SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FRACTIONAL BROWNIAN MOTION AND APPLICATIONS

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1 Communications on Stochastic Analysis Vol. 1, No Serials Publications SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FRACTIONAL BROWNIAN MOTION AND APPLICATIONS DAVID NUALART AND RAGHID ZEINEDDINE* Abstract. We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H < 1/. As applications, we study the asymptotic behavior of the trapezoidal weighted odd-power variations of the fractional Brownian motion and the fractional Brownian motion in Brownian time Z t := X Yt, t, where X is a fractional Brownian motion and Y is an independent Brownian motion. 1. Introduction Let X = X t t be a fractional Brownian motion fbm with Hurst parameter H, 1/. The purpose of this paper is to prove a non-central limit theorem for symmetric weighted odd-power variations of X and derive some applications. For any integers n 1 and j we will make use of the notation j,n X := X j+1 n X j n and β j,n := 1 X j n + X j+1 n. The main result of the paper is the following theorem. Theorem 1.1. Let X be a fbm with Hurst parameter H < 1/. Fix an integer r 1. Assume that f C r 1 R. Then, as n, we have n/ fβ j,n nh j,n X t r 1 Law σ r fx s dw s, t 1.1 where W is a standard Brownian motion independent of X, σ r is the constant given by σr = E[X1 4r ] + E[X 1 X 1+j X j r 1 ], 1. j=1 t and the convergence holds in the Skorokhod space D[,. The proof of this result is based on the methodology of big blocks-small blocks, used, for instance, in [5, 6] and the following stable convergence of odd-power Received ; Communicated by the editors. 1 Mathematics Subject Classification. 6F5; 6G15; 6G; 6H5; 6H7. Key words and phrases. Weighted power variations, limit theorem, Malliavin calculus, fractional Brownian motion, fractional Brownian motion in Brownian time. * Corresponding author. 37

2 38 DAVID NUALART AND RAGHID ZEINEDDINE variations of the fbm nh j,n X r 1, Xt n/ t Law σ rw t, X t t, 1.3 where σ r is defined in 1. and in the right-hand side of 1.3, the process W is a Brownian motion independent of X. The proof of the convergence 1.3 for a fixed t follows from the Breuer-Major Theorem we refer to [15, Chapter 7] and [6] for a proof of this result based on the Fourth Moment theorem. A rather complete analysis of the asymptotic behavior of weighted power variations of the fbm was developed in [13, Corollary 3]. However, the case of symmetric weighted power variations was not considered in this paper. On the other hand, motivated by applications to the asymptotic behavior of symmetric Riemann sums for critical values of the Hurst parameter, Theorem 1.1 was proved in [, Proposition 3.1] when H = 1 4r for a function of the form f r 1 and assuming that f C r 15 R is such that f and its derivatives up to the order r 15 have moderate growth. The proof given here, inspired by the recent work of Harnett, Jaramillo and Nualart [8], allows less derivatives and no growth condition. In the second part of the paper we present two applications of Theorem 1.1 First, we deduce the following convergence in law of the trapezoidal weighted odd-power variations of the fbm with Hurst parameter H < 1/. Proposition 1.. Let X be a fbm with Hurst parameter H < 1/. Fix an integer r 1. Then, if f C M R, where M > r + 1 H, as n, we have n/ 1 fx j n + fx j+1 n nh j,n X r 1 t Law σ r fx s dw s, t in the Skorokhod space D[,, where W is a Brownian motion independent of X. In the particular case r = and H = 1/6, this result has been proved in [16] with longer arguments and using in a methodology introduced in [1]. The limit in this case, that is σ t fx sdw s, is the correction term in the Itô-type formula in law proved in [16]. The asymptotic behavior of weighted odd-power variations of fbm with Hurst parameter H < 1/ has been already studied see [13] and the references therein. More precisely, it is proved that for H < 1/, for any integer r, and for a sufficiently smooth function f, we have nh n fx j n nh j,n X r 1 L µ r t f X s ds, t

3 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 39 where µ r := E[N r ] with N N, 1. By similar arguments, one can show that nh n which implies that fx j+1 n nh j,n X r 1 L µ r t f X s ds, nh n 1 fx j n + fx j+1 n nh j,n X r 1 L. 1.4 Thus, a natural question is to know whether it is possible to replace the normalization nh n by another one in order to get a non-degenerate limit in the convergence 1.4? Proposition 1. gives us the answer to this question. Our second application of Theorem 1.1 deals with the asymptotic behavior of weighted odd-power variations of the so-called fractional Brownian motion in Brownian time fbmbt in short when H < 1/. The fbmbt is defined as Z t = X Yt, t, where X is a two-sided fractional Brownian motion, with Hurst parameter H, 1, and Y is a standard one-sided Brownian motion independent of X. The process Z is self-similar of order H/, it has stationary increments but it is not Gaussian. In the case H = 1, where X is a standard Brownian motion, one recovers the celebrated iterated Brownian motion ibm. This terminology was coined by Burdzy in 1993 see [3], but the idea of considering the ibm is actually older than that. Indeed, Funaki [7] discovered in 1979 that ibm may be used to represent the solution of the following parabolic partial differential equation: u t = 1 4 u, t, x, R. 8 x We refer the interested reader to the research works of Nane see, e.g., [11] and the references therein for many other interesting relationships between iterated processes and partial differential equations. In 1998, Burdzy and Khoshnevisan [4] showed that ibm can be somehow considered as the canonical motion in an independent Brownian fissure. As such, ibm reveals to be a suitable candidate to model a diffusion in a Brownian crack. To support their claim, they have shown that the two components of a reflected two-dimensional Brownian motion in a Wiener sausage of width ϵ > converge to the usual Brownian motion and iterated Brownian motion, respectively, when ϵ tends to zero. Let us go back to the second application of Theorem 1.1, we have the following theorem on the convergence in law of modified weighted odd-power variations of the fbmbt.

