CENTRAL LIMIT THEOREMS FOR SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS

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1 The Annals of Probability 25, Vol. 33, No. 1, DOI / Institute of Mathematical Statistics, 25 CENTRAL LIMIT THEOREMS FOR SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS BY DAVID NUALART AND GIOVANNI PECCATI Universitat de Barcelona and Université de Paris VI We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes. 1. Introduction. In this paper, we characterize the convergence in distribution to a normal N, 1 law for a sequence of random variables F belonging to a fixed Wiener chaos, and with variance tending to 1. We show that a necessary and sufficient condition for this convergence is that the moment of fourth order of F converges to 3. If F is a multiple stochastic integral of order n of a symmetric and square integrable ernel f, for instance on [, 1] n, another necessary and sufficient condition for the above convergence is that, for all p = 1,...,n 1, the contractions of order p defined by f p := f p f converge to zero in L 2 [, 1] 2n p as tends to infinity. In general, we call central limit theorem CLT in the sequel any result describing the wea convergence of a sequence of nonlinear functionals of a Gaussian process or of a Gaussian measure toward a standard normal law. The reader is referred to Major 1981, Maruyama 1982, 1985, Giraitis and Surgailis 1985 and the references therein for results in this direction. Here, we shall observe that, in the above quoted references, the authors establish sufficient conditions to have a CLT in the general case of sequences of functionals having a possibly infinite Wiener Itô expansion. A common technique used in such a study is the method of moments [see, e.g., Maruyama 1985], requiring a determination of all moments associated to a given functional, usually estimated by means of the so-called diagram formulae [see Surgailis 2 for a detailed survey]. On the other hand, our techniques which are mainly based on a stochastic calculus result due to Dambis, Dubins and Schwarz [see Revuz and Yor 1999, Chapter V and Section 3.1] naturally bring the need to estimate and control expressions related uniquely to the fourth moment of each element of the sequence F. To this end, we apply extensively some version of the product formula for multiple stochastic integrals, such as the one presented in Nualart [1995, Proposition 1.1.3], and Received April 23; revised October 23. AMS 2 subject classifications. 6F5, 6H5. Key words and phrases. Multiple stochastic integrals, Brownian motion, wea convergence, fractional Brownian motion, Brownian sheet. 177

2 178 D. NUALART AND G. PECCATI perform calculations that are very close in spirit to the ones contained in the first part of Üstünel and Zaai Our results are specifically motivated by recent wors on limit theorems for quadratic functionals of Brownian motion and Brownian bridge [see Deheuvels and Martynov 24 and Peccati and Yor 24a, b], as well as Brownian sheet and related processes [see Deheuvels, Peccati and Yor 24]. We provide examples and applications, mainly related to quadratic functionals of a fractional Brownian motion, with Hurst parameter H> 1 2, and of a standard Brownian sheet. 2. The main result. Consider a separable Hilbert space H.Let{e : 1} be a complete orthonormal system in H.Foreveryn 1, we denote H n the nth symmetric tensor product of H.Forp =,...,n, and for every f H n,we define the contraction of f of order p to be the element of H 2n p defined by f p = i 1,...,i p =1 and we denote by f p s its symmetrization. In what follows, we will write f,ei1 e ip H p f,e i1 e ip X ={Xh : h H } H p, for an isonormal Gaussian process on H.This means that X is a centered Gaussian family indexed by the elements of H, defined on some probability space, F, P andsuchthat,foreveryh, h H, E XhXh = h, h H. For every n 1, we will denote by In X the isometry between H n equipped with the norm n! H n and the nth Wiener chaos of X. In the particular case where H = L 2 A, A,µ, A, A is a measurable space and µ is a σ -finite and nonatomic measure, then H n = L 2 s An, A n,µ n is the space of symmetric and square integrable functions on A n and for every f H n, In X f is the multiple Wiener Itô integral of order n off with respect to X, as defined, for example, in Nualart [1995, Section 1.1.2]. Our main result is the following. THEOREM 1. Let the above notation and assumptions prevail, and fix n 2. Then, for any sequence of elements {f : 1} such that f H n for every, and 1 lim n! f 2 + H n = lim E[I n X f ]=1, + the following conditions are equivalent: i lim + E[In Xf 4 ]=3;

