Nonsynchronous covariation process and limit theorems

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1 Sochac Procee ad her Applcao 121 (211) Noychroou covarao proce ad lm heorem Takak Hayah a,, Nakahro Yohda b a Keo Uvery, Graduae School of Bue Admrao, Hyoh, Yokohama , Japa b Uvery of Tokyo, Graduae School of Mahemacal Scece, Komaba, Meguro-ku, Tokyo 153, Japa Receved 3 Aprl 29; receved reved form 11 December 21; acceped 13 December 21 Avalable ole 22 December 21 Abrac A aympoc drbuo heory of he oychroou covarao proce for couou emmargale preeed. Two couou emmargale are ampled a oppg me a oychroou maer. Thoe amplg me pobly deped o he hory of he ochac procee ad hemelve. The oychroou covarao proce coverge o he uual quadrac covarao of he emmargale a he maxmum ze of he amplg erval ed o zero. We deal wh he cae where he lmg varao proce of he ormalzed approxmao error radom ad prove he covergece o mxed ormaly, or covergece o a codoal Gaua margale. A cla of coe emaor for he aympoc varao proce baed o kerel propoed, whch wll be ueful for acal applcao o hgh-frequecy daa aaly face. A a llurave example, a Poo amplg cheme wh radom chage po dcued. c 21 Elever B.V. All rgh reerved. Keyword: Dcree amplg; Hgh-frequecy daa; Margale ceral lm heorem; Noychrocy; Quadrac varao; Realzed volaly; Sable covergece; Semmargale 1. Iroduco Suppoe ha X ad Y are wo Iô emmargale. A obvouly kow, he mple quadrac form of creme U(I) = (X X 1 )(Y Y 1 ) coverge probably o he quadrac covarao [X, Y ] whe are deermc ad max{ 1 } alog a equece of paro I = ( ) of he erval [, ]. I alo kow ha b 1(U(I) [X, Y ]) coverge ably o a mxure of Gaua margale a for ome deermc calg Correpodg auhor. E-mal addre: akak@kb.keo.ac.p (T. Hayah) /$ - ee fro maer c 21 Elever B.V. All rgh reerved. do:1.116/.pa

2 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) coa b whe he equece I afe cera regulary codo; for example, he mple cae I = (/). Two aural queo are abou he weak covergece of uch a quadrac form. The fr oe doe he ame weak covergece ake place whe I co of oppg me? The ecod oe whe he creme X ( ) X ( 1 ) ad Y ( ) Y ( 1 ) are gve for wo dffere paro I = ( ) ad J = ( ) of [, ], poble o coruc a quadrac form U(I, J ) of hee creme ha perform lke U(I), apar from he dagoal quadrac form? A a mple maer o how, he awer o he fr oe egave geeral whou pug codo o he oppg me. Though U(I) eem o be a quadrac form of creme, o a real quadrac form becaue kerel 1 2 ( 1, ] pobly a fuco of X ad Y ad he U(I) ca poe a aure compleely dffere from he quadrac form. Really, o dcrmae bewee real quadrac form ad uperfcal oe, we eed a rog predcably codo for oppg me. The ecod queo more dffcul o awer. I requre a ew ype of fucoal of oychroou creme. Eve f we cofe our aeo o quadrac form, he coruco of he kerel o clear. Ideed, he ychrozao echque fal o gve a correc kerel; ee [9] for deal. The am of h arcle o awer hee bac queo. We wll coder a quadrac fucoal of oychroou creme, whch wa roduced by he auhor prevouly, ad prove he able covergece o a radom mxure of Gaua margale uder adardzao. Emao of he covarace rucure of he dffuo ype proce uder amplg oe of he fudameal problem he heory of acal ferece for ochac procee. Th problem had bee vegaed heorecally by may mahemacal aca each cocevable eg. There already a log hory of ude, bu amog hem we ca l [3, 21,22,26,18,2,5], ad alo refer he reader o he book by Prakaa Rao [23] for more referece. The amplg procedure reaed before had bee ychroou cheme he ee ha he compoe of he proce are oberved alog a gle equece of amplg me commoly, for all compoe. The ac codered here wa relaed o U(I) by he local rvaly of he ochac dffereal equao ad he ychrocy. The ychroou cheme f well o he adard formulao of ochac aaly. Theorecal udy of oychrocy eem o have almo bee lef behd. Mallav ad Maco [19] propoed a Fourer aalyc approach o h problem. Th a mpora work ha gve heorecal coderao o oychrocy. The curre auhor preeed a oychroou quadrac form [9]. 1 Th quadrac form, f regarded a a acal emaor, free from ay ug parameer becaue volve o erpolao ad o cu-off umber for a fe ere. Compuao eay ce he umber of erm he ummao of he ame order a he umber of he creme. Alo, he maxmum lkelhood emaor a bac eg ad hece aa hgh effcecy; ee [13]. The quadrac ype fucoal ha we wll vegae clude he oychroou covarace emaor. Thu, he lm heorem preeed below ca apply emao of covarace rucure baed o oychroou daa. Our udy amg a lm heorem whch gve a eeal exeo of he heory of acal ferece for ochac procee, o he ream decrbed above. However, a a mmedae applcao, our udy hould corbue o he rece red (or he revval wh ew 1 The auhor ackled he covarace emao problem by he ue of raday, hgh-frequecy daa, where wo ae prce are recorded rregularly ad oychroouly. Such a eup ha bee kow o be problemac; ee, e.g., [4]. Accordg o Google, our emaor referred o a he Hayah Yohda covarace emaor.

3 2418 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) obec) of reearch o he covarace emao problem, whch are que ofe dcued hgh-frequecy facal daa aaly. We wll gve ome comme o h maer Seco 9. Le u go back o our prmary queo. I h arcle, we wll defe a quadrac varao proce ad vegae he aympoc from probablc apec. To decrbe he depedecy of he radom amplg cheme, we wll aocae hem wh cera po procee, ad how he aympoc mxed ormaly, amely, a covergece of he ormalzed emao error of he oychroou covarao proce o a codoal Gaua margale. I hould be oed ha our reame of radom amplg cheme ew eve he ychroou cae of X = Y ad I = J. I [11], he auhor prevouly proved a CLT for he ame ac whe he amplg cheme are depede of he procee X ad Y. Sarg wh local margale a he uderlyg procee, Seco 3, we wll gve a ochac egral repreeao for he approxmao error. Oce he covergece of he quadrac covarao aumed, gve u he lm heorem (Propoo 3.1) whou ay rercve codo. Th work f he amplg cheme rval, uch a he hg me of a mple, parcular rucure; 2 however far from gvg a geeral oluo o he problem. The quadrac varao of he repreeg margale ll volve opoal ochac egral whch are ur egraed by a predcable egraor. We replace he opoal obec by predcable couerpar va Codo [B2]. If he predcable approxmao ufed, gve u Propoo 3.3. Thak o Codo [B2], he covergece of he quadrac varao of he ac verfed va ha of he emprcal drbuo fuco of he amplg me, ad o become a ba of praccal applcao; h reduco dcued Seco 4. Toward a geeral oluo, eeal o coruc a framework whch he real quadrac form ca be comprehevely reaed; a rog predcably codo wa roduced by Hayah ad Yohda [1] for h purpoe. Bede, Seco 5 aer Propoo 5.1 ha he rog predcably Codo [A2] eure Codo [B2], whch aoher mer of he rog predcably. The ma reul of h arcle wll be preeed Seco 6 for emmargale a well a local margale. The reader ca ump o h eco drecly f he/he whe o avod echcale a he fr readg. Seco 7 roduce he emprcal oychroou covarao proce ad prove lm heorem. Seco 8 wll be devoed o acal apec. We wll dcu Sudezao ad kerel emao for he radom aympoc varace. A llurave example wh radom chage po wll be preeed. Comme o facal applcao wll be provded Seco 9. Mo of he proof wll be pu Seco Obervao po procee ad he oychroou covarace proce Gve a ochac ba B = (Ω, F, F = (F ) R+, P), we coder wo couou local margale X = (X ) R+ ad Y = (Y ) R+, ad wo equece of oppg me (S ) Z+ ad (T ) Z+ ha are creag a.., S ad T, ad S =, T =. 2 For example, we ca coder a couou margale ampled whe quadrac varao croe po o a grd. A Browa moo oberved whe h grd po alo a example. More geerally, eay o rea he hg me a grd po for a rog Markov proce f we have uffce kowledge of he drbuo of he erval bewee hoe oppg me.

