10 1. CAA GAA CTA GTA CAT GAT CTT GTT 2. AAA TAA ATA TTA AAT TAT ATT TTT [0, 0] A. . [1, 0] T. 1° [L|spui¨±|3^F`F}`_~oy`li`_µj|3jknqps½p3£jk[]` !]yij 10 µw]_rq`lpsjknqo`li»p3£jk[]` -1 )jk[oy`E² pson¡]` t`_]`Vjk[oy¦29 ¦Z±·°±¦±XZ±XZÅXZ.[]`¢lp4psyYn¡L|3j`li9£¼psy9`|s[Çhw]_rq`lpsjknqo`F|3y` |3rq_worD|3j`l y`l_woyikn¡{`_r¡g½w]ikn¡ot 0.5) |si»ikj|3ykjkn¡ot ~,puikn¡jknqpsKÅ Z.[]`i`lvhw]`_]l`nqi y`|s£«yps^ rq`E£«j´jp¢ykn¡tu[hjŶxpsn¡µj(0.5, hwo^Vm`_y 3 lpsyky`li~,ps]oi.jp jk[]` !]yikj 3²³rq`_jkj`_y 1psy Åbjnqi rqph|3j`lPn¡jk[]`lpsyky`li~,ps]Yn¡otvµwL|sYy|3hjŶZ.[]`i`llps] 3²³rq`_jkj`_y-1psy AT G lpsyky`li~,ps]oijp»~,psn¡hj 4 |3]ip¨psKÅ T GC. Ò Ò Ù?]çBéEçA@.
(24) . 3 +?+ 0B6 . 5"),+-) . . 3 7 0
(25) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. . n¡tuwoy`Fy`_~oy`li`_µji£¼psyXn¡]ikj|3]l`¨jk[]`»m`_tun¡oon¡otp3£´jk[]`¨lps]ijkykw]_jknqps7p3£1jk[]`¬°±X} ² jky`l`¨p3£|i`lvµw]`_]l` GAGCACAGT GGAAGGG ÅFb jk[onqi!]tuwoy`uK`|s[Ã]pho` [L|sim`l`_ rD|3m`_r¡rq`lÀmhgn¡ji psyo`_y-p3£n¡]i`_ykjknqpsÇjp»^¬|3¹`jk[]`·`E®]|3^¨~orq`^Fpsy`y`|s]|3morq`uÅ Z.[]`^¬|3n¡ y`likwor¡jip3£pswoy¬~L|3~`_y|3y`jk[]`P£¼psr¡rqpn¡ot lpsµ{`_ykt`_]l`y`likwor¡jiº1jk[]`y|3Y² ops^ i`lvµw]`_]l` m,`_n¡otikwo~o~,pui`lPjp¨m,` |3yk¹p{4nD|3KŶb£ |3] o`_]psj`y`lik~`l_jkn¡{`_r¡g jk[]`¢rq`_otujk[ p3£¶jkU[]` ik[]psykj`likj·|3]Çp3£1jk[]`¢rqpsot`lij-moy|3][Ãp3`£¶jk[]`»°L±}²³jky`l`u,jk[]`_ ` / ln n 0 6
(26) 3"5 +B30-1
(27) . 5 3"1 .
(28) 3:6 .0. ª«Z.[]`lpsy`_^ Y6Å 5º¯EÅ9Çpsy`lpº{`_yn £ L / ln n o`_]psj`lijk[]` n¡]i`_ykjknqpsÃo`_~ojk[7|3] n £ nqiXjk[]`»rq`_otujk[Ãp3£´|woon £«psyk^¨r¡g []pui`_ y|3Y² D ops^ ~L|3jk[K4jk[]`_ D / ln n 0 M / ln n 6M
(29) 3"5 +B3 ) 2 5 740 7") 1 )/. . :0 6
(30)
(31) 6 .0. ª«Z.[]`lpsy`_^ ]Å65º¯EÅ ' 824 ¦ tun¡{`_ °±X} ²³jky`l`Pn¡jk[]pswojFn¡ji¨rD|3m`_rqiϪSn³ÅÆ`uÅ |tun¡{`_ ik[L|3~`Pp3£9jky`l`º¯»nqi `lvhwon¡{u|3rq`_µj¨jp |Çr¡nqikj p3£91psyoi¨n¡ jk[]`Pi`lvhw]`_]l`n¡jk[]pswoj¨jk[]`_n¡yFpsyo`_yÅ7psy`~oy`l_nqi`_r¡g ps]`Ç|3 |siip4_nD|3j`n¡jk[ |Ãik[L|3~`p3£¢°±X}²³jky`l`u | y`_~oy`li`_µj|3jknqps n¡ jk[]`Ïwoon¡jivhwL|3y` |si¢o`li_ykn¡m`lÃm`_rqp·ÅÀ n¡jk[§|3hg ]pho`p3£.jk[]`¬jky`l`ǪS[onq[§nqiVn¡7mon zk`l_jknqps7n¡jk[ |À´psy ¯EY´`V|siip4_nD|3j`jk[]`l`_µj`_y9p3£jk[]`Vlpsyky`lik~,ps]Yn¡otivµwL|3y` Sw w = U ...U n. n. n. n. n. n. n. 1. n. d. X w ; =<>=. d X. ` Uk d−k+1 2. +. X0 . 2d. .LpsyP`E®]|3^¨~orq`u . n¡tuwoy`çik[]pº iPjk[onqiÀip3²|3r¡rq`l
(32) -5
(33) :0 1 )3 5 3 5 3 3.0 ./)Ç£¼psy jk[]`»~L|3ykjknq_worD|3y¢1psy GAGCACAGT GGAAGGG Åpsy`lp{`_y .n¡tuwoy` P `_L|3morq`liVw]ijp vhwL|3r¡n¡j|3jkn¡{`_r¡g7lps^¨~L|3y`jk[]`psykn¡tun¡L|3r|3]Ãjk[]`]psyk^¬|3r¡n¡©`l °±X}y`_~oy`li`_µj|3jknqps]i¨ps |3 `E®]|3^¨~orq`uÅ e4`_{`_y|3r?y`likwor¡ji |3y`9¹h]pº ª«i`l`9[L|3~KŶ¢n¡À|3[o^Fpsw] '65* )S¯Eolps]l`_ykon¡ot¨jk[]`9[]`_n¡tu[µj jk[]`n¡]i`_ykjknqps o`_~ojk[ |3]§jk[]`~oyp !]rq`£«psy¨¤·e4Z(psmoj|3n¡]`l§£Syps^ ) B3 3B3.ikw]ll`liikn¡{` i`lvµw]`_]l`li-[L|l{4n¡ot jk[]` i|3^F` Ynqijkykn¡mowojknqpsKÅ bjPnqiP£³|3y£Syps^ pswoyikn¡jkwL|3jknqps []`_y` jk[]` iw]ll`liikn¡{`Fn¡]i`_ykj`lÃ1psyoi¨|3y`ikjkypsotur¡gÃo`_~,`_]o`_hj £Syps^ `|s[½psjk[]`_yÅ ¶|3yknqpsw]i¢y`liwor¡ji lps]l`_ykon¡otÇjk[]`¨ip3²|3r¡rq`lÃa.`_yk]pswor¡r¡n^Fp4o`_r-ªSmon¡L|3ykgjky`l`liKn¡]o`_~`_]o`_µj i`lvµw]`_]l`li¢|3] jk[]`Vj´p¬rq`_jkj`_yi-[L|l{`Vjk[]`Vi|3^F`V~oypsmL|3mon¡r¡n¡jg 5s¬p3£|3~o~`|3y|3]l`º¯|3Ïm`X£«pswo]Ïn¡Ï |3[Y² ^Fpsw] '65* )Ŧrqopsw]i-|3]Çeh[onq`_rqo(i '65 )K~oypº{`mµgÀ`_^¢m,`loYn¡otn¡Ïlpsµjkn¡hw]psw]ijkn¡^F`uojk[L|3jjk[]` []`_n¡tu[hj¬i|3jknqi !L`li H − log n → 0 n¡ ~oypsmL|3mon¡r¡n¡jgÅ ¦rqip ¤9yk^Fpsj,| 'ø* ) ~oypº{`liFjk[L|3j¨jk[]` []`_n¡tu[hjp3£ikw][À¤·e4Zi´nqi´lps]l`_µjky|3j`l 9 E[H − E(H )] nqi|sikg4^¨~ojpsjknq|3r¡r¡gFm,pswo]o`l£¼psy |3hg L > 0 Å .]psy1¤·e4Z lps]ikjkykw]_j`lF£Syps^¸n¡]o`_~,`_]o`_hj´i`lvhw]`_]l`li¶psP|3|3r¡~o[L|3m`_j¶n¡jk[ rq`_jkj`_yi n¡jk[§]ps]ikg4^¨^F`_jkyknqªSn³ÅÆ`uÅ]ps§`lvhwL|3r¶~oypsmL|3mon¡r¡n¡jknq`li¨ps½jk[]`¬rq`_jkj`_yiB¯·n³Å n³ÅÆ7psy» m|3yk¹p{4nD|3 ipswoyl`li.xn¡jkj`_r 'Ú 5)t`_jii`_{`_y|3r y`likwor¡jips jk[]`n¡]i`_ykjknqps o`_~ojk[ |3] ps jk[]`À[]`_n¡tu[µjÅ n. k=1. 2. n. n. L. ÐSÑÒЫÓ.
