Edgeworth type expansions for transition densities of Markov chains converging to diffusions
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(2) Universit´ es de Paris 6 & Paris 7 - CNRS (UMR 7599). ´ PREPUBLICATIONS DU LABORATOIRE ´ & MODELES ` ´ DE PROBABILITES ALEATOIRES. 4, place Jussieu - Case 188 - 75 252 Paris cedex 05 http://www.proba.jussieu.fr.
(3) Edgeworth type expansions for transition densities of Markov chains converging to diffusions V. KONAKOV & E. MAMMEN. JUILLET 2004. Pr´epublication no 923. V. Konakov : Laboratory of Probability and Statistics, Central Economics Mathematical Institute, Academy of Sciences, 47 Nakhimovskii avenue, 117418 Moscow, Russia. E. Mammen : Department of Economics,University of Mannheim, L 7, 3-5, 68131 Mannheim, Germany..
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(60) . ¥X¦. R O¨©T M B P+P 5 M T $ w P. M B P7P . 5 8 5 *M +Q . w M P+P S w S. e. u N U + UTB P: &v& & L © M U U + U B P U R R R N. U + UTB P U. M*+K w M P+P(I T $ w J S w S . ¥X¦. M*+Ps$ M B P $ , , / M B'U 5 % u N u L M U U + UB P L M © R R © R. R. N. 5MB PJ U . R O¨© M B P7P M# $ >P 8 8. ,. U. U + UTB P. R. , , Ir/ M*+ U Ps$Y/3M BU P J % . , ,. ¦. % M U z P M B <P I ¥ *M +Ps$ 5 5. . $ $. $6+P U. =3L5M Lo U L U + UTB P MA+Ps$ e . . ¦ / M B'U P)M R ¥X¦O¨© M B , , % MA+Ps$ M B P $ T $ $ z , % % 5 © 8 5 M*+l $. ¦. . u N. M U U + U B PªR R R. $. !#". ¨©. ¥X¦ M B. . N. ¥X¦. u N. $ $ 5 % ©. ¥. ¨©. u N u U + UTB P: L c© M U U + U B P_ L =M R R R R R o =3L3M L U L U + U B P u N M U U + U B PR R O¨© $ M M*+PR$ M B R R % e . ¦w¦. HL. R. U. u N U. . u N U + UTB P M $ nP =5L3M U U + U B P U R e e R R u N u N HL5M U U + TU B P $ 'L M U U + U B P U R R R R %. uL3M Ru. L. . ¥. R W K. '. $&. U.
(61) L'. ¥X¦ R. ©© M Ru. L. $. $. U. %. R. N. u N M © R U R U + UTB P. {¦. =3L5M Lo U L U + U B P $ nM n/M*+Ps$ nM B P7P9M e n% . ¦. ¥. U + UTB P¯R. . % . ¦. . M. $. !". ¨©. AP M*+P $ I M*+P % 6 6. ¦. 6. ¥X¦. ¦. ¦. ¦. 5 %. ¦. . ¥X¦. ¥. ¦. ¦. . ¥X¦. . MA+_Ps$ . . ¥. . . nM B P $ , , /3M BU PAM R O¨© M B P7P 5 M n% 5 % M*+ P $ M B P $ $ $ , , /3M B_U PAM R O¨¥© % % © 5 T $ $ $ n I nM*+P /MA+_ P $ ] n % % $ , , /3M B'U PAM R O¨© M B P+P 5 8. ¦. ¥ . . M*+P$ . . $ nP 5 8. . e. ¥. /M B P7P U. . 6 MB P ¥ MB PJ . M*+l w M P+P S w S. e. . 5 8 5 AM +l. M B P+P . w M P7P S w S. e. . ¥. nM B P . M B P J. . 5 MA+@ w M P7P S w S . e. . MA+PR$ M B P $ M B P 5 8 5 M*+P $ & ' % 5 % u N T $ L ¦M U U + UB PR R `© $ M nMA+_P$ n/M B P7P ,r, /3M*+ U © R R n'% % MA+l w M P7P<I T $ w'J S w S `U e u N T $ nM B P+PN, , Iw/ M*+ U Ps$6/3M B'U L' =M U U + UTB PR R 9© $ © R R n% % M*+Q w M P+P(I T $ wmJ S w S `U e u N T $ , , / M*+ U P $ L M U U + UTB Ps R ¨© $ H 8 M*+l © R R n% % % M n M* + PR$ n M B P+P w I T $ w J S w S `U u N u N u N L M U U + UB P L M U U + UTB P: &v& & L %M U U + U B P U $ $. . ¦. ¥. ¥. ¥. R. . ¦. ¦. . ¥. ¥. . . R. ¦. ¦. . ¦. ¦. ¦. R. R. . ¦. R. R. $ n/M*+PN, , ¦ 3/ MA+ U P ¥ n]% R R % ¦ $ 8 M*+K w M P+PAM nM*+Ps$ nM B P+P w I T $ w'J S w S i & e % ¦. HL =M. u N U. U + UTB Ps R. ¥. . T. $. . A¦. . P. PJ. w M P7P. e. .