4 4 DAVID NUALART AND RAGHID ZEINEDDINE Theorem 1.3. Suppose that H < 1 and fix an integer r 1. Let f CM R, where M > r + 1 H. Then, we have n 1 4 fztk,n + fz T k+1,n nh Z ZTk+1,n T r 1 k,n Law k= Yt σ r fx s dw s t t, 1.5 in the Skorokhod space D[,, where for u R, u fx sdw s is the Wiener- Itô integral of fx with respect to W defined in 4.1 and {T k,n : 1 k n t} is a collection of stopping times defined in 4.3 that approximates the common dyadic partition {k n : 1 k n t} of order n of the time interval [, t]. Theorem 1.3 completes the study of the asymptotic behavior of the modified weighted odd-power variations of the fbmbt in [19], where the case H 1/6 was missing. In addition, in Theorem 1.3 we have convergence in the Skorokhod space D[,, whereas in [19] we only proved the convergence of the finite dimensional distributions. We remark that in many papers see, for instance [] the authors use the uniform partition, but in this paper we work with dyadic partitions. Actually, Theorem 1.1 and Proposition 1. hold also with the uniform partition. However, the dyadic partition plays a crucial role in Theorem 1.3. The paper is organized as follows. In Section we give some elements of Malliavin calculus and some preliminary results. In Section 3, we prove Theorem 1.1 and finally in Section 4 we prove Proposition 1. and Theorem Elements of Malliavin Calculus In this section, we gather some elements of Malliavin calculus we shall need in the sequel. The reader in referred to [17, 15] for details and any unexplained result. Suppose that X = X t t R a two-sided fractional Brownian motion with Hurst parameter H, 1. That is, X is a zero mean Gaussian process, defined on a complete probability space Ω, A, P, with covariance function, C H t, s = EX t X s = 1 s H + t H t s H, s, t R. We suppose that A is the σ-field generated by X. For all n N, we let E n be the set of step functions on [ n, n], and E := n E n. Set ε t = 1 [,t] resp. 1 [t,] if t resp. t <. Let H be the Hilbert space defined as the closure of E with respect to the inner product ε t, ε s H = C H t, s, s, t R..1 The mapping ε t X t can be extended to an isometry between H and the Gaussian space H 1 associated with X. We will denote this isometry by φ Xφ. Let F be the set of all smooth cylindrical random variables of the form F = ϕx t1,..., X tl,

5 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 41 where l N, ϕ : R l R is a C -function such that f and all its partial derivatives have at most polynomial growth, and t 1 < < t l are some real numbers. The derivative of F with respect to X is the element of L Ω; H defined by D s F = l i=1 ϕ x i X t1,..., X tl ε ti s, s R. In particular D s X t = ε t s. For any integer k 1, we denote by D k, the closure of F with respect to the norm k F k, = EF + E[D j F H j ]. The Malliavin derivative D satisfies the chain rule. If φ : R n R is Cb 1 F 1,..., F n are in D 1,, then φf 1,..., F n D 1, and we have n φ DφF 1,..., F n = F 1,..., F n DF i. x i i=1 j=1 and if We denote by δ the adjoint of the derivative operator D, also called the divergence operator. A random element u L Ω; H belongs to the domain of the divergence operator δ, denoted Domδ, if and only if it satisfies E DF, u H c u EF for any F F. If u Domδ, then δu is defined by the duality relationship E F δu = E DF, u H,. for every F D 1,. For every n 1, let H n be the nth Wiener chaos of X, that is, the closed linear subspace of L Ω, A, P generated by the random variables {H n Xh, h H, h H = 1}, where H n is the nth Hermite polynomial. Recall that H =, H p x = 1 p exp x dp dx exp x p for p 1. The mapping I n h n := H n Xh.3 provides a linear isometry between the symmetric tensor product H n and H n. The relation. extends to the multiple Skorokhod integral δ q q 1, and we have E F δ q u = E D q F, u H q,.4 for any element u in the domain of δ q, denoted Domδ q, and any random variable F D q,. Moreover, δ q u = I q u for any u H q. For any Hilbert space V, we denote D k,p V the corresponding Sobolev space of V -valued random variables see [17, page 31]. The operator δ q is continuous from D k,p H q to D k q,p, for any p > 1 and every integers k q 1, that is, we have δ q u D k q,p C k,p u D k,p H q, for all u D k,p H q and some constant C k,p >. These estimates are consequences of Meyer inequalities see [17, Proposition 1.5.7]. We need the following result see [1, Lemma.1] on the Malliavin calculus with respect to X.

