Exact confidence intervals for the Hurst parameter of a fractional Brownian motion

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1 Exact cofidece itervals for the Hurst parameter of a fractioal Browia motio by Jea-Christophe Breto 1, Iva Nourdi, ad Giovai Peccati 3 Uiversité de La Rochelle, Uiversité Paris VI ad Uiversité Paris Ouest Abstract: I this short ote, we show how to use cocetratio iequalities i order to build exact cofidece itervals for the Hurst parameter associated with a oe-dimesioal fractioal Browia motio. Key words: Cocetratio Iequalities; Exact cofidece itervals; Fractioal Browia motio; Hurst parameter. 000 Mathematics Subject Classificatio: 60F05; 60G15; 60H07. 1 Itroductio Let B = {B t : t 0} be a fractioal Browia motio with Hurst idex H 0, 1. Recall that this meas that B is a real-valued cotiuous cetered Gaussia process, with covariace give by EB t B s = 1 sh + t H t s H. The reader is referred e.g. to [1] for a comprehesive itroductio to fractioal Browia motio. We suppose that H is ukow ad verifies H H < 1, with H kow throughout the paper, this is the oly assumptio we will make o H. Also, for a fixed 1, we assume that oe observes B at the times belogig to the set {k/; k = 0,..., + 1}. The aim of this ote is to exploit the cocetratio iequality proved i [10], i order to derive a exact i.e., o-asymptotic cofidece iterval for H. Our formulae hige o the class of statistics 1 S = B k+ k=0 B k+1 We recall that, as ad for every H 0, 1, see e.g. [8], ad also k=0, + B k H1 S 4 4 H, a.s.p, 1. Z = H 1 S 4 4 H 1.3 = 1 1 H B k+ B k+1 + B k 4 4 H Law = N0, c H, Uiversité de La Rochelle, Laboratoire Mathématiques, Image et Applicatios, Aveue Michel Crépeau, 1704 La Rochelle Cedex, Frace. jea-christophe.breto@uiv-lr.fr Laboratoire de Probabilités et Modèles Aléatoires, Uiversité Pierre et Marie Curie Paris VI, Boîte courrier 188, 4 place Jussieu, 755 Paris Cedex 05, Frace. iva.ourdi@upmc.fr 3 Equipe Modal X, Uiversité Paris Ouest Naterre la Défese, 00 Aveue de la République, 9000 Naterre, ad LSTA, Uiversité Paris VI, Frace. giovai.peccati@gmail.com 1

2 where N0, c H idicates a cetered ormal radom variable, with fiite variace c H > 0 depedig oly o H the exact expressio of c H is ot importat for our discussio. We stress that the CLT 1.4 holds for every H 0, 1: this result should be cotrasted with the asymptotic behavior of other remarkable statistics associated with the paths of B see e.g. [3] ad [4], whose asymptotic ormality may ideed deped o H. The fact that Z verifies a CLT for every H is crucial i order to determie the asymptotic properties of our cofidece itervals: see Remark 3.3 for further details. The problem of estimatig the self-similarity idices, associated with Gaussia ad o-gaussia stochastic processes, is crucial i applicatios, ragig from time-series, to physics ad mathematical fiace see e.g. [11] for a survey. This issue has geerated a vast literature: see [1] ad [6] for some classic refereces, as well as [5], [7], [8], [15], ad the refereces therei, for more recet discussios. However, the results obtaied i our paper seems to be the first o-asymptotic costructio of a cofidece iterval for the Hurst parameter H. Observe that the kowledge of explicit oasymptotic cofidece itervals may be of great practical value, for istace i order to evaluate the accuracy of a give estimatio of H whe oly a fixed umber of observatios is available. I order to illustrate the ovelty of our approach i.e., replacig CLTs with cocetratio iequalities i the obte! tio of cofidece itervals, we also decided to keep thigs as simple as possible. I particular, we defer to a separate study the discussio of further techical poits, such as e.g. the optimizatio of the costats appearig i our proofs. The rest of this short ote is orgaized as follows. I Sectio we state a cocetratio iequality that is useful for the discussio to follow. I Sectio 3 we state ad prove our mai result. A cocetratio iequality for quadratic forms Cosider a fiite cetered Gaussia family X = {X k : k = 0,..., M}, ad write Rk, l = EX k X l. I what follows, we shall cosider two quadratic forms associated with X ad with some real coefficiet c. The first is obtaied by summig up the squares of the elemets of X, ad by subtractig the correspodig variaces: M Q 1 c, X = c Xk Rk, k;.1 the secod quadratic form is Q c, X = c k=0 M X k X l Rk, l.. Note that Q c, X 0. It is well kow that, if Q 1 c, X is ot a.s. zero, the the law of Q 1 c, X admits a desity with respect to the Lebesgue measure this claim ca be easily proved by observig that Q 1 c, X ca always be represeted as a liear combiatio of idepedet cetered χ radom variables see [14] for a geeral referece o similar results. The followig statemet, whose proof relies o the Malliavi calculus techiques developed i [10], characterizes the tail behavior of Q 1 c, X. Theorem.1. Let the above assumptios prevail, suppose that Q 1 c, X is ot a.s. zero ad fix α 0 ad β > 0. Assume that Q c, X αq 1 c, X + β, a.s.-p. The, for all z > 0, we have z P Q 1 c, X z exp αz + β I particular, P Q 1 c, X z exp z αz+β ad. P Q 1 c, X z exp z. β