3 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 179 ii for every p = 1,...,n 1, lim + f p 2 H 2n p = ; iii as goes to infinity, the sequence {I X n f : 1} converges in distribution to a standard Gaussian random variable. As a consequence of this result we obtain COROLLARY 2. Fix n 2 and f H n such that E[In Xf 2 ]=1. Then the distribution of In Xf cannot be normal and E[I n Xf 4 ] 3. PROOF. IfIn Xf had a normal distribution or E[I n Xf 4 ]=3, then, according to Theorem 1, for every p = 1,...,n 1, we would have f p =. Thus, for each v H n p we obtain = f p,v v H 2n p = f,v H n p 2 H p, which implies f =. REMARK 1. The fact that a random variable with the form In X f, n>1, cannot be Gaussian is well nown. See, for example, the discussion contained in Janson [1997, Chapter VI]. To prove Theorem 1 in its most general form, we shall first deal with the case of X being the Gaussian family generated by a standard Brownian motion on [, 1]. To this end, some further notation is needed The Brownian case. For every n 1andT >, given a permutation π of 1,...,n,weset n π,t = { t 1,...,t n R n : T>t π1 > >t πn > }, and we define n π,t := n T ={t 1,...,t n R n : T>t 1 > >t n > } to be the simplex contained in [,T] n. Now, for a given n 1, tae f L 2 n T,dt 1 dt n = L 2 n T.Tosuchanf we associate the symmetric function on [,T] n, ft 1,...,t n = f t π1,...,t πn 1 n π,t t 1,...,t n, π where π runs over all permutations of 1,...,n. Forp =,...,n, the contraction of f of order p on [,T] is the application t 1,...,t 2n 2p f p,t t 1,...,t 2n 2p = [,T ] p fu 1,...,u p,t 1,...,t n p fu 1,...,u p,t n p+1,...,t 2n 2p du 1 du p

4 18 D. NUALART AND G. PECCATI and we note f p,t s the symmetrization of f p,t. In this section, we note W ={W t : [, 1]} a standard Brownian motion initialized at zero. Note that, in the terminology of the previous paragraph, the centered Gaussian space generated by W can be identified with an isonormal Gaussian process on H = L 2 [, 1],dt.Foreveryn, foreveryf L 2 n 1 and every t, 1], we put t sn 1 Jn t f = dw s1 dw sn fs 1,...,s n and also In t f= n!jn t f,sothat In 1 f= In X f, where X is once again the isonormal process generated by W. The following result translates the content of Theorem 1 in the context of this section. PROPOSITION 3. Let the above notation prevail, fix n 2 and consider a collection {g : 1} of elements of L 2 n 1 such that 2 lim g L + 2 n 1 := lim g n + 1 = 1. Then, the following conditions are equivalent: i lim + E[Jn 1g 4 ]=3; ii for every p = 1,...,n 1, lim + g p,1 2 = ; [,1] 2n 2p iii as, the sequence {Jn 1g : 1} converges in distribution to a standard Gaussian random variable; iv for every p =,...,n 2, 1 lim ds 1 ds 2n 1 p dt g p,t 3 2n 1 p,t s1,...,s 2n 1 p s =, 1 s 1 where, for every fixed t [, 1], g,t stands for the function on n 1 t given by s 1,...,s n 1 g t, s 1,...,s n 1. PROOF. We will prove the following implications: iv iii i ii iv. iv iii. Suppose that 3 holds. According to the Dambis Dubins Schwarz theorem [see Revuz and Yor 1999, Chapter V], for every there exists a standard Brownian motion W such that 4 Jn 1 g = W = W 1 dtjn 1 t g,t [n 1!] 2 1 dtin 1 t g,t.

5 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 181 We can expand the stochastic time-change in 4 by means of a multiplication formula for multiple Wiener integrals [see, e.g., Nualart 1995, Proposition 1.1.3] to obtain that 1 dt In 1 t g,t 1 2 = dt where c n 1,p = p! n 12 p, and therefore [ n 1 p= ] c n 1,p I2n 1 p t g p,t,t s, 1 dtin 1 t g,t n 2 [n 1!] 2 = g 2 n + p= c 1 n 1,p dti2n 1 p t g p,t,t s 5 1 [n 1!] 2. Moreover, thans to a stochastic Fubini theorem for multiple integrals, we obtain that for any p =,...,n 2, 1 dti2n 1 p t g p,t,t s [2n 1 p]! 1 = dt J2n 1 p t g p,t,t s = 2n 1 p 1 1 dw s1 dw s2n 1 p dt g p,t,t s 1 s1,...,s 2n 1 p s. Now tae the L 2 -norm of the right-hand side of 5, let go to infinity and observe that if 2 and 3 are verified, then the pair W, [n 1!] 2 1 dt In 1 t g,t 2 converges wealy to W, 1, so that the conclusion is immediately achieved by using, for example, Theorem 3.1 in Whitt 198, as well as formula 4. iii i. Trivial, given condition 2. i ii. Observe first that, for every, Now, as I 1 n g = n p= E[J 1 n g 4 ]=n! 4 E[I 1 n g 4 ]. n 2 p! I 1 p 2n 2p g p,1 s = n! g 2 [,1] n + I 1 2n g,1 n 1 s + p=1 n 2 p! I 1 p 2n 2p g p,1 s,