6 4 DAVID NUALART AND RAGHID ZEINEDDINE Lemma.1. Let q 1 be an integer. Suppose that F D q,, and let u be a symmetric element in Dom δ q. Assume that, for any r +j q, D r F, δ j u H r L Ω; H q r j. Then, for any r =,..., q 1, D r F, u H r belongs to the domain of δ q r and we have q q F δ q u = δ q r D r F, u r H r. r= Let {e k, k 1} be a complete orthonormal system in H. Given f H n and g H m, for every r =,..., n m, the contraction of f and g of order r is the element of H n+m r defined by f r g = f, e k1 e kr H r g, e k1 e kr H r. k 1,...,k r=1.1. Preliminary results. We will make use of the following notation: j n = 1 [j n,j+1 n ], ε t = 1 [,t], ε j n = 1 εj n + ε j+1 n. We need the following preliminary results. Lemma.. We fix two integers n > m, and for any j, define k := kj = sup{i : i m j n }. The following inequality holds true for some constant C T depending only on T : n T 1 j n, ε kj m C T m1 H..5 Proof. See Lemma., inequality.11, in the paper by Binotto Nourdin and Nualart []. In this paper the inequality is proved for ε kj m but the case ε kj m can be proved by the same arguments. Lemma.3. Let s < t. Then j= n s Proof. We can write j= n s = 1 H H j n, ε j n = 1 nh n t n s H..6 j n, ε j n j= n s = nh 1 H E [ X j+1 n X j n Xj+1 n + X j n] j= n s which gives the desired result. [ j + 1 H j H],

7 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM Proof of Theorem 1.1 In this section we provide the proof of Theorem 1.1. We will make use of the following notation: Φ n t = n/ fβ j,n nh j,n X r and Z t = σ r t fx sdw s, where we recall that W is a Brownian motion independent of X. In order to prove Theorem 1.1, we need to show the following two results: A Convergence of the finite dimensional distributions: Let t 1 < < t d be fixed. Then, we have Φ n t 1,..., Φ n t d Law Z t1,..., Z td. B Tightness: The sequence Φ n is tight in D[,. That is, for every ε, T >, there is a compact set K D[, T ], such that sup P [Φ n K c ] < ε. n 1 The proof of statements A and B will be done in several steps. Step 1: Reduction to compact support functions. As in [8] in the proof of A and B we can assume that f has compact support. Indeed, fix L 1 and let f L C r 1 R be a compactly supported function, such that f L x = fx for all x [ L, L]. Define Φ L nt = n/ f L β j,n nh j,n X r 1 3. and Zt L t = σ r f LX s dw s. For B, we choose L such that P sup t [,T ] X t > L < ε. Then, if K L D[, T ] is a compact set such that sup n 1 P [ Φ L n KL] c <, we obtain ε P [Φ n K c L] P [ Φ L n KL, c sup X t L t [,T ] ] + P sup X t > L < ε. t [,T ] With a similar argument, we can show that given a compactly supported function ϕ CR d, the limit lim E[ϕΦL nt 1,..., Φ L nt d ϕzt L 1,..., Zt L d ] = implies the same limit with Φ L nt i replaced by Φ n t i and Z L t i replaced by Z ti.

8 44 DAVID NUALART AND RAGHID ZEINEDDINE Step : Proof A assuming that f has compact support. The proof is based on the small blocks-big blocks approach. Fix m n and for each j we write k := kj = sup{i : i m j n }, that is, kj is the largest dyadic number in the mth generation which is less or equal than j n. Define Φ n,m t = n/ This term can be decomposed as follows m t 1 Φ n,m t = n/ k= fβ kj,m nh j,n X r fβ k,m + n/ fβ m t,m k+1 n m 1 j=k n m j= m t n m nh j,n X r 1 nh j,n X r 1 The convergence 1.3 implies that for any A -measurable and bounded random variable η, the random vector Φ n,m t 1,..., Φ n,m t d, η converges in law, as n tends to infinity, to the vector Y 1 m,..., Y d m, η, where m t i 1 Ym i = σ r fβ k,m k,m W + σ r fβ m t i,m W ti W m t m i k= for i = 1,..., d. Clearly, Ym i converges in L Ω, as m tends to infinity, to Z ti for i = 1,..., d. Then, it suffices to show that lim m lim sup d Φ n t i Φ n,m t i L Ω =. 3.4 i=1 Let c 1,r,..., c r,r will denote the coefficients of the Hermite expansion of x r 1, namely, r x r 1 = c u,r H r u+1 x. Then, we can write nh j,n X r 1 = = u=1 r c u,r H r u+1 nh j,n X u=1 r u=1 c u,r nhr u+1 δ r u+1 r u+1 j. 3.5 n Set w := wu = r u + 1. Substituting 3.5 into 3.1, yields Φ n t i = r u=1 n t i 1 c u,r fβ j,n n/+wnh δ w w j n.

