Yule process sample path asymptotics
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Yule process sample path asymptotics Arnaud de La Fortelle. N° 5577 May 17, 2005. ISSN 0249-6399. ISRN INRIA/RR--5577--FR+ENG. THÈME 1. apport de recherche.
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(24) ! "#$ . > ?!B9UB(¯V_UBen#QS?!B9UB~JV_b^ RLJSA1L; #=UL_a9QS~V_b V;1QSUL;b!JS~ QS~V_b!JkQS?L;QlL;UBJS~ seb^~¡°aML;bQS R a ?L;b^sgBK`6 QL;«~ b^sl~ bQVL_aKaKV_N^bQ1QS?!B C °!UJQCV_b!BKJMn2QS?^~JA1L;«gBªLk]^UV_]*V_USQS~V_b e QS?L;Q ~J"L_JRAr]^QV_QS~aML; RbN^ Ç>»N^USb^~ b^sªQS?^~J ]^UV_]*V_USQS~V_bV;bN^A²*B9U"~ bQVh]^UV_]*V_USQS~V_bV;QS~ ACBen V_b!BraML;a9N^L;QBKJ#BML_JS~ RkQS?L;Q"QS?!B!N^~`l ~ Ar~ Q²a9UVeJJBKJ QS?!BB9gB9 C L;Q QS~ ACB T = λ log 2 =QS?!BJV_ N^QS~V_b.V; C(e − 1) = C "( CFB9b!aKBQS?!B9UB~J L1°^O^BK`4QS~ ACB T `^VBKJb!V_Q `^B9]*B9b!` V_b C b!V_UV_b T "±V_U QS?!B L;USsgB#`^B9~L;QS~V_b!J B9?LM~V_UMnQS?!B9bªQS?!B ]^UV,aKBKJJ ¬aKV_bQS~ bN!BKJÈ®"¢~ QS? L;bL; ACVeJQ N^b^ACV`7~¡°BK` B9?LM~V_UM > ?^~J±*B9?LK,~V_U±~JN^b!`^B9UJSQL;b!`!L;^BF~¡BaKV_b!J~`^B9UQS?!B 2N^B ]^UV,aKBKJJL_JQS?!B bN^A²*B9UV; b!V,`^BKJ ~ bkLQSUBKBe NQ¢~JFBML_JS~B9UQV1L_aKaKB9B9UL;QB#QS?!B UB9]^ ~aML;QS~V_bV;±LrJSA1L; bN^A²*B9U\V;b!V,`^BKJ QS?L;b1LFL;USsgB bN^A²*B9UM CFB9b!aKB QS?!BseV_L;L_aKaKB9B9UL;QS~V_b1~JaKV_b!aKB9bQSUL;QBK`CV_brQS?!B *B9se~ b^b^~ b^s!n ¢?!B9b QS?!B9UB"L;UBFB9b!V,`^BKJM£> ?^~J¢B(O7]^L;bL;QS~V_bh~J ACBML;b^~ b^seBKJJ¢A1L;QS?!B9A1L;QS~aML; R =QS?!B9UB~J b!V#¬seV_L;£L_aKaKB9B9UL;QS~V_b7® " ^N^Q QS?^~J ~J¢QS?!B ~`^BML7 5.B"L;bL; R MMBK`hV_b^ RPQS?!B"L_aKaKB9B9UL;QS~V_baML_JBe 5?!B9bªaKV_Ar~ b^sCQVQS?!B `^BKaKB9B9UL;QS~V_b C → "(n^QS?!B" B9?LM~V_UF~J B9gB9b4JS~ Ar]^B9UM±QS?!B°!UJSQ QSUL;b!J~ QS~V_b~J JSQV_]^]*BK`L_J¢V_b^s L_J¢b!BKaKBKJJL;USRgn 0 QS?!B9bPQS?!B\]^UVaKBKJJ±sgV,BKJ V_b L; ACVeJSQb!V_USA1L; Rg2> ?