(34) . (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(35) 3 3 5 3 3.0 ./). 0 A 2 A 13. C 5. C. . . T. 4. 1. G. A. A. 12. 6. 3. A 7. G. A C 14. 8. C. G. A. T. 9 G. T 11 A 15. 10 G 16. } `_~oy`li`_µj|3jknqps p3£ 16 hw]_rq`lpsjknqo`liÄp3£ 6 1 ±X¦±·°´¦°´¦±XZ±-² ±X¦9¦±·±·± n¡7jk[]`°±X}²³jky`l`Ǫ«ps jk[]`¨rq`E£SjB¯·|3] ¡n kj []`
(36) ]psyk^¬|3r¡n¡©`l °±X} ª«ps jk[]`ykn¡tu[hjB¯EÅ. 1° [L|spuiϱ|3^F` }`_~oy`li`_µj|3jknqpsª«ps jk[]`rq`E£SjB¯|3]
(37) ]psyk^¬|3r¡n¡©`l 7°±X} ª«ps jk[]` ykn¡tu[µjB¯ p3£jk[]`*!]yikj 400000 µw]_rq`lpsjknqo`li9p3£°1[oyps^Fpuips^F`¢»p3£
(38) 20A )3 Å. Ò Ò Ù?]çBéEçA@.
(39) . 3 +?+ 0B6 . 5"),+-) . . 3 7 0
(40) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. ¤X`lik~on¡j`Vjk[]`Vn¡]o`_~`_]o`_]l` p3£jk[]`¢i`lvhw]`_]l`li xn¡jkj`_r#Æi91psyk¹Ïi`l`_^Fi-jpFm`·jk[]`¢_rqpui`likj9jp pswoyiº]|3]ips^F`~L|3ykji p3£pswoy ~oyp4p3£«i |3y`·n¡]ik~on¡y`lÀmµgPn¡jÅ e4ps^F`P~oyphp3£¼i n¡§jk[]`Ài`lvµw]`_r.w]i`P_rD|siiknq|3r.y`likwor¡ji¨ps jk[]`ÀYnqikjkykn¡mowojknqps p3£91psy p4E² _woy`_]l`liPn¡ |½y|3]ops^ i`lvµw]`_]l` p3£Vrq`_jkj`_yiêSn¡]o`_~`_]o`_µjpsyÀ |3yk¹p{4nD|3 i`lvµw]`_]l`liE¯EÅ a´rqps^ |3]PZ.[]psykmowoyk ' )Ktun¡{`9jk[]`Xt`_]`_y|3jkn¡ot¨£«wo]_jknqpsp3£ jk[]` !]yikjp4l_woy`_]l`p3£+| ´psy £¼psyn³Å n³ÅÆ?Åhi`lvµw]`_]l`liºµmL|si`l¬ps|y`l_woyky`_]l`-y`_rD|3jknqpspsjk[]` ~oypsmL|3mon¡r¡n¡jknq`liŶZ.[onqiy`likwor¡j1nqi `E®Yj`_]o`lPjp¨ |3yk¹p{4nD|3Ïi`lvhw]`_]l`limµg}psmon¡Ï|3]À¤|3w]Yn¡ 'Úu* )Ŷe4`_{`_y|3rikjkw]Ynq`li n¡Àjk[onqi ops^¬|3n¡§|3y`»mL|si`l ps t`_]`_y|3jkn¡ot£«wo]_jknqps]i£¼psy·`E®o|3^¨~orq`¨} _tuonq`_y 'Ú *)K}-`_n¡]`_ykj·`_jV|3r³Å 'Ú ).ehj`E£¼|3]pº{ |3] x+|3¹`li 'Úu7 )Ž¥-ps]`_jk[]`_rq`liiº¶psjk[]`_y¬|3~o~oyp|s[]`li|3y`Àlps]iknqo`_y`l 9Pps]` p3£´jk[]`»^Fpsy`»t`_]`_y|3rj`l[oonqvhw]`li·nqiXjk[]`¨ |3yk¹p{ [L|3n¡½`_^Vm`loYn¡ot^F`_jk[]ph n¡hjkyphYw]l`l mhg .ow '657 ) |3]ã«woykjk[]`_y¨o`_{`_rqps~o~,`l§mµ4g .ow |3] -pswojky|si '655 ) -pswojky|si '65* )Å ¦(^¬|3y² jkn¡otµ|3rq` |3~o~oyp|s[ ª«i`l`¢±X`_ykm`_y-|3] !n '657 ) !n '65º* )o n¡r¡r¡nD|3^F(i 'Ú 7)S¯.nqi|3|3r¡j`_ykL|3jkn¡{`¢jp jk[]` |3yk¹p{Ï[L|3n¡ `_^¢m,`loYn¡otÀ^F`_jk[]p4Çjpipsr¡{`¢~oypsmorq`_^Fi·|3ypswo] x `_o]`_g '65*)±·|3^F`uÅ Z.[]`li`¬j 1p |3~o~oyp|s[]`liF|3y`lps^¨~L|3y`l§n¡½xps©Yµgµ|3¹p{ `_j»|3r³Å 'Úu7 )ÅÏ [L|3j`_{`_y ^F`_jk[]p4 ps]`¬w]i`li jk[]`Ynqijkykn¡mowojknqps§p3£ jk[]` !]yikj¢p4l_woy`_]l`p3£|1psy§ikjkypsotur¡g7o`_~`_]oi¢ps½n¡ji pº{`_ykrD|3~o~on¡otikjkykw]_jkwoy`uÅ1Z.[onqi o`_~`_]o`_]l`·nqi|3jjk[]`·lpsy`Vp3£+pswoy~oyp4p3£«iºÅ Çpsy`lpº{`_y¶pswoy»y`likwor¡ji»g4nq`_rq |sigh^¨~ojpsjknqP~oyps~`_ykjknq`li¨ps jk[]`rq`_otujk[ p3£-jk[]`rqpsot`lij ykwoK[onq[ nqi¨|L|3jkwoy|3r psmYzk`l_j¨p3£ikjkw]YgÅ b§n³Å n³ÅÆ?Å|3] ikgh^¨^F`_jkyknqPi`lvhw]`_]l`li 8y ui |3] } _{ lik© 'Ú* )`likj|3mor¡nqik[ |3r¡^FpuikjFiwoy`y`liwor¡ji¬|3m,pswoj¨jk[]`Ptuypºjk[ p3£-jk[]`Prqpsot`lij¨ykwoKÅ Z.[]`li`Fy`likwor¡ji¨|3y`¬`E®Yj`_]o`l§jpÏ |3yk¹p{7[L|3n¡]i n¡ eY|3^¬|3ypº{s4| 'Ú* )|3] ±psyops§`_j»|3r³Å '65 )ik[]p)jk[L|3j jk[]`~oypsmL|3mon¡r¡nqikjknq¢m,`_[L|{hnqpswoy-p3£+jk[]`·rq`_otujk[Çp3£+jk[]`·rqpsot`likj ykwoÏnqi_rqpui`_r¡g |3~o~oyp®4n¡^¬|3j`lÏmµgPjk[L|3jp3£+jk[]`X^¬|N®4n¡^¢wo^ p3£ips^F`n³Å n³ÅÆ?Å]`E®Y~ps]`_µjknD|3rKy|3]ops^ {s|3yknD|3morq`liÅ Z.[]`P~L|3~,`_yFnqi¨psyktµ|3on¡©`l |si»£¼psr¡rqp iÅ b e4`l_jknqps 1`À`lij|3mor¡nqik[ jk[]`|siikwo^¨~ojknqps]i |3]]psj|3jknqps]iV1`»w]i` jk[oypswotu[]pswojÅFe4`l_jknqps nqiXo`_{psj`l jpÀ|3r¡^Fpuikj·ikwoy`»lpsµ{`_ykt`_]l` p3£jk[]`ik[]psykj`likjP|3] jk[]`Çrqpsot`likj¬moy|3][]`liPn¡ °±X} ²³jky`l`liÅ b e4`l_jknqps |sikg4^¨~ojpsjknq m`_[L|l{4nqpswoyXp3£jk[]`Vn¡]i`_ykjknqps o`_~ojk[nqi-ijkw]Ynq`l?Ŧ- |3~o~`_]Yn ®Ço`|3rqi9i`_~L|3y|3j`_r¡gn¡jk[jk[]` ops^¬|3n¡ p3£1o"` !]on¡jknqpsÃp3£1jk[]`¢t`_]`_y|3jkn¡otP£«wo]_jknqps p3£´|l`_ykj|3n¡ ´|3n¡jkn¡otÀjkn¡^F` y`_rD|3j`l jp jk[]`pº{`_ykrD|3~o~on¡otijkykw]_jkwoy`·p3£´psyoiÅ