9 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 45 On the other hand, 3.3 can be also written as Φ n,m t i = r u=1 n t i 1 c u,r fβ kj,m n/+wnh δ w w j n. With the help of Lemma.1 we can express these terms as linear combinations of Skorokhod integrals: Φ n t i = r u=1 c u,r w l= w Θ n l u,lt i and Φ n,m t i = r u=1 c u,r w l= w Θ n,m u,l l t i where n t i 1 Θ n u,lt i = n +wnh δ w l f l β j,n w l j ε n j n, j n l H, and n t i 1 Θ n,m u,l t i = n +wnh δ w l f l β kj,m w l j ε n kj m, j n l H. Then, it suffices to show that lim m lim sup d Θ n u,lt i i=1 Θ n,m u,l t i L Ω = for all 1 u r and l w. We can decompose the difference Θ n u,l t i Θ n,m u,l t i as follows where Θ n u,lt i n t n,m Θ u,l t i 1 i = n +wnh δ w l F n,m j,l w l j =: T n,m n i,l F n,m j,l = f l β j,n ε j n, j n l H f l β kj,m ε kj m, j n l H.

10 46 DAVID NUALART AND RAGHID ZEINEDDINE By Meyer s inequality T n,m i,l w l C n+nwh h= w l = C n+nwh h= w l C n+nlh h= n t i 1 n t i 1 j 1,j = n t i 1 j 1,j = D h F n,m kj,j,l w l j n L Ω;H w l+h [ ] E D h F n,m j, 1,l Dh F n,m j,l H h j1 n, j n w l D h F n,m j 1,l L Ω;H h D h F n,m j,l L Ω;H h ρ H j 1 j w l. We will consider two different cases: Case w l 1: We can make the decomposition F n,m and hence, we have j,l = f l β j,n ε l j ε l n kj, l m j n H l + f l β j,n f l β kj,m ε kj m, j n l H, D h F n,m j,l = f l+h β j,n ε h j ε l n j ε l n kj, l m +f l+h β j,n ε h j ε h n + f l+h β j,n f l+h β kj,m j n H l kj ε m kj m, j n l H ε h kj m ε kj m, j n l H. From the previous equality, and the compact support condition of f, we deduce that there exists a constant C >, such that D h F n,m j,l C ε j n h ε l L Ω;H h H j ε l H n kj l H m l j n l +C ε h j ε h H n kj ε m h kj m l H j n l H + f l+h β j,n f l+h β kj,m ε kj m j n l. H h+l H Using Cauchy-Schwarz inequality we get, for any natural number p 1 ε p j n ε p kj m H p ε j n ε kj m ε j n H p 1 i= C ε j n ε kj m H. i ε H kj m p 1 i H

11 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 47 Therefore D h F n,m j,l L Ω;H h C j n l H ε j n ε kj m H + f l+h β j,n f l+h β kj,m C lnh sup X t X s t s m + f l+h β j,n f l+h β kj,m. Because f l+h is uniformly continuous, for any given ε > there exists δ > such that x y < δ implies f l+h x f l+h y < ε. Therefore, we can write f l+h β j,n f l+h β kj,m ε + δ f l+h β j,n β kj,m and this leads to the estimate D h F n,m j,l C lnh sup X t X s + ε, L Ω;H h t s m which implies T n,m i,l C sup t s m w l X t X s + ε i= n t i 1 ρ H j w l. Then, the series ρ Hj w l is convergent because w l 1 and H < 1/, and we obtain lim T n,m i,l =, because ε is arbitrary. sup m n Case l = w. in this case we have T n,m i,w n+wnh C n+wnh C nh 1 n t i 1 F n,m j,w n t i 1 n t i 1 Finally, using.5 and.6, we obtain which implies ε j n, j n w H + ε kj m, j n w H ε j n, j n H + ε kj m, j n H. T n,m i,w C nh 1 m1 H, lim m lim sup T n,m i,w =.

