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Linear programming problems for $L_1$ optimal frontier estimation

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(1)Linear programming problems for L_1 optimal frontier estimation Stéphane Girard, Anatoli Iouditski, Alexander Nazin. To cite this version: Stéphane Girard, Anatoli Iouditski, Alexander Nazin. Linear programming problems for L_1 optimal frontier estimation. RR-5466, INRIA. 2005, pp.34. �inria-00070541�. HAL Id: inria-00070541 https://hal.inria.fr/inria-00070541 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Linear programming problems for L1 optimal frontier estimation Stéphane Girard — Anatoli Iouditski — Alexander Nazin. N° 5466 Janvier 2005. N 0249-6399. ISRN INRIA/RR--5466--FR+ENG. Thème NUM. apport de recherche.

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(14) |. L1  

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(17) w . . uLa!uZa¡pL•–gbcej–pLf ckpZckP†To—˜pL•–•sp’j–f†‡. α = (α1 , . . . , αN )T. (}. f9 3. αi. i=1. fbN (Xi ) ≥ Yi ,. log N , N h2. i = 0, . . . , N + 1 ,. αi 1{(j − 1)/mh ≤ Xi < j/mh } ≤ Cα h ,. j = 1, . . . , mh ,. i = 1, . . . , N,. ’P†T(ieT v<uKiyuKSUTceT(i m jma wbT(Ÿ†fRT"w7cep¢4TUckP†T”j–f/ckT"‡rT"i­v<uKikc)pr— S”u$_¤¢4T)—˜prieS”uK•s•–_”’iej–c¡ckT"f9ura&•sj–fRT>uKivRiepr‡LieurS §š,n€3 h. 1/h. žUO&PRjma prvbcej–SUjs›"uKckjsprfïvRikpL¢R•sT(S. f9Cyr3. , min 1TN α. JP∗. f9PdW3  9-Or3. i=1. 0 ≤ αi ,. $|  9>r~ 3  9 53. i = 1, . . . , N ,. |fbN0 (Xi )| ≤ Lf,β gmax Cβ (K, K 0 ) N X. . α. G‚rƒ ;‚:9-3 log N log N −L g C (K, K ) 1 ≤ Bα ≤ L g C (K, K ) 1 , Nh Nh ;‚r5‚ 3 D α ≤ C h1 , G‚rr„ 3 0 ≤ α. O&PRT"ikT”jma pLfRT”v<p/a¡j–ckjsrTUv†urieurSUTckT"i j–fïcePRT xpLf†a¡ckiyuKjsfLcyaH 9Pd53)uKf†w ;‚r‚53 j–cea uK•sgRT”’js•–•œ¢<T wRjsaexg†aeakT"w j–f¬]`T>xDckjsprf}†ž™Y¬prieT(prT"i"¥rckPRT)C—˜pL•–•sp’jsfR‡UfRpKcyuckjsprf<a&P†u$rT­¢<T"T(fjsfLceikpb wbg<xT"w  G‚K } 3 X , 0, X , 1, ;‚r5| 3 1 , (1, 1, . . . , 1) ∈ R ;‚Kr~ 3 A , kK (X , X )k.

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(21). dx ;W‚ Or3 D , k1{(j − 1)/m ≤ X < j/m }k ;W‚ yr3 Y , (Y , . . . , Y ) . subject to. Y. f,β. max. β. 0. N. 2.  r3. ≤ Aα ,. f,β. max. α. mh. T. β. 0. 2. N. α. 0. N +1. N. h. i. h. T. N. j. i,j=1,...,N. j. x=Xi i,j=1,...,N. h. 1. N. i. h. i=1,...,N ; j=1,...,mh. T. % ((   y! O&P†T­¢<urakjsx)uraeakgRSUvbckjsprf†a&pLf cePRT gRfR `fRp’f¢4prgRf†w†uKie_”—˜gRf†xckjsprf9uKieT . ɘæ%î,ÉËÊ.

(22) d. L1  

(23)      . rž 0 < f ≤ f (x) ≤ f < ∞ ¥b—˜priur•–• x ∈ [0, 1] ¥ Q ‚bž ¥`—˜pLiuK•s• x, y ∈ [0, 1] ¥’j–ckP L < ∞ urf†w 0 < β ≤ 1 ž |f (x) − f (y)| ≤ L |x − y| O&P†TZ—˜pL•–•sp’jsfR‡”uraea¡gRSUvbcej–pLf†a&prf ckP†T  rT(iefRT"•<—˜g†f†xDcej–pLf¬uKieTZjsf/ckiep`wRg†xT>w  =9rž < P r u  a U u  x L p S < v r u D x  c k a R g R v 4 v r p k i c ¥ K : R → [0, ∞) K(t) = [−1, 1]  supp ‚bž Z K(t) dt = 1 ¥ &„Rž K jsa&cePRieT(T­ckjsSUT"ax(prf/ckjsf`gRprg†ak•s_¤wb‚j FT(ieT(f/cejsur¢R•–TLž VXpKceTr¥ceP†uc —˜pri2uKf`_ gRfRjsSUpbwRuK•`T(LT(f© rT(iefRT"• SUT"TckjsfR‡ xpLf†wbj–ckjsprf†a =9&‚Rž  T&l/gRprckT j–iya¡c"¥2cePRT9a¡gRik—§urx(T pr—ckP†TieT(•muceT"w T"a¡ckjsSUuKckT>w cd’p v†ikT"•–jsSUgj–f†urik_#=ieT"2akgR•–cea prf ckPRTT>adcej–S”ucepri fb ž µ K(·) akgRvRv4prikc Sb jsaurvRvRiep bj–S”ucej–LT(•s_ P α ž]`T>xpLf†w'¥!ckPRTU—˜gRf<xDckjsprf fb jmaš,jma¡vFxyPRj–ck›"jsurf,ž n™iep`pK—§a urikT­v4pLa¡ckv4prfRT>w cep¤]`gR¢†akT"xckjsprf¬|blž 9Kž     r 

(24)  i    € X   0 < h < 1/4    P ~ W    

(25) !   œ¯    Q 9. min. max. β. f,β. f,β. t∈R. 1. −1. max. N i=1. N. 

(26) #"$&%     b

(27)  mh ,.  0/    1   e32#4.

(28) . N. N. i. = bh−1 c (' %^  )%^H  8

(29) *A. −2Cα Kmax h ≤. ¯  bŒ56. Z. 1 0. fbN (x) dx −. N X.

(30) H)%^HW  ;   5 +

(31) ,- AX)%^ . G„rƒ.  r3. αi ≤ 4Cα (gmax − 1)Kmax h .. i=1. 

(32) # 7 . 82 w^W :

(33) 9, 

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(36) * A >P ? 2 )%^w %:@%T i 5ACB*D$  :  /  /   5FE 9"XHG

(37) 

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(39) ) %^X  e_ ,2 ?  P 82Y=H5  5  ?  7 5   .   ^ % <  C  . .   /  > P+     I 4V) %^  K     g(x) > 0 ∀ x ∈ [0, 1] J%    K(·) 0  /  max , max |K(t)| ? X    W 8 2YP H    .

