Large Deviation Principle for a Markov Chain with a Countable State Space
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Large Deviation Principle for a Markov Chain with a Countable State Space Arnaud de La Fortelle et Guy Fayolle. N ˚ 3503 Septembre 1998 ` THEME 1. ISSN 0249-6399. apport de recherche.
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(52) Gó 1 ' '¼±<³. ar0. 0 . &± % % ±!± . 1 ijMÄ 0 % %½±. a. a. n [ ] r ª'§¨g] s }] s j}~ s }gpn r j}Hnh¤g]_¿Uj}]_}_] ' + L { 'Wª§½n [ ' + +gj} [ n [ hYn ' ' ' + + ¬ + ;74(T§¨_] L §¨g s ^!|6hMHn_¡>n [ ] r ]7]GkT§¨gng h5 s ¦ ] r §¨}ºgj}g]_¿Uj}]_}_] ' + Z[ ]_ s ¦ § s j}g½o ' ' + + ¡hM}~mg§¨_] §¨g' s ª¸] r g]_^`§½ÂÅ s ¼n§¨Uj s j}g R ¡ ' ¬ + ' ¬ + > ¨§¨ ^ "§! }¹ ' ' + ¬ + >74; O §¨}P S¹ ' ¬ +UE ª [ §¨ [ §gh! s ¼n r hM~}§¨Hn§ s {]Hn'j}g7~]H¯}] { ~ $ { A ' 3a EMMMlE +?[ ' ' + ¬ + C74 M rY s ^n [ ]©~}]H¯}§½n§ s s ¹ { ¡n [ ] r ]Sh r ]S]GklhMHn¨o ' a + [ ^`]ihMgj r ]_gmª'§½n [ hM¨ºhYn s ^`g ^Qj}½n§¨| ¨[ ]_g s ¹'Oa N hM}~©gjl|}| sMr nN§¨}_¨j}~]_~w§¨ 3a EMMMrE ¡hM}~ [ ]__]n [ ] r ]`h r ]`hYnº^ s gn ' a + ^C]is hMgj r ]_g§¨ [ { - sq n]²n [ hYn r zu{ s L s =@?s zu{rs L4[ { $ g§}]_¿UjhYn§ s 'ÔUé + hM~]_¿>jhYn§ 'ÔU +¡n ]º{¾ j}|}|] j }~f¹ ¨ ª'g7¹ ^«n ]§¨}]_¿UjhM¨§½n§¨]_g a ¨¸» q . zu{ LB2 > a ¨¸» q . ' z { + { 2 a l6¨ ' a + 4 0Í "M > 'ÔU j + ! w. . . $&% %+ Ç>Ë Ç 1 5 / , 4 >Ë ÊÇ ¤ Ë 5 Ì7 ± . ¢ #±`³Mà ! ± 1 &!$#&%½±;S³ ´ 0R< =T³"!Và 0 Ãw ° ! . = ´± GÃ6&± %¬ Gf³Mà ! . U³"! ± ! ! ³ % ±³ 1 ± ? A =T±HÃ#G 0 ®³ % % 0 T±_à _± . I J '¶ +HG §¨^ "§¨! }¹ a ¨¸» q I. O §}P ¹Q ' ¬ + 'ÔUö + Ë Ë {]Hn¬«]Q§ rr ]_~}j}_§¨¨]hM~ I J '¶ +NhM s |]_©g]Hn s ¹²hMhM_]_~S^`]ihMgj r ]_g ª§½n [ I S]5¯ r gpn'hMggj^`] ' +7hM}~s Tjl|} | ' +¸§¨g¸¯}§¨n]M [ q r s s s o | g§½n§ _ Wé @ª¸]w]Hnh g]_¿Uj}]_}_] "¹ ! >HoU_]_g ' +`ª'§½n ¯}§½n]wgjl|}| sMr n 'Åljl|}| ' r + s s , +gj} [ n [ qhYrn s s ' ' s + + [ r - s n]fn [ hYns n [ ]m¨]_}Mn [ s ¹ sM r ' + hM¨ª¸hioUgº ª'gNn §¨}¯}§½nÁo o | g§½n§ W½a¡<n ] ]Q]Gkl§¨gpngºhf hMzHoU_¨] !¹ ª§½n [ ¨]_gg7n [ hM n r hM}g§½n§ s }g_ [ ' èéå #
(53) _ÜOßGÿ7ß Ý-øYÝôßGçMååWß ãÔÜYæã ÍÝ-øYóôÝ ' èéå5õéßpÞ6Ý-àQåWÝôÿ¸è óôßGø_ã¶èöøiçYi ßGçYå m i m . i m i m _4. yaa H'. va. A. z {. F'. )5*. e. xa. e. * xa T,. l). ,. :a. a. a. ,. *. =). . {. .z {. . 2_C. A. t. Z. A. . . X>. ;. UM. . z. . . {. . W. . . Z. . 0. . `K÷ba4` .
