SUPPLEMENT TO STATISTICAL PROPERTIES OF MICROSTRUCTURE NOISE (Econometrica, Vol. 85, No. 4, July 2017, ) for j = (0 j) j = 0 1 (B.

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1 Ecoomerca Supplemeary Maeral SULEMENT TO STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE Ecoomerca, Vol. 85, No. 4, July 2017, JEAN JACOD Isu de Mahémaques de Jusseu, CNRS UMR7586 ad Uversé erre e Mare Cure 6 YINGYING LI Deparme of Iformao Sysems, Busess Sascs ad Operaos Maageme ad Deparme of Face, Hog Kog Uversy of Scece ad Techology XINGHUA ZHENG Deparme of Iformao Sysems, Busess Sascs ad Operaos Maageme, Hog Kog Uversy of Scece ad Techology AENDIX B: MORE RESULTS ON NUMERICAL STUDIES B.1. Supplemeary o Seco 4.1 IN THISSUBSECTION, WE REORT RESULTS ABOUT AUTOCOVARIANCES; hose abou auocorrelaos are gve he ma arcle. Fgure 14 compares he esmaes of scaled auocovaraces based o Theorem 3.1 wh he feasble esmaes based o he ose process ad he heorecal values. The esmaes are based o oe smulaed pah. More specfcally, we use red dashed curve o repor he esmaes of he scaled auocovaraces based o Theorem 3.1, amely, Rj := Uj /N for j = 0 j j = 0 1 B.1 as 3.9 wh = 1; blue doed curve o repor he feasble esmaes based o he ose process ε = γ T χ, amely, he auocovaraces esmaed from ε whch are o observed pracce; blac sold curve o repor he heorecal values, ha s, Rj 1 := Rj 1 as 3.4. The heorecal values Rj 1 are, wh ψ 0 = 1adψ j j= 1 2 as 4.39, M j 1 Rj 1 = σ 2 ψ ψ +j γ 2 α s s ds for j = 0 1 M B.2 =0 0 Fgure 14 shows ha for auocovaraces, our esmaes are also comparable o he feasble esmaes based o he ose process, boh of whch are close o he heorecal values. To exame he CLT Theorem 3.8, we plo he ormal quale-quale plos of N 1 Rj Rj 1 1 Ẑ j j 1 B.3 Jea Jacod: jea.jacod@gmal.com Ygyg L: yyl@us.h Xghua Zheg: xhzheg@us.h 2017 The Ecoomerc Socey DOI: /ECTA13085

2 2 J. JACOD, Y. LI, AND X. ZHENG FIGURE 14. Esmaes of scaled auocovaraces. The feasble esmaes based o he observed prces are compared wh he feasble esmaes based o uobservable ose process, ad wh he heorecal values. wh Ẑ j j gve by he secod formula 3.32, sce here α 1 1wh chose o be 3. Aalogously o Fgure 2, we show he plos for lags 1, 4, ad 7 Fgure 15. The ormaly s aga suppored. B.2.1. Colored Nose Wh Roudg B.2. Supplemeary o Seco 4.2 The seg cosdered s he same as Seco 4.1, amely, whe he processes X, γ, ad χ are specfed by equaos , excep ha he observed prces are gve by S = [ exp ] X T + ε / B.4 ha s, he coamaed observed prces Seco 4.1 furher rouded o ces. A sample pah of S s gve Fgure 16. Oe ca clearly see he roudg effec from Fgure 16. The proporo of fla radg, amely, zero raday observed reurs for hs FIGURE 15. Normal QQ-plos of B.3 for lags 1, 4, ad 7, based o 1000 replcaos.

3 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 3 FIGURE 16. A sample pah of he rouded prces S as B.4. parcular sample pah s 75%. Dffere sample pahs have smlar proporos of fla radg. Uder such a seg, he ose s ε = log S XT Our goal s o esmae he auocovarace ad auocorrelao of he ose ε based o he observaos S, or equvalely, based o Ỹ := log S To do so, we apply he same esmaors as Seco 4.1, amely, Rj := 1 Uj/N for 1 1 j = 0 j j = 0 1as 3.9 for esmag he auocovaraces, ad rj 1 as 3.12 for esmag he auocorrelaos. The ug parameer s chose o be 6 based o he heursc crero Seco We compare hese esmaes wh hose esmaed from he ose ε.fgure17 shows he comparso resuls for auocovaraces based o oe radom sample pah; hose for auocorrelaos are gve he ma arcle. We see fromfgure 17 ha our esmaes are close o he feasble esmaes based o he ose, dcag ha our mehod wors well eve he presece of roudg. B.2.2. Whe Nose Wh Roudg For he whe ose wh roudg case, we le he process χ be a sequece cossg of..d. N0σ 2 0 radom varables. The proporo of fla radg s abou 72%, slghly lower ha he proporo Seco B.2.1 where χ s colored. The esmaors are also he same as before, excep ha he ug parameer s reduced o 3, aga based o he heursc crero Seco Ths smaller choce of s due o he weaer depedece he ose; see also he dscussos Remar 3.6. Fgure 18 repors he esmao resuls for auocovaraces. Fgure 18 shows ha our esmaors also wor well hs case.