4 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) We wll regard he amplg cheme a a po proce. Accordg o h dea, we wll ue he followg ymbol hroughou he paper o decrbe radom erval: I = [S 1, S ), J = [T 1, T ), I = 1 [S 1,S ) (), J I () = [S 1, S ), r () = up I () up J (). N N = 1 [T 1,T ) (), J () = [T 1, T ), Here, deoe he Lebegue meaure, ad N = {1, 2,...}. I he precedg paper, he procee X ad Y were mplcly aumed o be obervable a ome fxed ermal me T. Th dfferece o eeal becaue caue o dfferece up o he fr-order aympoc reul. I alo poble o remove he aumpo ha boh ochac procee are oberved a =, whle we wll o purue h vero here for he ame reao. We wll refer o (I ) N ad (J ) N, or equvalely o (S ) Z+ ad (T ) Z+, a he amplg deg (or mply he deg) for X ad Y. Alo, he amplg deg opped a me, (I ()) N ad (J ()) N, may be referred o a he radom paro of [, ). For mplcy, whe we ay par () overlap wll mea eher I () J () (.e., he wo erval I ad J overlap by me ), or I J (.e., hey overlap by ay me), depedg o he uao. For procee V ad W, V W deoe he egral (eher ochac or ordary) of V wh repec o W f ex. Whe he egraor W couou, alway rue ha V W = V W, where V := lm V. For a ochac proce V ad a erval I, le V (I ) = 1 I ()dv. Wre I () = I [, ) for erval I, ad defe he proce V (I ) by V (I ) = V (I ()). The quay of ere he quadrac covarao [X, Y ], ad we wll vegae, a ample couerpar, he followg quay: Defo 2.1 (Hayah ad Yohda [9,1]). The oychroou covarao proce of X ad Y aocaed wh amplg deg I = (I ) N ad J = (J ) N he proce {X, Y ; I, J } = X (I ) Y (J ) 1 {I () J () }, R +. =1 The proce {X, Y ; I, J } o obervable from a acal po of vew. See Seco 7 for a ac correpodg o h proce. We wll wre mply a {X, Y } f here o fear of cofuo over amplg deg. I wa proved [9,6] ha for each R +, {X, Y } p [X, Y ] a provded r () p. I lgh of h reul, he auhor emphaze ha {X, Y } regarded a a geeralzao, he coex of oychroou amplg cheme, of he adard defo of he quadrac covarao proce for emmargale ochac aaly. For Iô procee X ad Y, we ca oba he ame coecy reul; ee he above paper for deal. 3. Sable covergece of he emao error The emao error of {X, Y } gve by M = {X, Y } [X, Y ] = L, (3.1)

5 242 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) = 1 {I () J () } = (I X) (J Y ) + (J Y ) (I X). Here he ) [X, Y ] = [X, Y ], a well a he defo of he quadrac covarao or Iô formula, ca be ued. We alo roduce he ymbol R () := S 1 T 1 ad R () := S T. L a couou local margale ha equal for R (), ar varyg a = R (), ad where equaly (I J ad L ay a he value L S T afer = S T. I ca vary, regardle of wheher he par () overlap. Lemma 3.1. L. Proof. Recall ha L couou. o +1 a = R () whe he par () overlap. So, L L = for R (). a ep fuco arg from a = ad ump = L R (). However, Now, he egrao by par of (3.1) ogeher wh Lemma 3.1 yeld M = L, (3.2) ad parcular, M a couou local margale wh V := [M, M ] = K [L, L ]. (3.3) Le V X, =,, X (I ) [X, Y ](J ) + Y (J ) [X, X](I ) ad V Y, = X (I ) [Y, Y ](J ) + Y (J ) [X, Y ](I ). I vew of he adard margale ceral lm heorem, we formally coder he followg codo. Deoe by (b ) a equece of pove umber edg o a. [A1 ] There ex a F-adaped, odecreag, couou proce (V ) R+ uch ha b 1 V p V a for every. [B1] b 1 2 V X, p ad b 1 2 V Y, p a for every. We deoe by C(R + ) he pace of couou fuco o R + equpped wh he locally uform opology, ad by D(R + ) he pace of càdlàg fuco o R + equpped wh he Skorokhod opology. A equece of radom eleme X defed o a probably pace (Ω, F, P) ad o coverge ably law o a radom eleme X defed o a approprae exeo (Ω, F, P) of (Ω, F, P) f E[Y g(x )] E[Y g(x)] for ay F-meaurable ad bouded radom varable Y ad ay bouded ad couou fuco g. We he wre X d X. A equece (X ) of ochac procee ad o coverge o a proce X uformly o compac probably (abbrevaed ucp) f, for each >, up X X p a.

6 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) We coder he followg codo: [W] There ex a F-predcable proce w uch ha V = w2 d. A am of h paper o prove he followg aeme: [SC] b 1 2 M d M C(R + ) a, where M = w d W ad W a oe-dmeoal Weer proce (defed o a exeo of B) depede of F. Propoo 3.1. Suppoe ha [A1 ], [B1] ad [W] are fulflled. The [SC] hold. Proof. Noce ha [M, X] = V X, ad [M, Y ] = V Y,. Sce [M, N] = for ay bouded margale N (o B) orhogoal o (X, Y ), we oba he able covergece of b 1/2 M from [15]. More precely, we apply Theorem 2-1 () here ogeher wh a characerzao of he F-codoal Gaua margale meoed (b) he proof of Propoo 1-1. I our uao, boh B ad G Theorem 2-1 vah. Therefore, he M-ba erm u dm (1.4) of h paper vahe ce u = M -a.e. from u d M = for all. Each expreo for V, V X, ad V Y, raher abrac; may be of lle help for explcly calculag he quadrac varao/covarao ad defyg he lmg drbuo of M. I h regard, aural he followg o purue a more cocree vero, epecally of V. Le V = (I I (J ) [X, X] J ) [Y, Y ] + (I J ) [X, Y ] (I J ) [X, Y ] (3.4) ad e V := V. 3 V deged o approxmae [L, L ] whe he erval legh I, I, J, ad J are uffcely mall, a wll be aed [B2] below. Throughou he re of he dcuo h eco, we wll poulae he followg hypohe. [B2] For every R +, b 1 (,, a. K ) [L, L ] = b 1 ( K ) V,, + o P (1) (3.5) Recall ha he lef-had de of (3.5) equal b 1V. A uffce codo for [B2] wll be provded Seco 5. Wre [X] = [X, X] ad [Y ] = [Y, Y ] a uual. Le V = [X](I ())[Y ](J ()) [X, Y ]((I J )()) 2. + [X, Y ](I ()) 2 + [X, Y ](J ()) 2 The followg propoo wll be ued for defyg he lm of he quadrac varao. I eable u o work wh V, a more racable proce ha b 1,, ( K ) V [B2]. See Seco 1 for a proof. 3 Sce [X, X] ow couou, I I ca be replaced by I I, for example.

7 2422 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Propoo 3.2. Suppoe [B2] hold. The V = V + o p (b ) a. We modfy [A1 ]. [A1] There ex a F-adaped, odecreag, couou proce (V ) R+ uch ha b 1 V p V a for every. By Propoo 3.2, we ca rephrae Propoo 3.1 a follow. Propoo 3.3. Suppoe ha [A1], [B1], [B2], ad [W] for V [A1] are afed. The [SC] hold rue. Sce he varace proce V much more covee o hadle ha V, Propoo 3.3 eeally mprove Propoo 3.1. However, Propoo 3.3 o he way o our goal. Fr, preferable o decrbe he lmg eergy proce V lgh of he amplg cheme elf. I Seco 4, we roduce cera amplg meaure o do h, followg Hayah ad Yohda [1]. Secod, Codo [B2] ll echcal. Ideed, h codo avod oe of he key ep o he awer. The HY emaor, or ay quadrac ype emaor, really quadrac oly whe he radom kerel of he quadrac form afe a kd of predcably codo. Oherwe, lm heorem wll fal. The auhor roduced a rog predcably codo o gve a ceral lm heorem [1] by verfyg [B2] uder mld regulary codo o he procee. Though he am of h paper o oba mxed ormal lm heorem, wll ur ou Seco 5 ha he ame rog predcably codo erve well for our purpoe. I ll rema o check he aympoc orhogoaly Codo [B1] a praccal eg. However, we wll how ha he ame kd of ak a olvg [B2], ad o addoal dffculy occur o do wh. 4. Covergece of he amplg meaure ad a repreeao of V I [8], he auhor roduced emprcal drbuo fuco of he amplg me gve by H 1 () = I () 2, H 2 () = J () 2, H 1 2 () = (I J )() 2, H 1 2 () = I () J (), where he Lebegue meaure. Clearly, he four fuco are (F )-adaped, odecreag, pecewe-quadrac couou fuco, whoe graph coa kk a he obervao oppg me. [A1 ] There ex pobly radom, odecreag, fuco H 1, H 2, H 1 2 ad H 1 2 uch ha each H k = hk d for ome radom proce hk, ad uch ha b 1 H k() p H k () a for every R + ad k = 1, 2, 1 2, 1 2. The, a exeo of Theorem 2.2 of [8] gve a follow. Propoo 4.1. Suppoe ha [A1 ], [B1] ad [B2] are fulflled, ad ha each [X], [Y ] ad [X, Y ] aboluely couou wh a càdlàg dervave. The [SC] vald wh w gve by w = [X] [Y ] h1 2 + ([X, Y ] )2 (h 1 + h2 h1 2 ). (4.1)