(41) % . # . . . # % . b |3r¡rjk[]`»i`lvhw]`_r³,jk[]`»i`lvµw]`_]l` U = U . . . U . . . nqiXikwo~o~pui`ljp¬m`¨|¬ |3yk¹p{[L|3n¡ p3£Kpsyo`_y 1 µn¡jk[jky|3]ikn¡jknqps^¬|3jkykn ® Q ªS[]`_y` Q o`_]psj`li¶jk[]`-jky|3]ik~pui`lF^¬|3jkykn ®¬p3£ Q¯ |3]n¡µ{u|3yknD|3µj^F`|siwoy`V|sin¡on¡jknD|3rYnqikjkykn¡mowojknqpsKÅ .]psy|3µg¢o`_j`_yk^¨n¡onqijknq.n¡ !]on¡j`i`lvµw]`_]l` srq`_jw]io`_]psj`´mµg jk[]`´1psy¢£¼psyk^F`l mµg jk[]` j !]yikj¶rq`_jkj`_yi.p3£ s µjk[L|3j´nqi1jp¢i|g s ;=< = ss . . . s h[]`_y` s so`_]psj`li´jk[]` i rq`_jkj`_y´p3£ Å `_j p(s ) o`_]psj` p(s ) ;=< = P(U = s , . . . , U = s ) Å-Z.[]`·]`l`lÏ£«psy-y`_{`_yikn¡otjk[]` s 1. t. n t. (j). (j). (j). (j). 1. 1. j. j. j. i. th. 1. ÐSÑÒЫÓ.
(42) . (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(43) 3 3 5 3 3.0 ./). ´ psy s lps^F`li1£«yps^¸jk[]`-lps]ikjkykw]_jknqpsp3£Kjk[]`X°±X}²³jky`l`uhmL|si`lps¬y`_{`_yi`li`lvhw]`_]l`li ª 5º¯Ežc-]`Ç|3 y`_^¬|3yk¹ jk[L|3j£«psyP|7in¡oturq`E²³rq`_jkj`_y´psy u p u o`_]psj`lijk[]`Çn¡µ{u|3yknD|3µj ~oypsmL|3mon¡r¡n¡j gp3£jk[]`·rq`_jkj`_y u Å Çpsy`lpº{`_y]1`Vo"` !]]`jk[]`·lps]ikj|3hji (j). n 1 o 1 : p s(n) > 0 , h+ ;=< = lim max ln n→+∞ n p s(n) n o 1 1 : p s(n) > 0 , lim h− ;=< = min ln n→+∞ n p s(n) 1 h 1 i . E ln lim h ;=< = n→+∞ n p s(n). ¤9w]`¢jp|3 |3yktuwo^F`_µj·p3£¶ikwomY² |soYn¡jkn¡{4n¡jg ª«i`l` xn¡jkj`_r 'Ú 5 )S¯ELjk[]`li`¢r¡n¡^¨n¡ji|3y`V´`_r¡ro`"!]]`l ªSn¡Ï£¼|s_j]n¡|»^Fpsy`¢t`_]`_y|3rjk[L|3|3yk¹pº{hnD|3 i`lvhw]`_]l`li£«y|3^F`_1psyk¹]¯EÅÇpsy`lpº{`_y,xn¡jkj`_r i[]p ijk[L|3j jk[]`_y`V`E®4nqikjj 1p¬n¡ !]on¡j`·i`lvhw]`_]l`lio`_]psj`lÀ[]`_y`mhg s |3] s iw][jk[L|3j 3 | ] ª¼¯ 1 1 1 1 h = lim ln h = lim ln , . +. +. n→∞. n. −. (n). p s+. n→∞. −. (n). n. p s−. LpsyÀ|3hg j ≥ 1 jk[]`Ç]psj|3jknqps T ;=< = T (W ) o`_]psj`liPjk[]` !]on¡j`jky`l`n¡jk[ j ]pho`li ªSn¡jk[]pswoj lpswohjkn¡ot¬jk[]`9yp4psjB¯EYmowon¡r¡j.£Syps^ jk[]` !]yikj j i`lvµw]`_]l`li W (1), . . . , W (j) Y[onq[ |3y`jk[]`ikw]ll`liikn¡{` y`_{`_yi`l~oy`"!o®o`li1p3£Kjk[]`-i`lvµw]`_]l` 4|si1o`"!]]`l¬n¡ ª 5º¯EÅ o`_]psj`li jk[]` jky`l` y`lYw]l`l jpjk[]` yphpsjÅVb ~L|3ykjknq_worD|3y?jk[]`»y|3]Uops^ jky`l`li·|3y` n¡]_y`|sikTn¡ot 9 T ⊂ Å T ... ⊂ T ⊂ ... ⊂ T `_jw]i o"` !]]` ªSy`lik~KÅ ¯.|si.jk[]`Xrq`_otujk[Ïp3£jk[]`ik[]psykj`likj~L|3jk[7ªSy`lik~KÅjk[]`Xrqpsot`lijB¯ £«yps^jk[]`yphpsjjp |9` £¼`|sikn¡morq``EL®Yj`_ykL|3rL]p4o` p3£?jk[]` jky`l` T (w) ÅÇpsy`lpº{`_y D o`_]psj`li jk[]`n¡]i`_ykjknqpso`_~ojk[Pp3£ W (n) n¡ T jpVmowon¡rq T Å .n¡L|3r¡r¡g M nqi1jk[]` rq`_otujk[p3£|·~L|3jk[ p3£ T oy|3]ops^¨r¡g|3]Àwoon £¼psyk^¨r¡g[]pui`_Ïn¡Àjk[]` n ~puiikn¡morq`X~L|3jk[]iÅ Z.[]`¬£¼psr¡rqpn¡ot y|3]ops^ {u|3yknD|3morq`li ~orD|lg§|Ϲ`_g7ypsrq`n¡ jk[]`¬~oyp4p3£¼iÅ .]psy»jk[]`iB|3¹`p3£ ~oy`l_nqinqpsKYrq`_j´w]i´y`l|3r¡rjk[L|3j s nqi.o`_j`_yk^¨n¡onqikjknquojk[]`9y|3]ops^¨]`lii.nqi.woonqvhw]`_r¡gYw]`9jpVjk[]` t`_]`_y|3jknqpsÀp3£jk[]`9i`lvhw]`_]l` U Å .n¡yikj´1`9o"` !]]`-£«psy.|3µgn¡ !]on¡j`-i`lvhw]`_]l` s |3]F£¼psy´|3hg j≥0 n£ nqi]psjn¡ T 0 s ´psy s nqi|3r¡y`|sYgPn¡]i`_ykj`lÀn¡ T } X (s) ;=< = max{k : ªS]psjknql`-jk[L|3j X (s) = 0¯E Å .]psy|3µg k ≥ 0 T (s) o`_]psj`li¶jk[]`9ikn¡©`p3£Kjk[]` !]yikj1jky`l`[]`_y` nqi n¡]i`_ykj`l 9 s .. j. j. n. 0. 0. 1. j. n. n. n−1. n−1. n. n. n. n. j. (k). 0. 1. j. (k) k. Tk (s) ;=< = min{j : Xj (s) = k}. Ò Ò Ù?]çBéEçA@. j.