4 4 J. JACOD, Y. LI, AND X. ZHENG FIGURE 17. Esmaes of auocovaraces for he case of colored ose wh roudg. The feasble esmaes based o he observed prces are compared wh he feasble esmaes based o he ose. B.3. Supplemeary o Seco 5 B.3.1. Resuls for Cgroup C o May 06, 2010 I hs seco, we aalyze he same soc, Cgroup C, bu focus o a specal day, May 06, 2010, he day o whch a flash crash occurred. The umber of rasacos o ha day s abou 524,000. Choosg. Followg he heursc crero Seco 5.1.2, s chose o be 60, whch yelds he plo Fgure 19, aalogous o Fgure 6. Dural Feaures he Nose. We sar wh he sze of ose. Fgure 20 shows he esmaed szes of ose ad esmaed volaly durg dffere half-hour ervals. We see ha whe he volaly s uusually hgh, so s he sze of ose. Nex, we exame he depedece durg dffere half-hour ervals. Fgure 21 gves he esmaed auocorrelaos durg dffere half-hour ervals. FIGURE 18. Esmaes of auocovaraces for he case of whe ose wh roudg. The feasble esmaes based o he observed prces are compared wh he feasble esmaes based o he uobservable ose.

5 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 5 FIGURE 19. Comparso of he wo esmaes of R0 1 Rj 1, oe based o Rj adj, he oher based o R0 1 Rj 1, for Cgroup C o May 06, FIGURE 20. Esmaed szes of ose ad volales for Cgroup C durg 13 half-hour ervals o May 06, FIGURE 21. Esmaed auocorrelaos of ose up o lag 50 for Cgroup C durg dffere half-hour ervals o May 06, 2010.

6 6 J. JACOD, Y. LI, AND X. ZHENG FIGURE 22. Esmaed auocorrelaos of ose up o lag 100 for Cgroup C soc o May 06, Esmag he Auocorrelaos of Nose. Fally, we esmae he auocorrelaos usg he whole-day daa. Fgure 22 shows smlar feaures o Fgure 11, amely, he auocorrelaos are posve ad slowly decay o 0. B.3.2. Resuls for Iel INTC I hs seco, we brefly repor he resuls for Iel INTC durg he same perod of Jauary The average umber of daly rasacos s abou 143,500. Choosg. = 20. Aalogously o Fgure 6, we have he plo Fgure 23 by usg Dural Feaures he Nose. We frs exame he sze of ose. Fgure 24 shows he esmaed szes of ose durg dffere half-hour ervals. The average curve suggess ha he ose eds o be larger owards he ed of radg hours. As o he volaly, we have he plos Fgure 25. FIGURE 23. Comparso of he wo esmaes of R0 1 Rj 1, oe based o Rj adj, he oher based o R0 1 Rj 1, for Iel INTC soc, o Jauary 3, 2011.

7 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 7 FIGURE 24. Esmaed szes of ose for Iel INTC soc, durg 13 half-hour ervals ad for dffere radg days. The frs fve curves plo he esmaes for he frs fve radg days Jauary The lower-rgh curve plos he szes of ose durg dffere half-hour ervals averaged over he 20 radg days Jauary FIGURE 25. Esmaed volales for Iel INTC soc, durg 13 half-hour ervals ad for dffere radg days. The frs fve curves plo he esmaes for he frs fve radg days Jauary The lower-rgh curve plos he esmaed half-hour volales durg dffere half-hour ervals averaged over he 20 radg days Jauary 2011.

8 8 J. JACOD, Y. LI, AND X. ZHENG FIGURE 26. Scaerplos of he esmaed szes of ose agas volales durg dffere half-hour ervals for Iel INTC soc, Jauary The rgh pael s a zoomed- verso of he lef pael wh he wo oulers o s far rgh removed. Aga, we see a U-shaped paer. Fally, we plo he esmaed sze of ose agas volaly durg dffere half-hour ervals Fgure 26. We observe a smlar paer o Fgure 9. The correlaos wo plos are 007 ad 013, respecvely, he former beg sgfca ad he laer beg sgfca a he 5% level. Nex, we exame he depedece durg dffere half-hour ervals. Fgure 27 shows he esmaed auocorrelaos durg dffere half-hour ervals. Esmag he Auocorrelaos of Nose. Fally, he esmaed auocorrelaos usg he whole-day daa are gve Fgure 28. Aga, we observe smlar feaures o Fgure 11, amely, he auocorrelaos are posve ad slowly decay o 0. FIGURE 27. Esmaed auocorrelaos of ose up o lag 50 for Iel INTC soc, durg dffere half-hour ervals. Each red curve represes he esmaes durg oe half-hour erval. Lef: auocorrelaos o Jauary 3, 2011; rgh: auocorrelaos averaged over he 20 radg days Jauary 2011.

9 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 9 FIGURE 28. Esmaed auocorrelaos of ose for Iel INTC soc Jauary Each curve s for oe radg day, ad plos he esmaed auocorrelaos of lags up o 50. AENDIX C: TECHNICALDETAILS OF THE ROOFS C.1. The Samplg Scheme ad he Localzao rocedure ROOF OF LEMMA A.1: For ay 0, we le F be he smalles flrao coag F ad for whch T0 T are soppg mes, so F 0 = F ad F = F. We wll prove by duco o he propery : ay F -margale s a F - margale. The proof s based o a cocree descrpo of F +1 erms of F ad + 1. Obvously, F +1 coas he class H of all ses B F such ha B {T+ 1>}= B {T+ 1>} ad B {T+ 1 }=B {T+ 1 } for some B F ad B F σ + 1. I s easy checg ha H s a σ-feld coag F ad creasg wh ad ha T+ 1 s a H -soppg me. Thus, we deed have F +1 = H. Suppose forsome 0, ad le M be a F -margale, hece a F - margale. Le s> 0adB F +1, wh whch we assocae B B as above, ad observe ha B ={ω : ω + 1ω B} for some F R-measurable subse B of Ω R R s he Borel σ-feld of R. We have EM s M 1 B = a + a,where a = E M s M 1 {+1 T} 1 B + 1 a = E M s M 1 B {+1> T} Now, we apply of Assumpo O, plus he obvous fac ha F T = FT.Byco- dog o hs σ-feld, sce M s M s F -measurable, ad wh F deog he F - T codoal law of,weoba T a = E E M s M 1 B x F T Fdx 0