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(43) <R QB  AS  €  9,=HW  YZB$  .      €  ~.      h → 0   N → ∞   =%Y>%T  lim. N →∞. log N = 0. Nh. fbN. ?  5T  . . ? 2U9VW

(44) <X,R. §„.  T9-3.  =Hr w ‚ €   82Q   N = N (ω)  %q)%^ $

(45)    2 N ≥ N >%T9  aI%J ' %T  S)%^      4 4 4  W    &

(46) *  )%^XHW  fbN P*)%^w     [0, 1]   ? : 5 j  

(47) . <. $4 LfbN. ,. max |fbN0 (x)|. ≤ 2Lf,β gmax Cβ (K, K 0 ). î! †ÐyÓŽŽ. §„/‚ G„r„r3  53. x∈[0,1]. log N . N h2.

(48) O.   . .   W 

(49) w . . ¯  bŒ5. M=b@A  ?   

(50)   >%T  C

(51)

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(60)  fb    >  n   /  N ? 2+9,XH&  àcuKvRv4T"uriea¤ckP†uKc cePRTïT"a¡ckjsSUuKckpLi b ¢<T"j–f†‡ ckP†Tïakpr•sgbckjsprf cep cePRT pLvbckjsSUj–›>ucej–pLf vRiepr¢R•sT(S  9"}r33 f9CO53 pri cep¬j¨cyaT>lLg†j–uK•sT(f/c©š¦nzrT"iefakNjsprf  9Cyr33 ;‚K„53wbTŸ<fRT"ackPRT¤ LT(iefRT(•oT"a¡ckjsSUuKckpLipK—&ckP†T a¡g†vRv<pLi¡c

(61). (x prT"ikjsfR‡ ur•–•cePRTUv<pLj–f/cea (X , Y ) urf†w'¥,urvRvRiep `jsS”uceT(•s_r¥FP†u$`j–f†‡ cePRT”a¡S”uK•s•sT"a¡c)akgRik—§urxTL¥'gRv7ckpckPRT ceT(ieS O(h) a¡v4T"x(j¨Ÿ†T>w jsf9š,T"S S”u9rž2Y9pLikT"prT"i"¥/xprf<adceieurj–f/ceaVf9"~53Y 9 d 3œpri*G‚^9C33$G‚L‚53œT(f†akgRieT fb ∈ ’j–ckPu v†uri¡cejsx(gR•muKi)š,jsv†akxyP†j¨ce› x(prf†a¡ceurfLc L ‡rjsrT(f j–fš,T"S S”u‚`ž©O&P†T x(prf†a¡ckiyuKjsf/c Σ (1, L ) —˜priuK•s• T(f<a¡gRieT"aceP†uc —˜priXur•–• a¡jsf†x(T)ckPRT ¢†uLa¡jmx­ rT"ikfRT"• α ≥0 jmaxyPRp/a¡T"fckp”¢4Ti =fRpr1,fRª;.fR.T".‡L,uKNckjsrT RckPRjmaXakT(T"SUafbckp”(x)¢4T≥f†u0cegRiyuK•'—˜pLix—˜gR∈f†[0, xckjspr1]f f (·) jmav4pLakj¨cej–LTrž µ jsf†uK•s•sK_r¥ f†pKckTckP†uKc™ckPRT­uK¢4prTwbT>akx(ikjs¢4T"wUT"a¡ckjsS”ucepri ;5| 3D¥  9-y533$G‚r5„ 32S”u$_¢4TceikT>uceT"w”ura2ckPRT­uKvRvRiep bj–S”uKckjsprf cep¬Y¬u bjsSgRS š,js rT(•sjsPRp`p`w ! a¡ckjsS”uckT ikT"•suKckT"w ckpcePRT”T"a¡ckjsS”ucej–pLf7—§urSUj–•sh _ ;W| 3 2akT(jT <*.">, a@"!$#  0 † ~ † ¥ | P , 2 ˜ — L p & i k c R P  T b w " T U S r p † f ¡ a k c y i  u e c – j L p , f ž W. i. i. N. [0,1]. fbN. fbN. i. N.  %  y% Í fUcePRT—˜pL•–•sp’jsfR‡ ckP†T(prieT(S¥rckPRT­xprf<a¡jmadceT(f†x(_ urf†w ckP†TZxpLf`rT(ie‡rT"f†xT&iyuceTpK—FckP†TXT"a¡ckjsS”ucepri™cep’uriew†a cePRT)ckiegRT­—˜ieprf/ckjsT(ijma&T"a¡ceur¢R•–jmakPRT"w ’j¨cePikT>a¡v4T"xcckp ckPRT L fRprieS prfcePRT [0, 1] jsf/ckT(ieuK•;ž

(62) ¯`³ Œ ¯   9,V)%^Y ? PYH    [ A   &  :MO

(63)  M v  

(64) ,R QB % s  q   )%^ 1.    €T A     C > 6f   α max lim inf N →∞.    x l h → 0   N → ∞   =%Y>%T . log N > ρ > 0, N h1+β. lim. N →∞. §„r}.  3. log N = 0. N h1+β/2. ' T%   HW S$,R Y*BI%^P_)%^S

(65) . ,.  Z/  2   7 & T04 . )%   .  ³ Œ ³ §bŒ °.   2+β kfbN − f k1 ≤ C12 (β)hβ + 2C4 (β)h−2 (log N/N ) 1+β (1 + o(1)). a.s.. C12 (β) , 2Lf,β gmax Cβ (K, K 0 ) + 4Cα (gmax − 1)Kmax 1{β = 1} . β ! 1+β 2   1+β 2C 1 f C4 (β) , 2Lf,β  + gmax Cβ (K, K 0 ) Lf,β ρ. . ' %^w IHr. . .  W .

(66)   P /    . %:7=%   /  W    .   β kfbN − f k1 = O (log N/N ) 1+β.  1 ! 1+β 2Cf . Lf,β . §„L|.  r3. §„L~.  r3. G„.  dW3. ? 2 %^  

(67) '. a.s.. ɘæ%î,ÉËÊ.

(68) y. L1  

(69)      .  wW    

(70)  h H 0/ )%^S

(71) . ,.  Z/  2   7I4. . %:7=%   . . . log N h ∼ ρe N. *>%T  5T ? : lim sup N →∞. . log N N. . YB . β − 1+β. 1  1+β. ,. G„.  5Or3. 1. 0 < ρe < ρ− 1+β ,. kfbN − f k1 ≤ C12 (β)e ρ β + 2C4 (β)e ρ −2. §„.  ryr3. a.s.. š¦TcXg<aPRjs‡rPR•sjs‡rP/cckP†uKc§„ryr3akPRp’a&ceP†uc fb ieT"uLxyPRT"a*§gRvckp u¤•–pL‡LuKiej–ckPRSUjmxX—§uLxDcepri3ckPRTS jsfRjsS”u  ieuKckT­—˜priš¦j–v†aexyPRj–ck›­—˜ieprf/ckjsT(i f ¥RakT(Tw\H."!.B+4$G^] GJI 'tR_+%B`T"U,.&Ih0º‚}W2;¥RO&PRT"prieT(S}<ž~9Lž~9Lž L Œ 5 ' %^_  U W      CB# HH<2 ? =Hr   r 