12 48 DAVID NUALART AND RAGHID ZEINEDDINE Step 3: Proof B assuming that f has compact support. We claim that for every s t T, and p >, there exists a constant C >, such that E [ Φ n t Φ n s p n t n s ] C n p n t n ph s + C. 3.6 Then, by the Billingsley criterion see [1, Theorem 13.5], 3.6 implies the desired tightness property. From the computations in the proof of A, we need to show that for any 1 u r and for any l w, where w = r u + 1, Θ n u,lt Θ n n t n s u,ls p C By Meyer s inequalities, n 1 n n t n H s + C. 3.7 n Θ n u,lt Θ n u,ls p = n +wnh C n +wnh w l h= j= n s = C n +wnh w l h= j= n s j= n s δ w l f l β j,n w l j n ε j n, j n l H f l+h β j,n ε h j w l n j ε n j n, j n l H f l+h β j,n ε h j w l n j ε n j n, j n l H p Lp Ω;H w l+h H w l+h As a consequence, since f has compact support, applying Minkowski inequality, there is a constant C such that 1 p. Θ n u,lt Θ n u,ls p w l C n+wnh h= j,k= n s f l+h β j,n f l+i β k,n ε h w l l j n, ε k n δj n, δ k n εj n, j n ε H k n, k n l H p C n+lnh j,k= n s We will consider two different cases: ρ H j k w l εj n, j n H l ε k n, k n l H.

13 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 49 Case w l 1: In this case, we obtain Θ n u,lt Θ n u,ls p C n j,k= n s C n t n s n C n t n s n. ρ H j k w l h= n s ρ H h w l because the series ρ Hj w l is convergent because w l 1 and H < 1/. This implies the inequality 3.7 in this case. Case l = w: We have Θ n u,wt Θ n u,ws p C n+wnh ε j n, j n C n+nh j= n s j= n s H w H ε j n, j n. Finally, applying.6 and the fact that n+nh 1, we obtain Θ n u,wt Θ n u,ws n t n H s p C. This completes the proof of part B. 4. Applications 4.1. The trapezoidal weighted odd-power variations of fractional Brownian motion. The trapezoidal weighted odd-power variations of the fbm is given in Proposition 1.. We give its proof below. Proof. By a localization argument similar to that used in the proof of Theorem 1.1, we can assume that f has compact support. Choose an integer N such that 1 1 H 1 < N M r 1, which is possible because M > r + H. Since f C M R, by Taylor expansion, we have for all x, y R and N M r 1, fy = f 1 x + y + 1 f 1 x + yy x N k k! f k 1 x + yy xk + R 1 N, k= fx = f 1 x + y + 1 f 1 x + yx y N k k! f k 1 x + yx yk + R N, k= n

14 5 DAVID NUALART AND RAGHID ZEINEDDINE where the residual terms R 1 N and R N are bounded by C y x N+1. We deduce that, for all integer N 1, 1 N fx+fy = f1 x+y+ 1 1 k k! f k 1 x+yy xk +R N x, y, 4.1 k=1 where R N x, y C y x N+1. Recall that β j,n := 1/X j n + X j+1 n and we also write j,n fx := 1 fx j n + fx j+1 n. Set Ψ n t = n/ j,n fx nh j,n X r 1 and let Φ n t be defined in 3.1. Then, in view of Theorem 1.1, it suffices to show that the difference Ψ n Φ n converges to zero in probability in the Skorokhod space as n. Using the expansion 4.1, we obtain Ψ n t Φ n t = n/ j,n fx fβ j,n nh j,n X r 1 N = n/ nhk 1 1 k f k β j,n nh j,n X k+r 1 k! k=1 + n/ =: A n t + B n t R N X j n, X j+1 n nh j,n X r 1 Thanks to Theorem 1.1, and taking into account that f k C k+r 1 R for all k N/ because N + r 1 M, we deduce that A n converges to in probability as n in D[,. Therefore, it is enough to prove the convergence in probability to of B n in D[,. This follows from the following estimates [ ] E sup t T B n t n T 1 C n nhn+1 E[ nh j,n X N+r ] C T n nhn+1, taking into account that HN + 1 > The weighted power variations of fractional Brownian motion in Brownian time. The so-called fractional Brownian motion in Brownian time fbmbt in short is defined as Z t = X Yt, t, where X is a two-sided fractional Brownian motion, with Hurst parameter H, 1, and Y is a standard one-sided Brownian motion independent of X. The process Z t is not a Gaussian process and it is self-similar of order H/ with stationary increments. When H = 1/, one recovers the celebrated iterated Brownian motion.

