ASYMPTOTIC BEHAVIOR OF WEIGHTED QUADRATIC AND CUBIC VARIATIONS OF FRACTIONAL BROWNIAN MOTION. BY IVAN NOURDIN Université Paris VI

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1 The Aals of Probability 2008, Vol. 36, No. 6, DOI: /07-AOP385 Istitute of Mathematical Statistics, 2008 ASYMPTOTIC BHAVIOR OF WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FRACTIONAL BROWNIAN MOTION BY IVAN NOURDIN Uiversité Paris VI The preset article is devoted to a fie study of the covergece of reormalized weighted quadratic ad cubic variatios of a fractioal Browia motio B with Hurst idex H. I the quadratic (resp. cubic) case, whe H<1/4 (resp. H<1/6), we show by meas of Malliavi calculus that the covergece holds i L 2 toward a explicit limit which oly depeds o B. This result is somewhat surprisig whe compared with the celebrated Breuer ad Major theorem. 1. Itroductio ad mai result. The study of sigle path behavior of stochastic processes is ofte based o the study of their power variatios ad there exists a very extesive literature o the subject. Recall that, a real κ>1beig give, the κ-power variatio of a process X, with respect to a subdivisio π ={0 = t,0 <t,1 < <t, = 1 of [0, 1], isdefiedtobethesum X t,k+1 X t,k κ. For simplicity, cosider from ow o the case where t,k = k/,for N ad k {0,...,. I the preset paper, we wish to poit out some iterestig pheomea whe X = B is a fractioal Browia motio ad whe the value of κ is 2 or 3. I fact, we will also drop the absolute value (whe κ = 3) ad we will itroduce some weights. More precisely, we will cosider (1.1) h(b k/ )( B k/ ) κ, κ {2, 3, where the fuctio h : R R is assumed to be smooth eough ad where B k/ deotes the icremet B (k+1)/ B k/. The aalysis of the asymptotic behavior of quatities of type (1.1) is motivated, for istace, by the study of the exact rates of covergece of some approximatio schemes of scalar stochastic differetial equatios drive by B (see [5, 10] ad [11]), besides, of course, the traditioal applicatios of quadratic variatios to parameter estimatio problems. Received May 2007; revised August AMS 2000 subject classificatios. 60F05, 60G15, 60H07. Key words ad phrases. Fractioal Browia motio, Breuer ad Major theorem, weighted quadratic variatio, weighted cubic variatio, exact rate of covergece, Malliavi calculus. 2159

2 2160 I. NOURDIN Now, let us recall some kow results cocerig the κ-power variatios (for κ = 2, 3, 4,...), which are today more or less classical. First, assume that the Hurst idex H of B is 1/2, that is, B is the stadard Browia motio. Let μ κ deote the κ-momet of a stadard Gaussia radom variable G N (0, 1). By the scalig property of the Browia motio ad usig the cetral limit theorem, it is immediate that, as : (1.2) 1 [ κ/2 ( B k/ ) κ μ κ ] Law N (0,μ 2κ μ 2 κ ). Whe weights are itroduced, a iterestig pheomeo appears: istead of Gaussia radom variables, we rather obtai mixig radom variables as limit i (1.2). Ideed, whe κ is eve, it is a very particular case of a more geeral result by Jacod [7] (seealso[13]) that we have, as : 1 h(b k/ )[ κ/2 ( B k/ ) κ μ κ ] Law 1 (1.3) μ 2κ μ 2 κ h(b s )dw s. Here, W deotes aother stadard Browia motio, idepedet of B. Wheκ is odd, we have this time, as : 1 h(b k/ )[ κ/2 ( B k/ ) κ ] (1.4) Law 1 0 h(b s ) ( μ 2κ μ 2 κ+1 dw s + μ κ+1 db s ) ; see [13]. Secod, assume that H 1/2, that is, the case where the fractioal Browia motio B has o idepedet icremets aymore. The (1.2) has bee exteded by Breuer ad Major [1], Dobrushi ad Major [3], Giraitis ad Surgailis [4] or Taqqu [16]. Precisely, four cases are cosidered accordig to the eveess of κ ad the value of H : If κ is eve ad if H (0, 3/4), as, (1.5) 1 [ κh ( B k/ ) κ μ κ ] Law N (0,σ 2 H,κ ). If κ is eve ad if H (3/4, 1), as, 1 2H [ κh ( B k/ ) κ μ κ ] Law Roseblatt r.v. If κ is odd ad if H (0, 1/2], as, (1.6) 1 κh ( B k/ ) κ Law N (0,σ 2 H,κ ). 0

