Auxiliary Results for Activity Signature Functions

Transcription

1 Auxiliary esults for Activity Sigature Fuctios Viktor Todorov ad George Tauche Jue 9, 29 Abstract This documet cotais additioal results pertaiig to the paper Activity Sigature Fuctios for High-Frequecy Data Aalysis, Todorov ad Tauche 29). The otatio is the same as i that paper. Departmet of Fiace, Kellogg School of Maagemet, Northwester Uiversity, Evasto, IL 628; v-todorov@kellogg.orthwester.edu Departmet of Ecoomics, Duke Uiversity, Durham, NC 2778; george.tauche@duke.edu. 1

2 Cotets 1 The Effects of Measuremet Error 3 2 Scaled log-power Variatio 3 3 Mote Carlo Mote Carlo Scearios Precisio of the Poit Estimator of Activity Size ad Power of the Test for Cotiuous Martigale Proofs of Theoretical esults Prelimiary esults Limit theorems for realized power variatio i pure-jump case Proof of Theorem 1 i the paper Proof of Theorem 2 i the paper

3 1 The Effects of Measuremet Error We aalyze the effect of measuremet error o our activity sigature fuctio. We will suppose that istead of the true process we observe the process with some error. For example, it is well kow that at very high-frequecy, fiacial data ca be cotamiated by microstructure oise; the literature o microstructure oise is vast, see Bardorff-Nielse ad Shephard 27) for a timely ad extesive review. We will focus oly o the i.i.d. case. The followig theorem gives the behavior of the realized power variatio i this situatio. Theorem 1 Let ɛ i ) i be a i.i.d. sequece of radom variables with fiite momets ad set Υ t = Υ t + ɛ t where Υ is oe of the processes X, Y or Z defied i the paper. The, if assumptio A1 holds, we have V p, Υ, ) t where K p is some costat depedig o p. u.c.p. K p t, As we ca see from the theorem, the asymptotic behavior of the realized power variatio is completely determied by the oise i the price process. Therefore, it is easy to see that for ay p ad Υ = Y, X or Z, β,t ] Υ, p) will diverge to ifiity as we start samplig more frequetly. Ituitively, the measuremet error becomes the most active part of the observed process, sice its magitude does ot decrease as we start samplig more frequetly. Proof of Theorem 1: First, usig a localizatio argumet we ca ad will) assume the stroger assumptio A1. Secod, by a stadard LLN we have [t/ ] i=1 ɛ i ɛ i 1) p P te ɛ u ɛ s p, for u s. Thus, to prove Theorem 1 it suffices to show [t/ V p, Υ, ] ) t ɛ i ɛ i 1) p This follows from i=1 u.c.p.. 1. the elemetary iequality x + y p x p K p x p 1 + y p 1) y, for arbitrary x ad y ad K p beig some costat that depeds o p. 2. the followig well-kow result see Jacod 28)) E i 1 i Z p K p/ applicatio of Hölder iequality. 2 Scaled log-power Variatio A alterative to usig differeces across time scales to determie the activity level is, from Woerer 26), to compute a direct scaled log-power variatio measure β,t ] X, p) = p l ) / l V p, X, k ) T ). 3

4 Scaled Log Power Variatio, Tempered Stable, β= β Figure 1: Scaled log power variatio for the symmetric tempered stable process with β = 1. ad other parameters give by Case TS1. of Table 1; the figures show B Υ, p, α) as a fuctio of p for the quatiles α =.25, α =.5, ad α =.75, readig bottom up. This measure asymptotically behaves as β,t ] X, p) i Sectio 4 of the text; however, Theorem 1 suggests a bias which is apparet i Figure 1. The figure shows three quatile plots of the fuctio B Υ, p, α) := β [αn] p), which is defied aalogously to the quatile fuctio i Sectio 4 of the paper. For the figure, Υ is simulated from case TS1.. Evidetally, B Υ, p, α) is off ceter ad the iterquatile rage is quite large. The bias evidet i Figure 1 is caceled out usig the the two-scale approach. 4