15 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 51 Let f : R R. Then, for any t and any integer p 1, the weighted p-variation of Z is defined as M n p 1 t = fzk n + fz k+1 n k,n Z p. k= where, as before, k,n Z = Z k+1 n Z k n. After proper normalization we may expect the convergence in some sense to a non-degenerate limit to be determined of N n p t = nκ 1 fzk n + fz k+1 n [ k,n Z p E[ k,n Z p ] ], k= 4. for some κ to be discovered. Due to the fact that one cannot separate X from Y inside Z in the definition of N n p, working directly with 4. seems to be a difficult task see also [1, Problem 5.1]. That is why, following an idea introduced by Khoshnevisan and Lewis [9] in the study of the case H = 1/, we introduce the following collection of stopping times with respect to the natural filtration of Y, denoted by T n = {T k,n : k }, n, 4.3 which are in turn expressed in terms of the subsequent hitting times of a dyadic grid cast on the real axis. More precisely, let D n = {j n/ : j Z}, n, be the dyadic partition of R of order n/. For every n, the stopping times T k,n, appearing in 4.3, are given by the following recursive definition: T,n =, and T k,n = inf { s > T k 1,n : Y s D n \ {Y Tk 1,n } }, k 1. As shown in [9], as n tends to infinity the collection {T k,n : 1 k n t} approximates the common dyadic partition {k n : 1 k n t} of order n of the time interval [, t] see [9, Lemma.] for a precise statement. Based on this fact, one can introduce the counterpart of 4. based on T n, namely, Ñ n p t = n κ 1 + fz fztk,n T [ k+1,n nh Z ZTk+1,n T p ] k,n µp, k= with µ p := E[N p ], where N N, 1 and for some κ > to be discovered. At this stage, it is worthwhile noting that we are dealing with a modified weighted p-variation of Z. In fact, the collection of stopping times {T k,n : 1 k n t} will play an important role in our analysis as we will see in Lemma Known results about the weighted power variations of fbmbt. The asymptotic behavior of Ñ n p t, as n tends to infinity, has been studied in [14] when H = 1/. For H = 1/, one can deduce the following finite dimensional distributions f.d.d. convergence in law from [14, Theorem 1.]. 1 For f Cb R and for any integer r, we have

16 5 DAVID NUALART AND RAGHID ZEINEDDINE n 4 f.d.d. k= 1 fztk,n + fz T k+1,n n 4 Z ZTk+1,n T r 1 k,n Yt fx s µ r d X s + t µ 4r µ r dw s, 4.4 t with µ n := E[N n ], where N N, 1, for all t R, t fx sd X s is the Stratonovich integral of fx with respect to X defined as the limit in probability of nh W n 1 f, t as n, with W n 1 f, t defined in 4.9, W is a two-sided Brownian motion independent of X, Y and for u R, u fx sdw s is the Wiener-Itô integral of fx with respect to W defined in 4.1. For H 1/, the second author of this paper has proved in [19] the following result with f Cb R f is infinitely differentiable with bounded derivatives of all orders, For 1 6 < H < 1 and for any integer r, we have n 1 4 fztk,n + fz T k+1,n nh Z ZTk+1,n T r 1 k,n t f.d.d. k= Yt β r 1 fx s dw s t, 4.5 where for u R, u fx sdw s is the Wiener-Itô integral of fx with respect to W defined in 4.1 and β r 1 = σ r, where σ r is defined in Theorem Fix a time t, for H > 1 nh L r! r! r k= Yt and for any integer r 1, we have 1 fztk,n + fz T k+1,n nh Z ZTk+1,n T r 1 k,n fx s d X s, 4.6 where for all t R, t fx sd X s is defined as in 4.4. As it has been mentioned in [19], the limit of the weighted r 1-variation of Z for H = 1 in 4.4 is intermediate between the limit of the weighted r 1- variation of Z for 1 6 < H < 1 in 4.5 and the limit of the weighted r 1- variation of Z for H > 1 in 4.6. A natural question is then to discovered what happens for H 1/6. The answer is given in Theorem 1.3. Remark 4.1. One can remark that, thanks to Theorem 1.3, 4.5 holds true for H 1/ Asymptotic behavior of the trapezoidal weighted odd-power variations of the fbmbt for H < 1/. The asymptotic behavior of the trapezoidal weighted oddpower variations of the fbmbt for H < 1/ is given in Theorem 1.3. Inspired by [9], the proof of Theorem 1.3, given below, will be done in several steps.

17 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 53 Step 1: A key lemma. For each integer n 1, k Z and real number t, let U j,n t resp. D j,n t denote the number of upcrossings resp. downcrossings of the interval [j n/, j + 1 n/ ] within the first n t steps of the random walk {Y Tk,n } k, that is, U j,n t = { k =,..., n t 1 : D j,n t = { k =,..., n t 1 : Y Tk,n = j n/ and Y Tk+1,n = j + 1 n/} ; Y Tk,n = j + 1 n/ and Y Tk+1,n = j n/}. The following lemma taken from [9, Lemma.4] is going to be the key when studying the asymptotic behavior of the weighted power variation V n r 1 f, t of order r 1, defined, for t, as: V n r 1 1 f, t = fztk,n + fz T [ k+1,n nh Z ZTk+1,n T r 1] k,n. k= 4.7 Its main feature is to separate X from Y, thus providing a representation of f, t which is amenable to analysis. V r 1 n Lemma 4.. Fix f : R R, t and r N. Then V n r 1 f, t = 1 fx j n + fx j+1 n j Z [ nh Xj+1 n X j n r 1] Uj,n t D j,n t. Step : Transforming the weighted power variations of odd order. By [9, Lemma.5], one has 1 { j<j n,t} if j n, t > U j,n t D j,n t = if j n, t =, 1 {j n,t j<} if j n, t < where j n, t = n/ Y T n t,n. As a consequence, j n,t 1 V n r 1 1 f, t = fx + + fx + X n,+ j n/ j+1 n/ j+1 r 1 Xn,+ j if j n, t >, V n r 1 f, t = if j n, t = and j n,t 1 V n r 1 1 f, t = fx + fx X n, j n/ j+1 n/ j+1 r 1 Xn, j if j n, t <, where X t + := X t for t, X t := X t for t <, X n,+ t := nh X + n t for t and X n, t := nh X for t <. n t