3 WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM 2161 If κ is odd ad if H (1/2, 1),as, H κh ( B k/ ) κ Law N (0,σ 2 H,κ ). Here, σ H,κ > 0 deotes a costat depedig oly o H ad κ, whichmaybe differet from oe formula to aother oe, ad which ca be computed explicitly. The term Roseblatt r.v. deotes a radom variable whose distributio is the same as that of Z at time oe, for Z the Roseblatt process defied i [16]. Now, let us proceed with the results cocerig the weighted power variatios i the case where H 1/2. I what follows, h deotes a regular eough fuctio such that h together with its derivatives has subexpoetial growth. If κ is eve ad H (1/2, 3/4), the by Theorem 2 i Leó ad Ludeña [9] (see also Corcuera, Nualart ad Woerer [2] for related results o the asymptotic behavior of the p-variatio of stochastic itegrals with respect to B) we have, as : (1.7) 1 h(b k/ )[ κh ( B k/ ) κ μ κ ] Law 1 σ H,κ h(b s )dw s, where, oce agai, W deotes a stadard Browia motio idepedet of B. Thus, (1.7) showsfor(1.1) a similar behavior to that observed i the stadard Browia case; compare with (1.3). I cotradistictio, the asymptotic behavior of (1.1) ca be completely differet from (1.3) or(1.7) for other values of H.The first result i this directio has bee observed by Gradiaru, Russo ad Vallois [6] ad cotiued i [5]. Namely, if κ is odd ad H (0, 1/2), we have, as : H 1 h(b k/ )[ κh ( B k/ ) κ ] L2 μ 1 κ+1 (1.8) h (B s )ds. 2 0 Before givig the mai result of this paper, let us make three commets. First, we stress that the limit obtaied i (1.8) does ot ivolve a idepedet stadard Browia motio aymore, as was the case for (1.3) or(1.7). Secod, otice that (1.8) agrees with (1.6) because, whe H (0, 1/2),wehave(κ +1)H 1 <κh 1/2. Thus, (1.8) with h 1 is actually a corollary of (1.6). Third, observe that the same type of covergece as (1.8) with H = 1/4 had already bee performed i [8], Theorem 4.1, whe i (1.8) the fractioal Browia motio B with Hurst idex 1/4 is replaced by a iterated Browia motio Z. It is ot very surprisig, sice this latter process is also cetered, self-similar of idex 1/4 ad has statioary icremets. Fially, let us metio that Swaso aouced i [15] that, i a joit work with Burdzy, they will prove that the same also holds for the solutio of the stochastic heat equatio drive by a space time white oise. Now, let us go back to our problem. I the sequel, we will make use of the followig hypothesis o real fuctio h: 0

4 2162 I. NOURDIN (H m ) The fuctio h belogs to C m ad, for ay p (0, ) ad ay 0 i m, we have sup t [0,1] { h (i) (B t ) p <. The aim of the preset work is to prove the followig result: THORM 1.1. Let B be a fractioal Browia motio with Hurst idex H. The: 1. If h : R R verifies (H 4 ) ad if H (0, 1/4), we have, as : (1.9) 2H 1 1 h(b k/ )[ 2H ( B k/ ) 2 1] L2 1 4 h (B u )du If h : R R verifies (H 6 ) ad if H (0, 1/6), we have, as : (1.10) 3H 1 L [ h(bk/ ) 3H ( B k/ ) h (B k/ ) H ] 1 0 h (B u )du. Before givig the proof of Theorem 1.1, let us roughly explai why (1.9) is oly available whe H<1/4 [of course, the same type of argumet could also be applied to uderstad why (1.10) is oly available whe H<1/6]. For this purpose, let us first cosider the case where B is the stadard Browia motio (i.e., whe H = 1/2). By usig the idepedece of icremets, we easily compute { h(b k/ )[ 2H ( B k/ ) 2 1] = 0 ad {( ) 2 h(b k/ )[ 2H ( B k/ ) 2 1] = 2 2 { { 1 0 h 2 (B k/ ) h 2 (B u )du. Although these two facts are of course ot sufficiet to guaratee that (1.3) holds whe κ = 2, they however roughly explai why it is true. Now, let us go back to the geeral case, that is, the case where B is a fractioal Browia motio of idex H (0, 1/2). I the sequel, we will show (see Lemmas 2.2 ad 2.3 for precise statemets) that { h(b k/ )[ 2H ( B k/ ) ] 2H [h (B k/ )],