^~J aML_JBF~J V_b^ R1JS«gB9Qa?!BK`P~ b Z]^]*B9b!`7~¡OvZ L_JFLaKV_ArACB9bQFV;B NL;QS~V_b ¯Z#Ôd "( > ?!BrJSQSUSN!a9QSN^UBrV;QS?!BUB9]*V_USQ~JQS?!B¯V_ Vm¢~ b^s!ZF]^]*B9b!`7~¡OkZJ?!VM J?!VMQS?!B#UBKJN^ Q aML;b *B?!B9N^US~JQS~aML; R1V_N^b!`P¢?^~ BF]^UV,V;J L;UBFse~ gB9bn7~ bª¨,BKa9QS~V_bo²¯V_UQS?!B\!N^~`P ~ Ar~ Q L;b!` ~ b4¨,BKa9QS~V_b %#¯V_U¢QS?!B"L;USsgB"`^B9~L;QS~V_b!J\L_JSR,Ar]^QV_QS~aKJM #$ $¥$S > ?!BF2N^B\]^UV,aKBKJJaML;b *BFaKV_b!J~`^B9UBK`ªL_JQS?!B\QS~ ACB(ÇUB9gB9UJBK`hV;ËQS?!BFi;i ∞ N!B9N!BF¢~ QS? b!V#B9bQSUL;b!aKBe> ?!B9UB(¯V_UBFA1L;bRCQV,V_J L;UBFJS~ Ar~ L;UM¤~ UJSQMn~ Q±~JBML_JS~ RCa?!BKa«gBK` QS?L;QMn¯V_U L; −λT. 0. 0. λt. c > −1 h(x, t) = (1 + ce−λt )−x. ~JJS]L_aKB(ÇQS~ ACB ?L;USACV_b^~a"" UM QM2QS?!B2N^B ]^UV,aKBKJJ¢sgB9b!B9UL;QV_UM2> ?^~JACBML;b!J ~JFLA1L;USQS~ b^sL;Be±> ?!B"`^BKaKV_Ar]*VeJS~ QS~V_b (1 + w). −x. (1 + ce−λt )−Yt. =. ∞ X (−1)n (x + n − 1)!. n=0. ÍËÍ Ódf!Ý(ÝÀÞ(Þ. n!. (x − 1)!. wn. −1.
(25) "/ X . R,~B9`^J£QS?!BA1L;USQS~ b^sL;BKJ V_U n = 1 L;b!` n = 2 . e−nλt Yt (Yt +1) . . . (Yt +n−1). i4VeJQN!JB(=N^!L;UBQS?!BA1L;USQS~ b^sL;BKJ. o7 D " o7Ôo " Y (Y + 1)e . NQ»~J£BML_JS~ R#a?!BKa«gBK`QS?L;Q£QS?!BKJBA1L;USQS~ b^sL;BKJ±L;UBN^b^~¡¯V_USAr R# V_N^b!`^BK`6n_?!B9b!aKB aKV_bgB9USsgB L7 JM L;b!`~ b NQ ~J B9 Ë«,b!VM¢b4QS?L;QQS?!B °!UJQFV_b!B"aKV_bgB9USsgBKJFQVPL;b4B(O7] V_b!B9bQS~L;»UM *¢~ QS? ]L;UL;ACB9LQB9UrD²~¡ Y = 1 ¨~ b!aKB²QS?!B²LK V; Y (n) ~J\B NL;»QVvQS?!B²JSN^AV; n ~ b!`^B9]*B9b!`^B9bQ 2N^B]^UV,aKBKJJBKJ\*B9se~ b^b^~ b^sL;Q 0 n*QS?!BA1L;USQS~ b^sL;B Y (n)e aKV_bgB9USsgBKJ"QV QS?!BJN^A¥V; n ~Ç ~Ç `6±B(O,]*V_b!B9bQS~L;UM *¢~ QS?