6 182 D. NUALART AND G. PECCATI due again to the multiplication formula, we obtain immediately E[Jn 1 g 4 ]= g 4 n + 2n! 1 n! 4 g,1 s 2 [,1] 2n + n! n 1 [ 4 p=1 n p! p ] 2 2n 2p! g p,1 s 2 [,1] 2n 2p. Now call 2n the set of the 2n! permutations of the set 1,...,2n, whose generic elements will be denoted by π, π, and so on. On such a set, we introduce the following notation: for p =,...,n, we write π p π if the set π1,...,πn π 1,...,π n contains exactly p elements. Note that, for a given π 2n,thereare n pn! permutations π such that π p π. Moreover, it is easily seen that if π π or π n π, then da [,1] 2n 1 da 2n g,1 aπ1,...,a π2n g,1 aπ 1,...,a π 2n = g 4 [,1] n = n!2 g 4 1 n, so that 2n! n! 4 g,1 s 2 [,1] 2n [ 1 = n! 4 g 4 2n! 1 n! + n! n π 2n π π π n π n 1 ] + da p=1 π p [,1] 2n 1 da 2n g,1 a π g,1 a π, π where a π = a π1,...,a π2n. Since, for p = 1,...,n 1, da [,1] 2n 1 da 2n g,1 a π g,1 n 2 a π = n! g p,1 p 2 [,1] 2n 2p, π p π we get 2n! n! 4 g,1 s 2 [,1] 2n = 2 g 4 n n p=1 g p,1 2 [,1] 2n 2p p!n p!,

7 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 183 and therefore E[J 1 n g 4 ]=3 g 4 1 n n p!n p! p=1 [ g p,1 2 [,1] 2n 2p + n 2p g p,1 n p s 2 [,1] ]. 2n 2p This yields in particular that, if E[Jn 1g 4 ] converges to 3, and 2 holds, then necessarily, for every p = 1,...,n 1, lim g p,1 6 2 [,1] 2n 2p =. ii iv. We introduce some notation: for any m 1, x m is shorthand for a vector x 1,...,x m R m, ˆx m = max i x i,anddx m stands for Lebesgue measure on R m. To conclude the proof, we shall show that, for p = 1,...,n 1, condition 6 implies necessarily that lim [,1] ds n p n p [,1] n p dτ n p 1 ŝ n p ˆτ n p dt g p 1,t,t s n p, τ n p =. Now, since for every t, s 2,...,s n [, 1] n, g,t s 2,...,s n = 1 [,t] n 1s 2,...,s n g t, s 2,...,s n, we obtain that 1 ds [,1] n p n p dτ [,1] n p n p 1 1 = dt [,t ] p 1 dv p 1 dt g p 1,t,t s n p, τ n p ŝ n p ˆτ n p dt [,t] p 1 du p 11 ûp 1 t,ˆv p 1 t ds [,t t ] n p n p g t, u p 1, s n p g t, v p 1, s n p 1 1 dt [,1] dv p 1 p 1 dt [,1] p 1 p 1 ds [,t t ] n p n p g t, u p 1, s n p g t, v p 1, s n p = ds [,1] n p n p dτ [,1] n p n p 1 dt ŝ n p ˆτ n p [,1] p 1 p 1 g t, u p 1, s n p g t, u p 1, τ n p.