(44) . 3 +?+ 0B6 . 5. 5"),+-) . . 3 7 0
(45) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. Sª ]psjknql`·jk[L|3j T (s) = 0¯EÅ Z.[]`li`Xj´p»{s|3yknD|3morq`li|3y`Xn¡ÀYwL|3r¡n¡j gn¡Pjk[]`9£¼psr¡rqpn¡ot¬i`_]i`9+ps]`X[L|si`lvµwL|3r¡n¡j gp3£jk[]` `_{`_hji ª ¯ {X (s) ≥ k} = {T (s) ≤ j} |3]lps]i`lvhw]`_µjkr¡g {T (s) = j} ⊂ {X (s) = k} ÅÇpsy`lpº{`_yYjk[]`y|3]ops^({u|3yknD|3morq` T (s) |3Ïm,`o`llps^¨~pui`lÏ|si.£¼psr¡rqp iº X #ª h¯ T (s) = Z (s), []`_y` Z (s) ; =<>= T (s) − T (s) nqi¢jk[]`Fhwo^Vm`_y p3£rq`_jkj`_yi jpÀy`|s½m,`E£¼psy`¨jk[]`¬moy|3][ jk[L|3j¢lpsyky`lik~,ps]oiVjp s n¡]_y`|si`li·mµg 1 Åb [L|3j£¼psr¡rqp iº Z (s) |3Ãm`»{hnq`_´`l½|sijk[]` .|3n¡jkn¡ot¬jkn¡^F` j p3£jk[]` !]yij phl_woy`_]l`¢p3£ s n¡Ïjk[]`i`lvµw]`_]l` 0. j. k. j. k. k. k. r. k. r=1. r. r. r−1. r. (r). . . . Uj+Tr−1 (s) Uj−1+Tr−1 (s) . . . U1+Tr−1 (s) s(r−1) ,. n³ÅÆ`uÅ Z (s) |3Ç|3rqip»m`o`"!]]`lÏ|si r.
(46) Zr (s) = min{j ≥ 1
(47) Uj+Tr−1 (s) . . . Uj+Tr−1 (s)−r+1 = s1 . . . sr }.. a.`l|3w]i`Vp3£jk[]` |3yk¹p{4nD|3on¡jgp3£+jk[]`^Fp4o`_r³ojk[]`Xy|3]ops^ {u|3yknD|3morq`li Z (s) |3y`·n¡]o`_~,`_Y² o`_hjÅ `_j·w]i·jk[]`_½n¡µjkyp4Yw]l` |si·m`_n¡otjk[]`¨.|3n¡jkn¡otÇjkn¡^F`Fp3£jk[]` !]yijVphl_woy`_]l`p3£ Y (s) ¡ n Ï k j ] [ · ` i l ` h v ] w _ ` ] l ` s r. r. (r). . . . Uj+Tr−1 (s) Uj−1+Tr−1 (s) . . . U1+Tr−1 (s) ,. jk[L|3jnqijp¨i|lg.
(48) Yr (s) = min{j ≥ r
(49) Uj+Tr−1 (s) . . . Uj+Tr−1 (s)−r+1 = s1 . . . sr }.. ¤9w]`.jpXjk[]`.~puiikn¡morq`pº{`_ykrD|3~o~on¡otVm`_j1`l`_jk[]`.~oy`"!o®Y`li¶p3£ |3]»jk[]`ikwY¨®o`lip3£ ´`[L|l{`Vjk[]`·n¡]`lvhwL|3r¡n¡jg Z (s) ≤ Y (s) Å´ehn¡]l`Vjk[]`·i`lvhw]`_]sl` (U ) nqi ikj|3jknqpsL|3ykgLsjk[]` lps]Yn¡jknqpsL|3r4Ynqikjkykn¡mowojknqps¢p3£ tun¡{`_ T (s) nqi?jk[]`Ynqikjkykn¡mowojknqpsVp3£Yjk[]` !]yikj phl_woy`_]l` p3£ jk[]`¶´psy s n¡»jk[]`1y`|3r¡n¡Y©º|3(s) jknqpsFp3£,|9|3yk¹pº{V[L|3n¡Fp3£ psyo`_y 1 sn¡jk[»jky|3]ikn¡jknqps»^¬|3jkykn ® 3 | ] P ¡ n k j Ï [ ¡ n o ¡ n k j D n 3 | K r Y q n k i k j k y ¡ n o m o w k j q n psÏjk[]`Xn¡h{s|3yknD|3hj^F`|sikwoy`uÅb P~L|3ykjknq_worD|3yjk[]`lps]Yn¡jknqpsL|3r Q Ynqijkykn¡mowojknqpsÏp3£ Y (s) tun¡{`_ T (s) nqin¡]o`_~`_]o`_µjXp3£ T (s) Å Z.[]`-t`_]`_y|3jkn¡otF£Swo]_jknqps Φ(s , t) ;=< = E[t ] nqi´tun¡{`_Àmµg}psmon¡|3]P¤X|3w]Yn¡ 'Úu )9 ª¼ ¯ Φ(s , t) = γ (t) + (1 − t)δ (t ) , (r−1). r. r. n n≥1. r. (r). (r). r−1. t. r. r−1. r−1. (r). (r). Yr (s). r. r. −1. −1. ÐSÑÒЫÓ.
(50) (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(51) 3 3 5 3 3.0 ./). 55. []`_y`·jk[]`X£Swo]_jknqps]i γ |3] δ |3y`y`lik~`l_jkn¡{`_r¡gÀo`"!]]`lÏ|si 1−t X m Q (s1 , sr )tm , γr (t) ; =<>= tp sr m≥1. δr (t. −1. ) ;=<>=. r X. 5 {sr ...sr−m+1 =sm ...s1 } , tm p s(m). ª¼¯. |3] []`_y` o`_]psj`lijk[]`.jky|3]in¡jknqps¨~oypsmL|3mon¡r¡n¡jg £Syps^ jp n¡ ikj`_~]iÅb »¦-~Y² ~`_]Yn ® ¦ 1Q` ijk(u,w]YgVv)jk[]`ops^¬|3n¡Fp3£,o`"!]on¡jknqpsp3£ Φ(s , t) []`_u t nqi+vn¡¬|-m]`_n¡tu[µmpswoyk[]php4 p3£ 1 Å ' 4 b jk[]`~L|3ykjknq_worD|3yÀ|si`[]`_ jk[]`Çi`lvµw]`_]l`Çp3£·µw]_rq`lpsjknqo`li nqi iwo~o~,pui`ljp»m`9n¡]o`_~,`_]o`_hj-|3]Pnqo`_µjknq|3r¡r¡gYnqikjkykn¡mowoj`l|sllpsyYn¡ot¬jp»jk[]`X](Ups)o`_t`_Y² `_y|3j`l½rD|l (p , p , p , p ) +jk[]`Fjky|3]ikn¡jknqps ~oypsmL|3mon¡r¡n¡j g Q (s , s ) nqi¢`lvhwL|3r´jp p(s ) |3][]`_]l` γ (t) = 1 Å m=1. m. (r). n n≥1. A. C. G. m. T. 1. r. r. 4:24 )8 r. 7 3'+ 3 3"540 ./) + 6?./) ) 0 . 1=340 . #3 " 3 Yr (s) 0, 1 + 7 3"5 3 ) 0 6 .0. ) B3 3B3. 0 (r) κ r s κp s [ )) 3 . 3 . 3 . 7 3 3 6 1 0-5 +B3 . 3"),+B3
(52) 0-1 3 . 7 3:./5 0 )/./)
(53) 0 ./5") $ 5 0 11. &('*),+)- .0/.),1. . Q . γ t ∈ [0, γ −1 [.