ª]L;UL;ACB9QB9UDe > ?!B.!N^~` ~ Ar~ Q~J4V_^QL;~ b!BK`RL;]^]^ R~ b^sVV_ J4~ b!B NL; ~ QRQVQS?!BA1L;USQS~ b^sL;B Y (n)e −n Yt e−λt ,. t. −2λt. t. 1. 0. t. −λt. t. −λt. t. \JS~ b^s. ". IP sup |Yt (n)e−λt − n| ≥ nα t≤T. p IE [|X|] ≤ IE [X 2 ] ". #. IE |YT (n)e−λT − n| ≤ . nα. L;b!`ªQS?!B A1L;USQS~ b^sL;BKJ o7 D "¢L;b!` o7Ôo "(n!V_b!B sgB9QJ. o7A% " > ?^~Jh~ b!B NL; ~ QR§~JhJSQSUV_b^sgB9UªQS?L;bQS?!BN!JSNL;!N^~` ~ Ar~ Q V_USA²N^L;QS~V_bn"¢?!B9UBlV_b!B a9L_JJS~aML; Rra?!V,VeJBKJ aKV_N^UJB QS?^~J±`^V,BKJb!V_Q2V,V_« ~ «gBa9L_JJS~aML;^!N^~`r ~ Ar~ QJmnJS~ b!aKB QS?!BJaML; ~ b^s~Jb!V_Q¢XαN^=B9U1 J¢JaML; ~ b^s!n7^N^QQS?^~J ~J]^UBKa9~JB9 R1¢?L;Q ~Jb!BKaKBKJJL;USR1¯V_UQS?!B\L;USsgB `^B9,~L;QS~V_b!JFL_JRAr]^QV_QS~aKJM i4V_UBKVmgB9UMn6V_b!B#aML;bJS?!VM QS?L;Q\QS?!BJaML;BK`]^UV,aKBKJJ aKV_bgB9USsgBKJ n (Y (n) − n) QVvQS?!B#!N^~`B NL;QS~V_b y = λ(1 + y) ¢~ QS?~ b^~ QS~L;2aKV_b!`7Z~ QS~V_68=b 7:9 y(0). > ! ? B J V_ N^QS~V_b~J =0 b4L²!N^~`h ~ Ar~ Q JaML;Ben^B JS?!V_N^`ªQS?!B9bh?LKgB Y (n) ' n(e − 1) + n = y(t) = e − 1 2 > ^ ? ~ ¢ J ~ ¢ J ^ ] UBKa9~JB9 RP¢?L;QFJSQL;QBKJ TUV_]*VeJS~ QS~V_b4o7 De ne y ¹2¹ w<x ¹
(26) % =º %
(27) <x #/ ,/.. α > 1/2 # ,/.
(28) . T ≥ 0 IP sup |Yt (n)e t≤T. −λt. − n| ≥ n. α. #. 1. ≤ n 2 −α .. −1. t. t. 0. λt. t. λt. λt. \B(°!b!B. ". lim IP sup |Yt (n)e−λt − n| ≥ nα = 0.. n→∞. J~ QS~V_bo7 D aML;b Bv68QS7:U9 L;b!JSL;QBK`QV. *BKaKV_ACBKJ. #. t≤T. n6¢?^~a?~J QS?!B¬~ A1L;sgB(®1V; 1> ?!B9bT2UV_] V; ~LQS?!BPUB9L;QS~V_b ÈDeÔo (likV_UBP]^UBKa9~JB9 Rgn o7A% ". Xk (n) = En + · · · + En+k Yt (n) " Xk (n) # "
(29)
(30)
(31)
(32) k
(33) ≥ λ−1 nα−1 ≤ n 21 −α . IP sup
(34)
(35) Xk (n) − λ−1 log n
(36) λT. o7 q ". n≤k≤e. Ê=ÌͻʯÎ.
(37) f.