8 184 D. NUALART AND G. PECCATI Now, since condition 6 holds by assumption, we now that A = ds [,1] n p n p dτ [,1] n p n p du [,1] p p g u p, s n p g u p, τ n p converges to zero, as goes to infinity. We may expand the above expression and write A = [,1] n p n p [,1] n p n p ŝn p ˆτ n p dt [,1] p 1 p 1 g t, u p 1, s n p g t, u p 1, τ n p + ds [,1] n p n p dτ [,1] n p n p 1 dt ŝ n p ˆτ n p du [,1] p 1 p 1 g t, u p 1, s n p g t, u p 1, τ n p + 2 ds [,1] n p n p dτ [,1] n p n p ŝn p ˆτ n p dt du [,1] p 1 p 1 g t, u p 1, s n p g t, u p 1, τ n p 1 dt ŝ n p ˆτ n p dv [,1] p 1 p 1 g t, v p 1, s n p g t, v p 1, τ n p = A 1 + A 2 + A 3. In particular, the Fubini theorem yields A 3 = [,1] p p 1 dv p 1 [,1] p ds [,1] n p n p 1 t t ŝ n p t t g t, u p 1, s n p g t, v p 1, s n p + 2 [,1] p p 1 dt dv p 1 [,1] p ds [,1] n p n p 1 t t ŝ n p t t g t, u p 1, s n p g t, v p 1, s n p dτ [,1] n p n p 1 ˆτ n p t t g t, u p 1, τ n p g t, v p 1, τ n p.

9 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 185 Since A 2 = dt du [,1] p p 1 dt dv p 1 [,1] p dτ [,1] n p n p 1 ˆτ n p t t g t, u p 1, τ n p g t, v p 1,τ n p, we obtain that, for every, A 2 + A 3, and therefore that A 1 must converge to zero as tends to infinity. This gives immediately the desired conclusion. REMARK 2. There exists an elegant explanation of the implication iii iv for the case n = 2. To do this, consider for simplicity a family {I2 1g } 1 of multiple integrals with symmetric ernels {g } 1 such that g [,1] 2 = 1. Then, by using the formula E [ exp iλi2 1 g ] [ = exp 1 j 2 j+1 2iλj Trg j ], j where g j is the jth power of the Hilbert Schmidt operator associated to the ernel g,, it is easy to show that if I2 1g converges in law to a standard Gaussian random variable as goes to infinity, then necessarily lim Trg j = for every j>2. In particular, by taing j = 4, we obtain that lim Trg 4 = lim dt du g t, rg u, r dr =. But some calculations analogous to the ones at the end of the proof of Proposition 3 yield dt du g t, rg u, r dr = 1 dt 1 t u du 1 1 r r + dr dr 1 1 r r + dr dr 1 1 r r + 2 dr dr g t, rg u, r dr dt g t, rg t, r r r dt g t, rg t, r dt g t, rg t, r r r thus giving the desired conclusion. r r dt g t, rg t, r,

10 186 D. NUALART AND G. PECCATI A slight refinement of Proposition 3 is the following. PROPOSITION 4. Let {Jn 1g : 1} be a sequence of iterated integrals as in the statement of Proposition 3 satisfying either condition i, ii, iii or iv. Suppose, moreover, that the sequence { 1 } dt In 1 t g,t : 1 converges to zero in probability as goes to infinity. Then, the pair J 1 n g, W converges in distribution to N, W, where N is a standard Gaussian random variable independent of W. PROOF. We shall only prove the asymptotic independence, which in this case is given by an application of an asymptotic version of Knight s theorem, such as the one stated, for instance, in Revuz and Yor [1999, Chapter XIII] Proof of Theorem 1. Let the assumptions and notation of Theorem 1 prevail. Since H is a separable Hilbert space, for every n 1 there exists an application i n from H n onto L 2 s [, 1]n,dt 1 dt n, such that, for every n, i n is an isometry and, moreover, the following equality holds for every f H n : n!j 1 n in f = n! 1 tn 1 dw t1 dw tn i n f t 1,...,t n law = In X f, where W is a standard Brownian motion on [, 1]. It is therefore clear that the sequence {f : 1} in the statement of Theorem 1 is such that: 1. lim + n! i n f 2 [,1] n = lim E[I X n f 4 ]=3 if, and only if, n! 4 lim E [ J 1 n in f 4 ] = For every p =,...,n 1, lim f p 2 = if, and only if, H 2n 2p lim i n f p 2 [,1] 2n 2p =. 4. For +, the convergence In X f law N, where N is a standard Gaussian random variable, taes place if, and only if, n!jn 1 in f law N. The proof is easily concluded.