5 3 Mote Carlo 3.1 Mote Carlo Scearios As oted i the paper, some readers, particularly i ecoometrics, prefer poit estimates ad formal test statistics to graphically-based iterval estimates. Therefore, this appedix examies i detail via Mote Carlo the properties of the poit estimator of the activity idex preseted i Sectio 4.3 of the paper. The estimator is a equi-weighted average of the activity sigature fuctio over a data-depedet iterval. Clearly there are may other weightig schemes that could be used to develop estimators of activity, ad explorig all of their properties is far beyod the scope of the paper ad this appedix. ather, our aim is to check if this particular easilycomputed estimator performs reasoably well over a wide variety of estimatio cotexts. We also examie the properties of the formal test of a jump diffusio agaist the alterative of a pure jump model developed i Subsectio 3.3 of the paper. The various scearios goverig the dyamics of the process Z are show i Table 1. The table shows the parameter settigs ad the implied populatio value of the activity idex, deoted β, which is always determied by the domiat compoet of the Z process. Cases A I ad N are Lévy processes. I A C the Lévy process is pure jump while i D it is a cotiuous/browia martigale. I cases E G rare jumps, i.e. compoud Poisso jumps, are added to a cotiuous martigale. Case H represets a particular challege where the activity idex is zero, as does I where the estimator faces the superpositio of two Lévy processes with idices 1. ad 1.5, respectively, with domiat compoet 1.5. Cases J M are Lévy-drive OU or CAMA1, ), with strog persistece ad very little mea reversio, where the backgroud drivig Lévy processes are Cases A D, respectively. Case N is a cotiuous/browia martigale plus a ifiitely active pure jump process. The values of the tuig parameters are set to be realistic for high-frequecy fiacial data ad were determied by kowledge gleamed from earlier studies, so the fidigs below eed to be iterpreted ad qualified accordigly. A very ituitive way to set up the samplig scheme is to thik of the basic time uit, or time period, as oe 24-hour day with observatios take at samplig itervals of 1-mi, 5-mi, 1-mi, ad 3-mi itervals. The first two are typical of may fiacial applicatios while the latter two are relatively coarse. These values correspod to M = 144, 25, 144, 72 observatios per time period, respectively, or samplig itervals of width =.7,.35,.69,.289, which rages over three orders of magitude. Each Mote Carlo pseudo data set is comprised of multiple days worth of data, which we take as N = 25, 15, 3, or 1-year, 5-years, ad 1-years worth of data. Thus M determies the samplig frequecy ad N the spa of the data set, ad the values here are typical of high frequecy fiacial data sets. Of course for a give sceario i Table 1 the outcome oly depeds upo the width of the samplig iterval ad the otios of days ad years are simple labels to guide ituitio. 5

6 Table 1: Mote Carlo Scearios: Dyamics for Z = Y + X Case β ρ σ 2 1 σ 2 2 Jump Specificatio X) A tempered stable with A = 1, β =.5 ad λ =.5 B tempered stable with A = 1, β = 1. ad λ =.5 C tempered stable with A = 1, β = 1.5 ad λ =.5 D oe E rare-jump with λ J =.4, τ = F rare-jump with λ J =.3333, τ =.7746 G rare-jump with λ J = 2., τ =.3162 H variace gamma: tempered stable with A = 1, β =. ad λ =.5 I superpositio of two idepedet tempered stables: oe with A = 1, β = 1.5 ad λ =.5 ad the other with A = 1, β = 1. ad λ =.5 J tempered stable with A = 1, β =.5 ad λ =.5 K tempered stable with A = 1, β = 1. ad λ =.5 L tempered stable with A = 1, β = 1.5 ad λ =.5 M oe N tempered stable with A = 1, β = 1.5 ad λ =.5 Note: With Z = Y + X the cotiuous compoet Y is otrivial if σ1 2 compoet X is otrivial if σ2 2 >. >, ad the jump 6

7 3.2 Precisio of the Poit Estimator of Activity For each sceerio ad M, N) pair we geerated 1 Mote Carlo replicates, computed the poit estimate of β o a period-by-period basis, i.e., day-by-day, ad took the media of the N values as the overall estimate for that replicate. This etails some loss i efficiecy because it does ot impose equality of β across time time periods, but it mimics actual practice where thigs are doe day-by-day to provide some robustess. Across the 1 Mote Carlo replicatios we compute the usual robust summary statistics give by the media, iter-quartile rage, ad media-absolute-deviatio about the the populatio value β. I order to assess the practical accuracy of the estimator, we also computed error rates as the proportio of the 1 replicates that the absolute deviatio of the overall estimator from the populatio value β is o more tha the pre-specified value α =.15,.1,.5. For the activity parameter β [, 2], estimatig it to withi.1 of the true value seems sufficietly accurate for may applicatios, while to withi.15 is a little loose ad.5 very precise. Tables 2 8 displays the summary measures from this Mote Carlo experimet. Oe immediately sees from the iter-quartile rages ad media absolute deviatios that the estimator is very tightly cocetrated aroud a value that is geerally, but ot always close to the true populatio variable. I other words, the samplig variace is early always very small ad the bias geerally small, or modest, but the bias ca be quite large i certai extreme circumstaces. The samplig distributio cocetrates aroud a value that is usually close to the correct value β, but i some circumstaces it cocetrates aroud a value somewhat distat from the correct value. The coclusios from Tables 2 8 are readily summarized. The key cotrol parameter is the samplig frequecy, M, or equivaletly the samplig iterval = 1/M. For samplig correspodig to 1-mi ad 5-mi, the poit estimator is geerally quite accurate across essetially all scearios ad data spas N. O the other had, for samplig at 1-mi ad coarser the estimator ca sometimes be quite off the mark as i Cases E, F, G, ad H, for example. The fact that reducig sharply decreases the bias while icreasig the spa N oly moderately reduces variace is ot surprisig because the asymptotic theory is always of the fill-i type where. A reasoable level of accuracy requires a large, dese, data set of the type that is commoly used i fiacial ecoometrics ad presumably other applicatios as well. Aother poit of ote is that for values of the idex β at or above.5 the estimator is geerally quite accurate at the higher samplig frequecies, while smaller values could require other approaches. Note that i case H β = the estimator cocetrates at the value τ =.1, which is the lower boud of the of iterval used to average the activity sigature fuctio. Keep i mid, however, that if β =, or is close to zero, the the plot of the activity sigature fuctio will reveal just that. 7