10 10 J. JACOD, Y. LI, AND X. ZHENG Sce 1 B x s F expecao above vashes, ad a = 0. A aalogous argume yelds a = 0, ad hus M -measurable ad M s a F -margale, he er codoal s a F +1 -margale. Therefore, mples +1 ad,sce 0 obvously holds rue, we see ha fac holdsforall. Now, provg he clam s easy. Frs, s eough o prove whe M s a bouded F -margale. From wha precedes s a F process M = M T as well. Sce F -margaleforall, hece he sopped resrco o he se { T} ad M -margale. Sce T -local margale, hece a F -margale Q.E.D. = F s cosa me afer T,wededucehaM s a F as, follows ha M s a F because s bouded. ROOF OF LEMMA A.2: By a classcal localzao procedure, s o resrco o assume ha τ 1 = Assumpo O, ad also ha V s bouded. a We frs prove 2.4; we se A = N L = T [/ ] [/ ] = =1 [/ ] S = =1 α T 1 We have S = [/ ] =1 ζ + [/ ],whereζ = /α T 1, ad 2.3 mples Eζ F T 1 K 3/2+κ ad Eζ 2 F T 1 K 2+κ,whereasζ s F T - measurable. Therefore, we deduce S, hece S ucp = as well, from E S K1+κ + K 2 1/2+3κ/2 By he subsequece prcple, for 2.4 s eough o show ha ay fe sequece coas a subsequece such ha A A for all, forallω ousde a ull se. Sce from ay subsequece oe ca exrac a furher subsequece such ha S holds, ousde a ull se aga, locally uformly me, s eough o show ha f S ω locally uformly for some gve ω, he A ω A ω. So below, we assume S ω locally uformly, ad om o meo ω. Thedefos of L ad S mply L = α 0 Hs ds. Equao 2.4 yelds s L +s L κ 1 S +s S; hece, by Ascol s heorem, from ay subsequece we ca exrac a furher subsequece such ha L coverges locally uformly o a couous odecreasg lm L. cg ay ε>0, we le 1 < 2 < be he mes a whch α has a jump of sze bgger ha ε, ad se B = 1 ε + ε] [0] ad B =[0]\B. The modulus of couy w ρ of α ε = α s s α 1 1 { s} o [0] sasfes lm sup ρ 0 w ρ ε, whereas L s L s locally uformly, so lm sup sup s B α L α L s s ε. Thus, for large eough, L α 0 L s ds s 2εS + K B ds, whch ur goes o 2ε + K s B ds Kε. Sce ε s arbrarly small, we ge L α 0 L s ds s 0. Aoher applcao of S s s for all s yelds α 0 L s ds s α 0 L s ds. ThusL = α 0 L s ds, sol s srcly creasg ad s verse L 1 s A, as defed by 2.4. Therefore, L s uquely deermed ad he orgal sequece L coverges o L = A 1. Now, he defos of A ad L mply ha hey are rgh-couous verses oe from he oher, hece A L 1 = A, ad he proof of 2.4 s complee. For a, remas o deduce A.2 from 2.4, ad whou loss of geeraly we assume d = 1. If V = sup s V s,wehave H 0 V s da s u V, whch goes o 0 by our

11 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 11 assumpos o u. The propery A A mples V 0 s da s V 9 s da s because V s càdlàg ad A s couous. We deduce he covergece A.2 for each, ad he local uform covergece easly follows, aga because A s couous. b Se H = HV = 1 N =0 V T αt + 1 H = H H If u > 0, we have K ρ, hece H Ku ρ 1/2 because V ad α are bouded. The u ρ 1/2 ucp 0yeldsH = 0. Heceforh, s eough o show he covergece of H, or equvalely suppose ha u 0. Wh ζ j = 1 VTα j T + 1,wehaveH j = N =0 ζj,adζ s F T+1 -measurable. Hece by Theorem IX-7-28 of Jacod ad Shryaev 2003, suffces o prove he followg covergeces probably, for all >0 ad all bouded F -margales M ad wh M = M T+1 M T : N =1 N =1 j E ζ F T 0 E j ζ 4 FT 0 N =1 N =1 E ζ j ζ m F T E ζ j M F T 0 V j V m s s 0 α s α s ds C.1 The frs ad hrd pars of C.1 readly follow from 2.4 ad A A.Nex, j E ζ ζ m F T α T VTV j m T K 1+κ so he secod par of C.1 follows from A.2 appled o αv j V m. For he las par, sce + 1 s F -codoally depede of F T,whereasM s F -measurable ad a F -margale by he prevous lemma, s lef sde s fac decally vashg. Q.E.D. ROOF OF LEMMA A.3: 1 Assume for a mome he exsece of a localzg sequece θ m of soppg mes ad, for each m, of a semmargale Xm,asamplg scheme m : 1, ad a ose process ε m : 1 0, wh whch we assocae N m ad T m as 2.2, α m ad α m as O, ad γ m as 2.7, such ha: a For each m, he famly [ X m m ε m b For each m, wehavelm Ω m = 1, where ] sasfes SHON Ω m = { T m = Tfor all wh T<θ m } c We have X m = X f <θ m ad ε m = ε f T m = T<θ m C.2 Each resul of Seco 3 s he covergece probably or sably law of a sequece of sascs Z oward some lm Z,whZ based o he resrco of [X ε ] o some me erval [0], ad he lm Z or s F-codoal law s always some fuco g α α γ oly depedg o he resrco of α α γ o [0].