8 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Remark. I he cae of perfec ychrocy (I J ), {X, Y } ohg more ha he realzed covarace baed o all he daa. I h cae, ce H 1 H 2 H 1 2 H 1 2 (=: H), he lmg varao proce reduce o V = [X] [Y ] + ([X, Y ] )2 H(d). Proof of Propoo 4.1. Lemma 4.1 below defe he lmg varao proce V Codo [A1] by (4.1). Thu Propoo 3.3 mple he aero. Lemma 4.1. Suppoe ha [X], [Y ], ad [X, Y ] are aboluely couou wh a càdlàg dervave. The [A1 ] mple ha () b 1 [X, Y ](I ()) 2 p ([X, Y ] )2 H 1 (d), () b 1 [X, Y ](J ()) 2 p ([X, Y ] )2 H 2 (d), () b 1 [X, Y ]((I J )()) 2 p ([X, Y ] )2 H 1 2 (d), (v) b 1 [X](I ())[Y ](J ()) p [X] [Y ] H 1 2 (d), a for every. The proof Seco Srog predcably ad Codo [B2] We preeed bac lm heorem, Propoo 3.1, 3.3 ad 4.1. They hold whou addoal codo for uch amplg cheme a he oe gve by cera hg me of he procee. I Propoo 3.3 ad 4.1, we aumed Codo [B2] Seco 3. Uder more geeral amplg cheme, however, Codo [B2] ll echcal. I h eco, we are gog o roduce a more racable codo o he amplg cheme o eure [B2]. Such a codo called rog predcably of he amplg me. I wa roduced [1], ad a movao for ha he fuure amplg me deermed wh delay praccal uao, e.g., ha caued whle a rader akg he broker o rade a facal marke. Le ξ ad ξ be coa afyg 4 5 ξ < ξ < 1. [A2] For every, N, S ad T are G () -oppg me, where G () = (G () ) R+ he flrao gve by G () = F ξ (b ) for R +. For real-valued fuco x o R +, he modulu of couy o [, T ] deoed by w(x; δ, T ) = up{ x() x() ;, [, T ], δ} for T, δ >. Wre H = up [,] H for a proce H. [A3] [X], [Y ], [X, Y ] are aboluely couou, ad for he dey procee f = [X], [Y ] ad [X, Y ], w( f ; h, ) = O p h 1 λ 2 a h for every, λ (, ), ad f < a.. [A4] r () = o p (b ξ ) for every R +. The followg he key aeme o he ma reul Seco 6. Propoo 5.1. [B2] hold rue uder [A2] [A4]. We gve a proof of Propoo 5.1 Seco 12.

9 2424 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Lm heorem for emmargale: ma reul I he prevou eco we focued o he cae where X ad Y are couou local margale. I h eco, we coder wo couou emmargale ad pree he ma reul of h arcle. Suppoe ha X = A X + M X ad Y = A Y + M Y are couou emmargale, where A X ad A Y are fe varao par, ad M X ad M Y are couou local margale par. Eve h cae, he oychroou covarao proce of X ad Y aocaed wh (I ) ad (J ) defed exacly by Defo 2.1. To go furher, we eed he wo addoal codo below. [A5] A X ad A Y are aboluely couou, ad w( f ; h, ) = O P (h 2 1 λ ) a h for every R + ad ome λ (, 1/4), for he dey procee f = (A X ) ad (A Y ). [A6] For each R +, b 1 a. I () 2 + b 1 J () 2 = O p (1) (6.1) Remark 6.1. Codo [A5] lghly roger ha (C4 ) of [7]. Codo [A1] doe o mply [A6]. Ideed, poble o coruc a amplg cheme ogeher wh procee (X, Y ) uch ha clude [ 7/1 ] erval of 4/5 legh, ha [X] ad [Y ] do o creae o he uo of hoe erval, ad ha [A1] hold. However [A6] break for b = 1 h example. O he oher had, [A1 ] mple [A6]. The Poo amplg cheme codered [7] a example. Here our ma reul. Theorem 6.1. Suppoe ha X ad Y are couou emmargale. (a) If [A1] [A6] ad [W] are afed, he [SC] hold. (b) If [A1 ], [A2] [A5] are afed, he [SC] hold for w gve by (4.1). I worhy of remark ha eher [A5] or [A6] eceary for local margale. Theorem 6.2. Suppoe ha X ad Y are couou local margale. (a) If [A1] [A4] ad [W] are afed, he [SC] hold. (b) If [A1 ], [A2] [A4] are afed, he [SC] hold for w gve by (4.1). Theorem 6.1 ad 6.2 are proved Seco Emprcal oychroou covarao proce The quay {X, Y } o alway obervable he acal coex, whch ceraly a dracg feaure from he vewpo of praccal applcao. The argume here a regard how o amed uch a mor flaw perag o he prevou coruco Defo 2.1. Defo 7.1. The emprcal oychroou covarao proce of X ad Y aocaed wh amplg deg I = (I ) N ad J = (J ) N he proce {X, Y } = =1 S T X (I )Y (J )1 {I J } R +.

10 Obvouly, T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) {X, Y } = =1 S T X (I ) Y (J ) 1 {I () J () }. Th he pecewe coa, càdlàg vero of he oychroou covarao proce ad {X, Y } = {X, Y } a (S ) (T ). Oherwe hey do o cocde geeral; however he dfferece eglgble. Suppoe ha X ad Y are he couou emmargale gve Seco 6. For M = {X, Y } [X, Y ], we wll how: [SCE] b 1 2 M d M D(R + ) a, where M = w d W ad W a oedmeoal Weer proce (defed o a exeo of B) depede of F. We oba he followg reul correpodg o Theorem 6.1 ad 6.2. See Seco 14 for he proof. Theorem 7.1. Suppoe ha X ad Y are couou emmargale. (a) If [A1] [A6] ad [W] are afed, he [SCE] hold. (b) If [A1 ], [A2] [A5] are afed, he [SCE] hold for w gve by (4.1). Theorem 7.2. Suppoe ha X ad Y are couou local margale. (a) If [A1] [A4] ad [W] are afed, he [SCE] hold. (b) If [A1 ], [A2] [A4] are afed, he [SCE] hold for w gve by (4.1). 8. Sacal applcao ad example 8.1. Sochac dffereal equao Suppoe ha X 1 ad X 2 are Iô emmargale decrbed by he ochac dffereal equao dx k = µ k d + σ k dw k (k = 1, 2) (8.1) where µ k are F-adaped procee, ad σ k are rcly pove, (F )-adaped, couou procee, k = 1, 2. The F-adaped Browa moo W k are correlaed wh d[w 1, W 2 ] = ρ d, where ρ a F-adaped proce. Th a ochac volaly model he face leraure. We coue o ue he ame ymbol I = (I ) N ad J = (J ) N a prevouly, for he amplg deg aocaed wh X 1 ad X 2, repecvely. [A3 ] () For every λ > ad R +, w( f ; h, ) = O p h 1 2 λ a h for f = σ 1, σ 2 ad ρ. () For ome λ (, 1/4) ad ay R +, w( f ; h, ) = O p h 1 2 λ a h for f = µ k, k = 1, 2.

