Gaussian Limits for Vector-Valued Multiple Stochastic Integrals

Universités de Paris 6 & Paris 7 - CNRS (UMR 7599) PRÉPUBLICATIONS DU LABORATOIRE DE PROBABILITÉS & MODÈLES ALÉATOIRES 4, place Jussieu - Case 188 - 75 252 Paris cedex 05 http://www.proba.jussieu.fr Gaussian limits for vector-valued multiple stochastic integrals G. PECCATI & C.A. TUDOR NOVEMBRE 2003 Prépublication no 861 G. Peccati : Laboratoire de Statistique Théorique et Appliquée, Université Paris VI, Case 158, 4 Place Jussieu, F-75252 Paris Cedex 05. C.A. Tudor : Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI & Université Paris VII, 4 place Jussieu, Case 188, F-75252 Paris Cedex 05. Gaussian limits for vector-valued multiple stochastic integrals Giovanni PECCATI Laboratoire de Statistique Théorique et Appliquée Université de Paris VI 175, rue du Chevaleret 75013 Paris, France email: [email protected] Ciprian A. TUDOR Laboratoire de Probabilités et Modèles Aléatoires Universités de Paris VI &VII 175, rue du Chevaleret 75013 Paris, France email: [email protected] October 31, 2003 Abstract We establish necessary`and sufficient ´ conditions for a sequence of d-dimensional vectors of multiple stochastic integrals Fkd = F1k , ..., Fdk , k ≥ 1, to converge in distribution to a d-dimensional Gaussian vector Nd = (N1 , ..., Nd ). In particular, we show that if the covariance structure of Fkd converges to that of Nd , then componentwise convergence implies joint convergence. These results extend to the multidimensional case the main theorem of [9]. Keywords – Multiple stochastic integrals; Limit theorems; Weak convergence; Brownian motion. AMS Subject classification – 60F05; 60H05 1 Introduction
For d ≥ 2, fix d natural numbers 1 ≤ n1 ≤ ... ≤ nd and, for every k ≥ 1, let Fkd = F1k , ..., Fdk be a vector of d random variables such that, for each j = 1, ..., d, F jd belongs to the nj th Wiener chaos associated to a real valued Gaussian process. The aim of this paper is to prove necessary and sufficient conditions to have that the sequence Fkd converges in distribution to a given d-dimensional Gaussian vector, when k tends to infinity. In particular, our main result states that, if for every 1 ≤ i, j ≤ d lim k→+∞ E Fik Fjk = δij , where δij is Kronecker symbol, then the following two conditions are equivalent: (i) F kd converges in distribution to a standard centered Gaussian vector N d (0, Id ) (Id is the d × d identity matrix), (ii) for every j = 1, ..., d, Fjk converges in distribution to a standard Gaussian random variable. Now suppose that, for every k ≥ 1 and every j = 1, ..., d, the random variable F jk is the multiple Wiener-Itô stochastic (k) n integral of a square integrable kernel fj , for instance on [0, 1] j . We recall that, according to the main h
4 i =3 result of [9], condition (ii) above is equivalent to either one of the following: (iii) lim k→+∞ E Fjk (k) (k) for every j, (iv) for every j and every p = 1, ..., nj − 1 the contraction fj ⊗p fj converges to zero in   2(n −p) L2 [0, 1] j . Some other necessary and sufficient condition for (ii) to hold is stated in the subsequent sections, and an extension is provided to deal with the case of a Gaussian vector N d with a more general covariance structure. 