12 12 J. JACOD, Y. LI, AND X. ZHENG Now, compue he same sasc, say Zm,wh[Xm m ε m ]. Equao C.2-a ad he assumpos of he lemma mply ha Zm coverges o a lm Zm whch s characerzed by g α m α m γ m wh he same g as above. Equao C.2-b,c mples ha Zm = Z ad also g α m α m γ m = g α α γ o he se Ω = m Ω {< m θ m },ad lm m lm f Ω m c lmm θ m + lm lm sup Ω c m m = 0 We deduce frs ha, resrco o { <θ m },wehavezm = Z case of covergece probably, or Zm ad Z have he same F-codoal laws. Secod, follows ha deed Z coverges o Z he approprae sese, ad he clam s proved. Therefore, remas o show ha we ca fd θ m ad [X m m ε m ] sasfyg C.2. Ths s acheved hrough several seps. 2 By he classcal localzao procedure see, e.g., Seco of Jacod ad roer 2012, here are a localzg sequece θ 1 ad processes m X m α m α m γ m sasfyg K wh 1/α m bouded, such ha X m α m α m γ m cocde wh X αα γ o [0θ 1. m Secod, he process X m = m Γ =1 1 { Γ m0<s } cocdes wh X o [0θ 2 m, wh he localzg sequece θ 2 = S m m fs : 1 Γ >m, ad sasfes of SHON. The localzg sequece θ m C.2 wll be θ m = m τ m θ 1 m θ2 m,adxm = X m + X m ad ε m = γ m χ T wh he same sequece χ as N: we have X m = X ad λ m = α ad α m = α ad γ m = γ for <θ m,adalsoε m = ε f T<θ m,whereas X m α m α m γ m ε m sasfy ad of SHON. 3 The cosruco of a samplg scheme m sasfyg O wh he processes α m α m ad m K q decally ad sasfyg C.2- s more delcae. Noe ha 1/α m α m C for a cosa C>0. Upo elargg he space f ecessary, we have a sequece of..d. varables, depede of F ad of all s ad χ s, ad whch are ceered wh varace 1 ad wh suppor C/2C ] for aoher cosa C > 0. Se also I = f : > ρ or T τ m. The wo sequeces m T m are defed by duco o, as follows: we sar wh T m 0 = 0ad se f <I m = + α m T m 1 f I α m T m 1 T m = T m 1 + m Sce C/2 C ad 1/α m ad α m are bouded, we have m C C for I, so he prevous duco defes a ew samplg scheme wh whch we assocae he flrao F m as O. I hs sep, we show ha hs ew samplg scheme sasfes SHON wh he assocaed processes α m α m 2.3. Sce < ρ f <I ad m C oherwse, we obvously have m K ρ for some cosa K. Bycosruco,Tm = Twhe <I, so he resrcos of he σ-felds F m T m ad F o he se {<I T } cocde, ad hs se s F - T measurable. Hece, for ay R F m T -measurable fuco f 0oR Ω ad ay m F -measurable varable Z 0, he varable B = EZf m F m T m 1 aes he

13 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 13 form E Zf 1 ρ {< } F T 1 + E Z1 ρ B = { }f α m T 1 E Zf α m T m 1 + α m T 1 F T 1 + α m T m 1 F m T m 1 o { 1 <I } o { 1 I } Sce he orgal scheme sasfes of O ad s depede of F ad of F m we deduce ha B s he produc of EZ F m ad Ef T m 1 m F m m m ad F are f m T m 1-codoally depede., T m 1,so T m 1 Sce s bouded, ceered wh varace 1, he above formula wh Z = 1ad fxω = x or fxω = xα m ω T 1 2 or fxω = x p shows us ha of O holds wh α m α m o he se { 1 I } wh τ 1 =.Toseehaholdsalsoo he compleme { 1 <I }, ad because T 1<τ m o hs se ad hs propery holds by hypohess for he orgal scheme, s clearly eough o prove ha, for ay r q 0, E q 1 ρ { } F T Krq q+r C.3 Usg Marov s equaly ad he hrd par of O-, we see ha he lef sde above s smaller ha κ mq+p1 ρ q+p1 ρ, ad upo ag p r/1 ρ. Therefore, he scheme m sasfes SHON. 4 I remas o show b of C.2. By our defo of our ew samplg scheme ad of θ m, hs wll be mpled by he propery B 1, where B ={ < ρ for = 1N m + 1}. Applyg C.3 whq = 0adr = 3, we ge B c [1/ N > 2 ] m 1/2 + ρ N > m 1/2 + K =1 ad N > m 1/2 0as by 2.4. Ths complees he proof. Q.E.D. C.2. Some Facs Abou Saoary Sequeces The proof of Theorem A.4 s raher volved. We beg wh some oao. By A.11 ad he Cauchy Schwarz equaly, 1 E Eξ H m < for ay m, whereasv s bouded, so he followg d-dmesoal varables U ad m M m are well defed, compoewse, as U j = m VTE j ξ j H m =m w + M j m = =0 V j j j T E ξ H m E ξ H 0 recall H m = F 0 G m, ad we wre M for he same varable as m M m,whξ subsued wh ξ. We also cosder he F 0 -measurable varables ν = N + w u Sce ξ s G +w -measurable, we have Eξ H ν = ξ whe 0 N u, hece G = M ν + U 0 U ν C.4