11 2426 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Now, defe he drbuo fuco aocaed wh he amplg deg I ad J by H 1 () = I 2, H 2 () = J 2, :S :T H 1 2 () = : I J 2, H 1 2 () = : I J, S T S T where = 1 {I J }. They are all obervable a ay. [A1 ] There ex pobly radom, odecreag, fuco H 1, H 2, H 1 2 ad H 1 2 uch ha each H k () = hk d for ome dey hk, ad ha b 1 H k () p H k () a for every R + ad k = 1, 2, 1 2, 1 2. The equvalece bewee [A1 ] ad [A1 ] ca be proved. Ideed, H k () H k () H k () + 2r () 2, for all [, ], k = 1, 2, 1 2, 1 2. We coder he cae k = 1 2 oly, for all he oher are raghforward. The fr equaly obvou by coruco. Moreover, for ay, wh, () H 1 2 () I () J(I )() I (J )() J () 2r () 2, H 1 2 where (, ) a uque par () uch ha [S 1, S ) ad [T 1, T ). Therefore up H 1 2 () H 1 2 () 2r () 2. [,] Hece, he ecod equaly alo hold. Sce b 1r () 2 p uder [A4] for example, we have aceraed ha he covergece of b 1 H 1 2 equvale o ha of b 1 H 1 2. The by he applcao of Theorem 7.1 we have: Theorem 8.1. Suppoe ha X ad Y are couou emmargale. Suppoe ha eher [A1 ] or [A1 ], ad alo [A2], [A3 ] ad [A4] are afed. The, for M = {X, Y } ρ σ 1σ 2d ad M = {X, Y } ρ σ 1σ 2 d, [SC] ad [SCE] hold for w gve by w = (σ 1σ 2)2 h ρ σ 1σ 2 2 (h 1 + h 2 h1 2 ). (8.2) We hall brefly dcu Sudezao. Coder w gve (8.2). I our coex, w o obervable ce coa ukow quae uch a ρ σ 1σ 2; eher w2 d. Suppoe we have a ac d uch ha w2 w 2d p w 2 d a. The, he able covergece aed Theorem 8.1 mple ha, for every >, b 2 1 {X 1, X 2 } ρ σ 1σ 2d d N(, 1) w2 d a wheever w2 d > a..

12 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Coruco of w2 d: a kerel-baed approach Le K (u) be a kerel fuco uch ha K (u)du = 1 ad K (u) for all u. K aumed o be aboluely couou wh dervave K afyg K () d <. For h >, le K h (u) = h 1 K (h 1 u). For every R +, le {X 1, X 2 } h = {X 1, X 2 } K h ( ) d, X k h = X k, X k h, k = 1, 2. Moreover, le w 2d = + Th quay obervable. {X 1 } h X 2 h b1 {X 1, X 2 } h b1 H 1 2 (d) H 1 + H 2 H 1 2 (d). Propoo 8.1. Uder he aumpo Theorem 8.1, w 2 ucp d w 2 d a, provded ha b 1 2 h 1. Proof. Le M = {X 1, X 2 } X 1, X 2 ; he clearly afe 1 b 2 Thu, by he egrao by par formula, {X 1, X 2 } h = [X 1, X 2 ] + M K h ( )d = [X 1, X 2 ] K h ( ) = + [X 1, X 2 ] K h( )d + O p b 1 2 K h ( ) d = ρ σ 1 σ 2 + w(ρσ 1 σ 2 ; h, ) + O p b 1 2 h 1 M up [,] = O p (1). uformly [, ] for ay >. Th ca be realzed by choog h uch ha b 1 2 h 1. Smlarly, {X k } h = (σ k )2 + w((σ k ) 2 ; h, ) + O p b 1 2 h 1 uformly [, ] for ay >. Therefore, obag he aero of he propoo mmedae.

13 2428 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) A pecal cae of he kerel-baed approach he followg aïve oe. For ay > ad h >, we may ue {X 1, X 2 } h = 1 {X h 1, X 2 } X 1, X 2, h {X k } h = X k, X kh, k = 1, 2, provded b 1 2 h 1. We wll refer he reader o [12] Poo amplg wh a radom chage po A a llurao for Seco 8.1, we hall dcu a Poo amplg wh a radom chage po. Th a mple model for ock prce, for ace, whoe radg ee vary a radom me uch a he me whe hey h a hrehold prce lke 1, ye. More precely, uppoe ha F -adaped procee ρ, µ k, σ k, W k, τ k, k = 1, 2, are gve o a ochac ba (Ω, F, F, P ). The procee X k are defed by (8.1). Le τ k (k = 1, 2) be F -oppg me. O a auxlary probably pace (Ω, F, P ), here are radom varable S, (T ), (S ), (T ) ha are muually depede Poo arrval me wh ey λ k = p k, p k (, ), k = 1, 2, repecvely for S, T ad wh λ k = p k, p k (, ), k = 1, 2, repecvely for (S ), (T ). We coruc he produc ochac ba (Ω, F, (F ), P) cog of Ω = Ω Ω, F = F F, F = F F P = P P. O he ew ba he aforemeoed radom eleme ca be exeded he uual way. Tha, W k ω, ω = W k ω, S ω, ω = S ω, ω, ω Ω, ad o forh. The amplg deg I = (S ) for X wll be made up of (S ) ad (S ) a follow. Se τ 1 = τ Defe S equeally by S 1 = f l N S = f l,m N S l S, l τ 1 <τ 1 + S1, S l S, 1 <S l τ 1 <τ 1 + Sm S 1 <τ 1+Sm, 2. Here, for a oppg me T wh repec o a flrao (F ) ad a e A F T, we defe T A by T A (ω) = T (ω) f ω A; T A (ω) = + oherwe. (T ) defed he ame way from T l ad T l, afer eg τ 2 = τ I he pree uao, he flrao G () co of G () = F ( ξ ) + F, R +. Lemma 8.1. S ad T are G () -oppg me. Proof. S l ad S l are G () -oppg me. Sce τ 1 a (F )-oppg me, τ 1 = τ a G () -oppg me; hece, τ 1 + Sm a G () -oppg me a well. Moreover, ce

14 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) {S l < τ 1} G(), S l {S l <τ 1} a G() -oppg me. Therefore, S 1 a G () -oppg me a S l well. Suppoe for ow ha S 1 a G () -oppg me. The rue ha {S 1 < τ 1 + Sm } G (). A he ame me, τ 1 {S1 < S l < τ 1 +Sm } = {S1 < S l } {S l < τ 1} G(). Thee fac S l mply ha S a G () -oppg me, a aered. Coequely, we have H 1 () p 2 p 1 (τ 1 ) + 2 p 1 ( τ 1 ) =: H 2 () p 2 p 2 (τ 2 ) + 2 p 2 ( τ 2 ) =: H 1 2 () p 2 p 1 + p 2 (τ 1 τ 2 ) + H 1 2 h 1 d, h 2 d, 2 p 1 + p 2 (τ 2 τ 1 τ 2 ) 2 + p 1 + p (τ 1 τ 1 τ 2 2 ) + 2 p 1 + p 2 ( (τ 1 τ 2 ) ) =: () p 2 p p 2 (τ 1 τ 2 2 ) + p p 2 (τ 2 τ 1 τ 2 ) 2 + p p 2 (τ 1 τ 1 τ 2 ) + p p 2 ( τ ) =: a for every. The, uder [A3 ], b 1/2 M d w d W, h 1 2 d, h 1 2 d, where (σ 1 σ 2 2 )2 p p 2 + (ρ σ 1 σ 2 2 )2 p p 2 2 p 1 + p 2 ( τ 1 τ 2 ) (σ 1 σ 2 2 )2 p p 2 + (ρ σ 1 σ 2 2 )2 p p 2 2 p 1 + p 2 1 {τ 1 τ 2 } w = + (σ 1 σ 2 2 )2 p p 2 + (ρ σ 1 σ 2 2 )2 p p 2 2 p p 2 {τ 1 >τ 2 } (τ 1 τ 2 < τ 1 τ 2 ) (σ 1 σ 2p1 2 )2 + 2p2 2 + (ρ σ 1σ 2)2 p p 2 2 p 1 + p 2 (τ 1 τ 2 < ) ad W a depede Browa moo. A example of uch τ k provded by he boudary hg me τ k = f{ > : X k > K k }, K k (, ), k = 1, 2.