1 Besides [9], our results should be compared with other central limit theorems (CLT) for non linear functionals of Gaussian processes. The reader is referred to [2], [5], [6], [7], [14] and the references therein for several results in this direction. As in [9], the main tool in the proof of our results is a well known time-change formula for continuous local martingales, due to Dambis, Dubins and Schwarz (see e.g. [12, Chapter 5]). In particular, this technique enables to obtain our CLTs, by estimating and controlling expressions that are related uniquely to the fourth moments of the components of each vector F kd . The paper is organized as follows. In Section 2 we introduce some notation and discuss preliminary results; in Section 3 our main theorem is stated and proved; finally, in Section 4 we present some applications, to the weak convergence of chaotic martingales (that is, martingales admitting a multiple Wiener integral representation), and to the convergence in law of random variables with a finite chaotic decomposition. 2 Notation and preliminary results Let H be a separable Hilbert space. For every n ≥ 1, we define H ⊗n to be the nth tensor product of H √ and write H ⊙n for the nth symmetric tensor product of H, endowed with the modified norm n! k·kH ⊗n . We denote by X = {X (h) : h ∈ H} an isonormal process on H, that is, X is a centered H-indexed Gaussian family, defined on some probability space (Ω, F, P) and such that E [X (h) X (k)] = hh, kiH , for every h, k ∈ H. For n ≥ 1, let Hn be the nth Wiener chaos associated to X (see for instance [8, Chapter 1]): we denote by InX the isometry between Hn and H ⊙n . For simplicity, in this paper we consider uniquely spaces of the form H = L2 (T, A, µ), where (T, A) is a measurable space and µ is a σ-finite and atomless measure. In this case, InX can be identified with the multiple Wiener-Itô integral with respect to the process X, as defined e.g. in [8, Chapter 1]. We also note that, by some standard Hilbert space argument, our results can be immediately extended to a general H. The reader is referred to [9, Section 3.3] for a discussion of this fact. Let H = L2 (T, A, µ); for any n, m ≥ 1, every f ∈ H ⊙n , g ∈ H ⊙m , and p = 1, ..., n ∧ m, the pth contraction between f and g, noted f ⊗p g, is defined to be the element of H ⊗m+n−2p given by Z f ⊗p g (t1 , ..., tn+m−2p ) = f (t1 , ..., tn−p , s1 , ..., sp ) × Tp ×g (tn−p+1 , ..., tm+n−2p , s1 , ..., sp ) dµ (s1 ) · · · dµ (sp ) ; by convention, f ⊗0 g = f ⊗ g denotes the tensor product of f and g. Given φ ∈ H ⊗n , we write (φ)s for its canonical symmetrization. In the special case T = [0, 1], A = B ([0, 1]) and µ = λ, where λ is Lebesgue measure, some specific notation is needed. For any 0 < t ≤ 1, ∆ nt stands for the symplex contained n n in [0, t] , i.e. ∆nt := {(t1 , ..., tn ) : 0 < tn < ... < t1 < t}. Given a function f on [0, 1] and t ∈ [0, 1], ft n−1 denotes the application on [0, 1] given by (s1 , ..., sn−1 ) 7→ f (t, s1 , ..., sn−1 ) . n n For any n, m ≥ 1, for any pair of functions f, g such that f ∈ L 2 ([0, 1] , B ([0, 1] ) , dλ⊗n ) := n m L ([0, 1] ) and g ∈ L2 ([0, 1] ), and for every 1 < t ≤ 1 and p = 1, ..., n ∧ m, we write f ⊗tp g for the pth contraction of f and g on [0, t], defined as Z f ⊗tp g (t1 , ..., tn+m−2p ) = f (t1 , ..., tp , s1 , ..., sp ) × 2 [0,t]p ×g (tn−p+1 , ..., tm+n−2p , s1 , ..., sp ) dλ (s1 ) · · · dλ (sp ) ; g = f ⊗ g. Eventually, we recall that if H = L2 ([0, 1] , B ([0, 1]) , dλ), then X coincides as before, f with the Gaussian space generated by the standard Brownian motion