(38) ! "#$ . ¢ $ $ ( . *+). . -/] _. 1 . ] [\R/WX.. ¤V_ VM¢~ b^s1QS?!B"?!B9N^US~JSQS~aKJV;Z]^]*B9b!`7~¡OhZ#n,V_ULse~ gB9baKV_b!JSQL;bQ L;b!`ªB"`^B(°!b!B QS?!B]^UV,aKBKJJ . b!B²a?!BKa«7J QS?L;Q. 687:9. f (n) =. n X. log. i=1. i+C , i. C≥0. n^B"JB9Q. J%7 D ". ∀n ≥ 0.. J%7Ôo. 687:9. ~JLA1L;USQS~ b^sL;B. Mt = exp {f (Yt − 1) − λCt} .. Mt. ". d IE [Mt+s |Ft ] = λYt ef (Yt )−f (Yt −1) − 1 − λC exp{f (Yt − 1) − λCt} ds C + Yt = λYt − 1 − λC Mt = 0 Yt. NQ~JQS?!B9bªa9L_JJS~aML;*QV`^B(°!b!B L#b!B9§]^UV_L;^~ ~ QR ACBML_JSN^UBen,QS?!B¬Q¢~JSQBK`,®]^UV_L;^~ ~ QRgn!R D N IP∗ [A] 6=7 9 IE M , ∀A ∈ F .. Fb!`^B9U2QS?^~J±Q¢~JSQBK`r]^UV_L;^~ ~ QRgn,QS?!B ]^UV,aKBKJJ Y ~J2JSQS~ iL;US«gVM,~L;b1¢~ QS?C~ bQB9b!JS~ QR {A}. £¯V_UL;bR V_N^b!`^BK` ¯N^b!a9QS~V_b h n Y). t. t. t. λ(C +. t. d d ∗ IE [h(Yt+s )|Ft ] = Mt−1 IE [h(Yt+s Mt+s )|Ft ] ds ds = λYt ef (Yt +1)−f (Yt ) h(Yt + 1) − h(Yt ) − λCh(Yt ). > ?^~JFACBML;b!JFQS?L;QMnN^b!`^B9U n ?L_JFQS?!B²JÀL;ACB"LK L_J N^b!`^B9U Z\J ¢~ » BN!JB(¯N^6L;QB9UMn^QS?^~J¢IPR,~B9`^YJMn^(1) aKV_A#^~ b^~ b^sv¢~ QS?ªQS?!B\!N^~`ªY~ b!(CB +NL;1) ~ QR−o7CA%7 IP # " J%7A % " − C − 1| < (C + 1) IP sup |(Y (1) + C)e = λ(C + Yt ) (h(Yt + 1) − h(Yt ))) ∗. ∗. t. t. −λt. t. α. t≤T. ". #. 1. = IP sup |Yt (C + 1)e−λt − (C + 1)| < (C + 1)α ≥ 1 − (C + 1) 2 −α .. ¤V_U QS?!B JL;«gB"V;JS~ Ar]^ ~a9~ QRgnB"`^B9b!V_QB"QS?!B L;USsgB"`^B9,~L;QS~V_bB9gB9bQFR t≤T. 687:9. E
(39) (T, C, α) =. ÍËÍ Ódf!Ý(ÝÀÞ(Þ. (. sup |(Yt (1) + C)e−λt − C − 1| < (C + 1)α t≤T. ). J%7 q ".