15 243 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Comme o he applcao o face The applcao o face o he prmary obec of h paper; however we gve ome comme h eco. I he la decade, raday facal me ere, o-called hghfrequecy daa, have bee becomg creagly avalable boh coverage ad formao coe. The ue of hgh-frequecy daa ha bee expeced o mprove facal rk maageme dramacally; oe uch applcao clude he emao of he varace covarace rucure of he facal marke, whch a eeal roue operao for all facal uo. I he leraure, adard o ue realzed volaly (or realzed varace) for emag egraed varace whe ae reur are regarded a beg ampled from dffuo ype procee. Lkewe, whe he egraed covarace of ere, he ue of realzed covarace farly commo. Neverhele, he adard realzed covarace emaor ha a fudameal flaw rucure whe appled o mulvarae ck-by-ck daa, where me ere are recorded rregularly, a oychroou maer. The commoly ued realzed covarace emaor volve a erpolao of rregularly ampled daa o equda daa. Hayah ad Yohda [9] proved ha uch a aïve mehod evably caue emao ba, whch had bee kow emprcally a he Epp effec, whe he defg regular erval ze mall relave o he frequecy of obervao. I he ame paper, he auhor propoed a way o crcumve uch ba by propog a ew emaor, whch owaday called he Hayah Yohda emaor, ad howed ha he emaor coe a he meh ze of obervao erval ed o zero probably. Th paper ha bee movaed by he que for a lm drbuo of he emao error. I he leraure, aympoc drbuo heore for realzed volaly ype quae have bee developed; ee, addoally o he leraure gve Iroduco, e.g., [14,16,1,2]. Dfferely from hem, h paper deal wh radom amplg cheme ha are depede o he uderlyg procee ad h far from raghforward. Raher ha demaded a ew e of dea ad echcal ool. Tha, we cao mply coduc aaly by codog o he amplg me all he way up o he fe fuure a a me, regardg hem all a deermc, a mo of he exg reul wh radom bu depede amplg cheme do. Reader may recall he fac ha eve he uvarae cae here a rkg carcy of ude whch ake uch depedecy o accou. Bede h, our reame of he bvarae cae ogeher wh oychrocy ew. I h paper, we dd o clude dcuo o he mcrorucure oe. I commo he leraure o far o apply a pre-averagg o ge back o a clacal ychroou amplg. However, he goal of h arcle le developg a ew mehodology o rea he oychrocy elf. Recely, Rober ad Roebaum [24] gave a ew gh o he HY emaor uder mcrorucure oe. See alo [25]. 1. Proof of Propoo 3.2 For compuaoal eae, we roduce he followg wo po procee: = 1 [R () R (), )() 1 [R (), )() = 1 [R () R (),R ())(), = (1 (R (), ) 1 [R (), )) (ochac egral) whch gve a orhogoal decompoo of.

16 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Lemma 1.1. (a) 1 {R ()<R (),R () }. = 1 {R ()<R (),R () }, (b),, ad are (F )-adaped procee wh = 1 {R () <R ()} ad = I addo, = + ad. (1.1). (1.2) Proof. Eay ad omed. Proof of Propoo 3.2. I lgh of (1.1), we decompoe he arge quay a ( K ) V,, ( K ) V + ( K ) V + 2 ( K ) V,,,,,, =: I + II + III. (a) Coder I fr. Recall ha defe he par () uquely,.e., So, Becaue oe ha K [ = ad = ]. I = = V = 1 [R () R (),R ()) V {V R () V R () R () }. V = {I [X]}{J [Y ]} + {(I J ) [X, Y ]}2, (1.3) V R () = [X](I (R () ))[Y ](J (R () )) + [X, Y ]((I J )()) 2. (1.4) O he oher had, V R () R () = [X](I (R () R () ))[Y ](J (R () R () )) Thu follow ha + {[X, Y ]((I J )(R () R () ))} 2 =. I = [X](I (R () ))[Y ](J (R () )) + {[X, Y ]((I J )())} 2. (b) Nex coder II. We decompoe a,, = +,,, : =, = +,,, : =, +,,, :, = The followg argume movaed by Hayah ad Yohda [9].. (1.5),,, :,

17 2432 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Cae 1: = ad =. Recall (1.3). Whe a par () doe o overlap,.e.,, he. Therefore V. Whe () overlap, V = V V R (). However, he ecod erm become (1.4). Coequely, V = [X] I () [Y ] J () [X] I R () [Y ] J R (). Cae 2: = ad. Accordg o (3.4), V = [X, Y ](I J )[X, Y ] I J, whch op varyg for R () R ( ). Noe ha whe eher par () or par ( ) K V. For wo par () ad ( ), <, ha overlap a doe o overlap, he ame me, K V = V Therefore,,,, : =, K = V VR ( ). V. Cae 3: ad =. By ymmery,,,, :, = K Cae 4: ad. Accordg o (3.4), V = [X, Y ] V. I J [X, Y ] I J. Hece, for V, boh par ( ) ad (, ) mu overlap, a he ame me,.e., mu be he cae ha ad K. I order ha K V, eceary ha ad K. However, hee wo codo are compable (.e., (), (, ), ( ), ad (, ) cao repecvely overlap a he ame me). Coequely, follow ha,,, :, K V. Pug he four ub-cae ogeher, we oba II = [X] I () [Y ] J () [X] I R () [Y ] J R (). (c) Coder III. We aga decompoe a (1.5) (b). Cae 1: = ad =. Recall ha = 1 [R () R (),R ()) ad = 1 (R (), ) 1 [R (), ). They are orhogoal whe = ad =,.e., K for = ad =. Hece, Cae 1 ( = ad = ) make o corbuo.

18 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Cae 2: = ad. Evdely, K for = ad <. Suppoe = ad >. The, a log a (or ( ) overlap). Noe ha V = [X, Y ](I J ) [X, Y ] I J. So, f ( ) overlap, he K V = og ha up I J R () V = V VR () R () = [X, Y ] (I J )() [X, Y ] I J (), up I J R () ad R () R () f(i J ). Clearly, he la expreo clude he cae whe ( ) doe o overlap becaue he boh he l.h.. ad he r.h.. are rvally zero. I follow ha K V = [X, Y ](I J ) [X, Y ] I J,,, :, : =, > = 1 [X, Y ] (I J )[X, Y ] I J 2, : = 1 [X, Y ] I J [X, Y ] I J [X, Y ](I J ) 2 = 1 [X, Y ](I ) 2 1 [X, Y ](I J ) Cae 3: ad =. By ymmery, K V = 1 [X, Y ] (J ) 2 1 [X, Y ] (I J ) ,,, :, = Cae 4: ad. Noe ha h cae, V = [X, Y ] I J [X, Y ] I J. Now, ha I J = or I J = mple ha, whch eal ha K V. O he oher had, due o he geomerc relaohp amog he,,, :, K four dc erval I, I, J, ad J, ha I J ad I J mple ha I J = or I J =, ad hece V, whch duce ha K V. Afer all, h cae, K V.

19 2434 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Gaherg he four ub-cae ogeher, we have III = [X, Y ](I ()) 2 + [X, Y ](J ()) [X, Y ] (I J )(). (d) Fally, we pu he hree compoe (a) (c) ogeher o oba I + II + III = V. 11. Proof of Lemma 4.1 Whou lo of geeraly, we may aume ha lm b 1 H 1() = H 1 () for all R + a.. For ay R + ad ϵ >, here are (radom) po ξ uch ha = ξ < ξ 1 < < ξ m = ad ha max m=1,..., m up [ξ m,ξ m+1 ) [X, Y ] [X, Y ] ξ m < ϵ. Le φϵ () = max{ξ m ; ξ m }. The where D ϵ := b 1 b 1 R ϵ (,, ) = [X, Y ](I ()) 2 b 1 2 [X, Y ] φ ϵ (S 1 ) I () 2 2 [X, Y ] φϵ(s1) I () R ϵ (,, ) + R ϵ (,, ) 2, (11.1) [X, Y ] [X, Y ] φ ϵ (S 1 I ) d. Le Ξ = {ξ m ; m = 1,..., m}. If [S 1, S ) Ξ =, he R ϵ (,, ) ϵ I (). There ex N uch ha #([S 1, S ) Ξ ) 1 for all for ay. Suppoe ha wha follow. If [S 1, S ) Ξ, he R ϵ (,, ) 2([X, Y ] ) I (). Thu, D ϵ b1 2([X, Y ] ) ϵ I () 2 + ϵ 2 I () 2 where I = {; [S 1, S ) Ξ }. Therefore + 8([X, Y ] ) 2 b 1 I () 2, I lm up D 2([X, ϵ Y ] ) ϵ + ϵ2 H 1 () ce b 1 I I () 2. Ideed, due o r ( + η), for every η >, lm up b 1 I () 2 I m m=1 H 1 (ξ m + η) H 1 (ξ m η), whch ca be a mall a we lke whe η. From (11.1), lm up b1 [X, Y ](I ()) 2 [X, Y ] 2 H 1 (d) 2([X, Y ] ) ϵ + ϵ2 H 1 2 () + [X, Y ] ξ m1 H 1 (ξ m ) H 1 (ξ m1 ) m

