Un Test de type Kolmogorov-smirnov pour processus de diffusion ergodiques
Texte intégral
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(118) MI`¡P. _ Og{ v Ogw E Z9|a{ Ogw E ZJo u u uo{ v Ow Z9|}{ Ow ZJo ) / v | )_Z2xkj dOnFo Z f f MIXQ4<RQ UK<oOPG>A.-CQ ,OP2%IKIK- < -I,87KAJ-C/./|GH2I OP- P f x O s Ogw 8 o"t $ Z!Z 8 $ E | G Ow Z G G { d Ogw Z9| { d Ow Z Ow Z W f W { d Ow Z G { d Ow Z 2*U' < 2I 792N/ _- ( { Ow Z!Z { d Ow Z x O ) $ 8 8 E $ E ~ -N,. O' 8 xnz O`X a d 8 Z!Z 8 E $ E ~ Ow Z x $ f / NB2%I9/c=?2%IP,J.A A- UK- <H-C/<H2%GR/u T/ $9IKGHDOP/u G>=?-NI9/|GH2I IK-N<H<>-C/<HGH= GH,.i- (V<>2A/ U -2p q s ( OP- u v l 2%U9 /|2I8, Q -N<H<>-C/o+O U I Q -NA.J, 4GHI 7KA.28Q -C/./|U9/ (*2 U ' -0/1, UKI\7KA.28Q -C/./|U9/_@ CU9/./|GH-NI QY-NI8,.A.-_ (COP-Q 24BiCA.G CI9QYs / Owjo"t&ZJ3 l 2%U9/H4<><H2IW/ =4GHI8,.-iI4I8,(7 A.2U B-NA= UK-P(o/12%U9/ ( u
(119) v xkj /u 3 l 2%U9/C<H<>2I9/792UKA QY-i< U ,.GH<>GR/|-NAUKI', M-N 2A -N' =?6- KKGH-iIXQ 2IKI8UjO U / A' V3?>-U KKG>I OQ3 3 c 3;SGH<><HGHI @/1<>-N`6 @ 4 DZJ3 c 2%UKA QY-N< N(P2%IXO )- $9IKG>,< c3 4= GH<H<>-O8- 3L2%I9QY,JGH2IWT/ O G ZAG LKM-N,^< c3L2I9QY,JGH2%I 7 4AOP G P @CnFo D `<¶>O~|jq `G @cnFo D M O$5u ZoZ x aM OC7u Z9| y V t { O$5u Zj|}{ O$5u Z G Oaa P @CnFo D `<¶>O~|jq `@cnFo D M O$5u ZoZ x aM OC7u Z9| 2{ OC7u Z u Oaa *2
(120) U ' -0/1,g< 2"7 -i A 4,J-NUKA@%A O8GH-NI8I, 3M`I u v x v O u v o y pO ) / v | )_Z!Z3 MIXi2 KK,.G>-NI8,(7 CTA C7K7K<>GRIQ C,.GH2%IOP- < 3L2AJ=UK<>-;OP- N68<H2A -O CI9T/ <¶> ~ CUX792G>IP, O!)bW *ow+ZAJ-N< CD ,JGHB-N= -NI, 4U'792GHI8=, O!)FoJwZJ(P*2 U ' #~b K (KUKI'B2%GR/1G>I 4@%- OP=- n-O CI96/ <¶> ~ P { OwZ x { Ogw+Z W { Ow+ZEu OW G " R Ow+Z *2 U ' R Ow+Z(-0/1, ^< &#= C,.AJGRQY- OP-C/ O -N A.G>B -N -0/ &7 4A.,JGH-N<><H-C/ /1-CQY2%I9OP-C=/ 4U 792%GHI8, ) ( *2 -U ' ) 47K&7 4A.,JGH-NI8, CU"/1-i@=?-NI8, OP-'OPAJ2GH,J - @ )Fo)lW FD3 4r2=?= - { OwZ]',.2U ,.-C/ /1-C/;O -N AJGHB -N -C/(&7 4A.,JGH-N<><H-C/ /1-CQY2%I9OP-C/ K92%A." I -N -0//1UKA ,.2U ,;QY2= 7 QY,MO8- ,(9UKIKGT3L2AJh= -N =?-NI,z-NI wj( G |jq (*V |±q s UKIKGT3L2A.= -N =?-NI8, /|UKA,.2%UK, QY2=?&7 QY, OP=- @CnFo D `<¶> ~ 3 A `@cnNo D e<¶> ~ ( <H-', M-N 2A -N' =?- BF3;B-O 4I9/ -N , CIP, QY2I8,.G>IPU -?792UKA < ',.2792%<H2@%GH-j 7KA.28 OPU GH, /1U 2 SG><H<>GHIK@F/1<H-i` 6 @ 4 Do= 2I8,.A.k- jU 2I F(n/12UW/ ( u v xkj /u x O u o 6 ZJ34r2%u=?/ =?-<>-c7 A. 28QY-C/J/1U9/ u -C/|,< D 7 3 )/ 3k ' ,.JA 0|-CQY,J2GHAJ-C/QY2%IP,JGHI8UK-C)/ (8-N] , { ~ E @cnNo D(K<H-;7KAJ28QY-C/./|U9/ -C/|,< D j7 3 )/ 3H' ,JA 0|-0QY,.2%GHA.-0/QY2%IP,JGHI8UK-C)/ 34r-CQY9G %Q -i' B-< \O -i =?2I9/|,.JA 4,.G>2IjOPU'!, "-N 2A -N' =?- 4 3 nF3 y p. . n.