(54) aJ. & ?. #. { ]Hn '}V + z ' '}V + +U[n [ ]_g]º6hMhM}_]_~^C]ihMgj r ]_g ¦ ] r §¨¹Vo n [ ] s }~}§¨n§ s }g s ¹ q(rs  | s g§¨n§ s fyj!hM~ a ¨¸» q z { '}V + C ' '}V + ¬ + ' '}V + + ¨; ' R ' + + M P '}V + '}V + ª§½n [ '}V +n [ ]¨]_}Mn [ s ¹ ' +(hM}~ R ' +n [ ] s }]UW>~}§¨^`]_g§ s hMgjl|}| sMr n s ¹ '¶g]_] ¾N]H¯§½n§ s !yé + »ô¹ ' R ' + + U¡¼ª¸]7nh ¥ ]h5Vh r ] r g s n [ hYnn [ §¨g ¿UjhM>n§½nÅo!§¨g(gpn r §¨Hn½o | s g§¨n§ ¦ ]Mt [ ]_qV ª¸] ¥ s ª '}V + hM}~ ¨§¨^ §}¹ a ¨¸» q z { '}V + C ¨§¨^ §¨}¹ ' '}V + ¬ + . . . P. "!. ' +. P. }V. P. "!. T §¨}_] §g( s >n§¨>j s j}g s A ' +zUo qrs | s g§½n§ s 50>½a¡Tn [ ]¨]_¹ng§¨~}]' s ¦ ] r ]_ghM~¡ I ]_§¨} s |]_¡}¹ sMr V4gjnmC_§¨]_>n½o h r ]M¡ '}V + Ig s n [ hYn a a q z { q ¨§{ ^ "§¨! }¹ ¨¸» . I2_C §¨^ "§¨! }¹ ¨¸» z { P '}V + C ' ¬ + 'ÔUö + . . . ' +. }V. P. - s ª ª¸]ª'§¨¨ r ]_hOk®n [ ]¯}§¨n]_}]_ggºhMggj^!|}n§ s K[ A ' + Z o qrs | s g§½n§ s 0>0>¡ [n ] r ] ]Gkl§¨gpng¸hQ "g]_! ¿Uj}]_}_] ' + A' + ª§½n [ ¯§½n]'gj}|}| sMr n( s ¦ ] r §¨}Qn s gj} [ n [ hYn ' ' "+ ¬ + ' ¬ +v( ¿UjhYn§ s 'ÔUö +7n [ ]_ [ s ~}g7¹ sMr hM¼o Ai' + {]Hn r ]_VhOk`hMhMggj}^`|}n§ s }g[ͧ¨¹ A ' +Ín [ ]_'¶g]_]'¹ s>s n s n]N + ' ¬ + hM~ ]_¿UjhYn§ s 'ÔUö + [ s ¨~g_ Z[ ]N| rsUs ¹ s ¹ n [ ] s ª¸] r s j}}~f§¨g7 s _¨j}~}]_~ _ $&%6 %" )( %Y) % - s s s }_]H|}n§¨g r ]ihM¨¨o }]Hª [ §¨s n [ §¨gfg]_Hn§ s ¡ª [ §¨ s [ §¨g¤r ^Cs ] r ]_½o sMgpnr hYnr ]_~ ¹ sMr n [ ]®gh ¥ ] s s ¹ ^`|¨]Hn]_}]_ggi(yHnjhM¨½o¡}§½n ~}g7§¨}~}]H|]_}~]_¼n½o ¹] ~}§¨_§¨nÅo n hM}g§¨]_}_]M¡hM}~hMg ¹ sMr HoT_¨§¨7 [ hM§¨}g_ Z[ ] s }½o!hMggj}^!|n§ s ! s }_] r }g(n [ ]§ rr ]_~}j}_§½§¨¨§½nÁo¡O§¨ sMr ~}] r n sºs }nhM§ n [ ]7 s ª¸] r s j}}~ T s ª] ¦ ] r ¡l] ¦ ]_`§¹}n [ ]¸jl|}|] r s j}}~ [ s ¨~g͹ sMr h r ]_~j}_§½¨] ¥ ] r }]_l¬!