20 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) [X, Y ] 2 H 1 (d) 2([X, Y ] ) ϵ + ϵ2 H 1 () + 2ϵ([X, Y ] ) H 1 (). Sce ϵ arbrary, we coclude ha lm b1 b 1 [X, Y ](I ()) 2 = [X, Y ] 2 H 1 (d). We have obaed (); () ad () ca be how mlarly. Le u prove (v). We have [X](I ()) [Y ] (J ()) = b 1 l L() =1 l=1 [X] [Y ] u I J u 1 Al (, u) ddu, (11.2) where A l := [, a l ) [, a l ), a l1 ), a l1 ), wh = a < < a l < < a L() = uable grd po uch ha max l up [al1,a l ) ( [X] [X] a l1 + [Y ] [Y ] a l1 ) < ϵ for gve ϵ >. Moreover, he r.h.. o (11.2) equal o b 1 [X] [Y ] u I J u 1 Al (, u) ddu (A) b 1 [X] a l1 [Y ] a l1 = b 1 = b 1 (B) b 1 l [X] a l1 [Y ] a l1 l I J u 1 Al (, u) ddu H 1 2 (a l ) H 1 2 (a l1 ) [X] a l1 [Y ] a l1 1 (al1,a l ]() l [X] [Y ] H 1 2 H 1 2 (d) (d) p [X] [Y ] H 1 2 (d), a, for every, uder [A1 ], where mea ha he dfferece goe o zero probably. I rema o valdae he approxmao (A) ad (B). For he proof, we ca adop a argume mlar o ha of Hayah ad Yohda [1]. 12. Proof of Propoo 5.1 Th eco devoed o he proof of Propoo 5.1. We eed wo kd of modfcao of amplg me a aed below. We wre r = b ξ ad for gve T >, prepare equece of oppg me Ŝ ad ˆT defed by Ŝ = S f ; max {S S 1 } r (T + 1)

21 2436 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) ad ˆT = T f ; max {T T 1 } r (T + 1). The Ŝ ad ˆT are G () -oppg me depedg o. Le Î = [Ŝ 1, Ŝ ) ad Ĵ = [ ˆT 1, ˆT ). Le Î = (Î ) ad J ˆ = (Ĵ ). The for a arbrary equece (υ ) of F-oppg me afyg υ T ad P[υ < T ] a, we have P[{X υ, Y υ ; Î, J ˆ} = {X, Y ; I, J } for all [, T ]] 1, accordg o [A4]. Thu, we may aume ha max{ I, J ; } r wha follow ad alo ha X ad Y afy propere characerzed by υ = T. We ake ξ uch ha ξ < ξ < ξ. Le G () = F ξ (b ad G () = ( G () ) + ) R+. We hall prepare a lemma o go o he ecod modfcao of oppg me. Lemma Suppoe ha max{ J ; } r ad ha b ξ r > b ξ. The M := up Z+ :T S T a G () -oppg me for each I. Proof. Fx I ad le (S r T = ) + o {T > S } T o {T S }. The radom me (S r ) + ad (T r ) + are G () -oppg me. Ideed, for R +, {(S r ) + } = {S + r } F (+ r b) ξ F ξ + (b = G () ) +. Therefore {T > S } G (), ad hece S {(S r ) +, T > S } = {(S r ) +, (T r ) + > (S r ) + > } {T > S, S r } G () becaue {(T r ) + > (S r ) + > } G () ad {T > S, S r (S r ) } G () + r = F G () G (). Afer all, {T } G (), ad for R +. Moreover, {T, T S } G () coequely all T are G () -oppg me. Sce up J r ad T =, here a T [(S r ) +, S ]. Therefore, we ee ha M = up Z+ :T S T = up Z+ T a G () -oppg me. We wll apply he reduco ued [7] o every realzao of (I ) ad (J ). Tha, we combe J o oe for J I, for each 1 (do ohg f here o uch J ), he relabel he dce from lef o rgh. Deoe he ewly creaed deg by (J ), wh he aocaed radom me T. We refer o he operao a J -reduco; I-reduco ca be carred ou he ame maer. We refer o he o operao a (I, J )-reduco. A formal coruco of T a oppg me a follow. Coder uffcely large. For each I, N := m Z+ :T S 1 T a oppg me wh repec o G (), ad herefore o G (). Accordg o Lemma 12.1, M are alo G () -oppg me. Whle ome of N, M ( N) have he ame value, we le hoe me up o oba T, whch we have ee above. Rouely, ur ou ha T are G () -oppg me; deed, T = ad T = f{n l {N l > T, 1 } Ml ; l N} for N. {M l > T 1 }

22 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Due o he bleary, {X, Y ; I, J } = {X, Y ; I, J } for I = (I ) ad J = (J ). I hould be oed ha r () vara uder hoe reduco. Le := 1 I () J (). A advaage of he reduco ha 3 ad 3. (12.1) Moreover, ce for each I (or J ), oe ca alway fd a erval I or a erval J ha cover, I (T ) 2 J (T ) 2 I (T ) 2 + J (T ) 2. Hece, Codo [A4] ad [A6] mpoed for he orgal deg (I, J ) wll rema vald for (I, J ). Sce each ummao (3.5) vara uder hoe reduco, order o how [B2] for (I, J ), uffce o verfy for (Ĩ, J ). The above argume eure ha f we ake ξ cloe o ξ, all he codo relaed o ξ are ll fulflled for ξ. Thu, we may aume hroughou he proof ha he (I, J )-reduco operao already carred ou. We wll coue o ue I = (I ) ad J = (J ) o expre he deg afer reduco, a well a ξ place of ξ. Hece (12.1) aumed o hold for from he begg. Moreover, r () r by he fr modfcao u before Lemma Accordg o he above dcuo, we may alo aume ha 4/5 < ξ < ξ < 1 he equel. Se β = ξ 2 3, ad α = ξ 2 3,. Le γ 1 9 ξ 4 5, ϵ 1, 1 2 ad c = b 3 4 γ. Defe υ by υ = f ; [X] > c f ; [Y ] > c f ; [X, Y ] > c X X r + Y Y r f ; up [,] ( r) 1/2ϵ > 1 T. (12.2) 1 (r,): r [(b ξ ) +,) By coruco ad from [A3], each υ a F-oppg me ad P[υ = T ] 1 a. Of coure, oce he localzao by υ appled o X ad Y, hey wll deped o hereafer; however he propere aumed for he orgal X ad Y are uchaged by h oppg, o we wll o wre o hem each me explcly. A oed before, we ake a uffcely large, deermc umber ad oly coder uch ha. I wha follow, for arbrarly gve ε, 3 8 γ, we ca aume he equaly w (X; r (T ), T ) + w (Y ; r (T ), T ) b 1 2 ξ ε (12.3) for all. Th becaue of he oppg by υ ad he fac ha r (T ) b ξ for all. The proof for Propoo 5.1 eeally ar wh he followg lemma. Lemma 12.2() wll be ued by Lemma 12.3(), whch wll ur be ued by Lemma 12.4(); he meame, Lemma 12.2() wll be ued by Lemma 12.3(), whch wll ur be ued by Lemma 12.4(). Lemma 12.4() a well a Lemma 12.5() wll be voked from Lemma 12.8, whle Lemma 12.4() a well a Lemma 12.5() wll be voked from Lemma Lemma 12.6 ad 12.8 coue he ma body of he proof of Propoo 5.1.

