Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

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Model 1

pp. 1–28 (col. fig: NIL)

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Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos Ciprian A. Tudor a , ∗, Nakahiro Yoshida b , c a Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France b Graduate School of Mathematical Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153, Japan c Japan Science and Technology Agency, CREST, Japan

Received 16 February 2018; received in revised form 8 September 2018; accepted 27 September 2018 Available online xxxx

Abstract We develop the asymptotic expansion theory for vector-valued sequences (FN ) N ≥1 of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence FN . Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of FN and we illustrate our results by several examples. c 2018 Published by Elsevier B.V. ⃝ MSC: 62M09; 60F05; 62H12 Keywords: Asymptotic expansion; Stein–Malliavin calculus; Quadratic variation; Fractional Brownian motion; Central limit theorem; Fourth moment theorem

1. Introduction The analysis of the convergence in distribution of sequences of random variables constitutes a fundamental direction of research in probability and statistics. In particular, the asymptotic expansion represents a technique that allows one to give a precise approximation for the densities of probability distributions (see, among many others, [6,13,17,15,16]). Our work concerns the convergence in law of random sequences to the Gaussian distribution. Although our context is more general, we will focus on examples involving elements of the ∗ Corresponding author.

E-mail addresses: [email protected] (C.A. Tudor), [email protected] (N. Yoshida). https://doi.org/10.1016/j.spa.2018.09.018 c 2018 Published by Elsevier B.V. 0304-4149/⃝

Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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Gaussian Wiener chaos. The characterization of the convergence in distribution of sequences of random variables belonging to a Wiener chaos of fixed order to the Gaussian law constitutes an important research direction in stochastic analysis since several years. This research line was initiated by the seminal paper [11] where the authors proved the famous Fourth Moment Theorem. This result, which basically says that the convergence in law of a sequence of multiple stochastic integrals with unit variance is equivalent to the convergence of the sequence of the fourth moments to the fourth moment of the Gaussian distribution, has then been extended, refined, completed or applied to various situations. The reader may consult the recent monograph [9] for the basics of this theory. In this work, we are also concerned with some refinements of the Fourth Moment Theorem for sequences of random variables in a Wiener chaos of fixed order. More concretely, we aim at finding the second order expansion for the density. This is a natural extension of the Fourth Moment Theorem and of its ramifications. In order to explain this link, let us briefly describe the context. Let H be a real and separable Hilbert space, (W (h), h ∈ H ) an isonormal Gaussian process on a probability space (Ω , F, P) and let Iq denote the qth multiple stochastic integral with respect to W . Let Z ∼ N (0, 1). Denote by dT V (F, G) the total variation distance between the laws of the random variables F and G and let D be the Malliavin derivative with respect to W . The Fourth Moment Theorem from [11] can be stated as follows. Theorem 1. Fix q ≥ 1. Consider a sequence {Fk = Iq ( f k ), k ≥ 1} of random variables in the qth Wiener chaos. Assume that lim EFk2 = lim q!∥ f k ∥2H ⊗q = 1.

k→∞ 22

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k→∞

(1)

Then, the following statements are equivalent: 1. The sequence of random variables {Fk = In ( f k ), k ≥ 1} converges in distribution as k → ∞ to the standard normal law N (0, 1). 2. limk→∞ E[Fk4 ] = 3. 3. ∥D Fk ∥2H converges to q in L 2 (Ω ) as k → ∞. 4. dT V (Fk , Z ) →k→∞ 0. A more recent point of interest in the analysis of the convergence to the normal distribution of sequences of multiple integrals is represented by the analysis of the convergence of the sequence of densities. For every N ≥ 1, denote by p FN the density of the random variable FN (which exists due to a result in [14]). Since the total variation distance between two continuous random variables F and G (or ∫ between the distributions of the random variables F and G) can be written as dT V (F, G) = 12 R | p F (x) − pG (x)|d x, we can notice that point 4. in the Fourth Moment Theorem implies that ∥ p FN (x) − p(x)∥ L 1 (R) → N →∞ 0 x2

36 37

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where p(x) = √12π e− 2 denotes, throughout our work, the density of the standard normal distribution. The result has been improved in the work [7], where the authors showed that if lim E∥D FN ∥−4−ε <∞ H

N →∞ 39

40

then the conditions 1.–4. in the Fourth Moment Theorem are equivalent to √ ⏐ ⏐ sup ⏐ p FN (x) − p(x)⏐ ≤ c (EFN4 − 3).