14 14 J. JACOD, Y. LI, AND X. ZHENG LEMMA C.1: Uder he assumpos of Theorem A.4, for ay >0 we have ad U ν 0 U 0 0 M ν M ν 0 C.5 C.6 ROOF: 1 We oly prove he frs covergece C.5, he secod oe beg smlar. We have U ν = B + C,where B = =1+ν VTE j ξ j H ν C = ν VTE j ξ j H ν =ν w + Sce V s bouded, we deduce from A.11, v>1, ad he Cauchy Schwarz equaly ha E B 2 F 0 K E E ξ H ν E ξ H j ν F 0 K j=1+ν hece B 0. Nex, we have E ξ 2 K by A.10, hece E C 2 s obvously smaller ha K w 2 because he sum defg C coas a mos w erms. Sce w 2 0, we deduce C 0, hece he frs covergece C.5. 2 Now we ur o C.6. Seg ξ = ξ ξ,wehavem ν M = ν ν =1 η, where η j = 0 V j j j T E ξ H E ξ H 1 s a margale creme, relave o he dscree me flrao H 0. Therefore, E M j ν M j 2 ν F 0 ν = =1 ν = E η j 2 F 0 =1 E ξ j E l 0 VTV j j j j Tl E ξ H E ξ l H j H 1 E ξ l H 1 F 0 The double seres l above s absoluely coverge almos surely, so we may permue he summao over l, he oe over, ad he codoal expecao E F 0,o ge E M j ν M j 2 ν F 0 = D ν D 0 where D = VTV j j Tl α l l 0 α l = E E ξ j H E ξ j l H F 0

15 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 15 If ρ = E ξ ξ 2, a applcao of A.8, A.9 ad he Cauchy Schwarz equaly gves us for l: Kρ / v l v f < α Kρ /l v f 0 <l l Kρ /l w v f + w <l Kρ oherwse use he propery Eξ H = ξ whe w. Thus, sce α = l α,wege l sup m D K ρ 1 + m + w m, hece, recallg A.5, E M j ν M j ν 2 F 0 Kρ C w o he se Ω Sce w ρ 0adΩ 1, we readly deduce C.6. Q.E.D. Nex, we observe ha M = m m ζ ζ j =1 = =0 where V j Tβ j βj = E ξ j H E ξ j H 1 C.7 LEMMA C.2: Uder he assumpos of Theorem A.4, for all >0 ad ε>0 we have ν E ζ j ζm H 1 a jm =1 ν E ζ 2 1 { ζ >ε} H 1 0 =1 0 V j V m s s da s C.8 ROOF: 1 The secod covergece s easy o prove. Sce β = β 0 θ for all Z, he varables β = Z β sasfy β = β 0 θ.apror, β could be fe; however, β = 0whe< w by A.10, so A.11 for ξ mples E β 2 <. The obvous esmae ζ D β for some D>0 ad saoary yeld E E ζ 2 1 { ζ >ε} H 1 F 0 D 2 E β 2 1 { β >ε/d } = D 2 γε where γε = E β { β 0 >ε/d } Now, γε 0as, because E β 2 0 <, ad he secod par of C.8 follows sce ν A by 2.4 ad u + w By vrue of he square-egrably of β, he d-dmesoal varables β = β 0 are well-defed, square-egrable, ad also β = β w+1 θ w 1 for all w + 1 hs fals whe 1 w. I hs sep, we show ha j E β βm j K ad f w + 1 he E β βm = a jm C.9

16 16 J. JACOD, Y. LI, AND X. ZHENG wh a jm gve by A.12. The frs esmae follows from β β = β 0 θ ad β 0 L 2. For he secod propery, by polarzao s eough o show he oe-dmesoal case d = 1, so below we om j m. Theβ l = Eξ l G Eξ l G 1 s G -measurable wh vashg G 1 -codoal mea, ad ξ +1 Eξ l+1 G = ξ Eξ l G 1 θ, hece Eβ β l = E Eξ G β l Eξ G 1 β l = Eξ β l Therefore, for ay L>2 we have = E Eξ ξ +1 G Eξ l G + E Eξ +1 G Eξ l ξ l+1 G L Eβ β l = l=0 L E Eξ 0 G Eξ l + ξ l+1 G l=0 L E Eξ L+1 G Eξ l + ξ l+1 G l=0 By A.8, he lh summad he las sum above s smaller absolue value ha K/L v always, ad ha K/L v l v whe l>2.scev>1, by leg L we oba ha E β 2 = E Eξ 0 G Eξ l + ξ l+1 G = E ξ Eξ 0 ξ l l=0 l=1 he las equaly followg from he fac ha w + 1, hece ξ 0 s H -measurable. The rgh sde above s A.12 he oe-dmesoal case, ad hus he las par of C.9 holds. 3 I hs sep, we se a jm = Eβ j βm G 1 ad prove ha ν B := =1 j E ζ ζm H 1 V j V m ucp T 1 T 1 a jm = 0 C.10 Leg η be he h summad above, we see ha η = η l 0 l,where η l = VTV j m V j V m j Tl T 1 T 1 E β βm G l 1 As see before, β = 0f< w ad E β 2 s smaller ha K/ 2v f > ad ha K always. Moreover, A.5 ad A.6 mply E V Tu V Tv 2 K v u ρ, whereas V s bouded; hece by A.11 ad he Cauchy Schwarz equaly, we oba for l: 0 f < w E η l K 1 ρ l f w <l l v K 1 ρ l + 1 f > v l v f w l K ρ/2 ad smlar esmaes hold for l. Sce oe ca always assume v 1 3/2, whch case 1 1 ρ / v K ρv 1,wegeE η K1+ρv 1. Therefore, for ay