23 2438 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) For oaoal mplcy, we roduce he ymbol R () := S 1 T 1 ad R () := S T, addo o R () = S 1 T 1 ad R () = S T already defed. Noce ha hey all are G () -oppg me wh obvou relaohp uch a R () R () ad R () R (). By coveo, gve a cla C of ube of Ω ad a e A Ω, we wre C A := {C A; C C}. We may uppoe ha < b < 1 hereafer. Lemma Suppoe [A2] ad le, [, T ] ad,,,, 1. () For, G () < R ( ) F R (). () For,, G () < R ( ) I () J () I () J () I () J () I () J () I () J () I () J () F R (, ). Proof. We wll ue repeaedly he mple fac ha for ay F-oppg me σ ad τ, F σ {σ τ} F σ τ = F σ F τ, ad parcular {σ τ} F σ F τ, ad ha F σ F τ f σ τ. () Suppoe. I uffce o how ha A B C D F u for A G (), B = { < R ( )} C = {I () J (), I () J () } D = {R () u}, u R +. We have C D = C D {R ( ) < u + 3b ξ } (12.4) due o he fr modfcao a he begg of h eco becaue he wo par () ad ( ) repecvely overlap a he ame me o C. Sce A G () ad C G, () we have (A C) B G () R ( ). Thu, A B C {R ( ) < u + 3b ξ } G() however, G () u+3b ξ u+3b ξ, = F 2/3 u+b (3b αbβ ) F u becaue α > β ad < b < 1. Th ogeher wh he fac ha {R () u} F u mple A B C D F u for ay u. () A argume mlar o ha of () ca be made. Whe four par (), (, ), ( ), (, ), (, ) repecvely overlap, a he ame me, he oal legh of he aocaed combed erval I I I J J mu be cofed a R ( ) R (, ) 4b ξ ; oe ha R (, ) = S 1 S 1 T 1 T 1. Th lead o a dey mlar o (12.4), from whch oe ca prove () he ame faho a (). Remark I ca alo be how ha G () G () { < S } F S 1, G () { < T } F T 1, { < R ()} {I () J () } F R ().

24 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Le H := I + J I J = (I J ), whch he dcaor of he uo of he erval I ad J up o me. Le Ξ = K J ad Λ = H. To mplfy he oao, we ofe wre X G whe a radom varable X G-meaurable. Lemma Suppoe ha [A2] afed. Le (Z ) ad (Z ) be G() -progrevely meaurable procee. Le, >, ad,,,, 1. The: () Λ Λ Z Z F R () for. () Ξ Ξ Z Z F R (, ) for ad. Proof. () Noe ha o I () J () =, { R ( )} {H for. Le = } {H =, ad parcular Λ = } {Λ Λ = } =, ad alo ha A(,,, ) = { < R ( )} {I () J () } {I () J () }. For ay, ad Borel meaurable e B, {Λ Λ Z Z B} = [{ B} A(,,, ) c ] [{Λ Λ Z Z B} A(,,, )] F R () by Lemma 12.2() becaue Λ Λ Z Z G() -meaurable by coruco. () A argume mlar o () ca apply wh Lemma 12.2() ead of (). Remark () mple ha Λ Z = H Z F R () by akg =, =. () mple ha Ξ Z = J K Z F R (, ) for, by akg =, =, ad =. By a argume mlar o ha he proof of Lemma 12.3, ca be how ha I Z F S 1, I J J Z F T 1, Z F R () ( ). For ad <, Λ Λ I J Z F R (), Z Z F R (, ). A mlar reul hold for he aeme (). For a F-adaped proce Z, we wre Z = Z (b ξ ) +. The Z clearly G () -adaped. Le X = (I X) (I X) for every ad. We oce ha X = for >. (12.5) Lemma Suppoe ha [A2] ad [A3] hold. Le, R +. () For,,, 1 wh ad, E[Λ Λ [X, Y ] [X, Y ] L ] =. L

25 244 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) () For all,, k, k,, l 1 wh k, E[Ξ Ξ kk l [Y ] [Y ] X X k k ] =. Proof. () Noe ha, for he overlappg par () ad (, ), < mple whle < mple ; hece we ca uppoe ha < ad < whou lo of geeraly. We fr clam ha, for every, ad, E[L F R ()] =. I fac, L = 1 {>R ()}L becaue L = for R () = S 1 T 1 by defo, ad hece he opoal amplg heorem mple ha E[L F R ()] = 1 {>R ()}E[L F R ()] = 1 {>R ()}L R () =. For < ad <, Lemma 12.3() mple ha Λ [X, Y ] F R () F R (, )- meaurable ad Λ [X, Y ] F R (, ) F R (, )-meaurable, for ay,. Moreover, becaue L F R ()-meaurable (oce ha L op a = R ()), F R (, )- meaurable for ay. I follow ha E[Λ Λ = E[Λ =. [X, Y ] [X, Y ] L Λ L F R ()] [X, Y ] [X, Y ] L E{L F R (, )} F R ()] () We may aume ha, k k due o (12.5), ad alo ha > k by ymmery. Lemma 12.3() or Remark 12.2 mple ha Ξ [Y ] F S 1-meaurable; he ame way Ξ kk l [Y ] F S k 1-meaurable. Sce X k k meaurable wh repec o F S k F S 1, E[Ξ Ξ kk l [Y ] [Y ] X X k k The opoal amplg heorem provde E X F S 1 = X S 1 =, whch coclude he proof. ] = E[Ξ Ξ kk l Lemma Suppoe ha [A2] ad [A3] are afed. () For every ad, up [,T ] L c b ξ for all. () For every ad, up [,T ] X c b ξ for all. [Y ] [Y ] X k k Proof. Sce L = (I X) (J Y ) (I J ) [X, Y ], we have E{X F S 1}]. L w(x; r (), )w(y ; r (), ) + w([x, Y ]; r (), ). 2 By (12.3), w(x; r (T ), T ) b 1 ξ ε ad w(y ; r (T ), T ) b 1 2 ξ ε ε, 3 8 γ. O he oher had, from (12.2), for all, where w([x, Y ]; r (T ), T ) c r (T ) c b ξ.

26 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Therefore we have obaed (). I he ame faho, from he equaly X w(x; r (), T ) 2 + w([x]; r (), T ), we deduce (). Here we hould oe ha (I X) (I X) + (I X) (I X) ( < ) X = (I X) (I X) ( = ) ( > ), ad o ha X adm a repreeao mlar o L ay cae. For he ma body of he proof for Propoo 5.1, le u coder he gap (3.5) whou calg ad decompoe a ( K ) [L, L ] ( K ) V,,,, where = 1, + 2, + 3,, (12.6) 1, = 2, = (,,, K ) ({(I X)(I X)} {(J J ) [Y ]}),,, ( K ) ({(I I ) [X]} {(J J (,,, ) ({(J ) [Y ]}), Y )(J Y )} {(I I ) [X]}),,, ( ) ({(J J ) [Y ]} {(I I ) [X]}) ad 3, = ( K ) ({(I,, X)(J Y )} {(J I ) [Y, X]}),, ( K ) ({(I J ) [X, Y ]} {(J I ) [Y, X]}) + (,, K ) ({(J Y )(I X)} {(I J ) [X, Y ]}),, ( K ) ({(J I ) [Y, X]} {(I J ) [X, Y ]}). Fr, we how ha b 1 1, aympocally eglgble. Le I (J ) = I ad J(I ) = J. Throughou he dcuo, for equece (x ) ad (y ), x y mea ha here ex a coa C [, ) uch ha x Cy for large. Lemma Uder [A2] [A4], hold ha b 1 1, p ad b 1 2, p a.

27 2442 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Proof. We oe ha J J wheever ad ha X = for >, o a o rewre 1, a 1, = 2, K J X [Y ]. Le R = [Y ] [Y ]. We have ( 1,T ) 2 = 4(I + II + III + IV), where I =, k k,l II =, k k,l III = IV = From [A3], T T T T T T, k k,l T T, k k,l up R = O p [,T ], Ξ Ξ Ξ Ξ b ξ 12 λ Ξ kk l Ξ kk l Ξ kk l Ξ kk l for ay λ >. O he oher had, T Ξ d J (T ) 9 hak o (12.1). Coequely, II, k k,l T T X X X X X k k [Y ] [Y ] dd, X k k [Y ] R dd, K T J (T ) 9T, Ξ (max, (X ) T )2 R T ( [Y ] ) T (c b ξ )2 O p b ξ 12 λ Ξ kk l X, X k k [Y ] R dd, X k k R R dd. T X k k [Y ] R dd T c 81T 2. Ξ 2 d Sce < γ < 1 9 ξ 4 5 ad ce λ > ca be ake arbrarly mall, b 2 II = O p b 2+2ξ +ξ 12 λ 9 4 γ O p b 2(ξ ξ) = o p (1). I a mlar maer, we ca how ha b 1 III = o p(1) ad b 1IV = o p(1).