5 ad, whe H (0, 1/4): WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM 2163 {( h(b k/ )[ 2H ( B k/ ) 2 1] H H { ) 2 [h (B k/ )h (B l/ )] h (B u )h (B v )dudv. [0,1] 2 This explais the covergece (1.9). At the opposite, whe H (1/4, 1/2), oe ca prove that {( h(b k/ )[ 2H ( B k/ ) 2 1] ) 2 σ 2 H { 1 0 h 2 (B u )du. Thus, whe H (1/4, 1/2), the quatity h(b k/)[ 2H ( B k/ ) 2 1] behaves as i the stadard Browia motio case, at least for the first- ad secodorder momets. I particular, oe ca expect that the followig covergece holds whe H (1/4, 1/2): as, (1.11) 1 h(b k/ )[ 2H ( B k/ ) 2 1] Law 1 σ H h(b s )dw s, with W a stadard Browia motio idepedet of B. I fact, i the sequel of the preset paper, which is a joit work with Nualart ad Tudor [12], we show that (1.11) is true ad we also ivestigate the case where H 3/4. Fially, let us remark that, of course, covergece (1.9) agrees with covergece (1.5). Ideed, we have 2H 1 < 1/2 if ad oly if H<1/4 (it is aother fact explaiig the coditio H<1/4 i the first poit of Theorem 1.1). Thus, (1.9) with h 1 is actually a corollary of (1.5). Similarly, (1.10) agrees with (1.5), sice we have 3H 1 < 1/2 if ad oly if H<1/6 (it explais the coditio H<1/6 i the secod poit of Theorem 1.1). Now, the rest of our article is devoted to the proof of Theorem 1.1. Istead of the pedestria techique performed i [6] (as their authors called it themselves), we stress the fact that we chose here a more elegat way via Malliavi calculus. It ca be viewed as aother ovelty of this paper. 2. Proof of the mai result Notatio ad prelimiaries. We begi by briefly recallig some basic facts about stochastic calculus with respect to a fractioal Browia motio. Oe 0

6 2164 I. NOURDIN may refer to [14] for further details. Let B = (B t ) t [0,1] be a fractioal Browia motio with Hurst parameter H (0, 1/2) defied o a probability space (, A,P). We mea that B is a cetered Gaussia process with the covariace fuctio (B s B t ) = R H (s, t),where (2.1) R H (s, t) = 1 2 (t2h + s 2H t s 2H ). We deote by the set of step R-valued fuctios o [0, 1]. LetH be the Hilbert space defied as the closure of with respect to the scalar product 1[0,t], 1 [0,s] H = R H (t, s). We deote by H the associate orm. The mappig 1 [0,t] B t ca be exteded to a isometry betwee H ad the Gaussia space H 1 (B) associated with B. We deote this isometry by ϕ B(ϕ). Let S be the set of all smooth cylidrical radom variables, that is, of the form F = f(b(φ 1 ),..., B(φ )) where 1, f : R R is a smooth fuctio with compact support ad φ i H. The Malliavi derivative of F with respect to B is the elemet of L 2 (, H) defied by D s F = i=1 f x i (B(φ 1 ),...,B(φ ))φ i (s), s [0, 1]. I particular D s B t = 1 [0,t] (s). Asusual,D 1,2 deotes the closure of the set of smooth radom variables with respect to the orm F 2 1,2 = [F 2 ]+[ D F 2 H ]. The Malliavi derivative D verifies the followig chai rule: if ϕ : R R is cotiuously differetiable with a bouded derivative, ad if (F i ) i=1,..., is a sequece of elemets of D 1,2,theϕ(F 1,...,F ) D 1,2 ad we have, for ay s [0, 1]: D s ϕ(f 1,...,F ) = i=1 ϕ x i (F 1,...,F )D s F i. The divergece operator I is the adjoit of the derivative operator D. If a radom variable u L 2 (, H) belogs to the domai of the divergece operator, that is, if it verifies DF,u H c u F L 2 for ay F S, the I(u)is defied by the duality relatioship for every F D 1,2. (F I (u)) = DF,u H,