17 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 17 η C > 0, B >η [C+1/ Ω c K ]+w +1 + E η Ω c K + η =1 η ρv 1 Sce Ω c 0, C.10 follows. 4 By he prevous sep, order o ge he frs par of C.8, we are lef o show ν =1 V j V m T 1 T 1 a jm a jl 0 V j s V m s da s C.11 The lef sde above s he egral of he càglàd fuco s V jv m s s wh respec o he radom measure F jm ν ds = =1 a jmδ T 1ds, whereδ x sads for he dela measure a x, so s eough o show ha F jm coverges probably o he measure a jm 1 [0] s da s, for he wea opology o he se of sged fe measures o R +.To hs am, s eough o prove he followg covergece of he cumulave dsrbuo fucos: ν s s =1 a jm a jm A s C.12 hs s obvous whe m = j, whch case F jm s a posve measure; whe m j, he absolue value of F jm s domaed by 1 jj F 2 + F mm,soagac.12 s eough. We recall ha β = β w+1 θ w 1 whe >w,mplyga = a w+1 θ w+1, so he ergodc heorem ad C.9ad a jm K f w mply ha 1 L L =1 a jm a jm 1 -a.s., as L Sce ν s A s, we readly deduce C.12. Q.E.D. ROOF OF THEOREM A.4: The proof heavly reles o Jacod ad Shryaev 2003, abbrevaed as [JS] below. 1 Le he par G H be as he saeme of he heorem. I ca be realzed as he value a me of a process G H defed o a exeso Ω F of Ω F, ad whch, codoally o F, s a couous ceered Gaussa margale, wh he covarace srucure gve by A.15 for all. The F -sable covergece G H G H amous o havg, for ay bouded F -measurable varable Y, ay couous bouded fuco f o R d,adayu R d : E Yf H e u G Ẽ YfH e u G u v s he scalar produc o R d. I vew of C.4 ad Lemma C.1, hssmpledby E Yf H e u M ν Ẽ YfH e u G C.13 2 Equao A.15 ad Lemma C.2 ad he fac ha he ζ s are margale cremes relave o he flrao H mply, wh he help of Theorems VIII.3.22 ad

18 18 J. JACOD, Y. LI, AND X. ZHENG VIII.5.14 of [JS], ha M ν coverge F 0 -sably law o G. C.14 O he oher had, Lemma A.2 ad our assumpo o u mply H coverge F -sably law o H. C.15 Equaos C.14 adc.15 are o eough for us; we eed a jo covergece. To hs ed, we eed o revs he covergece C.14. As for all specal semmargales, we ca wre e u M ν uquely as a produc gu Zu of a predcable càdlàg process wh fe varao gu called G u [JS] ad a margale Zu, boh relave o he flrao H ν 0, boh sarg a 1 a me 0, ad boh complex-valued. Aalogously, e u G = gu Zu,whgu predcable wh fe varao ad Zu a margale, 1 relave o he smalles flrao F o whch G s adaped ad such ha F 0 1 F 0. We do o eed o recall he explc form of gu ad gu alhough gu aes he smple form gu = exp 1 d 2 jm=1 uj u m a jm V jv m 0 s s α s ds, bu oly ha, accordg o he proof of Theorem VIII.2.4 of [JS], we do have, for all ad by Lemma C.2: gu gu C.16 Observe ha 2ε gu 1 for some cosa ε 0 1 depedg o ad u. The 2ε 2 H ν-soppg mes R = f{s : gu εor s gu 1 } sasfy R s ε 0by C.16 ad are predcable, so here are H ν-soppg mes S <R such ha S 0 as well, whereas ε gu 1 ad hus ε s ε Zu 1 for all s S s ε. The, parly reproducg he proof of Theorem VIII.5.16 of [JS], for ay uformly bouded sequece Y of F 0 -measurable varables, oe ca wre Ẽ Y e u M ν e u G KS + ẼY gu S Zu S gu Zu KS + Ẽ Y gu S gu Zu S + Ẽ Y gu Zu S Zu Sce EZu S F 0 = ẼZu F 0 = 1adY gu s F 0 -measurable, he las erm above vashes. We have Y Zu S K/ε, hece Ẽ Y e u M ν e u G KS + K ε E gu S gu Usg aga C.16, plus S 0, we deduce ẼY e u M ν e u G 0 C.17 3 Now, we are a poso o prove C.13. Frs, he characerzao of G gves us Ẽe u G F = gu. Nex, apply C.17whY = YfH o ge E Yf H e u Mν E Yf H e u G 0 Fally, Y = Ygu s bouded F -measurable, hece C.15yelds Ẽ Yf H e u G = E Yf H gu Ẽ YfH gu = Ẽ YfH e u G

19 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 19 where he las equaly comes from he depedece of H ad G, codoally o F. We he deduce C.13, ad he heorem s proved. Q.E.D. ROOF OF LEMMA A.5: We use a ufed approach, by seg case 1: F = G 1 ξ case 2: F = G 2 ξ case 3: F = G ξ = ξ 1 h = w 1 = ξ 2 +w 1 + = ξ 1 ξ 2 +w+ 1 h = w 2 h = + w 1 + w2 I all cases, E ξ 2 K, adξ s ceered cases 1 ad 2, whereas case 3, A.9 yelds Eξ K/v. The aoher applcao of A.9 also yelds l>h E ξ ξ { F 0 Kl h v +l cases 1 ad 2 K 2v + Kl h v case 3 Sce V s bouded ad N s F 0 -measurable ad depede of all ξ s, we deduce N E F 2 F 0 N j=0 E ξ ξ j Kh N + N Kh N m=1 + K 2v m v N 2 + N N m v m=1 cases 1,2 case 3 The resul s ow obvous. Q.E.D. C.3. Furher Auxlary Resuls ROOF OF LEMMA A.6: We se δ = Eδ H ad δ = δ δ ad, for j = 0w 1, so B l = B l B l = l =0 [l j/w δ ] B j l = δ j+ 1w + w 1 j=0 B j l. We clearly have Esup l B l + 1a. The summads δ j+ 1w are margale cremes, relave o he flrao H j+w 0, hece E sup l B j 2 l 4 [ j/w ] =1 E δ =1 [ j/w 2 ] 4 E δ =1 2 Ka w by Doob s equaly, so Esup l B j K l a /w. The lef sde of A.17 beg smaller ha Esup l B l + w 1 j=0 Esup l B j l, we deduce he resul. Q.E.D.