(62) $ I nM*+P n MA+Ps$ nM B P n M B P J n n % 5 5 % 5 ¦ , , ¦ / M*+ U P ¥ ¥ 5 M*+Q w M P+PAM T $ w P S w S U e ¦ u N T $ nM B P n 5 M B P , , Ir/ M*+ U Ps$Y/3M BU P J L =M U U + U B PR R ¥ ¨© $ n n % R R % 5 ¦ ¥ ¥ 5 M*+Q w M P7PAM T $ w P S w S U e T u N $ $ n/M*+P , , ¦ /3MA+ U P ¥ HL =M U U + UTB Ps R ¥ n]% R R % ¦ $ 8 M*+K w M P+PAM nM*+Ps$ nM B P+P w I T $ w'J S w S i & e 8 % ¦ M I M $ L M Q M X M U J M}| ZN\sQ J G T \sUt^FSPT J5L \qQ5Q 6 ^ M UwG J1M _*\ = 3 ^sT5X M U_T G Ut\qQ MYp J G I[SPT JjL M G J5LNM I M Q M X M U J _ M}| ZN\sQ J G V& E Z I JjLNM I X G I M O GI w 0 U z ^lS M ^ =M < U M M U z P^ 5 % ¦ 8 ¥ 5 % ¦ ¥9© S LNM I M M 1 P2X M \sU~_[\sU T5Ut^qT p \ J GI O ZU p J T GU 7
(63) *Mx Z J M*+P ¦ MA+P
(64) &v&& M*+P \sUt^ MA+P V \sUt^ M*+P V T O \ J Q M \_ J GU M G O:JjLNM[p G G I&^sT5U\ J1M _gG O T _vU M V\ J T M 7 M ^ =M < U M \sQh_ G J5LNMO G Q5Q GSPT5UWV O ZWU pJ T G U~_ L . ¥. u N M© U U + U B P R ¨© R R . . . T. $. .
(65) .