3 Proof. I this proof, we freely use the laguage of isoormal Gaussia processes ad Malliavi calculus; the reader is referred to [11, Chapter 1] for ay uexplaied otio or result. Without loss of geerality, we ca assume that the Gaussia radom variables X k have the form X k = Xh k, where XH = {Xh : h H} is a isoormal Gaussia process over H = R M, ad {h k : k = 1,..., M} is a fiite subset of H verifyig E[Xh k Xh l ] = Rk, l = h k, h l H. It follows that Q 1 c, X = I c M k=0 h k h k, where I stads for a double Wieer-Itô stochastic itegral with respect to X, so that the H-valued Malliavi derivative of Q 1 c, X is give by DQ 1 c, X = c M Xh k h k. Now write L 1 for the pseudo-iverse of the Orstei-Uhlebeck geerator associated with XH. Sice Q 1 c, X is a elemet of the secod Wieer chaos of XH, oe has that L 1 Q 1 c, X = 1 Q 1c, X. Oe therefore ifers the relatio k=0 DQ 1 c, X, DL 1 Q 1 c, X H = 1 DQ 1c, X H = Q c, X. The coclusio is ow obtaied by usig the followig geeral result. Theorem.. See [10, Theorem 4.1]. Let XH = {Xh : h H} be a isoormal Gaussia process over some real separable Hilbert space H. Write D resp. L 1 to idicate the Malliavi derivative resp. the pseudo-iverse of the geerator L of the Orstei-Uhlebeck semigroup. Let Z be a cetered elemet of D 1, := domd, ad suppose moreover that the law of Z has a desity with respect to the Lebesgue measure. If, for some α > 0 ad β 0, we have DZ, DL 1 Z H αz + β, a.s.-p, the, for all z > 0, we have z P Z z exp αz + β ad P Z z exp z. β Remark.3. Oe of the advatages of the cocetratio iequality stated i Theorem.1 with respect to other estimates that could be obtaied by usig the geeral iequalities by Borell [] is that they oly ivolve explicit costats. 3 Mai result We go back to the assumptios ad otatio detailed i the Itroductio. I particular, B is a fractioal Browia motio with ukow Hurst parameter H 0, H ], with H < 1 kow. The followig result is the mai fidig of the preset ote. Theorem 3.1. Fix 1, defie S as i 1.1 ad fix a real a such that 0 < a < 4 4 H. For x 0, 1, set g x = x log44x log. The, with probability at least [ ] ϕa = 1 exp a 71 a + 3 +, 3.1 3