8 Table 2: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 A A A A A A A A A A A A B B B B B B B B B B B B Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 8

9 Table 3: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 C C C C C C C C C C C C D D D D D D D D D D D D Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where β β > α. 9

10 Table 4: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 E E E E E E E E E E E E F F F F F F F F F F F F Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 1

11 Table 5: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 G G G G G G G G G G G G H H H H H H H H H H H H Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 11

12 Table 6: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 I I I I I I I I I I I I J J J J J J J J J J J J Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 12

13 Table 7: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 K K K K K K K K K K K K L L L L L L L L L L L L Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 13

14 Table 8: Summary Statistics for 1 eplicates of β Defied i Subsectio 4.3 Case N M β MED IQ MAD Error rates: α-away from the truth α =.15 α =.1 α =.5 M M M M M M M M M M M M N N N N N N N N N N N N Note: N is the umber of uits of time i a simulatio; M is the umber of high-frequecy observatios per uit of time. β is the populatio value. MED is the media, IQ the iter-quartile rage, MAD the media absolute deviatio, each computed over the 1 replicatios. Error rates are the proportio of replicates where ˆβ β > α. 14

15 3.3 Size ad Power of the Test for Cotiuous Martigale Tables 9 11 summarize the size ad power properties of the test for a cotiuous martigale proposed i Sectio 3.3 of the paper. The scearios are as i Table 1 above ad the samplig rate M as per Tables 2 8. For each sceario M pair, the test is coducted o each of 1 replicated periods i.e., days mimickig how such tests are coducted i actual practice. The coclusios from these three tables accord well with those of the precedig Sectio 3.2. For the highest samplig frequecy 1-mi) the size ad power properties are excellet, ad with 5-mi samplig the test is oly slightly udersized but with still quite good power properties. O the other had, at the courser samplig itervals there ca be large size distortios ad the test thereby ureliable. The size ad power appear appropriate, at least for samplig frequecies commoly used i fiace. These fidigs very likely carry over to other disciplies where large dese data sets are also available. 15

16 Table 9: Size ad Power of Test for Presece of Cotiuous Martigale Samplig Frequecy Percetage of ejectio of Null Hypothesis α =.1 α =.5 α =.1 Case A H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case B H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case C H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case D H true) M = mi) M = mi) M = mi) M = 48 3 mi) Note: The test is oe-sided ad is based o asymptotic distributio uder the ull of log β t Υ, p)) for p =.9, give i Theorem 3 i the paper. The umber of Mote Carlo replicatios is 1, uits of time ad α deotes the asymptotic) size of the test. 16

17 Table 1: Size ad Power of Test for Presece of Cotiuous Martigale Samplig Frequecy Percetage of ejectio of Null Hypothesis α =.1 α =.5 α =.1 Case E H true) M = mi) M = mi) M = mi) M = 48 3 mi) Case F H true) M = mi) M = mi) M = mi) M = 48 3 mi) Case G H true) M = mi) M = mi) M = mi) M = 48 3 mi) Case H H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case I H false) M = mi) M = mi) M = mi) M = 48 3 mi) Note: The test is oe-sided ad is based o asymptotic distributio uder the ull of log β t Υ, p)) for p =.9, give i Theorem 3 i the paper. The umber of Mote Carlo replicatios is 1, uits of time ad α deotes the asymptotic) size of the test. 17

18 Table 11: Size ad Power of Test for Presece of Cotiuous Martigale Samplig Frequecy Percetage of ejectio of Null Hypothesis α =.1 α =.5 α =.1 Case J H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case K H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case L H false) M = mi) M = mi) M = mi) M = 48 3 mi) Case M H true) M = mi) M = mi) M = mi) M = 48 3 mi) Case N H true) M = mi) M = mi) M = mi) M = 48 3 mi) Note: The test is oe-sided ad is based o asymptotic distributio uder the ull of log β t Υ, p)) for p =.9, give i Theorem 3 i the paper. The umber of Mote Carlo replicatios is 1, uits of time ad α deotes the asymptotic) size of the test. 18