7 WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM Proof of (1.9). I this sectio, we assume that H (0, 1/4). For simplicity, we ote δ k/ = 1 [k/,(k+1)/] ad ε k/ = 1 [0,k/]. Also C will deote a geeric costat that ca be differet from lie to lie. We first eed three lemmas. The proof of the first oe follows directly from a covexity argumet: LMMA 2.1. For ay x 0, we have 0 (x + 1) 2H x 2H 1. LMMA 2.2. For h, g : R R verifyig (H 2 ), we have (2.2) PROOF. {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1] = 1 4 2H {h (B k/ )g(b l/ )+o( 2 2H ). For 0 l, k 1, we ca write {h(b k/ )g(b l/ ) 2H ( B k/ ) 2 = {h(b k/ )g(b l/ ) 2H B k/ I(δ k/ ) = {h (B k/ )g(b l/ ) 2H B k/ ε k/,δ k/ H + {h(b k/ )g (B l/ ) 2H B k/ ε l/,δ k/ H + {h(b k/ )g(b l/ ). Thus, 2H {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1] = {h (B k/ )g(b l/ )I (δ k/ ) ε k/,δ k/ H (2.3) + {h(b k/ )g (B l/ )I (δ k/ ) ε l/,δ k/ H = {h (B k/ )g(b l/ ) ε k/,δ k/ 2 H + 2{h (B k/ )g (B l/ ) ε k/,δ k/ H ε l/,δ k/ H + {h(b k/ )g (B l/ ) ε l/,δ k/ 2 H. But (2.4) ε k/,δ k/ H = 1 2 2H ( (k + 1) 2H k 2H 1 ), ε l/,δ k/ H = 1 2 2H ( (k + 1) 2H k 2H l k 1 2H + l k 2H ).

8 2166 I. NOURDIN I particular, ε k/,δ k/ 2 H 1 4 4H (2.5) = 1 4 4H (( (k + 1) 2H k 2H ) 2 2 ( (k + 1) 2H k 2H )) Cosequetly, uder (H 2 ): 2H 3 4 4H ( (k + 1) 2H k 2H ) by Lemma 2.1. C {h (B k/ )g(b l/ ) ( ε k/,δ k/ 2 H 1 4 4H ) 1 2H Similarly, usig agai Lemma 2.1, we deduce ( (k + 1) 2H k 2H ) = C. ε k/,δ k/ H ε l/,δ k/ H + ε l/,δ k/ 2 H Sice, obviously C 4H ( (k + 1) 2H k 2H + l k 2H l k 1 2H ). (2.6) l k 2H l k 1 2H C 2H +1, we obtai, agai uder (H 2 ): 2H ( 2{h (B k/ )g (B l/ ) ε k/,δ k/ H ε l/,δ k/ H + {h(b k/ )g (B l/ ) ε l/,δ k/ 2 H ) C. Fially, recallig that H<1/4 < 1/2, equality (2.2) follows sice = o( 2 2H ). LMMA 2.3. For h, g : R R verifyig (H 4 ), we have (2.7) {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1][ 2H ( B l/ ) 2 1] = H {h (B k/ )g (B l/ )+o( 2 4H ).