(72) ¯  b ˜}/rƒ 3 log N lim <∞ Nh . %:7=% ~  _  & >%TS   /     5

(73)   l

(74) +     V  CB$ ,R CBJ  N. 1. 1+β/2. N →∞. %”  &O P†T vRikp`pr—!pK—2O&PRT"prieT(S)9 ’PRjmxyPjsa‡rjsrT(fjsf9]bgR¢†akT"xDcej–pLf¬|bž } jma¢†uLa¡T>w pLf¢<prckP9gRvRv4T(iZuKf†w•sp’œT"i ¢4prg†f†wRawbT(iej–LT"w“jsf ]`g†¢†a¡T>xDcej–pLf |bžº‚#urf†w ]`gR¢<a¡T>xDckjsprf |bž „#ieT"akv<T>xDckjsrT"•–_Lž  PRT"f vRiep`j–f†‡#ckPRT>a¡T ¢4prg†f†wRa"¥K’œTuLakakgRSUTckP†uKc%cePRTakT"l/gRT(f<xTpK—4ckPRTaeuKSUvR•sT X ª;v4prjsf/cea (X ) jma™uK•sieT"urwR_­jsf†x(ikT>urakT pLiewRT(ieT"w'¥`’j–ckPRpLgbcXxyP†uKf†‡rjsfR‡©f†pKceuKckjsprf —˜ikpLS X ckp X —˜pLi&ckPRTaeuK LTZpr—akjsS v†•–jmxj–cd_r¥`ceP†ucjma §T} 9-3 X ≤X , ∀i.  TT"aea¡T"fLcejsur•–•s_)uKvRvR•s_­ckPRTgRfRj–—˜prieS urak_`S vRckpKcejsxo¢<pLgRf†w O(log N/N ) prf ∆X , X −X v†ikpLT"w jsf9uKg bjs•–jmuKie_¤š,T"SUSUuU~bž T(—˜prieTZceP†uc>¥R’T­v†ikpLTZjsf¬]`gR¢†akT"xckjsprf9|blž 9ZcePRT)cd’p v†ikT"•–jsSUj–f†urik_”ieT"akgR•–cea"ž  

(75)  

(76)  !

(77) #"%$'&

(78) " ( Œ ³,³*)³* ) o¯   ( ,+ VprckT­ckP†uKcXwbTŸ<fRj¨cej–pLf†_a ;5| 33$§5~ 3jsSUvR•s_¤ckP†T)—˜pr•s•–p’jsfR‡UwbT"x(prSUv<p/a¡j–ckjsprf¦ž i i=1,...,N. i. i. (i). i+1. i. Z1 0. fbN (x) dx. =. 0. =. +. î! †ÐyÓŽŽ.   h 1−h Z1 Z Z  fbN (x) dx  + + h. i. ˜}`‚.  53. 1−h.   x − Xi 1 αi K dx h h 0 i=1   Z h Z 1 ! N X x − Xi g(x) − 1 + K dx . αi h h 1−h 0 i=1. Z1 X N. i−1. ˜}/„.  r3. ˜}L}.  3.

(79) >ƒ 9. ]bj–f†x(T.   . αi. urf†w ckPRT"ikT(—˜prieTr¥. 0. 0.   W 

(80) w . . urf†w  rT(iefRT"• K urikT)fRpLfbªàfRT(‡/uckjsrTL¥Lj–c—˜pr•s•sp’ackP†uKc §}/|.     Z N N X X 1 1 x − Xi x − Xi αi K dx ≤ αi dx = αi K h h h h i=1 i=1 i=1. Z1 X N. Z1. . fbN (x) dx −.  53. R. N X. αi. ≤. i=1.   h Z1 Z N X gmax − 1 1{|x − Xi | ≤ h}dx  r3 αi  + Kmax h i=1. ˜}/~. 0. N X. ≤ (gmax − 1)Kmax. 1−h. ˜}.  :dW3. αi 1{0 ≤ Xi ≤ 2h}. i=1. +. N X. αi 1{1 − 2h ≤ Xi ≤ 1}. i=1. !. ≤ (gmax − 1)Kmax 4Cα h .. §} §}ryr3  rOr3. O&P†Tj–f†T"l/g†uK•sj–cd_ ˜} y53&—˜pL•–•sp’a&—˜ieprS  9PdW3akj–f†x(T ¢<prckP¬jsf/ckT"ikuK•majsfK˜}:d 33$˜}rOr3uKieT pK—ckP†T•sT(fR‡rckP 2h urf†w ckP`g†a S”u$_¬¢4T¤xpLT(ieT"w¬¢`_9cd’pikT"•suKckT>w jsf/ckT(ieuK•ma)pK—ckPRTU—˜pLikS [(j − 1)/m , j/m ] jsfN 9Pd53Dž pLf†akT"l/gRT(f/ce•–_L¥/’T­P†u$rTZvRikpLT"wUcePRT)gRvRv4T(i¢4prgRf<w¤—˜pLiœcePRT wbj~4T"ikT"f†xT­j–f ckP†T­•sT—ËcP†urf†wa¡jmwbTH˜}L~ 3Dž O&P†T©•sp’œT"iX¢4prgRf<w jmaXv†ikpLT"w9jsf¬cePRT”akurS T©S”uKfRfRT"i"ž Íf<wbT(T>w'¥,wbT"x(prSUv4pLakj¨cej–pLn f ˜}`W‚ 33$˜}r } 3XjsSUvR•sj–T>a(¥ akjsf†xT­ckP†T)ckT(ieˆ S §}rr} 3jma&fRpLfbª;f†T(‡LuKckjsrTL¥ h. Z1 0. fbN (x) dx −. N X. αi. ≥ −. i=1. N X i=1. . αi . ≥ −Kmax. Z0. +. .    1 K x − Xi dx h h. 1+h Z 1. −h. N X. αi 1{0 ≤ Xi ≤ h}. i=1. +. N X i=1. ≥ −Kmax 2Cα h .. O&P†jsax(prSUvR•sTckT>ackPRT v†ikp`pK—%pK—2š,T"S S”u;9rž. h. αi 1{1 − h ≤ Xi ≤ 1}. !. G|rƒ.  r3. G|.  ^9-3. ;|r‚ ;|K„r3  53. ɘæ%î,ÉËÊ.

(81) 9r9. L1  

(82)      . ( Œ ³,³*))³ ) œ¯    E + tT"SUj–f†w ckP<uc)’œTUuraea¡g†S T;˜}a9C3Dž_¬urvRvR•s_/jsfR‡uKgbjs•–jmuKie_9š,T(SUS”u ~uKf<w š¦T(SUS”uHO©’T­Ÿ†iyadcXurikiej–LTZuKc G|K } 3 max |fb (x)| G|L5| 3 = max max |fb (x)|   ;|Kr~ 3 log N 1 (X − X ) max |fb (x)| ≤ L g C (K, K ) + max 0 N. x∈[0,1]. 0 N. 1≤i≤N +1 x∈[Xi−1 ,Xi ] f,β. ≤ Lf,β. ’j–ckP. max. 0. β. i. N h2. i−1. 2. x∈[Xi−1 ,Xi ] 8 1≤i≤N +1  2 log N log N 1 gmax Cβ (K, K 0 ) CX max |fbN000 (x)| , + 2 Nh 8 N x∈[0,1]. CX > 4Cf /fmin. 000 N. ž™O&PRT S”u bjsSgRS ceT(ieS jsfs;|rd53jsa&¢4prgRf<wbT"wura—˜pL•–•sp’a —˜pLiuKf`_ . |fbN000 (x)|. G|.  dW3. x ∈ [0, 1].