4 where [ ] + stads for the positive part fuctio, the ukow quatity g H belogs to the followig cofidece iterval: I = [I l, I r ] = 1 log S a log 1 log + 44 H ; 1 log log S a log 1 + log + 44 H. log Remark We have that lim g H = H. Moreover, it is easily see that the asymptotic relatio 1. implies that, a.s.-p, lim I l = lim I r = H, 3. that is, as, the cofidece iterval I collapses to the oe-poit set {H}.. I order to deduce from Theorem 3.1 a geuie cofidece iterval for H, it is sufficiet to umerically iverse the fuctio g. This is possible, sice oe has that g x 1 for every x 0, 1, thus yieldig that g is a cotiuous ad strictly icreasig bijectio from 0, 1 oto log 3/ log, +. It follows from Theorem 3.1 that, with probability at least ϕa, the parameter H belogs to the iterval J = [J l, J r ] = [ g 1 u ; g 1 Ir ], where u = max{i l ; log 3/ log }. Observe that, sice relatio 3. is verified, oe has that I l > log 3/ log, a.s.-p, for sufficietly large. Moreover, sice g 1 is 1- Lipschitz, we ifer that J r J l I r I l = H log log + a 4 4 H a so that, for every fixed a, the legth of the cofidece iterval J coverges a.s. to zero, as, at the rate O 1/ log. 3. We ow describe how to cocretely build a cofidece iterval by meas of Theorem 3.1. Start by fixig the error probability ε for istace, ε = 0, 05 or 0, 01. Oe has therefore two possible situatios: i If there are o restrictios o that is, if the umber of observatios ca be idefiitely icreased, select first a > 0 i such a way that a exp ε a + 3 esurig that ϕa 1 ε. The, choose large eough i order to have a 4 4 H < 1 ad H log log + a 4 4 H L, a where L is some fixed desired upper boud for the legth of the cofidece iterval. ii If is fixed, the oe has to select a > 0 such that exp a 71 a + 3 ε ad a < 4 4 H. If such a a exists that is, if is large eough, oe obtais a cofidece iterval for H of 1 legth less or equal to log log 44 H +a. 44 H a 4

5 4. The fact that we work i a o-asymptotic framework is reflected by the ecessity of choosig values of a i such a way that the relatio 3.3 is verified. O the other had, if oe uses directly the CLT 1.4 thus replacig Z with a suitable Gaussia radom variable, the oe ca defie a asymptotic cofidece iterval by selectig a value of a such that a coditio of the type expcst a ε is verified. 5. By a careful ispectio of the proof of Theorem 3.1, we see that the existece of H is ot required if we are oly iterested i testig H < H for a give H. Proof of Theorem 3.1. Defie X = {X,k : k = 0,..., 1}, where By settig X,k = B k+ B k+1 + B k. ρ H r = 1 r H + 4 r 1 H 6 r H + 4 r + 1 H r + H, r Z, oe ca prove by stadard computatios that the covariace structure of the Gaussia family X is described by the relatio EX,k X,l = ρ H k l/ H. Now let Z be defied as i 1.3: it easily see that Z = Q 1 H1/, X as defied i.1. We also have, see formula.: 1 Q H1/, X = 4H1 1 H1 1 H1 1 = H1 = X,k X,l ρ H k l H X,k X,l ρ H k l X,k + X,l ρ H k l 1 X,k ρ H k l H1 Z ρ H r H X,k k=0 ρ H r Z ρ H r + 3 = α Z + β 3.4 with α = ρ H r ad β = 6 ρ H r. 3.5 Sice Z 0, Theorem.1 applies, yieldig P Z > a exp a 4 ρ Hr a + 3 Now, let us fid bouds o ρ Hr that are idepedet of H. Fix r 3. Usig 1 + u α = 1 + k=1 αα 1... α k + 1 u k for 0 u < 1, k!