9 PROOF. Thus, WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM 2167 For 0 l, k 1, we ca write {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1] 2H ( B l/ ) 2 = {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1] 2H B l/ I(δ l/ ) = {h (B k/ )g(b l/ )[ 2H ( B k/ ) 2 1] 2H B l/ ε k/,δ l/ H + {h(b k/ )g (B l/ )[ 2H ( B k/ ) 2 1] 2H B l/ ε l/,δ l/ H + 2{h(B k/ )g(b l/ ) 4H B k/ B l/ δ k/,δ l/ H + {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1]. {h(b k/ )g(b l/ )[ 2H ( B k/ ) 2 1][ 2H ( B l/ ) 2 1] = {h (B k/ )g(b l/ )[ 2H ( B k/ ) 2 1] 2H I(δ l/ ) ε k/,δ l/ H + {h(b k/ )g (B l/ )[ 2H ( B k/ ) 2 1] 2H I(δ l/ ) ε l/,δ l/ H + 2{h(B k/ )g(b l/ ) 4H B k/ I(δ l/ ) δ k/,δ l/ H = 2H {h (B k/ )g(b l/ )[ 2H ( B k/ ) 2 1] ε k/,δ l/ 2 H + 2 2H {h (B k/ )g (B l/ )[ 2H ( B k/ ) 2 1] ε k/,δ l/ H ε l/,δ l/ H + 4 4H {h (B k/ )g(b l/ ) B k/ ε k/,δ l/ H δ k/,δ l/ H + 4 4H {h(b k/ )g (B l/ ) B k/ ε l/,δ l/ H δ k/,δ l/ H + 2 4H {h(b k/ )g(b l/ ) δ k/,δ l/ 2 H + 2H {h(b k/ )g (B l/ )[ 2H ( B k/ ) 2 1] ε l/,δ l/ 2 H 6 A i k,l,. i=1 We claim that, for 1 i 5, we have Ai k,l, =o(2 4H ).Letusfirst cosider the case where i = 1. By usig Lemma 2.1, we deduce that 2H ε k/,δ l/ H = 1 ( 2 (l + 1) 2H l 2H l k + 1 2H + l k 2H ) is bouded. Cosequetly, uder (H 4 ): A 1 k,l, C ε k/,δ l/ H. As i the proof of Lemma 2.2 [see more precisely iequality (2.6)], this yields A1 k,l, C. Sice H < 1/4, we fially obtai A1 k,l, =

10 2168 I. NOURDIN o( 2 4H ). Similarly, by usig the fact that 2H ε l/,δ l/ H is bouded, we prove that A2 k,l, =o(2 4H ). Now, let us cosider the case of A i k,l, for i = 5, the cases where i = 3, 4beig similar.agaibylemma2.1, wehavethat 2H δ k/,δ l/ H = 1 2 ( k l + 1 2H + k l 1 2H 2 k l 2H ) is bouded. Cosequetly, uder (H 4 ): A 5 k,l, C( k l + 1 2H + k l 1 2H 2 k l 2H ). But, sice H<1/2, we have ( k l + 1 2H + k l 1 2H 2 k l 2H ) = + p= C. ( p + 1 2H + p 1 2H 2 p 2H ) [( 1) ( 1 p) 0 ( p)] This yields A5 k,l, C = o(2 4H ),agaisiceh<1/4. It remais to cosider the term with A 6 k,l,. By replacig g by g i idetity (2.3) ad by usig argumets similar to those i the proof of Lemma 2.2, we ca write, uder (H 4 ): A 6 k,l, = 4H + o( 2 4H ). {h (B k/ )g (B l/ ) ε k/,δ k/ 2 H ε l/,δ l/ 2 H But, from Lemma 2.1 ad equality (2.4), we deduce ε k/,δ k/ 2 H ε l/,δ l/ 2 H H (2.8) C 8H ( (k + 1) 2H k 2H + (l + 1) 2H l 2H ). Thus, 4H ε k/,δ k/ 2 H ε l/,δ l/ 2 H H C 1 2H = o( 2 4H ), sice H<1/4 < 1/2. This yields, uder (H 4 ): A 6 k,l, = H {h (B k/ )g (B l/ )+o( 2 4H ), ad the proof of Lemma 2.3 is doe.

11 WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM 2169 We are ow i positio to prove (1.9). Usig Lemma 2.3, we have o oe had: 2 2H 1 { h(b k/ )[ 2H ( B k/ ) 2 1] (2.9) 4H 2 = {h(b k/ )h(b l/ ) [ 2H ( B k/ ) 2 1][ 2H ( B l/ ) 2 1] = {h (B k/ )h (B l/ )+o(1). Usig Lemma 2.2, we have o the other had: 2H 1 { h(b k/ )[ 2H ( B k/ ) 2 1] 1 h (B l/ ) 4 (2.10) = {h(b k/ )h (B l/ )[ 2H ( B k/ ) 2 1] = 1 2 {h (B k/ )h (B l/ )+o(1). 16 Now, we easily deduce (1.9). Ideed, thaks to (2.9) (2.10), we obtai, by developig the square: 2H 1 {( h(b k/ )[ 2H ( B k/ ) 2 1] 1 ) 2 h (B k/ ) 0, 4 as.sice 4 1 proved that (1.9) holds. 1 4 h (B k/ ) L2 l 10 h (B u )du as, we fially 2.3. Proof of (1.10). I this sectio, we assume that H (0, 1/6). We keep the same otatios as i Sectio 2.2. We first eed two techical lemmas. LMMA 2.4. (2.11) 3H For h, g : R R verifyig (H 3 ), we have {h(b k/ )g(b l/ )( B k/ ) 3 = 3 2 H {h (B k/ )g(b l/ )