(83) 3

(84)

(85) d

(86) αi

(87)

(88) 3 Kh (x, Xi )

(89)

(90) dx i=1

(91) X

(92) 3

(93) N

(94) ∂ αi 1{|x − Xi | ≤ h} ≤ sup

(95)

(96) 3 Kh (v, u)

(97)

(98) · u,v ∂v i=1. ≤. N X. −4 ≤ gmax LK · 3Cα h , e 00 h. akjsf†xT §akT(Tš¦T(SUS”u”|—˜pLi&ckPRTwRTceurj–•sT"wwbT"SUprf†a¡ckiyucej–pLfa3.  ^9-3.  2 4 3 gmax + 6Kmax gmax . LKe 00 , LK 00 + 3LK 0 Kmax gmax + LK gmax 3LK + 10Kmax. ]bgR¢†a¡ckj–ckgbcej–fR‡ ; |WO53¥"§~Lƒr3jsf/ckpYG| d 3_`j–T"•swRa max |fbN0 (x)|. x∈[0,1].  2 log N log N 3 C + g L C h X max K e 00 α N h2 8 N h2 log N gmax Cβ (K, K 0 ) N h2. ≤ Lf,β gmax Cβ (K, K 0 ) ≤ 2Lf,β. g†f†wbT(iuLwRwbj–ckjsprf†ur•,uraea¡g†S vRckjsprf[˜’PRjmxyPPRpr•mw¤ceikg†T)—˜priuK•s•,akgb£¤xjsT(f/ck•s_ •muKie‡rT h≥. O&P†T­ieT"akgR•–c—˜pr•s•–p’a"ž î! †ÐyÓŽŽ. 3CX2 Cα LKe 00 log N . 8Lf,β Cβ (K, K 0 ) N. G| G~rƒr3  5yr3. G~.

(99) 3

(100)

(101)

(102) −4

(103) sup

(104) 3 Kh (v, u)

(105)

(106) ≤ gmax LK e 00 h ∂v u,v. ’P†T(ieT. ;|.  WOr3. G~L‚.  53. G~r„ G~K} 3.  r3. 3. N . G~L|.  53.

(107) $‚ 9.   .     $    fbN  &   "  J ∗ P œ¯    9"€>%T A   &  

(108) ' %^  

(109) +% j   N' %T  . . . .   W 

(110) w . . .

(111)   2    . γ > Lf,β gmax Cβ (K). §~L~.  r3. ? €W  ω ∈ Ω ) %^ w=Hr          N1 = N1 (ω, γ)   %;>%T T

(112) _  N ≥ N1 ) %^ ? 9V 

(113) <XI,R Y*BIX V P ~      dW3 ∗ β  iW e . G~ ( Œ ³,³*)o³*)œ¯    #+ pLf†akjswbT"iouKie¢Rj–ckiyuKie_ ’ j¨ceP N (ω) —˜ikpLS š¦T(SUS”u~RžÍf/ckiepbwbg†xT —˜g†f†xDcej–pLf f (u) = f (u) + γh uKf†w v†akT(g†wRpKªàT"Na¡ck≥jsSUNuKckpL(ω) iea JP ≤ Cf + γh . 0. β. γ. 1 + δi1 α ei = 2. Z. Xi. 1 + δiN fγ (u) du + 2 Xi−1. Z. 0. Xi+1. fγ (u) du ,. i = 1, . . . , N. §~.  rOr3. Xi. ’ P†T(ieT adcyuKf†wRa—˜pri )ieprf†T"xy rT"ia¡_`S¢4pr•;ž T(•sp’ ’TwbT(SUpLf†adceieuKckT)ceP†ucZx(prf†wRj¨cej–pLfK§~r~r3T(f<a¡gRieT"a cePRTrT"xδckpLipK—Zv†a¡T"g†wbprª;T>adcej–S”ucepriya αe = (eα . . . , αe ) cep ¢4Turf urwbSUjmakakjs¢R•–Tv4prjsfLc©—˜pri©ckPRTš,n f9Cy53Y;‚K5„ 3¥b—˜priurf/_ a¡gb£¤x(j–T"fLce•–_ •surik‡LT žO&P†jsajsSUvR•sj–T>aa¡pL•–uK¢†j–•sj¨cd_ pK—%ckPRTš¦n1f9Cy53Y G‚K„ 3&uKf†w N . ij. 1. JP∗. š¦Tc. ≤. N X. 1. T. (f (u) + γhβ ) du = Cf + γhβ .. G~.  5yr3. 0. ž µ pLi&ckPRTaeuK LTZpr—%akj–SUvR•sjsx(j¨cd_L¥R’T­jsSUv<p/a¡T)cePRTurw†wbj¨cej–pLf†uK•,uLakakgRSUvbcej–pLf†a i=1. CX > 4Cf /fmin. α ei =. Z. N. log N ≤ min h ≤ ρN h β. ’P†jsxyPPRpr•mw ckiegRT­—˜priur•–• N •muKie‡rT­T(fRpLgR‡rP,ž. . fmax 1 , γ ρCX. . ,. ƒ.  d r3. ɘæ%î,ÉËÊ.

(114) >„ 9. L1  

(115)      . Lž µ jsiyadc>¥R’œT)vRieprT)xpLf†a¡ckiyuKjsfLcyaV 9>|53gRf†wbT"i 9. feN (x). ,. N X i=1. =. N +1 Z Xi X i=1. +. =. α ei Kh (x, Xi ). 1 2. 1 + 2 Z 1. fγ (u) du. Xi−1. Z Z. ¥. αi = α ei i = 1, . . . , N. ž µ priuKie¢Rj–ckiyuKie_. x ∈ [0, 1]. Kh (x, Xi ) + Kh (x, Xi−1 ) 2. X1. fγ (u) du (Kh (x, X1 ) − Kh (x, 0)) 0 1. fγ (u) du (Kh (x, XN ) − Kh (x, 1)) XN. +. i=1 X. +. 1 2. 1 + 2. Z Z. i−1. fγ (u). K‚.  d 53. K„. d r3. }. d 3. r|. 0. ZXi.  dJ9-3. d 53. fγ (u) Kh (x, u) du. N +1 X. ¥.   Kh (x, Xi )+Kh(x, Xi−1 ) − Kh (x, u) du 2. X1. fγ (u) du (Kh (x, X1 ) − Kh (x, 0)) 0 1. fγ (u) du (Kh (x, XN ) − Kh (x, 1)) .. K~. d r3 drdW3 dWOr3. XV p’ ’ Ta¡T"v†uKiyuceT(•s_­¢4prg†f†w T"urxyP pK—<ckPRTakgRSUS”uKf†w†abdK|533 d O53!—˜ieprS ¢<T"•–p’ ž Xg†Tœcepid 3¥KcePRTS”uKjsf ceT(ieS dr|53jma&¢<pLgRf†wbT>wura—˜pr•s•sp’a XN. . Z. î! †ÐyÓŽŽ. 1 0. fγ (u) Kh (x, u) du = f (x) + γhβ +. Z. 1. (f (u) − f (x)) Kh (x, u) du 0. ≥ f (x) + (γ − Lf,β gmax Cβ (K))hβ .. dWyr3. rƒ. O r3.