(40) "/ X @. *PO. 1 . W 1 RS_`3. ¨ ~ b!aKB M > 0 L7 Jmn IP L;b!` IP L;UB UBKa9~ ]^UV,aML; R L;!JV_ N^QB9 RaKV_bQS~ bN!V_N!J L;b!` IP [A] = D N V_UL; > ?!B9UB(V_UBB²`^B9US~ gB\=UV_A nV_UL;bRP~ bQB9sgB9U C ≥ 0 L;b!` IE A∈F L;bR UBML; TM> 0 D N J%7Ôd " IP [E
(41) (T, C, α)] = IE M ≥ IP [E
(42) (T, C, α)] inf M 6 > ?!B °!UJSQ QB9USA ~J¢*V_N^b!`^BK`ªR J%7A % "(S G V_N^b!`^JV_bªQS?!B²JBKaKV_b!`ªQB9USAªn^6 QS?!B"~ b7°!A²N^Aªn aKV_ACB ¯UV_A QS?!B *V_N^b!`7~ b^s1V; f £~ b!B NL; ~ QS~BKJZ D Nb!`^BKBK`6n!V_bªQS?!B"B9gB9bQ E
(43) (T, C, α) n J%7 " (C + 1 − (C + 1) )e − C < Y < (C + 1 + (C + 1) )e − C V_A#^~ b^~ b^s1¢~ QS?ª~ b!B NL; ~ QS~BKJZ D R,~B9`^J J%7 f " (C + 1 + (C + 1) )e + C ≤ λCT + 1 + (C + 1) + C. f (Y − 1) ≤ C log C > ?^~J ~ ArACBK`7~L;QB9 Rhse~ gBKJ¢QS?!BV_ VM¢~ b^s1*V_N^b!`ªV_b M J%7A @ " M ≥ exp {−(C + 1) − (C + 1) } inf ¤~ bL; RPB sgB9QQS?!B L;US6sgB"`^B9,~L;QS~V_b!J VMB9U*V_N^b!` J%7 " . log IP [E
(44) (T, C, α)] ≥ −(C + 1) − (C + 1) + log 1 − (C + 1) ∗. t. {A}. ∗. −1 t. t. ∗. −1 T. {E (T,C,α)}. α. λT. ∗. E (T,C,α). α. T. α. λT. −1 T. λT. α. T. T. E (T,C,α). −1 T. α. 1 −α 2. α. *
(45) . USU. . W 1 R`_S3. > ?!B\L;USsgB`^B9,~L;QS~V_b!J N^]^] B9U *V_N^b!`v~J¢V_^QL;~ b!BK` RCUB9gB9UJS~ b^srQS?!BB NL;QS~V_b J%7Ôd "(¨~ b!aKB QS?!B]^UV_L;^~ ~ QRª~J *V_N^b!`^BK`ªRDe J%7 DMp " IP [E
(46) (T, C, α)] ≥ sup M . E (T,C,α). −1 T. V_A#^~ b^~ b^s!nL_J V_U QS?!BVmB9U*V_N^b!`6n J%7 " ¢~ QS?6 ~ b!B NL; ~ QS~BKJ!¯Z# D " R,~B9`^J f (YT − 1) ≥ λCT + C − R1. J%7 DeD ". Ê=ÌͻʯÎ.
(47)
(48) ! "#$ . ¢~ QS?QS?!BUBKJSQ ¯V_^QL;~ b!BK`hN!JS~ b^s. . log(1 − x) ≥ −x/(1 − x). ( ". (C + 1)α−1 + (C + 1)−1 e−λT 1 − (C + 1)α−1 − (C + 1)−1 e−λT C2 + + log(C + 1) 2((C + 1 − (C + 1)α )eλT − C − 1) 1 C(C + 1)α Ce−λT ≤ + + log(C + 1) α α−1 C + (C + 1) 2 1−C − e−λT. 687:9. R1 (T, C, α) = C. J%7 DMo ". ≤ C α + Ce−λT + log(C + 1). L_JJSN^Ar~ b^s L;b!` ¢?^~a?¢~ ±*BCQS?!BvaML_JBCJS~ b!aKB L;b!` ¢~ *B QL;«gB9bk~ b!C`^B(°!b^~ ≤QB9 1/4 RªL;USsgBe Fe b!`^B9U≤QS?^1/4 ~J\aKV_b!`7~ QS~V_bnB²sgB9QFQS?!B"L;USsgB`^B9C~L;QS~V_b!JFTN^]^]*B9U *V_N^b!` J%7 D % " log IP [E