(66) . . . . . M*Z[P =5L3M. u T N R. e. U. R. U Z UB P U. M P R ¥X¦ M B P R ¥X¦O¨© i& e EHG I \sQ5Q u P¯N + \sU^ B9J5LzM O WZ U p J T G U T _^ M=< U M ^3\ _T5U M XgX*\ 7 > J5LNMvJ1M I X L¥ o ¦O¨© & E G I u ªNl$ T \sU^ 5 T1U(&&&lU lS M ^ M=< U M u N L M U U + UTB P Vi& R R 7. M I M \ V\qT5U ^ M UwG JyM _ . . s 3&Z!3(/: :%<:%<&)?u?@# g I 5#1 7&r-:1a /:o?u&K4 &A 21 o.zG)#%4 GA,340#¦ :%5 Agkt<c:%4 4 :F*t >73&"4 I5&A t :%<7 &! (7:9:1<_:=<Y&)?u? #K g I :15#1D=:8Eu#%3-zBC#%?u?@&rQ¥¡%¢o¢%¢[Xgst3&-3 I^&)(/&)3GA&j I k 5#¦n73 o 8 7 ?u&*&i,3 +&"5 21 &)(>:%(7-3&r(s#8.94 :%(>&rp`!5#=5 + I:oYg}s5&r *&"(/&)!34 #%G)& M*+P =3L5M L U L U + U B Pk9. =3L5M Lo U L U + U B P ¤#M 6$ >P AM +P6#13- IC 3&&rp`!5(+&A 7 / :o3 "<c:o( 'L U L © U e L © #%3- HL' & e s#).940:1(t&xp`!$#%3 + :1a<b:o( I +5&"<c:1404I:8*j 32<c:1(+?q,34 # s e :q 3 I &A3- *>&K,3 +&Z73&Z. ¦w¦. . u N =3L5M U U + U B Ps , /3M BU P M*+K R. . R. e. M P7P S X &. ¦. ¦. ¦. e <= >?>?@7C n LNM O G Q5Q G SPT5UWV GZWUt^ L GQ]^ _ PS T JjL \ p GU_ J \sU J L M u N%R U NN=R U + U B P W K Q A ¡1X R ¥ ¥X¦ ¨© ¥X¦ ¥R¦ M B $ +P _v 5 ToUG&&v& U U . . AF. ¦.
(67) L . Q A o X . . ¥ ¥ ¥R¦ ¥ ¥X¦ M B $6+P. M u N=R U NON%R U + UTB P W. " L M u N%R U NN=R U + U B P W . Q )A X Q FA X. u L3M u N%R U NN=R U + U B P W . Q 8A X . _v &5 T1U(&&&lU U. ¥ ¥ ¥X¦ ¥ ¥X¦ M B . . ¥ ¥ ¥X¦ ¥ ¥X¦ M B . . $6+P U $6+P U. L9P 6] J M u N=R U NON=R U + U B P I =3("L;M uL . e. T 6 M U P 6 ¥X¦w¥ ¥ ¥ ¥`© M B $6+P U O G Iv\sQ5Q u P N \qUt^ B& M I M \ V\qT5U L¥ o ¦O¨© & ®5:o( 5 U
(68) V G)4 #% ?u \Q A8 X#13-ªQ A¦ X5#)E1&'&)&r¸ +3:8*j¸ I T T M U P &v&&. W. :15#1D=:FE #%3-BC#1?u? &AQ¥¡%¢%¢o¢oXr`&)?u?@#q g A ¢`g}5:o(#!3(7:9:%<:1<73&A 7&KGA40#% I?@ |<c:o( yeV :1(
(69) yV #=3-a<c:1(73&Z!3(+:`:%<c t:=<tQ A ¡oX>#13-Q A8 X:o3&K!5(/: Gr&)&r-3 6 + ?u 4 #1(+4l.og . ?
(70) ; %. 2 . A;. . &5:8* GA:%?@&Z : 73&!5(/: :=<}:%<}st3&r:o(7&r? g A giBC#1 I :9:14I <c:1(j &!3(7:9:1<}3#8E1&'&)&A21 IE1&A IC9,53 7&rGA+ 0:13 g A ~ g g|5(7:1? &A? ? #1 t3g A #13- g ¡q*n&Z21&At75#% L T =M V`U)ToU + TU B R P $ =3 (;" h6 M V U)ToU + UTB P_ R © 6L3M*+ UB P U %. 6 . *j3&A(+& 6L3M*+ U B PK q#a<c,33GA7 :1*j I7 7,332o#%,3 7 + 0#% #% 4I ) ¥g &%gC<c:o(;G):1 +x#=[7 75#=. 6L$MA+ U B P W. U Mz I3:14 -3 . / 3eI $ z M*+$ B P © J &. ¯ l a&A?u?z# g A 3 I t21 IE%&) . ¥. ¦. =M V U8T%U + U B P$b=3L3M V U)T1U + U B Ps &v&& ¸R 9© L5M*+ UTB P U. Q FA X *j3&A(+&. . c $. . © L. 6% . L. =g ( " h 6 M V U8T1U + UB P$ $ =2 (iL " 6] M V3U)T1U + UTB P U %6 6% L L $ =2 ("L " h6 M V3U)ToU + U B P$ $ =2 ("L;M#" L®R ¥R¦ ¨© P h6 M V3U)ToU + U B P U ¦ 6% 6% L $ =3 ("LMq " L>®R ¥X¦O¨© ¦ P '6 M V U8T1U + UTB P$ =2 ("LM uL L[P 6] M V U8T1U + U B P U %6 . ¦. $. L. . . A¦.