(116) "} 9.   . .   W 

(117) w . . O&P†T ª;ckP”akgRSUS”uKf†w©—˜ieprS  d~53%jma™wbT>xpLS v4pLakT"wUurf†w©ckPRT"f ¢4prg†f†wbT"wU¢<urakj–fR‡)pLf©ceieurv<T"›(jsgRSz—˜pLikS©gR•mu T"ikiepri&i ura—˜pL•–•sp’a .  Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du fγ (u) 2 Xi−1  Z Xi  Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du ≥ fγ (x) 2 Xi−1 Z. . Xi. −. ZXi. Xi−1. OT9-3. /‚. O 53.

(118)

(119)

(120) Kh (x, Xi ) + Kh (x, Xi−1 )

(121)

(122) |fγ (u) − fγ (x)|

(123) − Kh (x, u)

(124)

(125) du 2. L„. O r3. r}.

(126) 2

(127)

(128) ∂ Kh (x, u)

(129)  O 3 (Xi − Xi−1 )3

(130)

(131) 1{|x − Xi | ≤ 2h} ≥ −(fmax + γh ) max

(132) 12 u∈[0,1]

(133) ∂u2 Z Xi  O 53 gmax LK −Lf,β [(u − Xi−1 ) + (Xi − u)] du. |u − x|β 1{|x − Xi | ≤ 2h} 2 2h Xi−1 β. /|. _ urvRvR•s_`j–fR‡”š¦T(SUS”u ~†¥`ckPRT)Ÿ<iea¡cckT"ikSjma¢4prgRf<wbT"wura—˜pL•–•sp’a K( } ) ≥ − C log N  N ru f†w ckPRTakT"x(prf†w prfRT)jma¢4prgRf†wRT"w¢`_ O. 2. fmax gmax LK 0 (Xi − Xi−1 ) 1{|x − Xi | ≤ 2h}, 6h3. X. g L| ( )≥−. O. max Lf,β. LK. 2h2. . log N CX 1{|x − Xi | ≤ 2h} N. Y¬prieT(prT"i"¥L—˜ieprS š¦T(SUS”u ~†¥bprfRT x"uKf9a¡P†p’“Ÿ†iyadcceP†uc urf†wa¡T>xprf<w¤ceP†uc. N +1 X. 1{|x − Xi | ≤ 2h}(Xi − Xi−1 ) ≤ 4h +. i=1. N +1 X. 1{|x − Xi | ≤ 2h}. ≤ ≤ ≤. Z. Xi. Z. Xi. |u − x|β du .. Xi−1. CX log N , N. OrOr3. |u − x|β du. Oryr3. |u − x|β du. x−2h−CX (log N )/N.  . 4h + CX. OJdW3. Xi−1. i=1 x+2h. Z. L~. O r3. log N N. log N 4h + CX N. . max. v∈[−2h−CX (log N )/N,2h]. . log N 2h + CX N. β. .. |v|β. rƒ. y r3 y^9-3. ɘæ%î,ÉËÊ.

(134) $| 9. L1  

(135)      . O&P`g†a"¥b’T)urikiejsrT)uc&ckPRT ¢4prg†f†w —˜pri&ckP†Ta¡gRS dK~53&ura—˜pL•–•sp’a . .  Kh (x, Xi ) + Kh (x, Xi−1 ) − Kh (x, u) du 2 i=1 Xi−1   CX fmax LK 0 log N log N log N 4 + CX ≥ −gmax CX Nh Nh 6 Nh β !  Lf,β LK β log N + h 2 + CX 2 Nh  2  5 log N ≥ − gmax CX CX fmax LK 0 + 3β+1 ρ−1 Lf,β LK . 6 Nh N +1 Z Xi X. fγ (u). /‚. y 53. r„. y r3. K}. y 3. L| Q c•mura¡c"¥bj–cjsaakjsS js•muKie•–_¤wbT"SUprf†a¡ckiyuceT"w ckP†uKc&¢4pKceP9a¡gRSUS”uKf<wRa*drd 3&uKf†wsdWO53&uKieT­¢<pLgRf†wbT>w ur¢<pLT ¢`_ O((log N/(N h)) ) ž µ pLij–f†a¡ceurf†xTL¥`—˜pri_ d5d 3¥RprfRT)pL¢bceurj–f†a. y 53. 2.

(136) Z

(137)

(138) X1

(139)

(140)

(141) fγ (u) du (Kh (x, X1 ) − Kh (x, 0))

(142) ≤ (fmax + γhβ )X1 |Kh (x, X1 ) − Kh (x, 0)|

(143)

(144) 0

(145) 2  y r3 log N . ≤ 2fmax gmax LK CX Nh. r~. O&P`g†a"¥`—˜ieprS. r~ j–c—˜pr•s•–p’aœ—˜pLiT"urxyP.  dJ9-3Yy 53. j = 1, . . . , N. ckP†uKc. feN (Xj ) ≥ f (Xj ) + (γ − Lf,β gmax Cβ (K))h + O β. —˜pLi™a¡gR£”x(j–T"f/ck•s_•muKie‡rT γ−Lf,β. N ≥ N0 (ω). . log N N h1+β/2. 2 . . CX fmax LK 0. ‚Rž]`j–SUjs•surik•s_r¥'x(prf†a¡ckiyuKjsf/ceaX 9"~r3PRpL•sw¬ckiegRT gRf†wbT"i α = αe ¥ ¥`’T fRp’ P<u$rTZckpU¢<pLgRf†w ckP†TuK¢†akpr•sgbckT)uK•sgRT pr— x ∈ [0, 1] i. jsf†a¡ckT>urw pK—bdJ9-3Dž g T"ikT. N X i=1. i. 2 !. N. α ei. 12LK + 5. i = 1, . . . , N. ≥ Yj. y dW3. . +3. β+1. .  Lf,β LK . ρ y5Or3. ž Í f†wbT"T"w'¥F—˜pLi­urik¢†j¨ceieurik_. X d e h (x, Xi ) Kh (x, Xi ) = α ei K dx i=1. e h (x, u) , ∂ Kh (x, u) K ∂x. î! †ÐyÓŽŽ. log N Nh. ’PRT"f©¢4pKceP”j–fRT>l/g†uK•sj¨cej–T>abdKƒ532urf†wcePRT&—˜pr•s•sp’j–fR‡)pLfRT&PRpr•mw©ckiegRT. 5 gmax Cβ (K) ≥ gmax CX 6. feN0 (x) =. . y5yr3. >ƒrƒ.  9 r3.

(146) >~ 9.   .   W 

(147) w . . . Ga¡T"T]`gR¢†akT"xckjsprf9~Ržl9C3’j–ckPckPRT)—˜pL•–•sp’jsfR‡©g†vRv<T"i¢4prgRf†w . >ƒ wRT"wbg†x(T"w —˜prieS f9(}/| 3¥" 9"}L~53ž gT"f†xTL¥RprfRT)S”u$_¤ieT(v4T"uKccePRTuKie‡rg†S T"f/ceapK—bdr‚W33$d Or3¢/_ xyP†uKfR‡Lj–f†‡ —˜pLi žO&PRT(ieT—˜pLikTL¥$ur•–•/cePRTœiyuceT"a¦—˜ikpLS‹OT9C33$y dW3¦akPRprg†•sw¢<T&wbjs/jmwbT>w¢`_ ¥’P†j–•sTocePRTuK¢†akpr•sgbceT K ur•–gRT)prK—!ecePRT S”uKjsf ceT(ieS pK—wbT"x(prSUv4pLakj¨cej–pLf,¥RwbgRT)cepqOr3¥Rjsa¢<pLgRf†wbT>wuLaœ—˜pr•s•–hp’a 

(148) Z

(149)

(150) Z

(151)

(152)

(153)

(154)

(155)  9>ƒLr‚ 3

(156) e (x, u) du

(157) =

(158) (f (u) − f (x)) ∂ K (x, u) du

(159) f (u) K

(160)

(161)

(162)

(163) ∂x  9>ƒr „ 3 ≤ L g C (K, K )h , jsf†a¡ckT>urwpr—B d y53YOLƒ53Dž2tT"SUj–f†wckP†T&wRTŸ†fRj–ckjsprfqy 33$ 9"rƒ 3,—˜pLi C (K, K ) ’PRjsxyP©—˜pr•s•sp’a,—˜ieprS f9"Tƒ 9C3Dž O&P`g†a"¥`—˜priakgb£¤xjsT(f/ce•–_ •surik‡LT N ≥ N (ω) urf†w —˜priT"uLxyP X ’T)urikiejsrTZuKc  

(164)

(165) }3 log N  9>ƒKJ log N

(166) e

(167) g C (K, K )h +O ≤L g C (K, K ) .