20 20 J. JACOD, Y. LI, AND X. ZHENG ROOF OF LEMMA A.7: Wh V = V α,wehave JV = D + B N u D = 1 TN u V s V TN s α s ds B = δ δ = 1 T+1 V s V T ds =0 T Equao A.5 ad he boudedess of V ad α mply D Ku ρ 1/2, whch goes o 0. I vew of A.5, B ucp N = 0 follows from he propery Esup u j [D/] B j 0foray cosa D. Ths ur follows from Lemma A.6 appled o δ, he assumpos of hs lemma beg fulflled wh H = F ad w T = 1 ad, by vrue of A.7, a = K 3/2 ad a = K2. Ths complees he proof. Q.E.D. ROOF OF LEMMA A.8: Sce rm K1 m v, we have for all j: E χ 2 1 j = r l 2 1 rm K Kf 2 m v v 0 l< 1 m=0 m=1 C.18 wh f v as he saeme of he lemma. Ths yelds he frs clam. Sce all momes of χ j are bouded j ad, by Hölder s equaly we also ge, for p>2adε>0: E χ p j Kpε f v 1 ε C.19 Nex, we deoe by Q he se of all o-empy subses Q of {1q}, he compleme of Q beg deoed as Q c, ad s cardal s Q. Wehave where U := q q χjm χ μ+2m 1 m=1 m=1 χ jm = Q Q V Q = χ jl χ μ+2l 1 l Q c l Q 1 Q V Q We fx Q Q ad le r 0 = max Q ad Q = Q \{r 0 }.Wehave V Q = V Qχ μ+2r 0 1 where V Q = χ jl χ μ+2l 1 l Q c l Q The varable V Q s G μ+2r 0 2 -measurable, wh all momes bouded, whereas χ μ+2r 0 1 s G μ+2r 0 1 -measurable, ceered, ad sasfes C.19. The C.18, C.19, A.9, ad he propery ha vf v 1/2 ε Kf v for ε 0 1/2] yeld E V Q K v fv E V Q 2 Kpε f v Sce 3.1 ad 3.26 yeld r ; j rj = EU, by summg up he esmaes above over all Q Q, we readly ge A.19. Q.E.D.

21 STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE 21 ROOF OF LEMMA A.9: 1 We wll focus o he secod par of A.21, he frs par beg aalogous, ad fac slghly smpler. We ae j j as above, ad se for ay eger : U = Ŷj Ŷ j +μ+2q+1 γt χj q χ j +μ+2q+1 =1 Suppose for a mome ha we have E sup j U K rqq j ρ 1/r + j 1/2 1+r/2r C.20 ρ/2r C.21 Observe ha, wh ζ = χj χj +μ+2q+1 ad u = u μ + 2q + 1,wehave U 4 j j U 4 j j KR + sup U N R = N =N u ζ Sce he varables ζ have a fe mome o depedg o ad are depede of F 0, by codog frs o hs σ-feld we see ha ER K q. O he oher had, o he se Ω we have N 1 + C/, so he lef sde of A.21 s smaller ha K plus he boud C.21 evaluaed a j =[1 + C/ ].Sceboh ad 1+r/2r ρ r/2r are smaller ha K 1/r ρ/r 1 by A.4, we deduce he secod par of A The proof of C.21 goes hrough several seps, he frs oe beg devoed o some esmaes. Se for u l w N: X T+u X + γ +l T+u γ T χ +u ζm = 1 1 γ ul T+l+j γ T χ +l+j f m = 1 j=0 γ T χ+u χ +l f m = 2 Upo usg A.5 ad A.6 wh V = X sce here X = X 0 + X orwhv = γ, he depedece of F 0 ad G, ad he fac ha χ has momes of all orders, we ge for p 2: E ζ1 p ul Kp ρ u + l + E ζ2 p ul Kp C.22 Moreover, A.6 ad he depedece of F 0 ad G yeld E ζ1 F G ul T K ρ u + l χ +u χ +l+m m=0 C.23 3 Sce Y Y = 4 +u +l m=1 ζm ul, ad wh he oao 1 l q u l = j l l l = μ + 2l 1 q<l q u l = μ + 2q j l q l l = μ + 2l + q