11 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS Examples and applications Quadratic functionals of a fractional Brownian motion. Consider a fractional Brownian motion fbm B H ={Bt H,t [, 1]} of Hurst parameter H, 1. Thatis,B H is a centered Gaussian process with the covariance function 7 R H t, s = 1 2 s2h + t 2H t s 2H. According to the so-called Jeulin s lemma [see Jeulin 198, Lemma 1, page 44], with probability 1, and for every H, 1 8 Bt H dt =+. t2h +1 In what follows, we shall use the results of the previous sections to characterize the speed at which the two quantities 1 1 F β = t 2β Bt H dt and L ε = Bt H dt 9 ε t2h +1 diverge to infinity, respectively, when β tends to H 1 2,andwhenε tends to. In particular, to be able to apply Theorem 1, we will mainly concentrate on the case H> 1 2. We will first find the Wiener chaos expansion of F β. Clearly, 1 EF β = 2β + 2H + 1. If D denotes the derivative operator [see Alòs and Nualart 23], we have D s F β = 2 1 s t 2β B H t and D r D s F β = 2 1 s r 2β+1. 2β + 1 Strooc s formula [see Strooc 1987] yields 1 F β = 2β + 2H β + 1 I 2 1 2β+1 1, where I 2 denotes the double stochastic integral with respect to B H. The following result is another CLT. dt 11 PROPOSITION 5. If H> 1 2, then 1 2β + 2H + 1 t 2β Bt H dt L 2 1, β H 1/2

12 188 D. NUALART AND G. PECCATI and 1 2β + 2H + 1 2β 1/2 + 2H + 1 t 2β Bt H Law 12 dt 1 c H N, β H 1/2 where N is a random variable with distribution N, 1 and c H is a constant depending on H. PROOF. We denote by E the set of step functions on [, 1]. LetH be the Hilbert space defined as the closure of E with respect to the scalar product 1[,t], 1 [,s] H = R H t, s = α H t s r u 2H 2 dudr, where α H = H2H 1. The mapping 1 [,t] B t can be extended to an isometry between H and the first chaos of B H. Let us compute E I 2 1 2β+1 = 2 1 β+1 2 H H = 2αH 2 1 s r 2β+1 1 t uβ+1 [,1] 4 Simple computations lead to t s 2H 2 r u 2H 2 dsdt dr du. lim 2β + 2H + β H 1/2 12 E I 2 1 2β+1 =, because EI 2 1 s rβ+1 behaves as 2β + 2H as β H 1 2. Therefore, 11 follows. In order to show 12, set G β = 2β + 2H + 1 1/2 2β + 2H + 1F β 1 2β + 2H + 11/2 = I 2 1 2β+1. 2β + 1 We have lim β H 1/2 EG2 β = lim 22β + 2H + 1α2 H β H 1/2 2β s r 2β+1 1 t uβ+1 [,1] 4 = α2 H 2H 2 lim 2β + 2H + 1 β H 1/2 t s 2H 2 r u 2H 2 dsdt dr du

13 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 189 [,1] 4s r2β+1 t uβ+1 = c 2 H. t s 2H 2 r u 2H 2 dsdt dr du Taing into account Theorem 1, it suffices to show that condition ii holds, that is, 2β + 2H + 1 lim β H 1/2 2β r 2β+1 [,1] 2 The above norm can be written as α 4 H 1 r β+1 r r 2H 2 dr dr 2 [,1] 8 1 s r 2β+1 1 t r β+1 1 s uβ+1 1 t u β+1 r r 2H 2 u u 2H 2 H H s s 2H 2 t t 2H 2 dr dr dudu dsds dt dt, =. and it is of the order of 2β +2H +1 1 as β H 1 2, which implies the desired result. Note that we have also the following noncentral limit theorem. PROPOSITION 6. For any H, 1, β + 2H + 1 t 2β Bt H dt L2 β + BH 1 2. PROOF. The convergence 13 follows from 1 and the fact that I 2 1 = B H REMARK 3. The combination of Propositions 5 and 6 generalizes results previously obtained for a standard Brownian motion and a standard Brownian bridge. In particular, analogs of formulae 11 and 13 for the case H = 1 2 are proved in Deheuvels and Martynov 24 and Peccati and Yor 24a, whereas a version of 12 for H = 1 2 is obtained in Peccati and Yor 24a.