t 7→ Wt := X 1[0,t] , t ∈ [0, 1] ⊗t0 2 and this implies in particular that, for every n ≥ 2, the multiple Wiener-Itô integral I nX (f ), f ∈ n L2 ([0, 1] ), can be rewritten in terms of an iterated stochastic integral with respect to W , that is: X In (f ) = In1 ((f )s ) = n!Jn1 ((f )s ), where Jnt ((f )s ) = Z t 0 ··· Z un−1 (f (u1 , ..., un ))s dWun ...dWu1 0 Int ((f )s ) = n!Jnt ((f )s ) , t ∈ [0, 1] . 3 d - dimensional CLT The following facts will be used to prove our main results. Let H = L 2 (T, A, µ), f ∈ H ⊙n and g ∈ H ⊙m . Then, F1: (see [1, p. 211] or [8, Proposition 1.1.3]) InX X (f ) Im (g) = n∧m X p=0    n m X (f ⊗p g) ; I p! p n+m−2p p (1) F2: (see [15, Proposition 1]) 2 (n + m)! k(f ⊗0 g)s kH ⊗n+m 2 2 = m!n! kf kH ⊗n kgkH ⊗m n∧m X nm 2 + n!m! kf ⊗q gkH ⊗n+m−2q q q q=1 (2) F3: (see [9]) n−1 h i X 4 2 4 E InX (f ) = 3 (n!) kf kH ⊗n + p=1 h 4 (n!) 2 2 kf ⊗p f kH ⊗2(n−p) (p! (n − p)!)   2n − 2p + (f ⊗p f )s n−p (3) 2 H ⊗2(n−p)  . 4 Let Vd be the set of all (i1 , i2 , i3 , i4 ) ∈ (1, ..., d) , such that one of the following conditions is satisfied: (a) i1 6= i2 = i3 = i4 , (b) i1 6= i2 = i3 6= i4 and i4 6= i1 , (c) the elements of (i1 , ..., i4 ) are all distinct. Our main result is the following. Theorem 1 Let d ≥ 2, and consider a collection 1 ≤ n1 ≤ ... ≤ nd < +∞ of natural numbers, as well as a collection of kernels  o n (k) (k) :k≥1 f1 , ..., fd (k) such that fj ∈ H ⊙nj for every k ≥ 1 and every j = 1, ..., d, and (k) 2 lim j! fj ⊗nj k→∞   H i h  (k) (k) InXl fl lim E InXi fi k→∞ Then, the following conditions are equivalent: 3 = 1, ∀j = 1, ..., d = 0, ∀1 ≤ i < l ≤ d. (4) (i) for every j = 1, ..., d lim (k) k→∞ fj (k) ⊗p fj H ⊗2(nj −p) for every p = 1, ..., nj − 1;   4  P (k) X (ii) limk→∞ E = 3d2 , and i=1,...,d Ini fi lim E k→∞ " 4 Y InXi l l=1  (k) fil  # =0 =0 for every (i1 , i2 , i3 , i4 ) ∈ Vd ;      (k) (k) (iii) as k goes to infinity, the vector InX1 f1 , ..., InXd fd converges in distribution to a ddimensional standard Gaussian vector Nd (0, Id );   (k) (iv) for every j = 1, ..., d, InXj fj converges in distribution to a standard Gaussian random variable; (v) for every j = 1, ..., d,   4  (k) lim E InXj fj = 3. k→∞ Proof. We show the implications (iii) ⇒ (ii) ⇒ (i) ⇒ (iii) and (iv) ⇔ (v) ⇔ (i)     (k) (k) [(iii) ⇒ (ii)] First notice that, for every k ≥ 1, the multiple integrals I nX1 f1 , ..., InXd fd are contained in the sum of the first nd chaoses associated to the Gaussian measure X. As a consequence, condition (4) implies (see e.g. [3, Chapter V]) that for every M ≥ 2 and for every j = 1, ..., d    M (k) sup E InXj fj < +∞ k≥1 and the conclusion is obtained by standard arguments. [(ii) ⇒ (i)] The key of the proof is the following simple equality  !  d   4 X (k)  InXi fi E i=1 = d X i=1   4  (k) E InXi fi +6 + X E (i1 ,...,i4 )∈Vd " 4 Y l=1 InXi l X 1≤i<j≤d  (k) fil    2  2 (k) (k) InXi fi E InXj fj #  . By the multiplication formula (1), for every 1 ≤ i < j ≤ d InXi  (k) fi  InXj  (k) fj     ni   X ni nj X (k) (k) q! Ini +nj −2q fi ⊗q fj = q q q=0 4 and therefore   ni    2  2  X   2 ni nj (k) (k) (k) (k) X X = (ni + nj − 2q)! fi ⊗q fj q! Inj fj E Ini fi q q s q=0 2 H ⊗ni +nj −2q . Now, relations (2) and (3) imply that  !  d   4 X (k)  = T1 (k) + T2 (k) + T3 (k) InX fi E i i=1 where T1 (k) = d X i=1 ( X T2 (k) = 6 2 3 (ni !) 