(121) "!w 57' & ['9O& 13-H9&0 ?C& l 2U9/;/12%=?=?-C/=#CGHI8,.-NICIP,-iIa=?-0/1UKAJ-\OP- =?-N,J,.A.-;-NI 7K<^QY-\UKI ,.-0/1,;7K<HUW/ 7 UKGR/J/!4I8,= U - QY-N<HU G O8U&&7 4A C@JA 4-7 K-\7 -A -C *Q -0 OP-NI8I, 3 E2%GH, g< -0/17 %QY-MOP-C/zOPG SnU9/|GH2IW/.A -i -N<H<>-C/-NAJ@28OPG^ UK-0/)(KO8-(<>2Gn/1,J4,.G>2IKI&4GHAJ-Oz( OP-M<H2G9G>IKGH,JG 4<>-l( O8-\QY2-6r QYG>E -NIP, OP- O -N A.G>B- T Ogw+Z(9-N, OP- QY2-6r QYG>-NIP,MO8- OPG SnU9/|GH2IX Ow+Z(9*2 U' IT OwZz-C/|,UKIK- 3L2%I9QY,.G>2I K92%-A -N <>GH-NIKI _- (YK92AJM I -N - /1UKAr,.2%UK,zQY2= 7 QY, OP6- <¶>H(-N,=X Ow+Zo-C/1,rUKI -3L2I9QY,JGH2%I?QY2%IP,JGHI8UK_- (/|,.AJGRQY,J-N=?-NI8, 792F/1GH,JGHB_- 3(I"IK2%,.-^ ( ^< &<>2GG>I9OPUKG>,.-?&7 4A\UKIK-'R ,J-N<H<>- OPG SnU9/|GH2I /|UK#A O Og<¶> J <¶>«Zo$T ZJ3YM}I O82I9QNP E x p O<¶> <¶>«ZJo$H o[ oIOT o6_ o X R Z~ Og<¶>«Z
(122) E Og<¶>«Z AO<¶>l <¶> Z 2*U ' Og<¶>«Zr-C/|,g< -NI9/1-im= KK<>-MOP-C/3L2%I9QY,.G>2I9M/ K92.A -N <HG>-NIKIK-C/ K92A."I -N -C/M/1UKAM,.2%UK,MQY2= 7 QY,MO8- <¶>T3 MI'IK2,J- E K (K<>-;/12%U9/1DL-C/1&7 QY- OP- E O8-C/OPG SnU9/|GH2%I9/ B -N AJ7G $*CIP,OP | G <HGH= 6OFV oIO`> o d<V Z!Z x ) o ) ~ ,o { )H~ ,=ko o . 9oz IO mZ x 9z O mZEu 2*U ' 6OFV oIO` oEd V ZZ -C/|,< -C/|,.G>#= 4,J-NUKA\O8- ) -O 4I9/ <H-X= 28+O -N' <H-j/|, 4,JGR/|,.G UK- 8K (MUKGMB -i A.7G $W-&<>-C/ 7 A.27 A.G -i -, -C #/ -N IK2%I9+Q -N -C6/ 4U O I- KKUK, OPU7 CA 4@%A C-7 K- 4 (V-N, 5 z Io )m~-,;\O -C /|GH@%IK- ^< 3 4=?G><H<>-OP-0/(<H2%GR/ /|, C,.GH2%IKI CGHA.-0/(OP-0/7KA.28QY-0/./1UW/(OP-;< -C/|7 QY- 8K 3 I -N -NM x OFMNG ZAGJILK G ?7KAJ2792/|GH,.G>2I/1UKG>iB 4I8,.-\79-NA.= -N,MOP-\,.-0/1,.-iIA (972%UKA;UKIK-\i2 K9/1-NAJiB 4,.G>2IOP2I M O82I8,< <H2%G. GHI9O8UKGH,J-;/1UK=A O Og<¶>8v <¶>«Zo$T Z47K&7 4A.,JGH-NI8=, ' EK (8g< c86872%!, -0' /1-QP K P ~ LK QY2I8,.AJ- P ~ K.