6 we ca write ρ H r = rh = rh + k=1 = r H + l=1 1 H H H 1 + H r r r r HH 1 H k + 1 k + 41 k + 4 k r k k! HH 1 H l l r l. l! Note that the sig of HH 1 H l + 1 is the same as that of H 1 ad HH 1 H l + 1 = H H 1 H l 1 H Hece, we ca write + ρ H r r H 4 l 4 l l=1 = 4r H log r l 1 1r r H log l 1!. 1 4 r sice log1 u = k=1 uk k 43 0 rh4 sice 4 log1 u log1 4u 43 0 u if 0 u r. Cosequetly, takig ito accout of the fact that ρ H is a eve fuctio, we get ρ H r ρ H 0 + ρ H 1 + ρ H + ρ H r = 4 4 H H 9 H H H 16 H Puttig this boud i 3.6 yields r=3 π = 17, , P Z > a exp a 71 a + 3 if 0 u < 1 ρ H r r= Note that the iterest of this ew boud is that the ukow parameter H does ot appear i the right-had side. Now we ca costruct the aouced cofidece iterval for g H. First, observe that Z = H 1 S 4 4 H. Usig the assumptio H H o the oe had, ad 3.7 o the other had, we get: P 1 log S log + log a 1 44 H g H 1 log log S log log + a H log 6

7 = P P 1 log S log log + 1 a 44 H log 1 4 log S log + log 4 4 H a log = P Z a 1 exp which is the desired result. a 71 a + 3 H log4 4H log H 1 4 log S log + log 1 log S a log 1 + log + 44 H log 4 4 H + a log Remark 3.3. The fact that Q H1/, X α Z + β see 3.4, where α 0 ad β > 0, Law is cosistet with the fact that Z = N0, c H, ad Q H1/, X = 1 DZ H, where DZ is the Malliavi derivative of Z see the proof of Theorem.1. Ideed, accordig to Nualart ad Law Ortiz-Latorre [13], oe has that Z = N0, c H if ad oly if 1 DZ H coverges to the costat c H i L. See also [9] for a proof of this fact based o Stei s method. Ackowledgmet. We are grateful to D. Mariucci for useful remarks. Refereces [1] J. Bera Statistics for Log-Memory Processes. Chapma ad Hall. [] Ch. Borell Tail probabilities i Gauss space. I Vector Space Measures ad Applicatios, Dubli, Lecture Notes i Math. 644, Spriger-Verlag. [3] J.-C. Breto ad I. Nourdi 008. Error bouds o the o-ormal approximatio of Hermite power variatios of fractioal Browia motio. Electroic Commuicatios i Probability 13, [4] P. Breuer ad P. Major Cetral limit theorems for oliear fuctioals of Gaussia fields. J. Multivariate Aal. 13 3, [5] J.F. Coeurjolly 001. Estimatig the parameters of a fractioal Browia motio by discrete variatios of its sample paths. Statistical Iferece for Stochastic Processes 4, [6] R. Fox ad M.S. Taqqu Large sample properties of parameter estimates for strogly depedet statioary Gaussia time series. A. Stat. 14, [7] L. Giraitis ad P.M. Robiso 003. Edgeworth expasio for semiparametric Whittle estimatio of log memory. A. Stat. 314, [8] J. Istas ad G. Lag Quadratic variatios ad estimatio of the local Hölder idex of a Gaussia process. Aales de l Istitut Heri Poicaré B Probabilités et Statistiques 334, [9] I. Nourdi ad G. Peccati 008. Stei s method o Wieer chaos. Probab. Theory Rel. Fields, to appear. [10] I. Nourdi ad F. G. Vies 008. Desity formula ad cocetratio iequalities with Malliavi calculus. Available at [11] D. Nualart 006. The Malliavi calculus ad related topics. Spriger-Verlag, Berli, d editio. 7

8 [1] V. Pipiras ad M.S. Taqqu 003. Fractioal calculus ad its coectio to fractioal Browia motio. I: Log Rage Depedece, , Birkhäuser, Basel. [13] D. Nualart ad S. Ortiz-Latorre 008. Cetral limit theorem for multiple stochastic itegrals ad Malliavi calculus. Stoch. Proc. Appl , [14] I. Shigekawa Absolute cotiuity of probability laws of Wieer fuctioals. Proc. Japa. Acad., 54A, [15] C.A. Tudor ad F. G. Vies 007. Variatios ad estimators for self-similarity parameters via Malliavi calculus. To appear i: A. Probab.. 8