19 4 Proofs of Theoretical esults 4.1 Prelimiary esults First, it is easy to see that the law of the pure-jump model X t, give with equatio 2.2), ca be geerated with the followig process defied o some differet probability space we will still call it X t ) X t = t t + b 2s ds + t + σ 2s κx)1 {y<as } µds, dx, dy) + σ 2s κ x)1 {y<as }µds, dx, dy), 4.1) where ow µ is a Poisso measure but o + + ad with compesator ds νx)dx dy; µds, dx, dy) is the compesated versio of µ. The process X t i 4.1) is writte as a stochastic) itegral with respect to a time-homogeous Poisso radom measure, but we ote that µ is defied o a three dimesioal space. Ituitively, the first dimesio is the time, the secod oe is the size of the jumps but ote that the jumps are multiplied by σ 2t ). The role of the third dimesio is to geerate thiig of the jumps accordig to the process a t ad thus create time-varyig itesity of the jumps i the process X t. I all of the proofs i this sectio we will assume the represetatio i 4.1) for X t, ad we will work with the correspodig probability space ad filtratio which support it. This is coveiet as whe chages we do ot eed to make a chage of the probability space i our proofs. We start with showig some prelimiary results. Deote with L covergece i law o the space D) of oe-dimesioal càdlàg fuctios: +, which is equipped with the Skorokhod topology. We have the followig two Lemmas. Lemma 1 Let Ψ t deote a Lévy process with characteristic triplet b,, F ) with respect to some trucatio fuctio κx), where F is a Lévy measure with F dx) = νx)1 { x ɛ} dx ad νx) is the desity defied i 2.4), while ɛ is a arbitrary positive umber. Further, let b = κx)1 { x ɛ}νx)dx if β < 1 ad b = if β 1. Fially, assume that ν is symmetric whe β = 1. Deote with L t aother Lévy process with characteristic triplet b 1,, F 1 ) with respect to the same trucatio fuctio). b 1 = κ x)ν 1 x)dx if β 1 ad b 1 = κx)ν 1x)dx for β < 1. F 1 is a Lévy measure with F 1 dx) = ν 1 x)dx where ν 1 x) is defied i 2.5). Fially, assume κx) is F 1 -a.s. cotiuous ad i additio that it is symmetric whe β = 1. The as we have: a) b) 1/β Ψ t L L t, 4.2) p/β E Ψ t p ) E L t p ), locally uiformly i t for some p < β. 4.3) Proof: Part a). Sice 1/β Ψ t is a Lévy process to prove the covergece of the sequece we eed to show the covergece of its characteristics, see e.g. Jacod ad Shiryaev 23), Corollary III.3.6. The case β < 1 is straightforward, we show the case β 1. We eed to establish the followig for κ 1/β ) x) 1/β κx) νx)1 { x ɛ} dx κ x)ν 1 x)dx, 4.4) κ 2 1/β x)νx)1 { x ɛ} dx κ 2 x)ν 1 x)dx, 4.5) 19

20 g 1/β x)νx)1 { x ɛ} dx gx)ν 1 x)dx, 4.6) where g is a arbitrary cotiuous ad bouded fuctio o, which is aroud. First, the covergece i 4.5) ad 4.6) trivially follow from a chage of variable i the itegratio ad the fact that the fuctio φ i 2.5) is bouded aroud the origi. Therefore, we are left with showig 4.4). For β = 1 this follows from the symmetry of the trucatio fuctio ad ν. We show 4.4) therefore oly i the case β > 1. First, we have = κ 1/β κ 1/β ) x) 1/β κx) ν 1 x)1 { x ɛ} dx ) x) 1/β x ν 1 x)1 { x ɛ} dx + 1 1/β x κx)) ν 1 x)1 { x ɛ} dx κ x)ν 1 x)dx. 4.7) Thus we will prove 4.4) if we ca show κ 1/β ) x) 1/β κx) ν 2 x)dx. 4.8) It suffices to show 4.4) for κx) = x1{ x 1}. For small eough we have ) 1/β κx) κ 1/β x) ν 2 x)dx = 1 1/β 1 1/β 1 1/β 1 1/β xν 2 x)dx + 1 1/β x β φx)dx + 1 1/β 1/β 1 1/β 1 xν 2 x)dx x β φx)dx. 4.9) For sufficietly small, usig the fact that φx) slowly varies aroud zero, has a fiite limit at ad is itegrable aroud, we have 1 1/β x β φx)dx K + Kφ 1/β ) 1/β β /β, 4.1) for some costat K. The same result obviously holds for 1/β x β φx)dx, ad from here 1 recall we are lookig at the case β > 1) we have the result i 4.8). Thus, combiig 4.7) ad 4.8), we have the result i 4.4) ad therefore 4.2) is established. Part b). First we establish the poitwise covergece for arbitrary t >. Give the result i Part a), we eed oly show that for some p such that p < p < β we have sup E p /β Ψ t p ) <. 4.11) I order to cosider the cases β 1 ad β < 1 together we make a slight abuse of otatio. I what follows we assume that κx) ad tur off the drift b whe β < 1, while whe β 1 we make o chage ad κx) cotiues to be a true trucatio fuctio i.e. it has bouded support ad coicides with the idetity aroud the origi). With this assumptio o κx), deotig with µ the jump measure associated with the process Ψ t ad with µ its compesated versio, we ca write 1/β Ψ t i all cases of β) as 1/β Ψ t = 1/β t κx) µds, dx) + 1/β 2 t κ x)µds, dx). 4.12)