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(124) R [ U* .10.1cL@#3'4O?)0D M *,3D OO<¶> <¶>«ZJo$TZPR q` 0 M. . . . <HGH= |/ UK K7 u v OwZ x snu kno f(' E 7 MI`{9w2~@cnNo Do{ v Ogw+ZAq a Ogw+Z(* D 73 /)(K2*U ' a Ogw+Zr-C/|,< c3L2%I9QY,.G>2IeOP-;A--N 7CA.,JGH,.G>2I O8-a<^ = -C/1UKAJ-XG>IPBi4AJG CIP,J- OP-a<^"OPG SnU9/|GH2I -NAJ@28OPG^ UK-M x OFMNGZAGJILK83 ?CUK,.AJ-a7&4A.,(b{9w ~ @CnFo Dgo`{ Ow+Z |+c q { Ogw+ZJ(/12%U9/ (z*2 U ' ) ~,3 c 4A O-) $WIKGH,JGH2I OP- E K ({ )Æ~/,o -wd~ @CnFo Dgo a Ogw+Z x { OwZJ3MI"Q 2I9QY<>UK,c3 %QYGH<>-N=?-NI8] , UK- /1U 7 KE E y p { v OwZ| { Ogw+Z |+c q s ( (f ' /|2U9/l3
(125) k* MIQY2I9/|GR*O -i' A.`- ?' IK2U B8- 4U^< 3 C=?GH<><H-lB x 5 OFM G Z GJILK o )`~ , x6D nNo6HW sz@+; OP7 A.28QY-C/J/1U9/M+O t(AJI9/1,J-NGHI D « <H-NNI K9-CQ O8-c&7 4A C= -N' ,JA.- )N(KIK2., - = 3 O!)_Z(PB -N AJ7G $*CIP,z<H-CM/ :M3?=3 E 3 RSM G x | ) M G RSVvW,R[ZG 8 . . | v. . . . . . . .
(126) FE 2GH, ) / v <g -C/1,JGH=#C,.-NUKA OPU =0,8GH= UK= OP-B8A4G /1-N=AKK<^4I9QY- OP- )N(42%IAO B2%GHAI(C7CA -I,8-N=?7K<>-_( 3 +3 :UKD ,J2C6N4I8,/H@ 4 D[ZHP v v v / y pTO ) | )_ZxdOgp E e K M G R RSV Z E p E R e K M G R[ZG E2%GHH, OaMN!G onN V h pkZrU IK-;Y2 K9/|-NA.iB 4,JGH2%jI 3"MI'B-NUK,,J-C/1,J-NA<c86P792%,!-C' /|-QP K:P OFMNG Z~ 5 OaM G Zo ) ~ , x6D nFoW sz@1;Lu MIjQ 2GR/|GH,H|_P <¶> |+q @CnFo Dg( | Ogw+ZxI- ,8*7 O!|Tw R Z3 MI"B -i A.7G $W- 3 %QYGH<>-N=?-NI=, UK-'<H-C/ 8 6872%!, -0' /1-C/\OPU QY2AJ2<>< 4G>A.- 4 3 na/|2I8,B-N A.G5$6-N -C/3 c 4A Q 2IKD N/ 8- UK-NI8I, (r<H-'7KA.28Q -C/./|U9/-N=?7 GHA.^G U - u v OwZhx y pO{ v Ogw+Z| { Ogw+Z!ZJo 9{ w ~ @CnFo Dg(.O 4I9/\<HQ /c*2 U ' g< i2 K9/1-iA.iB 4,JGH2I 7KAJ24B8GH-iIP, OP-?^< 3 C=?GH<><H-\O8-?7KA.28Q -C/./|U9/B ( QY2%IPB-iA.@- B-NA / U I 7 A.28QY-C/J/1U9/ @P4U9/J/1G>-NIQY2%IP,JGHI8U u / OP- QY2CiB 4A.