1≤i<j≤d 4 (k) fi + nX i −1 (ni !) 4 2 (p! (ni − p)!)     2ni − 2p (k) (k) + fi ⊗p fi ni − p s H ⊗ni  (k) ni !nj ! fi p=1 2 (k) fj H ⊗ni 2 H ⊗nj  (k) fi (k) ⊗p fi 2 H ⊗2(ni −p) H ⊗2(ni −p)  + "    ni 2   X ni nj (k) (k) + (ni + nj − 2q)! fi ⊗q fj q! q q s q=1     2 ni nj (k) (k) + ni !nj ! fi ⊗q fj , q q H ⊗nj +ni −2q and X T3 (k) = E (i1 ,...,i4 )∈Vd " 4 Y InXi l l=1 2 2 H ⊗ni +nj −2q #   (k) . fil But 3 d X 2 (ni !) (k) fi i=1 4 H ⊗ni +6 X ni !nj ! (k) fi 1≤i<j≤d 2 H ⊗ni (k) fj 2 H ⊗nj =3 " d X ni ! i=1 (k) fi 2 H ⊗ni #2 and the desired conclusion is immediately obtained, since condition (4) ensures that the right side of the above expression converges to 3d2 when k goes to infinity. [(i) ⇒ (iii)] We will consider the case H = L2 ([0, 1] , B ([0, 1]) , dx) (5) where dx stands for Lebesgue measure, and use the notation introduced at the end of Section 2. We stress again that the extension to a general, separable Hilbert space H can be done by following the line of reasoning presented in [9, Section 3.3.] and it is not detailed here. Now suppose (i) and (5) hold. The result is completely proved, once the asymptotic relation d X i=1 d    X  Law (k) (k) λi InXi fi λi ni !Jn1i fi ⇒ = k↑+∞ i=1 5 λd Rd × N (0, 1) is verified for every vector λd = (λ1 , ..., λd ) ∈ Rd . Thanks to the Dambis-Dubins-Schwarz Theorem (see [12, Chapter V]), we know that for every k, there exists a standard Brownian motion W (k) (which depends also on λd ) such that  !  Z 1 X d d     2 X (k) (k) = W (k)  λi ni !Jnt i −1 fi,t dt λi ni !Jn1i fi 0 i=1 " = W (k) d X λ2i i=1 X +2 Z  1 0 λi λj ni !nj ! Z 1 0 1≤i<j≤d Now, since (4) implies E i=1  2 (k) ni !Jnt i −1 fi,t dt h  (k) Jnt i −1 fi,t   (k) Jnt j −1 fj,t i  dt .   2  (k) ni !Jn1i fi → 1 k↑+∞ for every i, condition (i) yields – thanks to Proposition 3 in [9] – that d X λ2i i=1 Z 1 0   2 L2 (k) ni !Jnt i −1 fi,t dt → k↑+∞ λd 2 ℜd . To conclude, we shall verify that (i) implies also that for every i < j h    i (k) (k) t Z 1h Z 1 It  i   f I f ni −1 nj −1 i,t j,t L2 (k) (k) dt = dt → 0. Jn1i −1 fi,t Jn1j −1 fj,t k↑+∞ (ni − 1)! (nj − 1)! 0 0 To see this, use once again the multiplication formula (1) to write Z 1 h    i (k) (k) dt Int i −1 fi,t Int j −1 fj,t = 0 nX i −1 q=0 (ni + nj − 2 (q + 1))!q! × Z n +nj −2(q+1) ∆1 i Z    ni − 1 nj − 1 × q q 1 s1   (k) (k) dt fi,t ⊗tq fj,t when ni < nj , or, when ni = nj Z 1 h    i (k) (k) dt Int i −1 fi,t Int j −1 fj,t = Z 0 1 dtE 0 h × Int i −1 Z  s1 , ..., sni +nj −2(q+1)  dWs1 ...dWsni +nj −2(q+1) , 2  ni − 1 × + (2ni − 2 (q + 1))!q! q q=0 Z 1   
(k) t (k) s1 , ..., sni +nj −2(q+1) dt fi,t ⊗q fj,t −2(q+1) (k) fi,t n +nj ∆1 i s
 Int i −1  (k) fj,t i nX i −2 s s1 dWs1 ...dWsni +nj −2(q+1) . In what follows, for every m ≥ 2, we write tm to indicate a vector (t1 , ..., tm ) ∈ Rm , whereas dtm stands for Lebesgue measure on Rm ; we shall also use the symbol btm = maxi (ti ). Now fix q < ni − 1 ≤ nj − 1, 6 and observe that, by writing p = q + 1, Z Z ≤ Z n +nj −2(q+1) ∆1 i s1 Z dsni −p [0,1]ni −p ×   (k) (k) dt fi,t ⊗tq fj,t 1 "Z s1 , ..., sni +nj −2(q+1) s
2 ds1 ...dsni +nj −2(q+1) dτnj −p [0,1]nj −p Z 1 dt [0,t]p−1 b sni −p ∨b τnj −p (k) dup−1 fj
(k) t, τnj −p , up−1 fi = C (k) (t, sni −p , up−1 ) #2 and moreover 2 C (k) = (Z × Z 1 dt [0,1]p−1 0 "Z [0,t∧t′ ]ni −p dup−1 [0,t∧t′ ]nj −p Z 1 dt′ [0,1]p−1 0 (k) dsni −p fi "Z × Z dvp−1 1(bup−1 ≤t,bvp−1 ≤t′ ) (k) (t, sni −p , up−1 ) fi (t′ , sni −p , vp−1 )