(54)
(55)
(56) γr (t) − 1
(57) ≤ |1 − t| κ0 , 1 − γt 7 3"5 3 0 ) 0 . 7 3"5 6 .0. ) B 3 3B 3 . 0 κ r s . ) ) ) '5 : . 7 3"5 3. r X E Yr (s) ≤. M. ) ) B3 3 B3. . 5. j=1. r. 0. s . {sr ...sr−j+1 =sj ...s1 } p s(j). + M,. Z.[]`~oyp4p3£+p3£xyps~puikn¡jknqpsÇYÅ65nqi tun¡{`_n¡¦~o~`_]Yn ®À¦¢Å.
(58) ( ". b jk[onqii`l_jknqps ´`Ç|3y`Ïlps]l`_yk]`l n¡jk[ jk[]`|sikg4^¨~ojpsjknqm`_[L|l{4nqpswoyp3£jk[]`rq`_otujk[ ªSy`li~KÅ L ¯´p3£jk[]`·i[]psykj`likj¢ªSy`lik~KÅ+rqpsot`likjB¯.moy|3][Çp3£+jk[]`¢°±X} ²³jky`l`uÅ )' 4:24 n. `n a.s. 1 , −→ ln n n→∞ h+. Ò Ò Ù?]çBéEçA@. 0. Ln a.s. 1 . −→ ln n n→∞ h−. `n.
(59) . 3 +?+ 0B6 . 5. 5"),+-) . . 3 7 0
(60) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. ¦llpsyYn¡otFjp»jk[]`Xo`"!]on¡jknqpsÏp3£ X (s) Yjk[]`Xrq`_otujk[]i ` |3] L |3y`-£«wo]_jknqps]i p3£ X 9 |3] L = 1 + max X (s). ª³u¯ ` = 1 + min X (s), A A Z.[]`£¼psr¡rqpn¡ot ¹`_gÃrq`_^¨^¬|tun¡{`li¨|3 |sikg4^¨~ojpsjknqy`likwor¡j¨ps X (s) wo]o`_yFikwon¡j|3morq`À|si² iwo^¨~ojknqps]ips s Ŷc-woy~oyphp3£p3£+Z.[]`lpsy`_^ Y6Å 5XnqimL|si`lpsÀn¡jÅ n. n. s∈. ∞. n. n−1. n. n. s∈. n. n−1. ∞. n. . 3 .. 4 4. s. 7 3 . 6 7 . 7 0 . . 7 3"543 3 $ ) . . ª¼¯. 1 1 ;=< = h(s) > 0. lim ln n−→+∞ n p s(n). 8 7 3. Xn (s) a.s. 1 . −→ ln n n→∞ h(s). ' 4% `_j lps]iknqikjp3£¶y`_~`_jkn¡jknqps]ip3£´|Frq`_jkj`_y ÅZ.[]`_ nqi9jk[]` qr `_otujk[p3£jk[]`·moy|3]v˜[ ;=<|s= ivvip4_.nD.|3.j`ln¡jk[ v˜ n¡ T Å .]psy9iw][|Fi`lvµw]v`_]l`ª¼|3]ÏX`E®o(˜v_r¡)w]ikn¡{`_r¡g £¼psyXjk[]`_^¯9jk[]`»y|3]ops^¿{u|3yknD|3morq` Y (˜v) nqi·`lvhwL|3rjp T (˜v) ÅF°´ps]i`lvhw]`_µjkr¡g X (˜v) nqiXjk[]` rq`_otujk[ p3£ jk[]`rqpsot`likj¢ykwo p3£ # v # n¡ U . . . U Å []`_ (U ) nqi»|i`lvhw]`_]l`p3£ n³Å n³ÅÆ?Å jkyknD|3rqiº 8y ui-|3]} _{ lik © 'Ú* ) 8y ui9|3]À} _{ lik© ' *)]x`_jkyp{ 'Úu*) i[]p1`ljk[L|3j n. n. k. n. k. 1. n. n n≥1. Xn (˜ v ) a.s. 1 −→ , ln n n→∞ ln p1. []`_y` p ;=< = P(U = v) ŶZ.[onqi lpsµ{`_ykt`_]l`¢y`likwor¡jnqi|»~L|3ykjknq_worD|3y-|si`·p3£ `_^¨^¬| YÅÚYÅ '5 : 3
(61) 2
(62) 0 ehn¡]l` £¼psy ª«i`l`}`_^¬|3yk¹ ¯EYmµg^Fps]psjpsonqE² n¡j gP|3yktuwo^F`_hji]n¡jnqi ik wY¬_nq`_µXj-jp»(s)~oy=p{`Vk jk[L|3j n = T (s) i. n. k. ln Tk (s) a.s. −→ h(s). k→∞ k. c-mh{hnqpsw]ir¡gF£«yps^jk[]`9o`"!]on¡jknqpsp3£ T (s) 4´`[L|{` Å}-pswotu[or¡g 1 k. P ln Tk (s) < (1 − ε)kh(s). . P(Tk (s) = j) ≤ p s(k). Å `_j. . = P Tk (s) < exp((1 − ε)kh(s)) ≤ exp (1 − ε)kh(s) p s(k). 0<ε<. . psy`lp{`_y4|siikwo^¨~ojknqpsϪ¼¯ n¡^¨~or¡nq`li+jk[L|3jjk[]`_y``E®Ynqikji|9lps]ikj|3µj c o`_~`_]Yn¡ot·ps ε ikw][ jk[L|3jo£«psy|3r¡r k ≥ 1 1. p(s(k) ) ≤ c1 exp −(1 − ε2 )kh(s). . ÐSÑÒЫÓ.