(168) f (X )

(169) ≤ L N h ρN h VZuKSUT(•s_r¥`jsfRT"l/g†ur•–j–cd_ f9"ƒK5} 3œPRpL•sw†ackiegRT­ur•–SUp/adca¡g†ikT"•–_U—˜pLi&ur•–•4cePRpLakT N ≥ N (ω) a¡g†xyP ckP†uK_c dK5ƒ 3œjsa LT(iej¨Ÿ<T"wuKf†w      9>ƒLr| 3 log N log N 5 L C (K, K ) −1 ≥ g C h N 6 Nh     L L  9>ƒr 12L ~3 L + · C f +3 5 ρ ’P†T(ieHT Ga¡T"T š,T(SUS”u¤|—˜pri&cePRTwbTcyuKjs•–T>w wRT(SUprf†a¡ckiyucej–pLaf 3  9> ƒ d53 K . K , L ,L +L g L ,L +L g

(170)  

(171)  

(172) 

(173) 

(174)

(175)

(176) 0 x−u

(177)

(178) x − u

(179)

(180)

(181)

(182) e −2

(183)

(184)

(185) K ≤ h g K K (x, u) + g K

(186)

(187) h max

(188) max max

(189)

(190)

(191) . h h.  9 ^9P3. h. h. 1. 1. γ. h. h. 0. 0. max. f,β. β−1. 0. β. 0. β. 0. j. 2. 0 N. j. max. f,β. β−1. 0. β. max. f,β. 2 3. 0. β. 2. 0. 2. f,β. 0. β. max. β+1. X. 1+β/2. X max. K. K0. e K. max. max. e0 K. e K. e0 K. K 00. K0. max. β+1. f,β. e K. max. „†ž µ jsf†ur•–•s_r¥bcePRTxpLf†a¡ckiyuKjsfLcyaV 9PdW3’j¨ceP. >ƒ ur•sakp”PRpL•swceikg†TgRf†wRT(i = αe ¥ i = 1, . . . , N ž Íf†wbT"T"w'¥4¢`_š¦T(SUS”u¤~¤ckPRT—˜pr•s•–p’jsfR‡”jsfRT"l/g†ur•–j–ckjsT"a P†pr•mw u†ž a"ž%—˜priXur•–• N ≥ Nα (ω) urf†w —˜priT>urxyP j = 1, . . . , m ¥R’PRT"ikT m = bh c.  9 5O 3. Cα ≥ 6fmax. i. 0. N X. i. h. h. . −1. . α ei 1{(j − 1)/mh ≤ Xi < j/mh } ≤ (fmax + γhβ ) 1/mh + 2CX. log N N. . >ƒ.  9 5y 3. "ƒ g†f†wbT(i)urw†wbj¨cej–pLf†uK•uraeakgRSUvbckjsprf†aXdKƒ53DžO&P`g†a"¥'xpLf†adceieurj–f/cyaw 9 d 3ZuKieT—˜gR•–Ÿ†•s•–T>w¬gRf†wRT(i  9>ƒ5O53ZuK•sSUpLa¡c akgRieTr¥`—˜pLiuKf`_ akgb£¤xjsT(f/ck•s_¤•muKie‡rT N ž }<ž™]`j–f<xTuK•s• αe ≥ 0 ¥Rxprf<adceieurj–f/ceVa f9CO53P†pr•mw ceikg†Tr¥RuKf<wš¦T(SUS”uU„©jma&vRieprT>w'ž i=1. ≤ 6fmax h,.  9r9 3. i. ɘæ%î,ÉËÊ.

(192) 9 d. L1  

(193)      . ¯  bŒ5. . 2XA58 2  Z / 9, X< : 5   W W W $    &   30

(194) <XI  h  W } ,2WH. ‚  W  ) %^b  l   ? : PS  YX#I,R YX*"$V  :

(195) *D I,R 

(196) #D     5  9,   X, ?  Z/ A r& e  . >%  S^W     ν ∈ (1, 2)   W =)%^W.

(197). . N +1 X. . 3. 1{|x − Xi | ≤ 2h}(Xi − Xi−1 ) = o. i=1. . h logν N N2. .  9r959P3. %T  x)%^w  

(198) *)%^w  Y" <i    / e /  ? l . >%  T  )%^W &

(199) C"  €H x %J 2  #/ T W w  >T%   q ,2s  P >%T;  L

(200)   CX$&YX*"$[P .   P L

(201)   C

(202) *D    ,. IPZ/    .  & :%^ Z/ s  %   >%T K       ‚

(203) >%T ' %^    ' Ke^% WH  . %02 .  A7    P* >%T     0/ S)%J €  ?  e32:%T   0. 0.      $   fbN œ¯    5)%^    &     . .

(204) ' %T   <

(205) x

(206)   e W  ω ∈ Ω )%^  =Hr      ?      q % )  ^ % W  & 

(207)    2  W  N ≥ N (ω) N2 (ω)     x ∈ [0, 1]   

(208)  2. . )%  W      ( Œ ³,³*)³*) o ¯ . C4 (β) fbN (x) ≥ f (x) − h2. C4 (β) 5T  i  . CB ! . a¡g†xyP ckP<uc. N ≥ N6 (ω) ik ∈ {1, . . . , N }. urf†w. log N N. >‚.  2+β 1+β.  9r9 r3.  + š,T(c&g<a&ceur rT)g†akT­pr—š¦T(SUS”uidurf†wj¨cya priepr•s•surik_¤‚ jsf/ckiep`wRg†xjsfR‡ δy = Lf,β δxβ ,. O&P`g†a"¥4—˜pri)uKf`_. . urf†w urf`_. δx ,. 2Cf log N fmin Lf,β N. 1 ! 1+β. .. "„.  9r9 3. ckP†T(ieT©T`jma¡ceaX˜’j–ckP v†ikpL¢†uK¢Rjs•sj¨cd_prfRTP3Xurf¬jsfLceT(‡LT(i  9r9(J} 3 |x − X | ≤ δ. x ∈ [0, 1]. ik. x. VXp’ ¥`ckPRT T>adcej–S”uKckjsprfT(ieikpLi&uKcu©v4prjsf/c x x(urf ¢4T Tbv†urf†wbT"wura Yik ≥ f (Xik ) − δy .. f (x) − fbN (x). = [f (x) − f (Xik )] i h + f (Xik ) − fbN (Xik ) h i + fbN (Xik ) − fbN (x) .. >|.  9r9 r3. "~.  9r9 3  9r9Pd53  9r9CO 3. O&P†TZceT(ieS jsfckPRT iej–‡LP/c&P<uKf†wa¡jmwbT  9r9"~53S”u$_¤¢4T ¢4prgRf†wRT"wura—˜pL•–•sp’a. |f (x) − f (Xik )| ≤ Lf,β |x − Xik |β ≤ Lf,β δxβ ,. î! †ÐyÓŽŽ.  9r9Cy 3.