(k) dτnj −p fj (k) t, τnj −p , up−1 fj ≤ Ci (k) × Cj (k) # ′ t , τnj −p , vp−1
#)2 where, for γ = i, j Cγ (k) Z = 1 dt [0,1]p−1 0 "Z Z [0,t∧t′ ]nγ −p Z dup−1 Z 1 dt ′ 0 [0,1]p−1 dvp−1
dsnγ −p fγ(k) t, snγ −p , up−1 fγ(k) ′ t , snγ −p , vp−1
#2 and the calculations contained in [9] imply immediately that both C j (k) and Ci (k) converge to zero whenever (i) is verified. On the other hand, when q = n i − 1 < nj − 1 Z n −ni ∆1 j ≤ Z Z   (k) (k) dt fi,t ⊗tni −1 fj,t 1 s1 s1 , ..., snj −ni s
2 ds1 ...dsnj −ni dτnj −ni [0,1]nj −ni × "Z Z 1 dt (k) duni −1 fj [0,t]ni −1 τbnj −ni
(k) t, τnj −ni , uni −1 fi = D (k) and also (t, uni −1 ) #2 2 D (k) ≤ D1 (k) × D2 (k) where D1 (k) = Z 1 dt 0 × "Z Z [0,1]ni −1 duni −1 [0,t∧t′ ]nj −ni Z Z 1 dt ′ [0,1]ni −1 0 (k) dτnj −ni fj dvni −1
(k) t, τnj −ni , uni −1 fj 7 ′ t , τnj −ni , vni −1
#2 and D2 (k) = Z 1 dt 0 = (k) fi Z [0,1]ni −1 4 duni −1 Z 1 dt ′ Z [0,1]ni −1 0  2 (k) (k) dvni −1 fi (t, uni −1 ) fi (t′ , vni −1 ) H ⊗ni so that the conclusion is immediately achieved, due to (4). Finally, recall that for n i = nj Z 1 0 Z h    i (k) (k) dtE Int i −1 fi,t Int i −1 fj,t = (ni − 1)! Z (k) (k) dt duni −1 fj (t, uni −1 ) fi (t, uni −1 ) ni −1 0 [0,t] Z (k) (k) 2 dtduni −1 fj (t, uni −1 ) fi (t, uni −1 ) = ((ni − 1)!) 1 n =  (ni − 1)! ni ! 2 ∆1 i h    i (k) (k) InXi fj → 0 E InXi fi k↑+∞ again by assumption (4). The proof of the implication is concluded. [(iv) ⇔ (v) ⇔ (i)] This is a consequence of Theorem 1 in [9].  In what follows, Cd = {Cij : 1 ≤ i, j ≤ d} indicates a d × d positive definite symmetric matrix. In the case of multiple Wiener integrals of the same order, a useful extension of Theorem 1 is the following Proposition 2 Let d ≥ 2, and fix n ≥ 2 as well as a collection of kernels n  o (k) (k) f1 , ..., fd :k≥1 (k) such that fj ∈ H ⊙n for every k ≥ 1 and every j = 1, ..., d, and (k) 2 lim j! fj ⊗n k→∞ h    H i (k) (k) lim E InX fi InX fj k→∞ = Cjj , ∀j = 1, ..., d = Cij , ∀1 ≤ i < j ≤ d. (6) Then, the following conditions are equivalent:      (k) (k) (i) as k goes to infinity, the vector InX f1 , ..., InX fd converges in distribution to a d-dimensional Gaussian vector Nd (0, Cd ) = (N1 , ..., Nd ) with covariance matrix Cd ; (ii)  lim E  k→∞ and  X i=1,...,d 4   d   X  (k) X   In fi Cii + 2 =3 i=1 lim E k→∞ " 4 Y l=1 InX  (k) fil  # X 1≤i<j≤d =E " 4 Y l=1 2  Cij  = E  Nil d X i=1 Ni !4  , # for every (i1 , i2 , i3 , i4 ) ∈ Vd ;   (k) (iii) for every j = 1, ..., d, InX fj converges in distribution to Nj , that is, to a centered Gaussian random variable with variance Cjj ; 8 (iv) for every j = 1, ..., d,   4  (k) X 2 lim E In fj ; = 3Cjj k→∞ (v) for every j = 1, ..., d (k) lim fi k→∞ (k) ⊗p fi H ⊗2(n−p) = 0, for every p = 1, ..., n − 1. Sketch of the proof – The main idea is contained in the proof of Theorem 1. We shall discuss only implications (ii) ⇒ (v) and (v) ⇒ (i). In particular, one can show that (ii) implies (v) by adapting the same arguments as in the proof of Theorem 1 to show that  !  d   4 X (k)  = V1 (k) + V2 (k) + V3 (k) InX fi E i=1 where d X V1 (k) = i=1 ( 3 (n!) + n−1 X 4 (n!) 2  H ⊗n p=1 (p! (n − p)!)     