¡>§½n ihMf] r ]H¯]_~¡ThM~fn [ ]N s }gpn r j}Hn§ ¦ ]N| rsUs ¹ s ¹<n [ ]5 s ª¸] r s j}~ [ hMg¸n s ]N~ s ]hMhM§¨< tS]© s }g§¨~}] r [ ] r ]ihM¹n] r n [ ]~]_ s ^!| s g§½n§ s 0§¨>n s _hMgg]_g s ¹ qrs | s g§¨n§ s WN hM~ q(rs | s g§½n§ s / s ¦ r. [ [ [ 0>öU# ª] h ] A ' + A ' +ª'§¨n ª ] ] §¨g¤n [ ] r hY| [ s ¹ ¬ s w_hMgg' u sMr ] si¦ ] r ª¸]hMggj}^`]5n [ hYnN¹ sMr hM¨ 4 n [ ] r ]]GkT§¨gngh!|6hYn [ ª§½n [ | s g§¨n§ ¦ ]| rs hY§¨¨§¨nÅo]_}~}§¨}QhY n 4 w n [ ] r ª'§¨g]'ª] s j}¨~ r ]_ ^ sJ¦ ]5hM¨ s TÂÁhM__]_gg§½¨] ]_]_^`]_¼ngª'§½n [ s jln7 [ hM}§¨}`hM>o | rs |] r nÅo s ¹n [ ]ºuwh r¥si¦ [ hM§¨{]Hnjg§¨>n rs ~j}_]n [ ] ¹ s ¨ s ª'§¨} X sUs '~ Z`^`]ihMgj r ]_g_ . . . . .
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(56) 7 )_±va !¶³ % e ±³l) 1 a± a±&% ³ . ! < ±&%éÄ .$0 ³5ap±9' 1 , ! #% ±5´±a ñ&%¬ !y)³ #³ % ³MÃ(,G±H' e ±³l) 1 a± ` 4 '4D +'° ! . = '4D + A ' +HG) 1 ,= . =U³ . R 0 . .1 . 0 0 . 0. * . l= aC± ³)_±% %2. D 0 G °)il=±va ±v!Wa³ ± % e ±!y³l) ) 1 T= a±±C) Ãap³ ±%í³ ap. !³ < %_±*U%öÄ ³:a .$0 !W³ ¬% ? ab'¼±va !VÃp à =l±5,%í³:)9)_±v)?w±5'¼±Hà ± #_Ä /²Ç 1 A =l± )i±va !W³ % e ±³l) 1 ap±v)Q³:a± ,=U³:ap³-, . ±va !O±H' #_Ä . =T± 0 % % 0 ° !Kà p±, 1 ! < ³ %½±HÃ(,G± ? ! P } ' V +. E 'ÔU½aJ + * =@? + » q z { }V + M ' P
(57) K3 Ç#5 S Ë 5 Ì 3 K3 Ë65 »ô¹ ËË A ' +5n [ ]_ n [ ] r ] ]Gkl§¨gpng! h¹ sMr §~}~}]_Sn r hMT g§½n§ s B '¶§¶·]M +¡ hM~@¹ sMr V gjfmC_§¨]_>n½o©h r ] '}V + B '}V + C 0 'Wª'§½n [ a §¨n s j¨~ [ h ¦ ]m]_]_ h® [ s gn!n r hM}g§½n§ s +ºg s n [ hYn!» q z { P '}V + U Z[ §¨g §¨g!h s >n r hM~}§Hn§ s &T]_}_]M¡}]__]_ggh r §¨½o¡ A ' +7hM}~ = ` 4 '4D +¸ª§½n [ '4D + [ [ {]Hn_E " ! hM}~ gj} n hY n B hM}~ B '¶§Ô·]Mo4 QhM~54 + yg '}V + P ¡Tª]~} ]_~}j}_]'n [ hYn²n [ ] r ]]Gkl§½ngV gj [ n [ hYn¹ sMr V C7V ¡ B '}V + hM~ B '}V + U Z[ §¨g¤^`]ihM}gfhS|hYn [ '}4 R MMM94 { +C rs gg]_g E hM}~ ª'§½n [ | s g§½n§ ¦ ] | rs hY§¨ §½nÅo¡ [ ]_}_] sMr Z[ ] r ]_¹ sMr ] §¨g7hºn s nhM sMr ~] rr ]_hYn§ s s fn [ ] g]Hn s ¹ ¹ sMr ª [ §¨ [ 4 U¡hM}~ §¨g7g] r §hM¶ Ç Ç Ji È Ë 5 Ì 3 K3 Ë65 Yz§ r gpn¨o®hMggj}^`] [ hMg¯}§½n]!gj}|}| sMr n_ Z[ ]_©n [ ]`~}]_ s ^! s| g§¨n§ s s ¹ hMgg] r §VhMº^`]ihMgj r ]S§¨g ¯}§½ n]ShM}~ §¨ng~}]_ s ^!| s g§¨n§ s §¨>n s HoT_¨]_ghMg s [ ] r ]_¹ sMr ]fn [ ] r ]]G kT§¨gng T]_}_]M¡¸ª¸]ihM ª r §½n] ` QSR 4 z ' +Qª'§½n [ R Z hS|hYn [ R MMM R ª'§¨n [ | s g§½n§ ¦ ]f| rs hY§¨¨§¨nÅo¡¸gj} [ n [ hYn R ]_§¨}g s hM~ ]_~}g s R Z[ ]_0ª¸]®nh ¥ ]®n [ ]©ghM^`] r hYn§ s hM5hY|}| rs kT§^ChY n§ s }gmhMg§¨ n [ ]®| rs>s ¹ s ¹N{]_^`^Ch W 0ShM}~ n [ ]m|6hYn [ '}V + R P R [ P b MMM P [ hMgC| s g§½n§ ¦ ] | rs hY§¨§½nÅo ÍY §hM¨½o '}V +`~ $ z ' '}V + + s ¦ ] r ]_g® n s ª¸h r ~ ª' §¨n [ V hM~ » qz { P '}V + U T]_ s } ~}½o¡l§¨¹ [ hMg§¨l¯§½n]'gj}|}| sMr n_¡Tn [ ] r ]5]Gkl§¨gpng(' ¦ §hQn [ ]5~}]_ s ^!| s g§¨n§ s f§¨mHoU_]_g jg]_~m§¨ qrs | s g§½n§ s 0>0 +7h!g]_¿Uj}]_}_] s ¹Íg] r §VhM ' +²ª'§¨n [ ¯}§½n]Ngjl|| sMr n s ¦ ] r §¨}`n s Y sMr ]ihM [ ' +n [ ] r ]]Gkl§¨gpnghºg]_¿>j]_}_] ' EbV + Ai' +ª'§¨n [ | s g§½n§ ¦ ]| rs hY§¨¨§½nÁo s ¦ ] r §¨Cn s ' +{]Hn ' + ' EbV { +ª'§½n [ V { gj} [ n [ hYn R b ' ' +UE ' + +> aON - n [ ]5g]_¿Uj}]_}_] ' + s ¦ ] r ]_gn s hM}~ [ hMg'h| s g§½n§ ¦ ]| rs hY§¨¨§¨nÅo 5 Ç Ì 3 4 Ç ¼} Ç 7 ± . ¢ #±³ S Ç>Ë Ç 1L ; ³:a´ 0R< ,>U= ³"!Ké° ! . = ³5a±H' 1 ,&!$#&%½±´±vaGÃ6±&% * . / . 0 e 1 . < $ . 0 A ¬G ³MÃF' T= ±X)_± )_±va !¶³ % ±³:) ap±v)5a±&% ³ ! ±&%éÄ ¬? T= ± *U³"!xam± e *!ya !T,³ % e ±³l) 1 a± c ÛÜYÝîièéåKãÅæøMóÁÝ5 èéåãÔÜYÝ >Ý _ðîJèéåKãÅæøYóôÝzèöøUh [ ,fÇ.-. . Silk m . `K÷ba4` .
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