18 54 DAVID NUALART AND RAGHID ZEINEDDINE Let us now introduce the following sequence of processes W r 1 ±,n : n/ t 1 W r 1 1 ±,n f, t = fx ± j n +fx± X n,± j+1 n j+1 Xn,± j r 1, t We then have, { 4.8 W n r 1 W r 1 +,n f, t if t f, t := W r 1,n f, t if t <. 4.9 V n r 1 f, t = W n r 1 f, Y T n t,n. 4.1 Step 3: A result concerning the trapezoidal weighted odd-power variations of the fbm. We have the following proposition. Proposition 4.3. Let H < 1. Given an integer r 1 then, for any f CM R, where M > r + 1 H, t n 4 W r 1 Law n f, t σ r fx s dw s, 4.11 t R t R in DR, where W n r 1 f, t is defined in 4.9, W is a two-sided Brownian motion independent of X, Y, and t fx sdw s is defined in the following natural way: for u R, u fx s dw s := { u fx+ s dw + s if u u fx s dw s if u <, 4.1 where W + t = W t if t > and W t = W t if t <, X + and X are defined in Step, and u fx± s dw ± s must be understood in the Wiener-Itô sense. Proof. We define, for all j, n N, β ± j,n := 1 X± j n +X±. Let us introduce j+1 n the following sequence of processes: M ±,n f, t = n/ t 1 f β± j,n X n,± j+1 Xn,± j r 1, t, { M+,n f, t if t M n f, t := M,n f, t if t < Then, by the same arguments that have been used in the proof of Proposition 1., we have n 4 Mn f, n 4 W r 1 n f, n +, in probability in DR. So, in order to prove 4.11 it is enough to prove the following result t n Law 4 Mn f, t σ r fx s dw s, 4.14 t R t R in DR. The proof of 4.14 will be done in two steps, first we prove the convergence in law of the finite dimensional distributions and later we prove tightness.

19 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM Convergence in law of the finite dimensional distributions. Our purpose is to prove that t n f.d.d. 4 Mn f, t σ r fx s dw s, t R t R which is equivalent, by 4.13, to prove that t n 4 M±,n f, t σ r fx s ± dw s ± t f.d.d t The proof of 4.15 uses arguments similar to those employed in part A of the proof of Theorem 1.1, the main ingredient being the small blocks/big blocks approach. Fix m n and for each j we denote by k := kj = sup{i : i m/ j n/ }. Define M ± n,mf, t = n/ t 1 f β ± kj,m X n,± j+1 r 1. Xn,± j It is known that see 3.5 in [18] and part a in the proof of Proposition 5.1 in [19] n/4 ± f.d.d. M n,m f, t L ± mt, t t where m/ t 1 L ± mt := σ r f β ± k,m W ± W ± k+1 m/ k m/ k= +σ r f β ± m/ t,m W ± t W ± m/ t m/, with W t + = W t if t > and Wt = W t if t <, where W is a two-sided Brownian motion independent of X, Y. From the theory of stochastic calculus for semimartingales, we deduce that L ± L mt t σ r fx± s dw s ± as m. Then, it is enough to prove that, for all t, lim m lim m lim sup n 4 M+,n f, t n/4 + M n,m f, t L Ω =, lim sup n 4 M,n f, t n/4 M n,m f, t L Ω =. The proof of the last claim is similar to the proof of 3.4 and is left to the reader.. Proof of Tightness. The distribution of the sequence n 4 M n f, is n N tight in DR. To prove this claim we will show that for any T > and for every T < s t < T, and p >, there exists a constant C >, such that E [ n 4 Mn f, t n 4 Mn f, s p] n t n s C n p n t n ph s +C To do so, we distinguish three cases, according to the sign of s, t R: n

20 56 DAVID NUALART AND RAGHID ZEINEDDINE i. Suppose that s t. In this case we can write E [ n 4 Mn f, t n 4 Mn f, s p] = E [ n 4 M+,n f, t n 4 M+,n f, s p] n t n p s n t n ph s C + C, n where the proof of the last inequality is the same as the proof of 3.6. ii. Suppose s t. Then, we have E [ n 4 Mn f, t n 4 Mn f, s p] = E [ n 4 M,n f, t n 4 M,n f, s p] n s n p t n s n t C + C n n t n s = C n p n n n t n ph s + C, n ph where the proof of the second inequality is the same as the proof of 3.6 and we get the last equality since for any x <, x = x 1. iii. Suppose s < < t. Then, we can write E [ n 4 Mn f, t n 4 Mn f, s p] C E [ n 4 Mn f, t n 4 Mn f, p] +E [ n 4 Mn f, s n 4 Mn f, p] = C E [ n 4 M+,n f, t n 4 M+,n f, p] +E [ n 4 M,n f, s n 4 M,n f, p] n p t n ph s C + C n n t + n s + 1 C n n t n s = C n p n p + C n s + n t + 1 n n t n ph s + C, where we have the third inequality by i and ii. Finally, we have proved 4.16 which proves the tightness of n 4 M n f, n N in DR. Step 4: Convergence in law of Y T. As it was mentioned in [9], { n/ Y n,n Tk,n : k } is a simple and symmetric random walk on Z. Observe that for all t, Y T n t,n = n/ n/ Y T n t,n = n/ l= n/ Y Tl+1,n Y Tl,n, where n/ Y Y Tl+1,n T l,n are independent and identically distributed random variables following the Rademacher distribution. By Donsker theorem, we get l N that n ph