(209) 9-O.   . uLa&’œT"•–•,uLackPRT)ceT(ieS. .   W 

(210) w . .  9r9CO53. $‚Kƒ.

(211)

(212)

(213)

(214) b

(215) fN (Xik ) − fbN (x)

(216) ≤ LfbN |x − Xik | ≤ LfbN δx ,. ’j–ckP©uš,jsv†aexyPRj¨ce›oxpLf†a¡ceuKf/c L —˜pLi,ckP†T™—˜gRf†xckjsprfT>adcej–S”uKckpri wRgRT)ckYp  9$W| 3pr_i ;‚K5ƒ 3ž™O&P`g†a(¥, 959$|W3œj–SUvR•sjsT"a fbN. fbN (x).  9 3. ž%tT"S jsf†w)ceP†uc. f (Xik ) − fbN (Xik ) ≤ (Yik + δy ) − Yik = δy .. pLS¢RjsfRjsfR‡”uK•s•FckPRT>a¡T ¢4prg†f†wRa&’T­pL¢bceurj–f—˜ieprS. >~ œckP†uKc&—˜priuK•s•. f959 53. N ≥ N6 (ω). f (x) − fbN (x) ≤ δy + Lf,β δxβ + LfbN δx .. O&P†T(ieT—˜prieTr¥!uKvRvR•s_`j–f†‡¬š,T(SUS”u9‚uKf†wakgR¢†a¡ckj–ckgbcej–f†‡¬TbvRieT"aea¡jsprf<aXf959>„53Z—˜pri •sT"uLw¤cep ckPRT •sp’œT"i&¢<pLgRf†w. δx. fbN (Xik ) ≥ Yik. $‚.  9 :9P3. ¥ uKf†w. $‚r‚ jsf/ckpK 9$‚r‚W3.  9 r3. δy.   ≥ f (x) − 2Lf,β δxβ + LfbN δx. fbN (x). $‚K„.  9 3. $‚} —˜pLiœurf/_ a¡gR£”x(j–T"f/ck•s_ •muKie‡rT N GadcyuKikckjsfR‡ —˜ieprS ieurf†wbpLS uRž a"ž¦Ÿ†f†j¨ceTXjsfLceT(‡LT(i>¥r’PRjmxyP wbp`T"aofRpKcwbT(v4T(f†w pLf x3DžO&PRTŸ†iea¡cj–fRT>l/g†uK•sj¨cd_ jsfG„K}r3!P†uLa%¢<T"T(fUuKvRvR•sjsT"wP†T(ieTœjsfpLiewbT"i¦cep­akjsS v†•–j–—˜_ ckP†T•sp’T(i%¢<pLgRf†w'ž š¦T(SUS”u } jmavRieprT>w'ž C4 (β) ≥ f (x) − h2. . log N N.  2+β 1+β.  

(217)    

(218)   9Lž™]`j–f<xT ¥`ckP†T L ªàfRprieSpK—%T"a¡ckjsS”uckjsprfT"ikiepri&x(urf ¢4T Tbv†urf†wbT"wuLa |u| = u − 2u1{u < 0} .  9 J3. . 1. kfbN − f k1. =. Z. + 2. 1 0. Z. 0. h 1. i fbN (x) − f (x) dx h. i n o f (x) − fbN (x) 1 fbN (x) < f (x) dx.. $‚r|  9$‚K~r3  9 53. ‚Rž Q Rv vR•s_`j–f†‡¤š,T(SUS”uraV9 urf†w „ cep ckPRT iej–‡LP/c&P<uKf†wa¡jmwbTif9>‚r|53_/jsT(•mwRa lim sup h. −β. Z. 1. h. i  b fN (x) − f (x) dx ≤ γ + 4Cα (gmax − 1)Kmax 1{β = 1} a.s.. >‚. f9 dW3. VXpKceTr¥`ckP<ucprf†T­S”u$_”Ÿ  γ = 2L g C (K) ¥b—˜prijsf†a¡ceuKf<xTrž „†ž ÍfpLiewRT(ickp”pL¢bceurj–fuUa¡jsSUj–•muKiieT"akgR•–c—˜pLi&ckPRT)ceT(ieSˆf9>‚K~W3¥RfRprckT)ckP<ucXš,T"SUSUu }UjsS v†•–jsT"a N →∞. 0. f,β. max. β. h i bN (x) ≤ C4 (β) < ∞ a.s. ζN (x, ω) , ε−1 (N ) f (x) − f LB. ɘæ%î,ÉËÊ.

(219) 9-y. L1  

(220)      . g†fRj¨—˜pLikSU•s_¤’j–ckPikT>a¡v4T"xc&ckp”¢4pKckP. x ∈ [0, 1]. uKf†w. N ≥ N2 (ω). ¥R’j¨ceP. $‚ gXT(f†x(Tr¥,pLfRT SUu$_¬urvRvR•s_ µ uKckpLg •sT(SUS”uR¥'cyuK `j–f†‡ jsf/ckpurx(x(prgRf/c­ckP†uKc jma u x(prf/ckjsf`gRprg†a"¥ u1{u > 0} SUpLfRpKceprfRT­—˜gRf†xckjsprf  Z h o i n  9$W‚ yr3 lim sup ε (N ) f (x) − fb (x) 1 fb (x) < f (x) dx 1 εLB (N ) , 2 h. . log N N.  2+β 1+β.  9 WO 3. .. 1. N →∞ 1. −1 LB. N. N. 0. "„Lƒ  9"„T9-3 ≤ C (β) < ∞ a.s. }<ž9O&P/g<a(¥%ckP†T pr¢RceuKjsfRT>wikT"•suKckjsprf†a)cepr‡LTckP†T(i’j–ckP f9>‚L|W3urf†wN 9$‚K~53)jsSUvR•s_ G„L|W3ž O&PRT"prieT(S 9 jsa v†ikpLT"w'ž Z. ≤. f9 r3. lim sup ζN (x, ω) 1{ζN (x, ω) > 0} dx N →∞. 0. 4. ï Íf ]`gR¢†akT"xckjsprf©~†ž~9o’œTœT"a¡ceuK¢†•–jma¡P©a¡pLS TœvRieprv4T(ikckjsT"a,ieT(•muceT"w)cepcePRTx(prieikT>xDceT"w) rT"ikf†T(•;ž%]`gR¢<a¡T>xDckjsprf©~Ržº‚ v†ikT>a¡T"fLcya,akprSUT™uKg bjs•–jmuKie_X•sT(SUS”ura¦’PRjmxyP P†u$rT¢4T(T(fg†a¡T>w­ckpv†ikpLTO&P†T(prieT(M S 9rž µ jsf†ur•–•s_r¥$’T™xpr•s•sT"xDc jsf¬]`gR¢†akT"xckjsprf~Rž „Ua¡pLS T)•sT(SUS”urawbT>wbjmx(uceT"w ckpUcePRT vRiep/pr—!pr—%tT"SUurik |`ž   