21 The, we ca proceed with the followig decompositio 1/β Ψ t p K 1/β +K +K 1/β 1/β t t t p κx) µds, dx) x >s 1/β p κx) µds, dx) x s 1/β p κ x)µds, dx), 4.13) where K is some costat. Obviously, give the fact that κx) if β < 1, for the first two terms o the right had side of 4.13) we look oly at the case β 1. For the secod itegral o the right-had side of 4.11) for some 1 q 2 such that q > β we have E 1/β K t q/β E p κx) µds, dx) E x s 1/β t κx) q dsνx)dx x s 1/β 1/β )) p /q t κx) µds, dx) x s 1/β q) p /q K, 4.14) where the first iequality follows from the fact that p < q; the secod oe is a applicatio of Burkholder-Davis-Gudy iequality see e.g. Protter 24)); ad the third oe uses the fact that q 2, the properties of the fuctio φx) ad a similar argumet as i 4.1). For the first itegral o the right-had side of 4.13) if β > 1 we ca pick some q such that max{p, 1} q < β ad make exactly the same steps as i 4.14) to get E 1/β t p κx) µds, dx) K. 4.15) x >s 1/β Whe β = 1, ν is symmetric so the itegratio with respect to µ is the same as the itegratio with respect to the compesated measure µ aroud the origi. Thus, whe β = 1, for the first itegral o the right-had side of 4.13) we ca write E 1/β = p /β p /β E t p κx) µds, dx) x >s 1/β t t p E κx)µds, dx) κx)dsνdx) x >s 1/β x >s 1/β t ) κx) p νds, dx) + K K, 4.16) x >s 1/β where for the first iequality we made use of the fact that p 1. Fially, for the last itegral o the right-had side of 4.13), sice the jump measure has a bouded support i.e. there are o jumps higher tha ɛ i absolute value) the expectatio exists ad for the case β 1 we trivially have t p E κ x)µds, dx) K 1 p /β. 4.17) 1/β 21

22 Whe β < 1 recall κx) = i this case) we split t κ x)µds, dx) = t x s 1/β xµds, dx) + t x >s 1/β xµds, dx), ad use the fact that for arbitrary < m 1 ad some a ad b we have a + b m a m + b m, to prove E 1/β t p κ x)µds, dx) K. 4.18) Combiig 4.14)-4.18) we get 4.13), ad thus the poitwise covergece. To prove the covergece of p/β E Ψ t p ) for the local uiform topology o t we eed oly show that this sequece is relatively compact. For this we use the Ascoli-Arzela s Theorem see e.g. Jacod ad Shiryaev 23), Theorem VI.1.5). The relative compactess of p/β E Ψ t p ) follows from the fact that the bouds i 4.14)-4.17) are cotiuous i t. This proves part b) of the Lemma. Lemma 2 Cosider the probability space Ω, F, P) o which the process X t is defied via 4.1). Let u s, l s ad h s be some processes o this probability space which are adapted, bouded, ad have càdlàg paths. Assume that A2 holds for νx) ad that νx) is symmetric whe β = 1. Fially, if β < 1 set κx) ad whe β 1 let κx) be a trucatio fuctio, which is symmetric i the case β = 1. The, for arbitrary ɛ > ad some q 1 > β ad p q 2 < β we have E K p/β x ɛ E u s κx)1 {ls <y<h s } µds, dx, dy) + + )) p/q1 sup { u s q 1 h s l s } + K p/β s where the costat K does ot deped o. E u s κ x)1 {ls <y<h s }µds, dx, dy) x ɛ + )) p/q2 sup { u s q 2 h s l s }, s 4.19) Proof: The proof of the Lemma is very similar to the proof of Lemma 1, part b). First we make the decompositio x ɛ = + u s 1 {ls <y<h s }κx) µds, dx, dy) + s 1/β < x ɛ x s 1/β ɛ u s 1 {ls <y<h s }κx) µds, dx, dy) + + u s 1 {ls <y<h s }κx) µds, dx, dy). 4.2) We proceed with boudig the p-th absolute momet of the two itegrals o the right had side of the above equatio for the case β 1. For the last oe we choose some 1 q 2, such that p 22