14 19 D. NUALART AND G. PECCATI Next, simple calculations yield that the Wiener chaos decomposition of L ε,for any ε>, is given by L ε = log 1 ε 1 2H I 2 1 ε 2H, where the second summand on the right-hand side is the double stochastic integral, with respect to B H, of the symmetric function This yields the following CLT. s, t 1 ε t s 2H. 14 PROPOSITION 7. If H> 1 2, then log 1 1/2 L ε log 1 Law H N, ε ε ε where N is a standard Gaussian random variable, and H depending exclusively on H. is a real constant PROOF. We eep the notation in the proof of Proposition 5. Since for every t>the Wiener chaos decomposition of Bt H is given by B H t = t 2H + I 2 1 t, it is easily verified that 1 E[L 2 ε ]= dt 1 ds ε t2h +1 ε s 2H +1 E[BH t Bs H 2 ] = log dt 1 ds ε 2 t2h +1 s 2H +1 t2h + s 2H t s 2H and consequently that E[L 2 ε ] E[L ε] 2 = E[L 2 ε ] log 1 ε ε behaves lie log1/ε when ε tends to zero. Thans to Theorem 1, to prove 14 it is now sufficient to show that lim[log1/ε] 2 ε 1 r ε 2H 1 r ε 2H r r 2H 2 dr dr [,1] 2 =. ε 2 H H

15 SEQUENCES OF MULTIPLE STOCHASTIC INTEGRALS 191 Now, it is clear that 1 r ε 2H 1 r ε 2H r r 2H 2 dr dr [,1] 2 = αh 4 1 s r ε 2H 1 t r ε 2H [,1] 8 1 s u ε 2H 1 t u ε 2H r r 2H 2 u u 2H 2 s s 2H 2 t t 2H 2 dr dr dudu dsds dt dt, 2 H H and one can show that the latter quantity is asymptotic to log1/ε for ε converging to zero. This concludes the proof. REMARK 4. An analog of Proposition 7 for the case H = 1 2 Peccati and Yor 24b. is proved in 3.2. Quadratic functionals of a Brownian sheet. We now extend the results of Deheuvels and Martynov 24 and Peccati and Yor 24a to the case of a Brownian sheet W on [, 1] n,thatis, W ={Wx 1,...,x n : x 1,...,x n [, 1] n } is a centered Gaussian process with covariance function n E[Wx 1,...,x n Wy 1,...,y n ]= y i x i. Note that the Gaussian space generated by W can be identified with an isonormal Gaussian process on L 2 [, 1] n,dx 1 dx n. In particular, we are interested in the limiting behavior of the two functionals, defined, respectively, for vectors β = β 1,...,β n such that β i > 1 andfor ε>, n A β = dx [,1] n 1 dx n Wx 1,...,x n, B ε = x 2β i i dx 1 dx n [ε,1] n x 1 x n Wx 1,...,x n, when β converges to 1,..., 1 and ε converges to zero. We recall that, due again to Jeulin s lemma, with probability 1, lim β 1,..., 1 A β = lim ε B ε =+.

16 192 D. NUALART AND G. PECCATI REMARK 5. The law of the random variable A β, for a fixed vector β and for n = 2, is studied in detail in Deheuvels, Peccati and Yor 24. Now, for every β and for every ε, we write the Wiener chaos decompositions of A β and B ε which are given by n n A β = 2β i β i + 1 I n 2 1 xi y i β i+1, B ε = log 1 n n + I 2 xi y i ε 1 1, ε where I 2 now stands for a double Wiener integral with respect to W. Then, calculations analogous to those performed in the previous section yield, thans to Theorem 1, another CLT. PROPOSITION 8. Let the above notation and assumptions prevail. i When β 1,..., 1, n n 2β i + 2 A 1/2 β 2β i n n = 2β i + 2 1/2 2β i + 2A β 1 converges in distribution to 2 n N, 1, where N, 1 indicates a standard Gaussian random variable, and also n 2β i + 2A β 1 converges to zero in L 2. ii When ε, log 1 n/2 B ε log 1 n ε ε converges in distribution to 2 n N, 1. Acnowledgments. Part of this paper was written while Giovanni Peccati was visiting the Institut de Matemàtica of the University of Barcelona IMUB, in February 23. This author heartily thans David Nualart and Marta Sanz-Solé for their support and hospitality. REFERENCES ALÒS, E. and NUALART, D. 23. Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep

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