(63) (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(64) 3 3 5 3 3.0 ./). 5:. |3]jk[]`_y`E£«psy` P ln Tk (s) < (1 − ε)kh(s) ≤ c1 exp −kh(s)(−ε2 + ε) ,. [onq[nqi.jk[]`t`_]`_y|3r?j`_yk^ p3£|»lpsµ{`_ykt`_hj9i`_yknq`li.[]`_ ε nqiik^¬|3r¡rK`_]pswotu[KÅÂÃ`o`lYw]l` £«yps^ a.psy`_r ²á°.|3µj`_r¡r¡n `_^¨^¬|¨jk[L|3j9|3r¡^Fpuikj ikwoy`_r¡g lim inf. ln Tk (s) ≥ (1 − ε)h(s). k. °´psh{`_yi`_r¡gL |3yk¹p{Pn¡]`lvhwL|3r¡n¡jgPg4nq`_rqoi k→∞. P ln Tk (s) > (1 + ε)kh(s) ≤ e−(1+ε)kh(s) E[Tk (s)]. # Zr (s) ≤ Yr (s). a´gjk[]`o`llps^¨~,puin¡jknqps§ª h¯|3]Pjk[]`n¡]`lvhwL|3r¡n¡jg E[Tk (s)] ≤. k X. ops]`[L|si. E[Yr (s)].. ]yps^ |sii`_ykjknqpsn¡n¡nT¯p3£+xyps~puikn¡jknqpsÇYÅ65u |3]ikn¡]l` .. |3] ]! L|3r¡r¡g. E[Tk (s)] ≤. X k r=1. r p(s(r) ). . r=1. + kM ≤. p(s(j) ) ≥ p(s(r) ). o£«psy|3hg. j≤r. . 1 k(k + 1) + kM, 2 p s(k). 1 k(k + 1) P ln Tk (s) > (1 + ε)kh(s) ≤ + kM e−(1+ε)kh(s) . 2 p s(k). ]yps^ jk[]`»|siikwo^¨~ojknqps ª¼¯ELjk[]`_y` `E®4nqiji9|¬lps]ikj|3µj c o`_~`_]Yn¡otps ε iw][Ïjk[L|3j-£¼psy |3r¡r k ≥ 1 p(s ) ≥ c exp −(1 + ε )kh(s) , [onq[rq`|soi jp .. 2. (k). 2. 2. 2 P ln Tk (s) > (1 + ε)kh(s) ≤ M 0 k 2 e−kh(s) ε−ε .. b j nqijk[]`·t`_]`_y|3rj`_yk^ p3£¶|Flpsh{`_ykt`_µji`_yknq`li9|si-ip4psÇ|si a.psy`_r ²á°.|3hj`_r¡r¡n `_^¨^¬|Y lim sup. ln Tk (s) ≤ (1 + ε)h(s), k . [onq[Ïlps]_r¡w]o`li-jk[]`X~oyphp3£p3£rq`_^¨^¬| YÅÚYÅ k→∞. Ò Ò Ù?]çBéEçA@. ε<1. Å°´ps]i`lvhw]`_µjkr¡g,Yw]`Vjp.
(65) 3 +?+ 0B6 . 5 '5 : . . 8 7 3 543
(66). ¶vµwL|3jknqps½ª³u¯.ghnq`_rqoi 8. . 5"),+-) . . 3 7 0
(67) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. b j nqin¡]i~on¡y`lÀ£«yps^ xn¡jkj`_r Ú' 5)Å.°1rq`|3ykr¡gPjk[]`Vo`"!]on¡jknqpsÏtun¡{`_Çn¡. 3| ] L ≥ 1 + X (s ) ª«o`"!]on¡jknqps]i-p3£ s |3] s ´`_y`tun¡{`_n¡Ç¯EÅÁ9`_]l`uomhg `_^¨^¬| YÅÚ |YÅÆiÅ L 1 1 ` ≤ , lim inf ≥ lim sup `n ≤ 1 + Xn−1 (s+ ). +. n. n−1. −. n. n. •. . 5 : -5. ln n. n→∞. h+. n→∞. ln n. 9. ]psy-|3µgP`n¡hj`_t`_y r Y´`[L|l{`Vjk[]`Xn¡]`lvµwL|3r¡n¡jknq`li .. n. −. h−. ª ¯ []`_y`-jk[]`X|3m,pº{`9iwo^Fi´|3y`-j|3¹`_pº{`_y.jk[]`-i`_j A p3£ 1psyoi´n¡jk[rq`_otujk[ r ªT£¼psy.|·~oyps~,`_y ^F`|3on¡ot p3£jk[onqi £¼psyk^VworD|Y¶ps]`i[]psworq½y`_~orD|sl` s mhg½|3µg½n¡ !]on¡j`P1psy§[L|l{4n¡ot s |si ~oy`"!o®?-n¡ mpsjk[ phl_woy`_]l`liE¯EÅÂÃ` |3mow]i` p3£¢jk[onqiP]psj|3jknqps £«yps^ ]p¿psKÅehn¡]l`£¼psy Yjk[]`9t`_]`_y|3jkn¡ot¨£«wo]_jknqps]i , t) |3y`o`"!]]`l£¼psy |3µg j ≤ r ª«i`l` t ∈ [1, 1 + κp(s )] ¦9ii`_ykjknqpsÀnT¯1n¡Çxyps~,puin¡jknqpsÇYÅ65º¯E]`|s[Çj`_yk^ p3£+jkΦ(s []`ikwo^ ª ¯´|3m`lpsµjkypsr¡rq`lÏmµg P(`n ≤ r) ≤. X. s(r) ∈Ar. P(Xn−1 (s) ≤ r − 1) =. X. s(r) ∈Ar. P(Tr (s) ≥ n),. r. (r). (r). j. P(Tr (s) ≥ n) ≤ t−n E[tTr (s) ] ≤ t−n. r Y. Φ(s(j) , t).. b ~L|3ykjknq_worD|3y,mpswo]Yn¡otP|3mp{` |3r¡rjk[]`Vpº{`_ykrD|3~o~on¡ot£Swo]_jknqps]i 5 n¡7ª¼¯E]1`·o`lYw]l`X£«yps^ ª¼ ¯|3]£«yps^ ¦9ii`_ykjknqpsÀn¡nT¯.p3£xyps~puikn¡jknqpsÇYÅ65jk[L|3j j=1. {sj ...s1 =sr ...sr−j+1 }. . `_j. P(Tr (s) ≥ n) ≤ t 0<ε<1. −n. r Y. 1 κ0 1 + (1 − t) + tν p(s(ν) ) 1 − γt ν=1. ÅZ.[]`_y`V`E®4nqikji-|»lps]ikj|3µj j=1. p(s(j) ) > c2 αj ,. ªT£¼psy·jk[]`i|3¹`Fp3£moy`_{hn¡j g c c |3] ÂÃ`Xjk[]`_Ï[L|l{` 1. P(Tr (s) ≥ n) ≤ t. −n. r Y. j=1. X j. c2 ∈]0, 1[. !−1. mhg. .. o`_~,`_]Yn¡otpsor¡gPps ε ikw][Ïjk[L|3j. with α ;=< = exp(−(1 + ε2 )h+ ) c2. 1. o`_]psj`Yn&3?`_y`_µj lps]ikj|3µji¨|3r¡r1|3rqpsotjk[]`¨j`E®YjB¯EÅ 1 − (αt)−j. κ0 1 + (1 − t) + c2 (αt − 1) 1 − γt. !−1. .. ÐSÑÒЫÓ.
(68) . (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(69) 3 3 5 3 3.0 ./) 5. °1[]p4puikn¡ot L[]`_y` κ nqi jk[]`¢lps]ikj|3µjXo`"!]]`lÇn¡Ï¦-ii`_ykjknqpsÏnT¯ p3£xyps~,puikn ² jknqpsYÅ65uopst]`=t1`_j+i c κα r. 2. P(Tr (s) ≥ n) ≤ t−n. r Y. αj − (1 + c2 καr )−j αr c2 κκ0 1 − καr−j − α(1 + c2 καr ) − 1 1 − γ(1 + c2 καr ). psy`lp{`_y-ikn¡]l`·psmµ{4nqpsworqikg. j=1. !−1. .. 1 αj − (1 + c2 καr )−j = , j→∞ α(1 + c2 καr ) − 1 1−α c2 κκ0 / 1 − γ(1 + c2 καr ) r λ L j r lim. |3] nqi woon £«psyk^¨r¡g½m,pswo]o`l n¡ +jk[]`_y``E®Ynqikj»j´p~,puikn¡jkn¡{` lps]ij|3µji |3] n¡]o`_~`_]o`_µj9p3£ |3] ikw][jk[L|3j P(Tr (s) ≥ n) ≤ (1 + c2 καr )−n L. r Y. b Ï|soYn¡jknqpsKojk[]`~oyp4Yw]_j|3m`Xmpswo]o`lÏ|3m,pº{`mµg ´° ps]i`lvµw]`_hjkr¡g .Lpsy r = b(1 − ε) []`_y`. r Y. j=1. 1 − λα. r−j. −1. ≤. ∞ Y. j=0. j=1. 1 − λαj. 1 − λαr−j. −1. −1. .. = R < ∞.. P(Tr (s) ≥ n) ≤ LR(1 + c2 καr )−n .. ln n h+ c. |3] ε ik^¬|3r¡rK`_]pswotu[KLjk[]`_y`·`E®4nqiji |¨lps]ikj|3µj R ikw][Àjk[L|3j 0. P(Tr (s) > n) ≤ R0 exp(−c2 κnθ ), θ = ε − ε 2 + ε3 > 0. ÅÂÃ`Xjk[]`_Ïo`lYw]l`X£«yps^ ª ¯´jk[L|3j. P(`n ≤ r) ≤ 4r R0 exp(−c2 κnθ ),. [onq[nqijk[]`t`_]`_y|3rj`_yk^ p3£+|¨lpsh{`_ykt`_µjXi`_yknq`liŶc-]l`V|3tµ|3n¡KLa´psy`_r ²á°.|3hj`_r¡r¡n `_^¨^¬| |3~o~or¡nq`li9|3] |YÅÆiÅ ` 1 ≥ lim inf n. •. . 5 : -5. Ò Ò Ù?]çBéEçA@. n→∞. Ln. ln n. h+.