2n − 2p (k) (k) + fi ⊗p fi n−p s X V2 (k) = 6 4 (k) fi 2 1≤i<j≤d  2 (n!) 2 (k) fi (k) H ⊗n fj 2 H ⊗n (k) fi 2 H ⊗2(n−p) H ⊗2(n−p)  +   2 !2   (k) (k)  q! n + (2n − 2q)! fi ⊗q fj q s q=1 #)  2 2 n (k) (k) 2 + (n!) fi ⊗q fj q H ⊗2n−2q X D (k) (k) E2 2 + 12 (n!) fi , fj n−1 X 2 H ⊗2n−2q H ⊗n 1≤i<j≤d and X V3 (k) = 2 (k) ⊗p fi E (i1 ,...,i4 )∈Vd " 4 Y InX l=1  (k) fil  # . But (6) yields 3 (n!) 2 d X i=1 → 3 k↑+∞ d X i=1 (k) fi 4 H ⊗n Cii2 + 6 X +6 1≤i<j≤d X 1≤i<j≤d  i=1 (k) 2 (n!) fi 2 Cii Cjj + 2Cij and the conclusion is obtained, since  !4  d d X X   E Ni Cii2 + 6 =3 i=1  X 1≤i<j≤d 2 H ⊗n (k) fj 2 H ⊗n + 2 (n!) 2 D (k) (k) fi , fj E2 H ⊗n    2 + Cii Cjj + 2Cij 9 X (i1 ,...,i4 )∈Vd E " 4 Y l=1 # Nil .  Now keep the notations of the last part of the proof of Theorem 1. The implication (v) ⇒ (i) follows from the calculations therein contained, implying, thanks to (6), that the quantity Z d X 1 0 t λi n!Jn−1 i=1  (k) fi,t  !2 dt P P 2 converges in L2 to i=1,...,d λi Cii + 2 1≤i<j≤d λi λj Cij , and therefore the desired conclusion. The remaining details can be easily provided by the reader.  4 Applications In this section, we will present some consequences of our results. We mention that our list of applications is by no means exhaustive; for instance, the weak convergence results for quadratic functionals of (fractional) Brownian motion given in [9], [10] and [11] can be immediately extended to the multidimensional case. An example is given in the following generalization of the results contained in [11]. Proposition 3 Let W be a standard Brownian motion on [0, 1] and, for every d ≥ 2, define the process t 7→ Wt⊗d := Z t 0 ··· Z Then: (a) for every d ≥ 1 the vector q 1 log 1ε Z ε 1 da ⊗2 W , a2 a sd−1 dWsd ...dWs1 , 0 Z ε 1 da ⊗4 W , ..., a3 a Z 1 ε t ∈ [0, 1] . da W ⊗2d ad+1 a  converges in distribution, as ε → 0, to   p √ N1 (0, 1) , 2 3!N2 (0, 1) , ..., d (2d − 1)!Nd (0, 1) where the Nj (0, 1), j = 1, ..., d, are standard, independent Gaussian random variables; (b) by defining, for every d ≥ 1 and for every j = 0, ..., d, the positive constant c (d, j) = (2d)! , (d − j)! 2d−j for every d ≥ 1 the vector Z 1 Z 1 1 da 2 1 1 da 4 q Wa − c (1, 0) log , Wa − c (2, 0) log , ... 2 3 a ε a ε ε ε log 1ε  Z 1 1 da 2d Wa − c (d, 0) log ..., d+1 ε ε a converges in distribution to a Gaussian vector (G1 , ..., Gd ) with the following covariance structure: ′ E [Gk′ Gk ] = k X j=1 c (k, j) c (k ′ , j) j 2 (2j − 1)! for every 1 ≤ k ′ ≤ k ≤ d. 10 Proof. From Proposition 4.1 in [11], we obtain immediately that for every j = 1, ..., d, Z 1 da 1 (d) p q Wa⊗2j → j (2j − 1)!Nj (0, 1) , j+1 log 1ε ε a and the asymptotic independence follows from Theorem 1, since for every i 6= j Z 1  Z 1 Z 1 Z 1 db  ⊗2j ⊗2i  da db da ⊗2i ⊗2j E W W E Wa Wb = a b j+1 i+1 j+1 i+1 ε b ε a ε b ε a = 0. To prove point (b), use for instance Stroock’s formula (see [13]) to obtain that for every k = 1, ..., d Z ε 1 k X da c (k, j) Wa2k = k+1 a j=1 Z 1 da 1 W ⊗2j + c (k, 0) log , aj+1 a ε ε so that the result derives immediately from point (a). In what follows, we prove a new asymptotic version of Knight’s theorem – of the kind discussed e.g. in [12, Chapter XIII] – and a necessary and sufficient condition for a class of random variables living in a finite sum of chaos – and satisfying some asymptotic property – to have a Gaussian weak limit. Further applications will be explored in a subsequent paper. More specifically, we are interested in an asymptotic Knight’s theorem for chaotic martingales, that is, martingales having a multiple Wiener integral representation (we stress that there is no relation with normal martingales with the chaotic representation property, as discussed e.g. in [1, Chapter XXI]). To this end, take d ≥ 2 integers 1 ≤ n1 ≤ n2 ≤ ... ≤ nd , and, for j = 1, ..., d and k ≥ 1 take a class  φtj,k : t ∈ [0, 1] of elements of H ⊙nj , such that there exists a filtration {Ft : t ∈ [0, 1]}, satisfying the usual conditions and such that, for every k and for every j, the process