(71) . $ . . L. 6% . L. $. . !&A(7&"*&! , . =g (iL;M uL L[P h6 M V U8T1U + UB P $ $. . L. 6% . = Ld(KLKM uL> LoP 6] M V U8T1U + UTB P U. 3= L ("LZM uL% LoP '6 M V U8T%U + U B e P $ $. 6% ¦ M U L U + UTB PRM# $ |Pe *e ¦ M U L U + UTB P &. L. 6% . =3 Ld(iL " L '6 M V U8T%U + U B P &. &; :F*«-3 +GA,5 / 73&Z#1 /.`?u! := GK'&A3#FE9 I:o,3(t:%< 3&"7&A(7?u µ6 + 32Cst3&):1(+&A? ¡9g ¦ I 6:13#1D%:FEa#%3-CB]#%?u?u&)Q¥¡=¢o¢o¡1X*&K2o&rj 3#=. 4_ X xtJ G J T p J I M \ J X M U J G O9J5LNM9J1M I X /. st3&A:1(7&r?. A gA. A. ¦ U(&&v&IU. . . g. #1 - 3&\(+&A?@#%(7DH<c:14 4I:F*t 32. T 6L5MA+ U B P U R © e e *j3&A(+& 6L3M*+ U B PK q#a<c,33GA7 :1*j I7 7,332o#%,3 7 + 0#% #% 4I ) ¥g &%gC<c:o(;G):1 +x#=[7 75#= . ¦. T. 7. . . . =3("LZM# $ P © =3("LKM V3U)ToU + UTB P_. =R. U z. I3:14 -3 . / 3eI $ z M*+h$ B P © J. L3M*+ UTB P W #%3- *t3&A(7&KM# U L U + UTB P. 6b % " ' 6 M U L U + UTB P & /4_ X xtJ G J T p:J I M \ J X M U J G O JjLNM4JM IYX 7 &*t 4 4' /3:F* + 3&i<c:14I40:8*j 2;&p !3#13 7 I:1z:1< © © c< :o(# GA:13 + #%o yeV #13-\<c:1( wyeV 7? #14I4_&A3:1,32oYg Q 8A X . © $ . $b 3 = ("L " '6 M V U8T%U + UTB P %6 e. ¥X¦ ¦ P h6 M V U8T1U + UB P. $ b =g(iL;M" L R O¨©. ¦w¦. 6% e. . $ b =2("L;M " L © P h6 M V U)ToU + U B P %. ¦. 6 e $ b =2" ( L;M#" L © ¦ P h6 M V U)ToU + U B P %6 e $ b =3("L;M#q " L ¦ P '6 M V U)T1U + U B P W KR ¥X¦w¥§ M B $ +P & 6% e. 5:o(j 3&7&)(/?@ i:1]75&4I&A<b65#13-H 7 I-3&:%<iQ 8A X*>&q*t I4 4 73:F* 3&<c:14 4 :8*j I32@':1,3 -5 i*j I7# GA:13 + #%o yeV g. ¦w¦. ¥R¦ ¦ "P 6] M V`U NN=R U + U B P. =3("L;M L R O¨¥©. Q 8A X. W. e. 6 R. . $ =3("L " '6 M V U NN=R U + U B P . T T e M U P (& &'&. . T N M U )P M P h6 R AF. ¥X¦ ¨© 6 8 ¦ M B. $6+P U.