(70) . 3 +?+ 0B6 . 5. 5"),+-) . . 3 7 0
(71) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. ZpFlps^¨~orq`_j`Vjk[]`X~oyphp3£k]ps]`X]`l`loi jp¨i[]pjk[L|3j. Y| ÅÆiÅ ¦-tµ|3n¡K4ikn¡]l` X (s) = k £¼psy n = T (s) µmhg¨^Fps]psjpsonq_n¡jg|3yktuwo^F`_hji.n¡j1ikwY¬l`li´jp¢ik[]p jk[L|3j |YÅÆiÅ ln T (s) lim inf min ≥h k `_j Å+¦-in¡¨jk[]`´~oy`_{4nqpsw]i~oyphp3£ £«psy+jk[]`ik[]psykj`lijmoy|3][]`lin¡jikwY¬l`lijp-mpswo] |3mp{0` < ε < 1 lim sup n→∞. n. 1 Ln ≤ ln n h−. k. k. k→∞. −. s. P min Tk (s) < expkh− (1−ε). . mhg·jk[]`´t`_]`_y|3rYj`_yk^#p3£,|-lpsµ{`_ykt`_hj¶i`_yknq`lijpX|3~o~or¡g¢a.psy`_r ²á°.|3µj`_r¡r¡n `_^¨^¬|YŶc-mh{hnqpsw]ir¡g s. P. °1[]p4puikn¡ot []`_y` p3£+jk[]`. X min Tk (s) < expkh− (1−ε) ≤ P Tk (s) < expkh− (1−ε) .. s(k) ∈Ak. t ∈ [0, 1]. £¼psy. jk[onqin¡^¨~or¡nq`li jk[L|3j. s(k) ∈Ak. P Tk (s) < expkh− (1−ε) ≤ P tTk (s) > tn ,. +Å Z.[]`o`llps^¨~puikn¡jknqps ª#h¯Ejpst`_jk[]`_y´n¡jk[Fjk[]` n¡]o`_~`_]o`_]l` ]ghnq`_rq. n ;=< = exp(kh− (1 − ε)) Zj (s) 1≤j≤k. k Y P tTk (s) > tn ≤ t−n E tZj (s) . j=1. b ¬psyo`_y´jp·mpswo]|3mp{`-jk[]` j`_yk^ ªSn¡jk[ ¯Ehrq`_j m`[]pui`_K []`_y` c nqi|3hgÀlps]ikj|3µj9ikw][jk[L|3jot £«psy|3r¡r j0≥<1t < 1 t ;=< = (1 + c/n) ª 5¯ ≤ cβ , where β ;=< = exp(−(1 − ε )h ). p s Z.[]`t`_]`_y|3jkn¡ot £«wo]_jknqps§p3£ Z (s) nqi tun¡{`_§mhg7}psmon¡ |3]§¤X|3w]Yn¡ 'Úu* ) |3]½ijkypsotur¡g o`_~`_]oi9psÇjk[]`¢p{`_ykrD|3~o~on¡otikjkykw]_jkwoy` p3£jk[]`·´psy s Å .Lpsy 0 < t < 1 jk[onqi £«wo]_jknqps nqi 1`_r¡ro"` !]]`lÏ|3]nqitun¡{`_mµgê«i`l`¦9ii`_ykjknqpsÀnT¯p3£+xyps~puikn¡jknqpsÇY6Å 5º¯ n. (j). −1. j. 2. j. E tZj (s) = 1 −. −. (j). tj p. sj. . (1 − t) , γj (t) + (1 − t)δj (t−1 ). ÐSÑÒЫÓ.
(72) º. (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(73) 3 3 5 3 3.0 ./) 5. []`_y` γ (t) |3] δ (t) |3y`o`"!]]`l§n¡ ª¼¯EÅpsy`lp{`_y£«yps^ ¦9ii`_ykjknqps½n¡nT¯·p3£xyps~puikn ² jknqpsYÅ65un¡jnqipsmh{hnqpsw]i1jk[L|3jjk[]`_y` `E®Ynqikji¶|Xlps]ikj|3hj θ n¡]o`_~`_]o`_µj.p3£ j |3] s iw][Fjk[L|3j £¼psy t = (1 + 1/n) j. j. −1. γj (t) ≤ 1 + θ(1 − t).. ehn¡]l`9jk[]`£Swo]_jknqps x 7→ (A + x)/(B + x) n¡]_y`|si`li´[]`_ ª 5¯.[]psrqoiYjk[]`t`_]`_y|3jkn¡otF£«wo]_jknqpsÏi|3jknqi4!L`li E[tZj (s) ] ≤ 1 −. []`_y`. qk (s). βj. . 1 1−t. B≥A. o|3]ikn¡]l`9n¡]`lvhwL|3r¡n¡jg ª 55º¯. c−1 , + θ + c−1 + qk (s). Lo`_~,`_]Yn¡ot¬psÏjk[]`p{`_ykrD|3~o~on¡otPikjkykw]_jkwoy`·p3£ s onqi o`"!]]`lmµg (k). qk (s) ; =<>= max. 1≤j≤k. j−1 X 5. {sm ...s1 =sj ...sj−m+1 } β. j−m. «ª o`"!]on¡jknqps p3£ δ ´`_y`tun¡{`_§n¡ ¯EÅ [L|3j`_{`_y jk[]`p{`_ykrD|3~o~on¡ot½ikjkykw]_jkwoy`nqi q (s) nqi lpshjkypsr¡rq`lmhg ª 5¯ β . 0 ≤ q (s) ≤ 1−β Z.[hw]i m=1. j. k. k. k Y. E[t. Zj (s). j=1. X k ] ≤ exp − ln 1 − j=1. −1 c−1 . β j (1 − t)−1 + θ + c−1 + qk (s). ehn¡]l`»jk[]` £«wo]_jknqps nqiXn¡]_y`|sin¡ot] |N£«j`_y|Plps^¨~L|3yknqipsÃm,`_j 1`l`_7iwo^ |3]n¡µj`_tuy|3r|3]|N£«jx`_y 7→jk[]ln`1/(1 [L|3ot−`Vx)p3£+{s|3yknD|3morq` y = β (1 − t) + θ]ps]`psmoj|3n¡]i x. k Y. E[t. Zj (s). . 1 ] ≤ exp − ln β −1. Z. (1−t)−1 +θ. βk. −1. −1 dy c−1 . ln 1 − y + c−1 + qk (s) y (1−t)−1 +θ . Z.[onqi´n¡hj`_tuy|3r,nqi.lpsh{`_ykt`_µjn¡À|·]`_n¡tu[hm,pswoyk[]p4phÀp3£ +∞ µ[]`_]l`9jk[]`_y`9`E®4nqikji.|¢lps]ikj|3µj Yn¡]o`_~,`_]o`_hj-p3£ k |3] s ikw][Àjk[L|3j C Z dy ª 5:¯ Y 1 c E[t ] ≤ C exp − . ln 1 − j=1. k. +∞. Zj (s). j=1. Ò Ò Ù?]çBéEçA@. ln β −1. β k (1−t)−1 +θ. −1. y+. c−1. + qk (s). −1. y.