23 q > β ad apply the Burkholder-Davis-Gudy iequality to get p E u s 1 {ls <y<h s }κx) µds, dx, dy) x s 1/β ɛ + q) p/q E u s 1 {ls <y<h s }κx) µds, dx, dy) x s 1/β ɛ + )) p/q K E u s q κx) q 1 {ls <y<h s }νx)dsdxdy x s 1/β + ) p/q K p/β E sup { u s q h s l s }, 4.21) s where the costat K is chagig from lie to lie. For the secod itegral i 4.2) we proceed similar to 4.15) ad 4.16) to get p E u s 1 {ls <y<h s }κx) µds, dx, dy) s 1/β < x ɛ + )) p/q K p/β E sup { u s q h s l s }, 4.22) s for some q such that max{p, 1} q < β if β > 1 ad q = p if β = 1. Similar trasformatios as i 4.17)-4.18) ad usig the fact that we itegrate over the bouded regio x ɛ give p E x ɛ K p/β E + u s 1 {ls <y<h s }κ x)µds, dx, dy) sup { u s p h s l s } s ) + K p/β E )) p/q sup { u s q h s l s }, s 4.23) where β < q < 1. Combiig iequalities 4.21)-4.23) we prove the Lemma. 4.2 Limit theorems for realized power variatio i pure-jump case Usig the prelimiary results we ow prove the followig theorem. Theorem 2 For the process X t defied i 2.2) assume that A1b), A2 ad A3 hold. Deote with L s a pure-jump Lévy process defied o some probability space) which has Lévy measure ds ν 1 x)dx ad drift { b = κ x)ν 1 x)dx if β 1 κx)ν 1x)dx if β < 1, with respect to the trucatio fuctio κ used i the defiitio of X t. If g p s) = E L s p ), the for p < β we have 1 p/β u.c.p. V p, X, ) t t g p a s ) σ 2s p ds. 4.1) Proof: We prove the theorem uder the followig stroger assumptio A1 : Assumptio A1. I additio to assumptio A1 assume that the processes a s, b 2s ad σ 2s are bouded. 23

24 To prove that if the claim i the theorem is true uder the stroger assumptio A1, it holds also uder assumptio A1, we ca use exactly the same localizatio argumet as i the proof of Lemma 4.6 i Jacod 28). Therefore we omit this part of the proof ad proceed with provig Theorem 1 uder the stroger assumptio A1. Also as for the lemmas i the previous subsectio we will use the equivalet i probability) represetatio of X give i 4.1). First, we eed some otatio. ecall that for X t we work with its represetatio i 4.1). For arbitrary ɛ > we set t Xɛ) t = X t σ 2s x1 {y<as }µds, dx, dy), X ɛ) t = X t Xɛ) t. 4.2) x >ɛ + Further, deote with S 1, S 2,..., S m,... the sequece of successive jump times of X ɛ) i.e. the jumps of the process X t which after scalig by σ 2Sm are higher tha ɛ i absolute value). Furthermore, if Ω T ) deotes the set of ω-s for which each iterval [, T ] i 1), i ] cotais at most oe S m ad i additio i Xɛ) Kɛ for some costat K recall the process σ 2t is bouded pathwise), the Ω T ) Ω as. Fially, for each S m, we set m = i Xɛ), where i 1), i ] is the iterval cotaiig S m. With this otatio o the set Ω T ) we have V p, X, ) t V p, Xɛ), ) t = { m + X Sm p m p } <S m t K S m t X Sm p + K, t T, 4.3) where K is some costat ad the sum i 4.3) is of course well defied ad almost surely fiite the iequality is trivial for p 1 ad for p > 1 it from a Taylor expasio). Therefore, sice Ω T ) Ω we have Thus, to fiish the proof we eed oly show 1 p/β {V p, X, ) t V p, Xɛ), ) t } u.c.p.. 1 p/β u.c.p. V p, Xɛ), ) t t g p a s ) σ 2s p ds. 4.4) We make a slight abuse of otatio for κx), as i the proof of Lemma 2 part b), i order to aalyze the differet cases for β together. I particular, we set κx) whe β < 1, which correspods to o compesatio of the small jumps, ad set b 2t see assumptio A3). We start the proof of 4.4) by makig the followig decompositio. 1 p/β V p, Xɛ), ) t t A 1 = 1 p/β g p a s ) σ 2s p ds = A 1 + A 2 + A 3, 4.5) [t/ ] i=1 ξ i, ξ i = { ξ i1 p ξ i2 p }, ξ i1 = i Xɛ), ξ i2 = i i 1) i + i 1) x ɛ x ɛ σ 2i 1) 1 {y<ai 1) }κx) µds, dx, dy) + σ 2i 1) 1 {y<ai 1) }κ x)µds, dx, dy) + 24