22 22 J. JACOD, Y. LI, AND X. ZHENG a smple calculao shows us ha U = =0 ξ,where q 2 q ξ = ζm +2+2 u ζ2 l l l +2+2 u l l l l=1 m=1 So, f Q s he se of all paros Q = Q 1 Q 2 of {1q } such ha Q 1, wehave ξ = ηq where Q Q l=1 ηq = ηq 1 1 ηq 2 2 ad ηq m m = l Qj ζm +2+2 u l l l I remas o prove ha, for each Q Q, he sequece M = =0 ηq sasfes C We sar wh he case where Q 1 s a sgleo, say Q 1 ={l} for some l {1q }, so ηq = ζ1 u ηq l l 2 2.ByC.22, C.23, successve codog, Hölder s equaly, ad he facs ha ηq 2 2 l s G-measurable wh bouded momes of all orders ad u + l l K l q, he assumpos of Lemma A.6 are sasfed by he varables δ = ηq,whw = μ + 2q + 1 ad H = H ad, for ay r>1, a K qq ρ a K rqq ρ 1/r The, sce ρ 0, A.17 mples ha MQ sasfes C.21. I all oher cases of Q Q, here are a leas wo dsc egers l ad l Q 1. The ηq = ζ u ζ1 l l l +2+2 u l l ζ l,whereζ has fe momes of all order by C.22. The C.24, C.22, ad Hölder s equaly yeld E ηq K r ρ 1/r. Thus C.21 holds aga. Ths complees he proof. Q.E.D. ROOF OF LEMMA A.10: 1 We beg by provg ha E F T αt Kφ E 2 F T K C.24 Se λ = 1 α T λ 2 λ 3 = T+ 1 + T+ 1 1 α T = T+ T 1 α T λ 4 = λ 1 λ 3 2 λ 5 = T+ T φ Frs, 2.3 wh τ 1 = yelds E λ 1 F T K 1/2+κ E λ 1 4 F T K E λ 1 2 α FT T Kκ α 2 T C.25

23 Nex, we have λ 1 = STATISTICAL ROERTIES OF MICROSTRUCTURE NOISE m=1 j=0 ζ mj ζ mj = E λ 1 +j F T+j f m = 1 λ 1 +j E λ 1 +j F T+j f m = 2 1/α T+j 1/α T f m = 1 The process 1/α sasfes K, hece A.6, C.25, ad 3.5 yeld for r = 1 2: E λ r F T K 1/2+κ + E λ r 4 F T K E λ r 2 F T K 1+2κ K C.26 Upo expadg he square he defo of λ 4, hs for r = 2adC.25 adalso A.6 for he process α/α 2 yeld E λ 4 α + F T T K κ + 1 E λ FT K C.27 α 2 T Observe ha λ 5 = α T /1 + α T λ 2 b,whereb = φ α T /.LeD be a cosa such ha α D, adp = 2 4. Expadg x 1 aroud 0 for x 1,ad 1+x p 2 usg C.26whr = 2, we ge sce φ / 0 by 3.14, we ca assume b < 1 : 2 E λ 5 p FT p α T αp T λ 2 b p > 1/2D F p+1 T + Kα TE λ 4 p FT 2D2 p φ p E λ 4 2 FT + KE λ 4 p K 2 f p = 2 φ 2 F T K K f p = 4 Sce = λ λ 5 2 ad 3 4 /φ4 0, he above esmae ad C.27yeldC Now we ur o our clam. The assumpos of Lemma A.6, for he sequece δ = V T α, are sasfed wh T H = F ad w T = ad a = Kφ ad = K, byc.24. Hece A.22 readly follows from A.17. Q.E.D. a φ 4 ROOF OF LEMMA A.11: 1 We prove 3.19 oly, he oher wo clams beg proved aalogously a slghly smpler way. The secod par of 3.19 s a obvous cosequece of he frs par ad of 2.4, so we focus o he frs par, ad we le j j J + wh μ = μj ad μ = μj ad q = q + q ad μ = μ + μ. By vrue of A.2, A.4, ad Lemma A.8, s eough o show ha B := U 3 j j r ; jr ; j N 1 μ 2q +3 =0 α T γ q T 0

24 24 J. JACOD, Y. LI, AND X. ZHENG Wh he same oao χj as A.20, we se U = δ =0 U = δ =0 U = =0 δ where δ = Ŷj +2+2 Ŷ j +2+μ+2q+3 γt χj q +2+2 χ j +2+μ+2q+3 δ = γq T χj +2+2 χ j +2+μ+2q+3 r ; jr ; j δ = r ; jr ; j γ q T α T We wll prove ha, for ay r>1, E U K r j ρ 1/r + j 1/2 1+r/2r E E sup j sup j sup j U K j v + j U K jφ + j ρ/2r C.28 Suppose deed ha C.28 holds.wehave B sup U [1+C/] + U resrco o he se Ω, whose probably goes o 1. Subsug j wh [1 + C/ ] C.28, ad sce φ 0 ad uder our assumpos, we readly deduce B 0. 2 We are hus lef o provg C.28. The hrd esmae s A.22 wh V = γ q.the frs oe s proved exacly as C.21, oce oced ha U here has he same srucure as C.20, wh, each summad, a shf by of he dces ad he addoal mulplcave erm whch sasfes C.24. Fally, wh he oao χ j = χj r ; j, wehave δ = γq T χj +2+2 χ j +2+μ+2q+2 + r ; j χ j +2+2 Sce he varables χ j are ceered by defo of r ; j, we deduce from A.8 ad he facs ha χj +2+2 s G +2+μ+2q+1 -measurable ad χ j +2+μ+2q+2 s G +2+μ+2q+2 -measurable ha E Eδ H K/v.Theδ sasfes he assumpos of Lemma A.6, wh H = F ad w T = 1 + μ + 2q + 3 ad a = K ad a = K/ v, ad he secod par of C.28 follows from A.17. Q.E.D. REFERENCES JACOD, J.,AND. ROTTER 2012: Dscrezao of rocesses. Berl: Sprger-Verlag. [12] JACOD, J., AND A. SHIRYAEV 2003: Lm Theorems for Sochasc rocesses Secod Ed.. Berl: Sprger- Verlag. [11,17] Co-edor Lars eer Hase hadled hs mauscrp. Mauscrp receved 29 December, 2014; fal verso acceped 13 February, 2017; avalable ole 2 March, 2017.