12 2170 I. NOURDIN 1 8 3H {h (B k/ )g(b l/ )+o( 2 3H ). PROOF. For 0 l, k 1, we ca write 3H {h(b k/ )g(b l/ )( B k/ ) 3 = 3H {h(b k/ )g(b l/ )( B k/ ) 2 I(δ k/ ) = 3H {h (B k/ )g(b l/ )( B k/ ) 2 ε k/,δ k/ H + 3H {h(b k/ )g (B l/ )( B k/ ) 2 ε l/,δ k/ H + 2 H {h(b k/ )g(b l/ ) B k/ = 3H {h (B k/ )g(b l/ ) B k/ ε k/,δ k/ 2 H + 2 3H {h (B k/ )g (B l/ ) B k/ ε k/,δ k/ H ε l/,δ k/ H + 3 H {h (B k/ )g(b l/ ) ε k/,δ k/ H (2.12) + 3H {h(b k/ )g (B l/ ) B k/ ε l/,δ k/ 2 H + 3 H {h(b k/ )g (B l/ ) ε l/,δ k/ H = 3H {h (B k/ )g(b l/ ) ε k/,δ k/ 3 H + 3 3H {h (B k/ )g (B l/ ) ε k/,δ k/ 2 H ε l/,δ k/ H + 3 3H {h (B k/ )g (B l/ ) ε k/,δ k/ H ε l/,δ k/ 2 H + 3 H {h (B k/ )g(b l/ ) ε k/,δ k/ H + 3H {h(b k/ )g (B l/ ) ε l/,δ k/ 3 H H {h(b k/ )g (B l/ ) ε l/,δ k/ H Bk,l, i. i=1 We claim that Bi k,l, =o(2 3H ) for i = 2, 3, 5, 6. Let us first cosider the cases where i = 2adi = 6. Usig Lemma 2.1 ad equality (2.4), we have ad ε l/,δ k/ H 2H ( (k + 1) 2H k 2H + l k 1 2H l k 2H ) ε k/,δ k/ H 2 ε l/,δ k/ H C 6H ( (k + 1) 2H k 2H + l k 1 2H l k 2H ).

13 This yields, uder (H 3 ): WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM 2171 B 2 k,l, C1 H = o( 2 3H ) sice H<1/6 < 1/2, B 6 k,l, C1+H = o( 2 3H ) sice H<1/6 < 1/4. Similarly, we prove that Bi k,l, =o(2 3H ) for i = 3ad5. It remais to cosider the terms with Bk,l, 1 ad B4 k,l,. From Lemma 2.1 ad equality (2.4), we deduce (2.13) ε k/,δ k/ H H 2H ( (k + 1) 2H k 2H ), (2.14) ε k/,δ k/ 3 H H C 6H ( (k + 1) 2H k 2H ). Thus, sice H<1/6, H 3H This yields, uder (H 3 ): ε k/,δ k/ H H 1+H = o( 2 3H ), ε k/,δ k/ 3 H H C 1 H = o( 2 3H ). B 4 k,l, = 3 2 H B 1 k,l, = 1 8 3H ad the proof of Lemma 2.4 is doe. LMMA 2.5. (2.15) 6H {h (B k/ )g(b l/ )+o( 2 3H ), {h (B k/ )g(b l/ )+o( 2 3H ), For h, g : R R verifyig (H 6 ), we have {h(b k/ )g(b l/ )( B k/ ) 3 ( B l/ ) 3 = 9 4 2H {h (B k/ )g (B l/ ) H {h (B k/ )g (B l/ )