25 [t/ ] A 3 = + [t/ ] A 2 = g p a i 1) ) σ 2i 1) p i=1 i=1 t g p a s ) σ 2s p ds, { i σ 2i 1) p p/β 1 {y<ai 1) }κx) µds, dx, dy) i 1) x ɛ + p } 1 {y<ai 1) }κ x)µds, dx, dy) g p a i 1) ). x ɛ + We will show that each of the three terms o the right-had side of 4.5) coverges i probability, uiformly i time, to. We start with A 1. I what follows E i 1 ) is shorthad for E F ) i 1). We wat to derive boud for E i 1 ξi. If p 1 we ca make use of the basic iequality a + b p a p + b p for arbitrary a ad b. If p > 1, we ca use first-order Taylor expasio ad the apply the Hölder s iequality. Both cases lead to where E i 1 ξ i K E i 1 ζi p) 1 p 1 E i 1 ξi1 p) p 1 p + K E i 1 ζi p) 1 p 1 E i 1 ξi2 p) p 1 p, ζ i = i i 1) i + i 1) x ɛ x ɛ σ 2s 1 {y<as } σ 2i 1) 1 {y<ai 1) })κx) µds, dx, dy) + σ 2s 1 {y<as } σ 2i 1) 1 {y<ai 1) })κ x)µds, dx, dy). + Usig Lemma 2 ad the fact that the processes σ 2s ad a s are bouded, we have E i 1 ξ i K p 1 β To boud E i 1 ζ i p we first use the followig decompositio E i 1 ζ i p) 1 p ) σ 2s 1 {y<as } σ 2i 1) 1 {y<ai 1) } = σ 2s 1 {y<as } 1 {y<ai 1) } )+σ 2s σ 2i 1) )1 {y<ai 1) }, ad the apply Lemma 2 to get E i 1 ζ i p K p/β E i 1 sup i 1) s i σ 2s σ 2i 1) p +K p/β E i 1 sup a s a i 1) i 1) s i +K p/β +K p/β E i 1 E i 1 ) p/q sup σ 2s σ 2i 1) q i 1) s i ) p/q sup a s a i 1), i 1) s i where q is some umber higher or at most equal to p. But [t/ ) ] E sup σ 2s σ 2i 1) r i 1) s i = t E i=1 sup σ 2u σ 2[u/ ] r [u/ ] u [u/ ] + 25 ) du,

26 for some arbitrary r. The for arbitrary u > we have lim E sup σ 2u σ 2[u/ ] r [u/ ] u [u/ ] + ) CE σ u r ). Therefore, usig the Lebesque s covergece theorem ad the fact that σ 2s is a bouded càdlàg process uder A1 ), we have [t/ ) ] E sup σ 2s σ 2i 1) r. i 1) s i i=1 Similar result holds for a s. Therefore we have [t/ ] i=1 E i 1 ξ i, u.c.p. ad thus A 1. For A 2 usig iema itegrability we have that A 2 coverges poitwise i ω ad locally uiformly i time to. Fially, for A 3 first ote that i i 1) x ɛ + 1 {y<ai 1) } κx) µds, dx, dy)+ i i 1) x ɛ + 1 {y<ai 1) } κ x)µds, dx, dy), is equal i distributio to L a i 1), where L is the Lévy process defied i Lemma 1 it is defied o some possibly) differet probability space from the origial oe o which X t is defied, but the choice of the costat ɛ is of course the same). The we ca use the covergece result i 4.3), which recall is locally uiform i time. Therefore, sice a s is bouded, we have i E i 1 p/β 1 {y<ai 1) }κx) µds, dx, dy) i 1) x ɛ + i p + 1 {y<ai 1) }κ x)µds, dx, dy) g p a i 1) ) i 1) x ɛ + f ), where f ) is some fuctio such that lim f ) = ad therefore A 3 u.c.p Proof of Theorem 1 i the paper We prove all three parts together. O a set Ω Ω, β,t ] Υ, p) is a cotiuous trasformatio of V p, Υ, k ) T ad V p, Υ, ) T. Therefore we will be doe if we ca show that for a fixed T > ad some p l < p u which for part b) ad c) of the Theorem are such that β is outside [p l, p u ]) we have Π p) := 1 p/β Υ,T V p, Υ, k ) T 1 p/β Υ,T V p, Υ, ) T ) ) k p/β Υ,T 1 C T p) C T p) P ) uiformly o [p l, p u ], 4.1) where C T p) is oe of the limits i 3.3) ad 3.5) depedig o whether Υ Y, Z or Υ X respectively ad the covergece i 4.1) is i C[p l, p u ], 2 +) the space of 2 +-valued cotiuous fuctios o [p l, p u ]) equipped with the uiform metric. The proof of 4.1) cosists of establishig fiite-dimesioal covergece ad tightess of the sequece o the left-had side of 4.1). The 26