21 SYMMETRIC WEIGHTED ODD-POWER VARIATIONS OF FBM 57 YT nt,n t law Y t t in D[, Step 5: Last step in the proof of Theorem 1.3. Thanks to Proposition 4.3, to 4.17, and to the independence of X, W and Y, we have n 4 W r 1 law n f,, Y T n,n n + σr fx s dw s, Y in DR D[, t Let us define B t t R as follows B t := σ r fx sdw s. Since x, y DR D[, + x y D[, + is measurable see M16 at page 49 in [1] for a proof of this result and since B Y is continuous, then, by 4.18 and Theorem.7 in [1], it follows that n 4 W r 1 law n f, Y T B Y = σ n,n r n + Y fx s dw s, in D[, +. The proof of Theorem 1.3 follows from 4.1 and the last convergence in law. Acknowledgment. The first author was supported by the NSF grant DMS The first drafted version of this paper has been done when the second author was a member of the Research training group 131, working in the Technical University of Dortmund. He is thankful to the financial support of the DFG German Science Foundations Research Training Group 131. He is also thankful to the financial support of FRIAS-USIAS program. References 1. Billingsley, P.: Convergence of Probability Measures, nd ed., Wiley, New York, Binotto, G., Nourdin, I. and Nualart, D.: Weak symmetric integrals with respect to the fractional Brownian motion, Ann. Probab , no. 4, Burdzy, K.: Some path properties of iterated Brownian motion, in: Seminar on Stochastic Processes, 199, , Birkhaüser, Boston. 4. Burdzy, K. and Khoshnevisan, D.: Brownian motion in a Brownian crack, Ann. Appl. Probab , no. 3, Corcuera, J. M., Nualart, D. and Podolskij, M.: Asymptotics of weighted random sums, Communications in Applied and Industrial Mathematics 6 14, no. 1, e Corcuera, J. M., Nualart, D. and Woerner, J.: Power variation of some integral fractional processes, Bernoulli 1 6, no. 4, Funaki, T.: Probabilistic construction of the solution of some higher order parabolic differential equation, Proc. Japan Acad , Ser. A., Harnett, D., Jaramillo, A. and Nualart, D.: Symmetric integrals with respect to a class of self-similar Gaussian processes, J. Theor. Probab., to appear. 9. Khoshnevisan, D. and Lewis, T. M.: Stochastic calculus for Brownian motion on a Brownian fracture, Ann. Appl. Probab , no. 3, Khoshnevisan, D. and Lewis, T. M.: Iterated Brownian motion and its intrinsic skeletal structure, in: Seminar on Stochastic Analysis, Random Fields and Applications 1999, 1 1, Birkhaüser Verlag, Basel. 11. Nane, E.: Higher order PDE s and iterated processes, Trans. Amer. Math. Soc. 36 8, Nourdin, I. and Nualart, D.: Central limit theorems for multiple Skorokhod integrals, J. Theor. Probab. 3 1, no. 1,

22 58 DAVID NUALART AND RAGHID ZEINEDDINE 13. Nourdin, I., Nualart, D. and Tudor, C. A.: Central and non-central limit theorems for weighted power variations of fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat. 46 1, no. 4, Nourdin, I. and Peccati, G.: Weighted power variations of iterated Brownian motion, Electron. J. Probab. 13 8, paper no. 43, Nourdin, I. and Peccati, G.: Normal Approximations using Malliavin Calculus: from Stein s Method to the Universality, Cambridge University Press, Cambridge, Nourdin, I., Réveillac, A. and Swanson, J.: The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6, Electron. J. Probab. 15 1, paper no. 7, Nualart, D.: The Malliavin Calculus and Related Topics, nd ed., Springer-Verlag, Berlin, Zeineddine, R.: Fluctuations of the power variation of fractional Brownian motion in Brownian time, Bernoulli 1 15, no., Zeineddine, R.: Asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time, J. Theor. Probab. 17 DOI 1.17/s David Nualart: The University of Kansas, Lawrence, Kansas 6645, USA address: nualart@ku.edu Raghid Zeineddine: Freiburg Institute For Advanced Studies and University of Strasbourg Institute for Advanced Studies FRIAS-USIAS, Freiburg 7914, Germany address: raghid.z@hotmail.com