^G 4I9Q - P { { s / Ow±oZt&Z x s Ow±oZt&Z| ) OwZ T z OAX a d x Z9| ) OUt-ZpT z OAX a d Z!Z f { { W ) OwZ ) OUt-Z NB-CQ ) s Ow±oZt&Z x W e I- ,8*7 O!)[u R Z OUu+Z x O\uZR u±o wjo"tA~}@cnNo D f 2*U ' g< 2%}I X792N/ - OUu+Zxi v | O\w&Z- ,P*7 O!| )[w R ZUR w3(I Ogw+Zx w -N, O B2G>IA ( 7 CA\I- ,8-N=?7K<>_- ( 3 + 3 : U ,.2CN6 4I8,6/ @ f 4 D[Z(*T x o x f )F3 EG9< Y2 KW/1-NAJYB C,.G>2I?7KAJ2CBPG>-NIP,5+O UKI 7KA.28QY-0/./1UW«/ O pgG Z -NA.@%28OPG UK_- (8OP-<>2Gn/1J, 4,JGH2I I 4G>A.-O9z (PjI ;47K7 CA.D ,J-NI CIP,7 %/ \' < c3 C=?GH<><H-8B I(K,.-N< U -c< -C/|,.G>#= 4,J-NUKA ) / v C7K7K<>G MU A- #' O pgG ZzQY2I8B-NAJ@-;7KA.-0!/ UK-;/1U A.-ND = -NIP =, OQ -?/|-NA '<H- IQ / /1G86i w R RLz Ow+ZNx shZ( <H- /1U 7KA.-N= UK= /1UKA @cnF( DzOPU 7 A.28QY-C/J/1U9/ -i=?7KGHAJG UKO u
(127) v Ogw+Z ow ~ @cnFo DZJ(QY2%I9/1G O -N .A -e QY2= =?-7 A.28QY-C/J/1U9/ -NhI pH(,J-NI9O"j7 3 / 3)B-iA/ g< GHFI $WIKgG (r<H2%A!/ UKp q s 3 = -i A.G UK-C/9QYG>DO8-C/./|2U9/V=?2%IP,JA.-NI8,V<>-zQY2=?792%A.,.-i=?-NI8,9OPU ,J-C/1, 3 l 2%U9j/ NB2I9/o=?G /-NI G -C/o-C/J!/ 4G / I8UK 2-iUKBPAJ-<H-z,.-C/|, NB-CQUKI -M/1GH= UK^< 4,.G>2I +O UKI\7 A.28QY-C/J/1U9/= 3 =(4-N, IK2U9/ NB2IW/Y2 K9/|-NA.B - UKI K92I QY2=?D 792%A.,.-i=?-NI8,MOPUa,J-C/1, 3 l 2UW/ < ?NB2IW/ -NIW/1UKG>,.-\=?G /-NI2-NUKB8A.A- NB-CQ OP-C/(i2 K9/1-NAJiB 4,.G>2I9/7KAJ2CB-NI 4I8, O8-(=?28*O -N' <H-C/rOP- OPGTSVUW/1GH2%I?7K<>U9/2%U?=?2G>I9/ OPG /1J, 4I8,/ \O8U=?28*O -N' <H-O= 3 (O82I8,<H-0/5<>2G /5G>IPiB CA.G CIP,J-C/ %OP=?-N,J,.-NI8,rOP-0/=?2%=?-NI8, +O 2A OPA.-;O8-N-U ,`OQY2%I9OPGH,JGH2%I" I -0 QY-C/./ 4G>A.-792U MA UK-g< -C/1,JGH#= C,.-NU A OPU'#= 8,PD v / G>=U = OP-MBPJA 4GR/|-NA= KK< CI9QY- ) OPU 7 CA 4= -N' ,.AJ- ) C7K7K<>G MU =- ;' QY-C/ Y2 K9/|-NA.iB C,.GH2%I9/Q 2I8B-NA.@%-j7 3 )/ 37ZJ3 G -C/ /|GH= UK< C,.G>2I9/ 2I8`, -N ., - - SV-0QY,."U -i -CM/ NB-CQ(<H-0/7 4JA 4= -N' ,JA.-C/z/1UKG>iB 4I8,/lP I O 2A OPA.- Z
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