28 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Nex, we wll evaluae E[I]. I lgh of Lemma 12.4(), he erm corbue o he um oly whe = k. Thu, from (12.2), wh he ad of Lemma 12.5(), we have E[I] c 2 b2ξ c 2 E T T Ξ Ξ,k,l dd. Now,, k,l T T However, for each,, : Hece,, k,l T Ξ Ξ d = T T Thu follow ha b 2 Ξ, k,l Ξ k l dd = T, : T Ξ 2 d. J K d 3 J(I )(T ). : Ξ k l dd 9 E[I] b2 c 2 b2ξ c 2 bξ 9 max J(I )(T ) 2 J(I )(T ) J(I )(T ) 9(3r (T ))(3T ) = 81r (T )T b ξ. = b3(αγ ) a. Afer all, we coclude ha b 1 1,T = o p (1). By ymmery, we alo oba b 1 2,T = o p (1). A he la ep for Propoo 5.1, we are gog o how ha b 1 3, aympocally eglgble. The expreo for 3, ca be mplfed a below. Lemma , = 2 ( H L ) [X, Y ]. Proof. By he ue of aocavy ad leary of egrao a well a egrao by par, oe ha 3, = {( K )(K L + L )} [X, Y ].,, The ummao break dow o four cae. Cae 1:,. Wheever < ad >, boh () ad (, ) cao overlap a he ame me; hece K. The cae of > ad < mlar. Whe < ad < (ad whe () ad (, ) repecvely overlap a he ame me), f (, ) overlap, he rvally ; moreover, K K becaue K = for R (, ) = T 1 bu K for R (, ) < R (, ) = T T 1. The cae whe ( ) overlap ead ca be deal wh mlarly.

29 2444 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) The cae of > ad > ca be how by ymmery. I follow ha (K,, :, K )(K L + L ). Cae 2: =,. Whe < (ad whe () ad ( ) repecvely overlap a he ame me), becaue K = for R ( ) = T 1 bu for R () < R () = T T 1 ; herefore ( K )(K L + L ) = ( )( L + L ) = L. Whe >, by ymmery, ( K I follow ha (K,, : =, = )(K L + L ) = L. K K,, : =, < = 2 K,, : =, < )(K L + L ) L + K,, : =, > L = 2, < K L L by ymmery ad by he fac ha (Lemma 1.1). Cae 3:, =. Lke he above cae, whe <, K, o ( K whle for >, ( K I follow ha (K,, :, = = 2 )(K L + L ) = ( K )(K L + L ) = K K L, )(K L + L ) = K L. K K,, : <, = Cae 4: =, =. Evdely, (K,, : =, = K )(K L + L ) K K L = 2, : < )(K L + L ) = 2 K K L. K,, : =, = L = 2 L.

30 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Pug all he four cae ogeher, 1 2 Becaue,, ( K = K L +, <, : < )(K L + L ) K K L + L. L for < ad K L for <, he r.h.. equal o, : K L +, : =, = L +, L I + K K L + K L L J L L L = (I + J I J )L = H L, ad herefore, he aero obaed. Lemma Uder [A2] [A4], b 1 3, p a. Proof. Le R = [X, Y ] [X, Y ]. We apply Lemma 12.7 o ge ( 3,T ) 2 = 4(I+II+III+IV), where I = II = III = IV = T T Λ Λ, T T Λ Λ, T T Λ Λ, T T Λ Λ, [X, Y ] [X, Y ] L [X, Y ] R L R [X, Y ] L R R L L L L L dd. dd dd dd From Lemma 12.5(), we have up [,T ] L c b ξ for all. Alo, 12 λ up R w([x, Y ] ; b ξ, T ) = O p [,T ] b ξ

31 2446 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) for ay λ > by [A3]. Sce H T Λ d 3 T Coequely, we coclude ha II (c b ξ )2 O p b ξ I I + J, I (T ) λ d + c 36T 2, T J J (T ) 6T. d o b 2II = o p(1). Lkewe, we oba b 2III = o p(1) ad b 2IV = o p(1). By he uform boudede [X, Y ] c ad Lemma 12.5(), we oba E[I] c 2 b2ξ c 2 E T T 1 {= or = } Λ Λ, dd. Here we remark ha he cae (, ) make o corbuo o he um, hak o Lemma 12.4(). We have T T 1 {= or = } Λ Λ dd +,,, =: A 1 + A 2. Sce ad A 1 = we ee ha T T 2 Λ d T T Λ d I d + J d 3 I (T ) + J(I )(T ) 4 J(I )(T ), A 1 16 J(I )(T ) 2 16 max J(I )(T ) 16(3r (T ))(3T ) = 144r (T )T b ξ. J(I )(T ) By ymmery, A 2 b ξ. Pug all of h ogeher, we oba E[A 1 + A 2 ] b ξ, ad a a reul, b 2 E[I] b2 c 2 b2ξ c 2 bξ a. Lemma 12.8 ha bee proved. = b3(αγ ) = o(1)

32 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) Proof of Propoo 5.1. The dered reul follow from he decompoo (12.6) ad Lemma 12.6 ad Proof of Theorem 6.1 ad 6.2 Th eco devoed o he proof of Theorem 6.1 ad 6.2. Wh B 1, = A X (I ) M Y (J ), B 2, = A Y (J ) M X (X ), B 3, = A X (I ) A Y (J ), we have he decompoo {X, Y } = {M X, M Y } + B 1, + B 2, + B 3,. The lmg drbuo of he fr erm ha bee foud he prevou eco. We ow clam ha he re are, afer beg caled, aympocally eglgble. We hall maa he ame eup a decrbed Seco 12. Lemma Suppoe ha [A2] [A6] are afed. The b 1/2 Bl,T p a for l = 1, 2, 3. Proof. Th me, place of (12.2), we wll ue he radom me υ defed by υ = f{; [X] > c } f{; [Y ] > c } f{; [X, Y ] > c } M X Mr X f ; up + MY MY r [,] ( r) 1/2ϵ > 1 1 (r,): r [(b ξ ) +,) f{; (A X ) > d } f{; (A Y ) > d } T for ome ζ 13, α 3 4 γ 13, 1. A meoed Seco 12, wh d = b ζ/2 we ca aume ha X ad Y are opped by υ. Th υ here o greaer ha υ (12.2); however h doe o maer ce P[υ = T ] 1. Though wll o be wre explcly o he procee, hey deped o afer localzao. Furher, we ca aume ha he amplg deg have bee modfed by (I, J )-reduco. We oly coder B 1,T. The oher cae ca be how he ame way. For oaoal mplcy, we drop X ad Y from A X ad M Y. Sce = 1 for R () f he par () overlap, ad = oherwe, whle he proce A(I )M(J ) ar o vary a ad beyod = R (), oe ha B 1, = {A(I ) M(J ) } = {A(I ) M(J )} + {M(J ) A(I )} =: I + II.

33 2448 T. Hayah, N. Yohda / Sochac Procee ad her Applcao 121 (211) The proce I clearly a couou local margale wh he quadrac varao [I] = {( K (I A) (I A) (J J )[M] ) }, due o [A3]. Each ummad vahe wheever, ad alo (I A) A I () by [A5]; hece [I] [M] A 2 I () J () K I (). Sce I (), J () 3r () ad K I () 3r (), a well a [M] c by he localzao already doe, oe ha [I] 9c A 2 r () 2, ad herefore b 1 3 [I] 9 c O p (1) b 1 +2α = O p b 2(αγ )+ 5 4 γ = o p (1). The Leglar equaly mple ha b 1/2 up T I p a dered. Nex we coder II. Sce {b 1/2 II} 1 C-gh (cf. Defo VI.3.25 of [17]) by Lemma 13.2 below, uffce o how ha b 1/2 II = o p (1) for every o coclude uform covergece. We rewre II a II = A T 1 =: II 1, + II 2,. I (J M) d + I (J M) (A A T 1 )d (13.1) Fr we clam ha b 1/2 II 1, = o p (1) a. If <, he clearly A A T 1 T 1 F T 1-meaurable; bede, ca be verfed ealy ha K K u I I u F T 1-meaurable due o a vara of Lemma 12.3(). Therefore, E[II 2 1, ] = 2E E A T 1 A T 1,, < (J M) u F T 1 ddu + E (A T 1 ) 2,, = E (A T 1 ) 2,, 4d 2 c E J (T ) K u I I u (J M) K u I I u E[(J [M])2 F T 1]ddu K u I I u (J [M] v) ddu 2 T I (T ) 4d 2 c E[9T r (T ) 2 ] b α 3 4 γ ζ = o(b ), ad hece b 1/2 II 1, p for every T.