27 fiite-dimesioal covergece is trivial. It follows from the poitwise i p) covergece results of Sectio 3.1 i the paper. Therefore we eed oly prove tightess. To this ed, deote the modulus of cotiuity o C[p l, p u ], 2 +) see e.g. Jacod ad Shiryaev 23), VI.1) Set wπ, θ) := sup{ sup u,v [p,p+θ] [T/ ] UΥ, p, q, ) T = i=1 With this otatio for sufficietly small θ we have Π u) Π v) : p l p p + θ p u }. 1/β Υ,T i Υ p 1/β Υ,T i Υ q. 4.2) wπ, θ) UΥ, p l + θ, p l, ) T + UΥ, p u, p u θ, ) T +UΥ, p l + θ, p l, k ) T + UΥ, p u, p u θ, k ) T. 4.3) The we ca use the iequality [T/ ] UΥ, p, q, ) T K p q i=1 1/β Υ,T i Υ p q ɛ + 1/β Υ,T i Υ p q+ɛ), 4.4) for arbitrary small ɛ ad a costat K. Therefore the Ascoli-Arzela s criteria for tightess is satisfied see Jacod ad Shiryaev 23), VI.3.26ii)) ad together with the fiite-dimesioal covergece, this implies 4.1) ad hece the result of the Theorem. 4.4 Proof of Theorem 2 i the paper Part b) of the theorem follows from Bardorff-Nielse ad Shephard 24, 26) ad Bardorff- Nielse et al. 26), hece we show oly part a). We proof the result i 3.13) uder the stroger assumptio A1, the extesio to the weaker assumptio A1 follows by a localisatio argumet. Usig the fact that p 1 we have trivially V p, Z, ) t V p, Y, ) t V p, X, ) t. 4.1) At the same time a straightforward applicatio of Jese s iequality yields Combiig 4.1) ad 4.2) we have p BG T X) ε ) 1 E i 1 i X p K for ay ε >. 4.2) 1/2 p/2 V p, Z, ) t V p, Y, ) t u.c.p., 4.3) uder the coditios of the theorem. Therefore it is eough to prove 3.13) for Υ Y. But this trivially follows from the followig applicatio of Theorem 7.3 i Jacod 27) ) 1 k ) 1 p/2 t V p, Y, k ) t µ p σ 1u p du 1 p/2 t V p, Y, ) t µ p σ 1u p du L s Ξ t, 4.4) where the process Ξ t is defied o a extesio of the origial probability space, is cotiuous, ad coditioally o the σ-field F is cetered Gaussia with variace-covariace matrix give by t σ 1u 2 pdu kµ 2p µ 2 p) k 1 p/2 µk, p) kµ 2 p k 1 p/2 µ p k) kµ 2 p µ 2p µ 2 p The the covergece i 3.13) follows trivially from the result i 4.4) ad a Delta method ote that the covergece i 4.3) is stable). ). 27

28 efereces Bardorff-Nielse, O. ad N. Shephard 24). Power ad Bipower Variatio with Stochastic Volatility ad Jumps. Joural of Fiacial Ecoometrics 2, Bardorff-Nielse, O. ad N. Shephard 26). Ecoometrics of Testig for Jumps i Fiacial Ecoomics usig Bipower Variatio. Joural of Fiacial Ecoometrics 4, 1 3. Bardorff-Nielse, O. ad N. Shephard 27). Variatio, jumps, Market Frictios ad High Frequecy Data i Facial Ecoometrics. I. Bludell, T. Persso, ad W. Newey Eds.), Advaces i Ecoomics ad Ecoometrics. Theory ad Applicatios, Nith World Cogress. Cambridge Uiversity Press. Bardorff-Nielse, O., N. Shephard, ad M. Wikel 26). Limit Theorems for Multipower Variatio i the Presece of Jumps i Fiacial Ecoometrics. Stochastic Processes ad Their Applicatios 116, Jacod, J. 27). Statistics ad High-Frequecy Data. Semstat Course i La Maga. Jacod, J. 28). Asymptotic Properties of Power Variatios ad Associated Fuctioals of Semimartigales. Stochastic Processes ad their Applicatios 118, Jacod, J. ad A. N. Shiryaev 23). Limit Theorems For Stochastic Processes 2d ed.). Berli: Spriger-Verlag. Protter, P. 24). Stochastic Itegratio ad Differetial Equatios 2d ed.). Berli: Spriger- Verlag. Todorov, V. ad G. Tauche 29). Activity Sigature Fuctios for High-Frequecy Data Aalysis. Workig paper, Duke Uiversity ad Northwester Uiversity. Woerer, J. 26). Aalyzig the Fie Structure of Cotiuous-Time Stochastic Processes. Workig paper. 28