t 7→ Mj,k (t) = InXj φtj,k , t ∈ [0, 1] , is a Ft - continuous martingale on [0, 1], vanishing at zero. We note hM j,k , Mj,k i and hMj,k , Mi,k i, 1 ≤ i, j ≤ d, the corresponding quadratic variation and covariation processes, whereas β j,k is the DambisDubins-Schwarz Brownian motion associated to M j,k . Then, we have the following Proposition 4 (Asymptotic Knight’s theorem for chaotic martingales) Under the above assumptions and notation, suppose that for every j = 1, ..., d, hMj,k , Mj,k i (d) → k→+∞ Tj , where t 7→ Tj (t) is a deterministic, continuous and non-decreasing process. If in addiction   lim E hMi,k , Mj,k it = 0 k→+∞ for every i 6= j and for every t, then {Mj,k : 1 ≤ j ≤ d} converges in distribution to {Bj ◦ Tj : 1 ≤ j ≤ d} , where {Bj : 1 ≤ j ≤ d} is a d dimensional standard Brownian motion. 11 (7) (8) Proof. Since
Mj,k (t) = βj,k hMj,k , Mj,k it , t ∈ [0, 1] , and hMj,k , Mj,k i weakly converges to Tj , we immediately obtain that Mj,k converges in distribution to the Gaussian process Bj ◦ Tj . Thanks to Theorem 1, it is now sufficient to prove that, for every i 6= j and for every s, t ∈ [0, 1], the quantity E [Mj,k (s) Mi,k (t)] converges to zero. But   E [Mj,k (s) Mi,k (t)] = E hMi,k , Mj,k it∧s and assumption (8) yields the result. Remark – An analogue of Proposition 4 for general martingales verifying (7) can be found in [12, Exercise XIII.1.16], but in this case (8) has to be replaced by hMj,k , Mi,k i (d) → k→+∞ 0 for every i 6= j. Since chaotic martingales have a very explicit covariance structure (due to the isometric properties of multiple integrals), condition (8) is usually quite easy to verify. We also recall that – according e.g. to [12, Theorem XIII.2.3] – if condition (7) is dropped, to prove the asymptotic independence of the Brownian motions {βj,k : 1 ≤ j ≤ d} one has to check the condition lim hMi,k , Mj,k iτ k (t) = lim hMi,k , Mj,k iτ k (t) = 0 k→+∞ k→+∞ j i in probability for every i 6= j and for every t, where τ jk and τik are the stochastic time-changes associated respectively to hMj,k , Mj,k i and hMi,k , Mi,k i. We conclude the paper by stating a result on the weak convergence of random variables belonging to a finite sum of Wiener chaos to a standard normal random variable (the proof is a direct consequence of the arguments contained in the proof of Theorem 1). (k) Proposition 5 Let 1 ≤ n1 < ... < nd , d ≥ 2, and let fj ∈ H ⊙nj , for every k ≥ 1 and 1 ≤ j ≤ d. Assume that 2 (k) nj ! lim fj = 1, j = 1, ..., d, (9) ⊗n k↑+∞ H and lim k↑+∞ (k) Define moreover Sd = (k) (i) the sequence d−1/2 Sd to infinity; X E j " 4 Y InXi l l=1 (i1 ,...,i4 )∈Vd  (k) fil  # ≥ 0. (10)   (k) X f I . Then, the following conditions are equivalent: j j=1,...,d nj P converges in distribution to a standard Gaussian random variable, as k tends (ii) for every j = 1, ..., d, lim k↑+∞ (k) fj (k) ⊗p fj 2 H ⊗2(nj −p) = 0, p = 1, ..., nj − 1;   (k) (iii) for every j = 1, ..., d, InXj fj converges in law to a standard Gaussian random variable, as k goes to infinity. 