(221)

(222)   *&  # ,# š¦Tc&ckP†T ¢†urakjsxZ rT"ikf†T(•4—˜gRf†xckjsprf K ¢<T wbT(Ÿ†fRT"wuLaœjsf¬]`T"xckjsprf}R¥†urf†w¤cePRT)¢†uKf<wb’jswbckP h ∈ (0, 1/2) ž tT(SUj–f<w cePRT T"a¡ckjsS”ucepri fb wbT(Ÿ†fRT>w js[f G5| 3&uLa—˜pr•s•–p’a . . . N.  N X   b fN (x) = αi Kh (x, Xi ) i=1   αi ≥ 0, i = 1, . . . , N,. ’P†T(ieTZcePRT  rT"ikf†T(•4—˜gRf†xckjsprf. Kh (x, t) = h−1 K((x − t)/h). ’P†j–•sT. Kh (x, t) = h. urf†w. Kh (x, t) = h. î! †ÐyÓŽŽ. −1. −1. K((x − t)/h). K((x − t)/h). Z. Z. ∀ x ∈ (h, 1 − h). x/h. K(t) dt −1. 1. K(t) dt (x−1)/h. >„L‚.  9 r3. !−1. !−1. ∀ x ∈ [0, h]. ∀ x ∈ [1 − h, 1] .. "„r„. f9 3. >„K}.  9 3. "„/|.  9 53.

(223) ‚rƒ.   . .   W 

(224) w . . &O P`g†a"¥'cePRT” rT"ikfRT"•%—˜gRf†xDcej–pLf K (x, t) jmawbT(Ÿ†fRT>w¬—˜pLiuKf`_ (x, t) ∈ [0, 1] × R ¥!uKf<w¬cePRT”T"a¡ckjsS”uckpLi f9"„/‚W3™jmawbTŸ†f†T"w”—˜pLiurf`_ `jmu cePRTZ LT(iefRT(• K (x, t) xprieieT"xDceT"w”uKcœcePRT¡¢4prgRf†w†uKiej–T>aKž p fRT S”u$_¤T>urakj–•s_”pr¢<a¡T"ikLTXceP†uc x ∈ [0, 1] Z f9"„Lr~ 3 K (x, u) du = 1 ∀ x ∈ [0, 1] urf†w'¥Rx(prf†akT"l/gRT"fLce•–_L¥RwbgRT)cep T RxyP†uKfR‡Lj–f†‡cePRT j–f/ceT(‡riyuK•'urf†w ckPRTwbT"ikjsucej–LTr¥ Z f9"J„ dW3 ∂ K (x, u) du = 0 ∀ x ∈ [0, 1] ∂x VXpKceTr¥bceP†ucXT>l/g†ucej–pLsf f9" „ dW3SUu$_uK•ma¡pU¢4T rT"ikj–Ÿ†T"wwbjsieT"xDce•–_Lž µ pLij–f<adcyuKf†x(Tr¥RpLf cePRT•sT—ËcX¢4prg†f†wRuKie_r¥ j;ž Trž—˜pri x ∈ [0, h] ¥b’œT)P†u$LT h. h. 1. h. 0. 1. h. 0. Kh (x, t) = h. −1. K((x − t)/h)g(x) ,. g(x) =. ZT(f†pKckjsfR‡. Z. x/h. K(t) dt −1. !−1. .. >„.  9 5yr3. e h (x, u) = ∂ Kh (x, u) , K ∂x. ’T­ckP`g†aP<u$rT g 0 (x) = −. urf†w. Z. x/h. K(t) dt −1. !−2. >„.  9 5O 3. h−1 K(x/h) = −g 2 (x)h−1 K(x/h). e h (x, u) = h−1 K((x − u)/h)g 0 (x) + g(x)h−2 K 0 ((x − u)/h) K       x−u x−u −1 −1 0 = g(x)h −K Kh (x, 0) . h K h h. "}Lƒ.  9 r3. (}  9"}/‚r3  9 a9-3. XT(f†x(Tr¥bcePRT)j–f/ceT(‡riyuK• g. "}L„ T>l/g†uK•ma›"T(iep9—˜pri x ∈ [0, h] ž Q a¡jsSUj–•muKi©vRiep/pr—S js‡rP/c ¢<TikT"v<T>uceT"w—˜pLi x ∈ [1 − h, 1] ž µ j–f<uK•s•–_L¥ T>l/g†uK•sj¨cd_  9"„L~W3PRpr•mwRa&ceikg†TZ—˜pLiuK•s• x ∈ (h, 1 − h) ckp`p†¥Rakjsf†xT g(x) ≡ 1 prT(icePRjsaj–f/ckT"ikur•Gž Íf’P†uKc&—˜pr•s•–p’a"¥b’œT)g†akT)SUprieT­‡LT(fRT"ieur•4—˜prieSgR•muraVG|r33$§~r3jsf†a¡ckT>urw pK—b 9"„rOW3D¥bceP†ucjma Z. 1. 0.   2 e h (x, u) du = − g (x) K x K h2 h. Kh (x, t) = h. −1. Z. K((x − t)/h) g(x) ,. 1. K. 0. . x−u h. g(x) =. Z. . du +.   u=1 g(x) x−u −K h h u=0. x/h. K(t) dt (x−1)/h. !−1. ,. x ∈ [0, 1] ..  9 3. "}r}.  9 3. ɘæ%î,ÉËÊ.

(225) ‚^9. L1  

(226)      . O&P†T(ieT—˜prieTr¥buLa—˜pr•s•–p’a—˜ieprS f9"„ry53D¥"f9(}L}53œ—˜priuKf`_. x ∈ [0, 1]. ¥. e h (x, u) = h−1 K((x − u)/h)g 0 (x) + g(x)h−2 K 0 ((x − u)/h) K       x−u g(x) 1 0 x − u = +K (Kh (x, 1) − Kh (x, 0)) . K h h h h. (}`|  9"}L~ 3  9 53. O&P†TZ—˜pL•–•sp’jsfR‡”š,T(SUS”u vRieprT"a&š¦j–v†aexyPRj–ck›(ª;•sj– LT­x(prf†a¡ceurf/cea&jsf[f9"ƒJdP3&uKf†w[§~T9C33$§~/‚W3Dž œ¯    9" r   K r7   ;   ,R   w)%^i A   &  

(227) ,R QB,V  .  W>%. . %TA. . %TA. ?  5T  . h. ? 2 

(228) <B#XI &' ^%   )%^S

(229) . <.  Z/. 9  Ke n  h ∈ (0, 1/2)  , 

(230)

(231) 

(232) e

(233) 2 ,

(234) Kh (x, u)

(235) ≤ gmax h−2 LK + gmax Kmax

(236)

(237)

(238)

(239)

(240) e h (x, u)

(241) ≤ gmax h−3 L e , K K

(242) ∂u

(243). Y>T %  ?  ? r^ : P  %    4. "}.  9 Jd53.

(244) 2

(245)

(246)

(247)

(248) e h (x, u)

(249) ≤ gmax h−4 L e 0 , K K

(250) ∂u2

(251).   L = L 00 + L 0 g K LK K max max K e = LK 0 + LK gmax Kmax  e0 K

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