(72) QO¡%¢oX. QO¡ A X. QO¡1¡1X. *j3&A(+&. ¦. =3(iL;M HL' © " P 6 M V U NON%R U + UTB P$ =3("L " h6 M V U NON=R U + UTB P . e. 6. W. R. T T M U P &(&'&. . M U T AP M N P '6 e ¥`© ¨¥© 6 8 ¦ M B $ +P U R. {¦. =3(iL;M " L' © P 6] M V U NON%R U + UTB P$ =3("L " h 6 M V U NON=R U + UTB P e 6 T T e T N ' 6 ¨¥© 6 8 M B $ + P U M P M A P M P W U &(&'& U R R. ¦. ¥R¦. ¦. =2("LMj"q H L P '6 M V U NON%R U + UTB Pe$ =2("L" 6 M V U NON%R U + UTB P . e. 6 W. . R. T T M U P &(&'&. . M U T AP M N P '6 ¥Re ¦ ¨¥© 6 8 ¦ M B $ +P U R. o *M +P 3 &(&'& M*+P " S V U'&(&'&rU S ¦
(73) ©¦ &'&(& © M¥NQ$ u P N=Re U M*+P I T + © 5 J ¥R¦ , /
(74) . . V U. . "!$#&%('*),+.-!0/1&!3254%%,67%,698;:=<>@?A8;:CBD>9E,#FG8IH,JD>LKNMO15!3254%%,67%,6QPSRTE,+VUW7X8YH:(>[Z@8IHH\>]E\4;! %\U+.//^!SF_K`Ma1!b7^!APCRVE,+VU7c6%R.RV%d'feg07+.Uh+.RTE,4cE,4)\iUW!S#D/^7jEk7l8;:dB>[Zb8IH,JD>@KnmoRVR PSRVE\+VUW7aE,4!Q24%=-,!CF +p/!S4^E,/^+p-!bR.gegq+V#&Fi5Pb/^+V%#rK`s[#t/^15!A+.#5F5iP@/+.%\#t7;/^!C257u/^15!Aev%\i5#F57O),+.-\!S#t+V#w8:dBD>[Z@8YH\H\> E,4!oi57!CF 6%\4 9xzy{x}|X~f Ko%,/!/1E(/Q+V#8;:d<\>O/^15!l/^!C4U7%\#5R.gXE,252 !SE\46%4 y0z| K"! #%d'z7/LE,4;/ eg0254%d-+.#)$8:=B\>aE,#F"8YH\J>@K
(75) Ma15!254^%%,6%\6u8;:=<> 'O+.RTR e !A)+p-\!C#XEk6/^!S4;'aE,4F57CK ^= W`!A4;7/254^%d-\!q8:dB> 6%4a z K %4a/^15+.7O25i4;2 %7;!A'u!A'j4^+p/^! Q Ik | I @¡ ^¢ I£&¤ y&¥,| ¤¦§¤^¨ ¢ f| I O© ® ¯° « ¤ ª «Cª ¬5 'j15!S4! ¥k¸ 8IH,±> ¶;£\¤[· ¤^¦§¤^¹ ¢º »« 5 .· ¥k¸ ¤@¼ ¥,¸ ¤¹§¤¨ ¢C½ ¾ « ¹ ¢¿À¾ « ¨ ¢Á «° |© ³² ¬5´`µ È È ÂAà « ¨ ¢ÄÅ ¶ I£&¤[· ¥(¸ ¤^¦§¤¹ ¢lÆW© È Ç º »«LÊ5ËIÌ ¶Í .· ¥,¸ ¤@¼ ¥,¸ ¤^¹§¤¨ ¢ Î ¹ ¢n¿ Î ¨ ¢¢ µ ¬ É È È Â{ÐÏ È © Ç º »«^ÊËYÌpÊ5ËYÒ ¶ .· ¥,¸ ¤L¼ ¥,¸ ¤ ¹ ¤¨ ¢ ^Ó Ñ ¹ ¢¿ Ó ÑI¨ ¢^¢[Ô Ñ ¬ É Õ"ÖA×ØdÙGF5ÖbÚCØ\ÛÜ Ø,Ý^ÖQÞß5ÖÝ;àÛÛE\×5F « ° ³@© âdã@ä ³© â(ã;é ° «á ¸ Ï µæådåCå^ç ¹ Âè¸ Ï µêåSådå^ç ¹§¤ ëB.
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