12 An interesting consequence of the above result is the following (k) ∈ H ⊙nj , k ≥ 1 and 1 ≤ j ≤ d. Assume moreover that (9)   (k) is verified and that, for every k, the random variables InXj fj , j = 1, ..., d, are pairwise independent. Corollary 6 Let 1 ≤ n1 < ... < nd , d ≥ 2, fj (k) Then, the sequence d−1/2 Sd , k ≥ 1, defined asbefore,  converges in law to a standard Gaussian random (k) X variable N (0, 1) if, and only if, for every j Inj fj converges in law to N (0, 1). Proof. We know from [15] that, in the case of multiple stochastic integrals, pairwise independence implies mutual independence, so that condition (10) is clearly verified.   (k) Remarks – (i) If we add the assumption that, for every j, the sequence I nXj fj , k ≥ 1, admits a weak limit, say µj , then the conclusion of Corollary 6 can be directly deduced from [4, p. 248]. As a matter of fact, in such a reference R the following implication is proved: if the d probability measures µj , j = 1, ..., d, are such that (a) xdµj (x) = 0 for every j, and (b) µ1 ⋆ · · · ⋆ µd , where ⋆ indicates convolution, is Gaussian, then each µj is necessarily Gaussian. (ii) Condition (10) is also satisfied when d = 2 and n 1 + n2 is odd. References [1] Dellacherie, C., Maisonneuve, B. and Meyer, P.A. (1992), Probabilités et Potentiel, Chapitres XVII à XXIV, Hermann, Paris. [2] Giraitis, L. and Surgailis, D. (1985), “CLT and Other Limit Theorems for Functionals of Gaussian Processes”, Z. Wahr. verw. Gebiete 70(2), 191-212. [3] Janson, S. (1997), Gaussian Hilbert spaces, Cambridge University Press, Cambridge. [4] Lukacs, E. (1983), Developments in characteristic functions, MacMillian Co., New York. [5] Major, P. (1981), Multiple Wiener-Itô Integrals, Lecture Notes in Mathematics 849, Springer Verlag, New York. [6] Maruyama, G. (1982), “Applications of the multiplication of the Ito-Wiener expansions to limit theorems”, Proc. Japan Acad. 58, 388-390. [7] Maruyama, G. (1985), “Wiener functionals and probability limit theorems, I: the central limit theorem”, Osaka Journal of Mathematics 22, 697-732. [8] Nualart, D. (1995), The Malliavin Calculus and Related Topics, Springer, Berlin Heidelberg New York. [9] Nualart, D. and Peccati, G. (2003), “Central limit theorems for sequences of multiple stochastic integrals”, to appear in The Annals of Probability. [10] Peccati, G. and Yor, M. (2003a), “Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge”, to appear in the volume: Asymptotic Methods in Stochastics, American Mathematical Society, Communication Series. [11] Peccati, G. and Yor, M. (2003b), “Hardy’s inequality in L 2 ([0, 1]) and principal values of Brownian local times”, to appear in the volume: Asymptotic Methods in Stochastics, American Mathematical Society, Communication Series. 13 [12] Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion, Springer, Berlin Heidelberg New York. [13] D. W. Stroock (1987), “Homogeneous chaos revisited”, in: Séminaire de Probabilités XXI, Springer, Berlin, LNM 1247, 1-8. [14] Surgailis, D. (2003), “CLTs for Polynomials of Linear Sequences: Diagram Formula with illustrations”, in: Theory and Applications of Long Range Dependence, Birkhäuser, Boston. [15] Üstünel, A. S. and Zakai, M. (1989), “Independence and conditioning on Wiener space”, The Annals of Probability 17 (4), 1441-1453. 14