(74) . 3 +?+ 0B6 . 5. b£. 5"),+-) . . 3 7 0
(75) ?) 2 . 7 03 ) ) 1 1,0 6) 6 -1,0 0 3. o`_]psj`liPjk[]`_rD|siiknq|3rXYn¡rqpstµ|3ykn¡jk[o^-ps]`Ç[L|si ][onq[Ïrq`|soi jp»jk[]`X£¼psyk^VworD|. P k 2 Li2 (z) = k≥1 z /k 1 y log(1 + v/y) Z. +∞. c−1 ln 1 − y + c−1 + qk (s) . −1. d dy. Li2 (− yv ) =. dy qk (s) 1 + cqk (s) = Li2 − − Li2 − . y ak cak. psy`lp{`_yLn¡Ï| ]`_n¡tu[µmpswoyk[]php4Ïp3£ −∞ ak. ª 5 h¯ [onq[Àghnq`_rqoiºhwo]o`_y.jk[]`|siikwo^¨~ojknqps]i a → 0 |3] q (s) → 0 []`_ k j`_]oi.jp¢n¡ !]on¡j g 1 Li2 (x) = − ln2 (−x) + O(1), 2 k. Z. +∞. c−1 ln 1 − y + c−1 + qk (s) . −1. k. dy 1 qk (s) = Li2 − + ln2 ak + O(ln ak ). y ak 2 5: q√ k (s) (k) s qk (s) < exp − k. .Z [hw]iojk[]`m`_[L|l{4nqpswoyp3£+jk[]`n¡µj`_tuy|3r n¡7ª ¯.o`_~`_]oi psjk[]`V|sikg4^¨~ojpsjknqli p3£ . n¡yijrq`_j w]i lps]iknqo`_y jk[]`|si`·p3£jk[]`X´psyoi iw][jk[L|3j √ rq`_j z ; =<>= exp − k Å .Lpsyikw][Ï1psyoiojk[]`¢|3m,pº{`·`lvµwL|3r¡n¡j gPn¡^¨~or¡nq`li ak. k. Z. +∞. c−1 ln 1 − y + c−1 + qk (s) . |3]. dy 1 ≤ ln ak ln zk − ln2 zk + O(ln zk ). y 2 ak = β k (1 − t)−1 + θ ∼ exp(−kh− (ε − ε2 )). °´ps]i`lvµw]`_hjkr¡g][]`_ ak. −1. Å. k Y. ]ps]`t`_ji. E[tZj (s) ] ≤ C exp −. ε k 3/2 + O(k) . 2(1 + ε). Z.[]`_y` |3y` 4 1psyoip3£,rq`_otujk[ k s[]`_]l`{`_ykgVypswotu[or¡gµmµg j|3¹hn¡otXjk[]`ikwo^pº{`_yjk[]`.1psyoi p3£+rq`_otujk[ k ikw][jk[L|3j q (s) < z ]|3]ikn¡]l` t nqimpswo]o`l? j=1. k. k. X. −n. k. P Tk (s) < exp. kh− (1−ε). . . ε 3/2 k + O(k) , ≤ 4 exp − 2(1 + ε) k. [onq[nqijk[]`t`_]`_y|3rj`_yk^ p3£|Flpsµ{`_ykt`_µjXi`_yknq`liÅ bjy`_^¬|3n¡]ijp§ikjkw]Yg jk[]`|si`Ç[]`_ Å L. psyjk[]`li`Ç1psyoiº rq`_j¬w]iPpsor¡g lps]inqo`_y jk[]`n¡]`lvhwL|3r¡n¡jgê 5¯|3]Àjk[]`_ q (s) ≥ z s(k) ∈A. k|. qk (s)<zk. k. k Y. j=1. E[t. Zj (s). ] ≤ C exp − 0. 1 ln β −1. Z. +∞. β k (1−t)−1 +κ. k. ln 1− . −1 dy c−1 . y + c−1 + β(1 − β)−1 y. ÐSÑÒЫÓ.
(76) . (),+-) .0 1%2%340-546"7 8543 3 0 7 0 0
(77) 3 3 5 3 3.0 ./). ehn¡]l`. x ≤ log(1 − x)−1 k Y. 5. ]|N£«j`_yips^F`X´psyk¹Àp3£n¡hj`_tuy|3jknqpsK. c−1 E[tZj (s) ] ≤ exp − kh ε(1 − ε) + o(k) . − ln β −1. Z.[]`9L|3jkwoy|3rvhw]`likjknqps|3yknqikn¡ot¨]pº nqi.[]p ^¬|3µg1psyoi s |3y`iw][Àjk[L|3j `_j w]i o"` !]]`
(78) o n
(79) . E ; =<>= s
(80) q (s) ≥ e psy`lp{`_y _rq`|3ykr¡g¬£«yps^ jk[]`·o"` !]on¡jknqpsÇp3£ q (s) j=1. (k). (k). k. √ − k. k. qk (s) ≥ zk. k. j−1
(81)
(82) n X 5
(83) (k)
(84) |Ek | ≤
(85) s
(86) ∃j ≤ k :. `_j w]i o`"!]]`jk[]`·i`_j4£«psy . Sj (t) ;=< =. m=1. n. j−m. ≥ e−. j≤k. n. s. j−1
(87) X 5
(88). (k)
(89). c-]`·[L|sijk[]`X£¼psr¡rqpn¡otFn¡]_r¡w]iknqpsK j−1 \. {sm ...s1 =sj ...sj−m+1 } β.
(90) 5
(91) s(k)
(92). m=1. {sm ...s1 =sj ...sj−m+1 } β. {sm ...s1 =sj ...sj−m+1 }. j−m. √ o
(93) k
(94). o ≥t .. β j−` o , = 0 ⊂ Sjc 1−β. []`_y`´jk[]`´]psj|3jknqps B o `_]psj`li+jk[]`.lps^¨~orq`_^F`_µj|3ykg¨i`_jp3£ B n¡ A Åehn¡]l` £¼psy ` ; =<>= j − √k/ ln β m=`. c.
(95) .. k. e−. √ k. −1. Ek ⊂. k j−1 [ [ n. n¡L|3r¡r¡gojk[]`·µwo^Vm`_y-p3£+1psyoi .. j=1 m=`. s(k).
(96) 5
(97) s(k)
(98). {sm ...s1 =sj ...sj−m+1 }. ikw][Ïjk[L|3j. qk (s) ≥ zk. j−1 k X
(99)
(100) X 4 √
(101) Ek
(102) ≤ 4j−m ≤ k4 k/ ln 3. b jy`_^¬|3n¡]i jpF|3~o~or¡gÀa.psy`_r ²á°.|3µj`_r¡r¡n `_^¨^¬|F|3] j=1 m=`. lim sup n→∞. Ò Ò Ù?]çBéEçA@. 1 Ln ≤ ln n h−. |YÅÆiÅ. o = 1. .. nqi m,pswo]o`lÏ|3mp{`·mµg β −1. . .. =. β j−` 1−β.
(103) u. 3 +?+ 0B6 . 5"),+-) . . 3 7 0
Documents relatifs
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
usual route to school and provide us with some details about their travel routine (e.g., schedule, mode of transportation, people accompanying them, etc.); 3) recent interventions
La question de l’extension des limites de Paris jusqu’à cette ligne avait été discutée mais ne fut tranchée que par un décret impérial du 9 janvier 1859 qui décidait de
C'est aussi le résidant pas chez leurs parents fréquentent deux fois parativement à la province, les étudiants de Nanterre ne lement ce moindre attachement à la ville
In this paper, we study the performance of an enhanced channel estimation technique combining estimation using an autocorrelation based method and the Expectation-
Selon Laeven et Valencia (2012) , le coût budgétaire de la crise bancaire s’est respectivement élevé à 44 et 41 points de PIB en Islande et en Irlande, ce qui place ces deux
Dans ce travail, nous comparerons les entreprises coopératives, détenues par leurs membres, dont le produit risqué est la ressource critique apportée et dont l’objectif est