(49) (T, C, α)] ≤ −C + C + Ce + log(C + 1). −λT. α−1. α. *. −λT. ]W. 3/.+]cb+1 _S[ L;QS?!B9US~ b^sr V_N^b!`^J J%7 "L;b!`J%7 D % "(n!V_b!B\sgB9QJ¢QS?!B\L;USsgB `^B9,~L;QS~V_b!JL_JSR,Ar]^QV_QS~aKJM2ikV_UB. . '. ]^UBKa9~JB9 Rgn7¯V_UL; . C : R+ 7→ Z+. QB9b!`7~ b^sCQVC~ b7°!b^~ QRgn^V_UL; . 1/2 < α < 1. n. J%7 D9q. 1 log IP [E
(50) (T, C(T ), α)] = −1. T →∞ C(T ) lim. ". b!B#aML;blJS~ Ar]^ ~¡¯RhQS?!B#B9gB9bQ L;b!`4«gBKB9]4QS?!BJL;ACB# ~ Ar~ QMF> ?!BACV,`7~¡°BK` VmB9U\ V_N^b!`~JFV_^QL;~ b!BK`4Rªa?L;b^Ese
(51) ~ b^(T,s αC,L;α)b!`QS?!B"N^]^]*B9U*V_N^b!`ªRª`7~ UBKa9QS RªACV`7~¡¯R,~ b^s QS?!B*V_N^b!`^J J%7 DMp " J%7 D % "(vi4V_UBKVmgB9UMn»BCaML;bBML_JS~ RkUB9L{OQS?!BCaKV_b!`7~ QS~V_b.V_U C QV *B ~ bQB9sgB9UM±> ?!B9UB(V_UB B"sgB9Q QS?!BV_ VM¢~ b^sC]^UV_]*VeJS~ QS~V_b y ¹ 2¹ w <x ¹
(52) z^y µ º µ =z7x ¹ w # , .
(53) . C : R 7→ R 0 !$1
(54) ! # ! ! ''& . . . #/ ,/.. 1/2 < α < 1 . .
(55). +. . +. " # 1 log IP sup |(Yt (1) + C(T ))e−λt − C(T )| < C α (T ) = −1. lim T →∞ C(T ) t≤T. J%7 DMd ". Z J B N!B9V;QS?^~J ]^UV_] VeJ~ QS~V_b~Jmn UV_N^se?^ R§JSQL;QBK`6nQS?L;Q QS?!B]^UV_L;^~ ~ QRQVUBML_a? `^BKa9UBML_JBKJªL_J e >ËL;«,~ b^s C(T ) = e nV_b!BksgB9QJ QS?!BkUB( Y ' C(T )e JN^ Q ÈDeÔd "( 5?L;QBvsL;~ b!BK`¢~ QS?QS?^~J`^B9ACV_b!JSQSUL;QS~V_b~J"°!UJSQL;bB(O7]^L;bL;QS~V_bV; QS?!B V_USA©V;QS?^~JUB9L;QS~V_b!JS?^~ ] *B9QBKB9b ~ b!a9UBML_JBFUL;QBFL;b!`PB(O7] V_b!B9bQS~L;*`^BKa9UBML_JBUL;QB ¯V_U ,/! & T. ÍËÍ Ódf!Ý(ÝÀÞ(Þ. λT. −C(T ). (α−1)λT.
(56) DMp. "/ X . QB9b!`7~ b^sQV~ b7°!b^~ QRgn¢L;b!` JBKaKV_b!`Bh«b!Vm QS?!B LMRQS?^~JrL;USsgBª`^B9,~L;QS~V_bB9gB9bQ ,V aKa9N^UJFRACBML;bV;2QS?!Ba?L;b^sgBV;2ACBML_JSN^UBe NbQS?!B]^UBKJB9bQaML_JBen BaKV_N^`kB9gB9blJSQL;QB QS?!BvaKV_bgB9USsgB9b!aKB V;¢QS?!B1aKV_b!`7~ QS~V_bL;]^UV_L;^~ ~ QS~BKJMn£^N^Q²QS?!BC]^UVV; V_N^`.B9b^seQS?!B9bQS?!B ]L;]*B9U#L;b!`.`^V,BKJ b!V_Q"^US~ b^sª=N^USQS?!B9U²JN^!JSQL;bQS~L;N^b!`^B9UJSQL;b!`7~ b^s4V;¢QS?!Br*B9?LK,~V_U²V; QS?!B ]^UV,aKBKJJM v ( #$ # $!$
(57) ¢ * (
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(81) Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(82). . ISSN 0249-6399.
(83)
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