14 2172 I. NOURDIN PROOF H {h (B k/ )g (B l/ ) H {h (B k/ )g (B l/ )+o( 2 6H ). For 0 l, k 1, we ca write 6H {h(b k/ )g(b l/ )( B k/ ) 3 ( B l/ ) 3 = 6H {h(b k/ )g(b l/ )( B k/ ) 3 ( B l/ ) 2 I(δ l/ ) = 6H {h (B k/ )g(b l/ )( B k/ ) 3 ( B l/ ) 2 ε k/,δ l/ H + 6H {h(b k/ )g (B l/ )( B k/ ) 3 ( B l/ ) 2 ε l/,δ l/ H + 3 6H {h(b k/ )g(b l/ )( B k/ ) 2 ( B l/ ) 2 δ k/,δ l/ H + 2 4H {h(b k/ )g(b l/ )( B k/ ) 3 B l/ = 6H {h (B k/ )g(b l/ )( B k/ ) 3 B l/ ε k/,δ l/ 2 H + 2 6H {h (B k/ )g (B l/ )( B k/ ) 3 B l/ ε k/,δ l/ H ε l/,δ l/ H + 6 6H {h (B k/ )g(b l/ )( B k/ ) 2 B l/ ε k/,δ l/ H δ k/,δ l/ H + 3 4H {h (B k/ )g(b l/ )( B k/ ) 3 ε k/,δ l/ H + 6H {h(b k/ )g (B l/ )( B k/ ) 3 B l/ ε l/,δ l/ 2 H + 6 6H {h(b k/ )g (B l/ )( B k/ ) 2 B l/ ε l/,δ l/ H δ k/,δ l/ H + 3 4H {h(b k/ )g (B l/ )( B k/ ) 3 ε l/,δ l/ H + 6 6H {h(b k/ )g(b l/ ) B k/ B l/ δ k/,δ l/ 2 H + 9 4H {h(b k/ )g(b l/ )( B k/ ) 2 δ k/,δ l/ H = 6H {h (B k/ )g(b l/ )( B k/ ) 3 ε k/,δ l/ 3 H + 3 6H {h (B k/ )g (B l/ )( B k/ ) 3 ε k/,δ l/ 2 H ε l/,δ l/ H + 9 6H {h (B k/ )g(b l/ )( B k/ ) 2 ε k/,δ l/ 2 H δ k/,δ l/ H + 3 6H {h (B k/ )g (B l/ )( B k/ ) 3 ε k/,δ l/ H ε l/,δ l/ 2 H H {h (B k/ )g (B l/ )( B k/ ) 2 ε k/,δ l/ H ε l/,δ l/ H δ k/,δ l/ H H {h (B k/ )g (B l/ ) B k/ ε k/,δ l/ H δ k/,δ l/ 2 H

15 WIGHTD QUADRATIC AND CUBIC VARIATIONS OF FBM H {h (B k/ )g(b l/ )( B k/ ) 3 ε k/,δ l/ H + 6H {h(b k/ )g (B l/ )( B k/ ) 3 ε l/,δ l/ 3 H + 9 6H {h(b k/ )g (B l/ )( B k/ ) 2 ε l/,δ l/ 2 H δ k/,δ l/ H H {h(b k/ )g (B l/ ) B k/ ε l/,δ l/ H δ k/,δ l/ 2 H + 3 4H {h(b k/ )g (B l/ )( B k/ ) 3 ε l/,δ l/ H + 6 6H {h (B k/ )g(b l/ ) B k/ ε k/,δ l/ H δ k/,δ l/ 2 H + 6 6H {h(b k/ )g(b l/ )( B k/ ) 2 δ k/,δ l/ 3 H + 9 4H {h(b k/ )g(b l/ )( B k/ ) 2 δ k/,δ l/ H. To obtai (2.15), we develop the right-had side of the previous idetity i the same way as for the obtetio of (2.12). The, oly the terms cotaiig ε k/,δ k/ α H ε l/,δ l/ β H,forα, β 1, have a cotributio i (2.15), as we ca check by usig (2.5), (2.8), (2.13) ad(2.14). The other terms are o( 2 6H ).Details are left to the reader. We are ow i positio to prove (1.10). Usig Lemmas 2.4 ad 2.5,wehaveo oe had 3H 1 [ {( h(bk/ ) 3H ( B k/ ) h (B k/ ) H ]) 2 (2.16) 6H 2 = {[ h(b k/ ) 3H ( B k/ ) h (B k/ ) H ] [ h(b l/ ) 3H ( B l/ ) h H ] (B l/ ) = {h (B k/ )h (B l/ )+o(1). O the other had, we have, by Lemma 2.4: 3H 1 { [h(b k/ ) 3H ( B k/ ) ] 2 h (B k/ ) H (2.17) 3H 2 = h (B l/ ) l { h(b k/ )h (B l/ ) 3H ( B k/ ) 3

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