(2)

(3)

x∈R

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We also refer to [3,4] for other references related to the density convergence on Wiener chaos. Our purpose is to find the asymptotic expansion in the Fourth Moment Theorem (and, more generally, for sequences of random variables converging to the normal law). By finding the asymptotic expansion we mean to find a sequence ( p N ) N ≥1 of the form p N (x) = p(x) + r N ϕ(x) such that r N → N →∞ 0 and ϕ is a deterministic function not depending on N , which approximates the density p FN in the following sense ⏐ ⏐ sup|x α | ⏐ p FN (x) − p N (x)⏐ = o(r N ) (4)

1 2 3 4 5 6 7

8

x∈R

for any α ∈ Z+ . This improves the main result in [7] and it will also allow us to generalize the Edgeworth-type expansion (see [8,1]) for sequences on Wiener chaos to a larger class of measurable functions. As mentioned before, we develop our asymptotic expansion theory for a general class of random variables, which are not necessarily multiple stochastic integrals, although our main examples are related to Wiener chaos. Let us briefly explain our strategy. The main idea to get the asymptotic expansion is to analyze the asymptotic behavior of the characteristic function of FN . Then, we find the approximate density p N by Fourier inversion of the dominant part of( the characteristic function. Our main ) assumption is that the vector-valued random sequence FN , γ N−1 (Λ(FN ) − C) converges to a d + d × d-dimensional Gaussian vector, where (γ N ) N ≥1 is a deterministic sequence converging to zero as N → ∞, Λ(FN ) is the so-called Stein–Malliavin matrix of FN whose components are ( j) Λi, j = ⟨D FN(i) , D(−L)−1 FN ⟩ H for i, j = 1, ..., d (L is the Ornstein–Uhlenbeck operator) and (i) ( j) Ci, j = lim N →∞ EFN FN . In the one-dimensional case the assumption becomes ( ) FN , γ N−1 (⟨D FN , D(−L)−1 FN ⟩ H − 1) (5) converges in distribution, as N → ∞, to a two-dimensional centered Gaussian vector (Z 1 , Z 2 ) with EZ 12 = EZ 22 = 1 and EZ 1 Z 2 = ρ ∈ [−1, 1]. The condition (5) is related to what is usually assumed in the asymptotic expansion theory for martingales, which has been developed in the last decades and it is based on the so-called Fourier approach. Let us refer, among many others, to [6,17,13,15,16] for few important works on the martingale approach in asymptotic expansion. Recall that if (M N ) N ≥1 denotes a sequence of random variables converging to Z ∼ N (0, 1) such that M N is the terminal value of a continuous martingale and if ⟨M⟩ N is the martingale’s bracket, then one assumes in (e.g. [17]), [15] that ⟨M⟩ N → N →∞ 1 in probability

11 12 13 14 15 16 17 18 19 20 21

22

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and the following joint convergence holds true ) ( ⟨M⟩ N − 1 →(d) MN , N →∞ (Z 1 , Z 2 ) γN

32

(6)

where (Z 1 , Z 2 ) is a centered Gaussian vector as above. The notation “→(d) ” stands for the convergence in distribution. In our assumption (5), the role of the martingale bracket is played by ⟨D FN , D(−L)−1 FN ⟩ H which becomes q1 ∥D FN ∥2H when FN is an element of the qth Wiener chaos. The choice is natural, suggested by the identity EFN2 = E⟨F⟩2N which plays the role of the Itˆo isometry and by the fact 2 that ∥D FqN ∥ − 1 converges to zero in L 2 (Ω ), due to the Fourth Moment Theorem. A similar assumption with (5) in terms of the joint convergence of FN and

9 10

∥D FN ∥2 q

− 1 is also assumed

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in [8] for exact Berry–Ess´en asymptotics of sequences in the qth Wiener chaos. In particular, from our Corollary 1 we can recover the result in Theorem 3.1 in [8]. We organized our paper as follows. In Section 2, we fix the general context of our work, the main assumptions and some notation. In Section 3, we analyze the asymptotic behavior of the characteristic function of a random sequence that satisfies our assumptions while in Section 4 we obtain the approximate density and the asymptotic behavior via Fourier inversion. In Section 5, we illustrate our theory by several examples related to Wiener chaos.

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2. Assumptions and notation

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We will here present the basic assumptions and the notation utilized throughout our work. Let H be a real and separable Hilbert space endowed with the inner product ⟨·, ·⟩ H and consider (W (h), h ∈ H ) an isonormal Gaussian process on the probability space (Ω , F, P). In the sequel, D, L and δ represent the Malliavin derivative, the Ornstein–Uhlenbeck operator and the divergence integral with respect to W . See the Appendix for their definition and basic properties.

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2.1. The main assumptions

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Consider a sequence of centered random variables (FN ) N ≥1 in Rd of the form ( ) FN = FN(1) , . . . , FN(d) . Denote by Λ(FN ) = (Λi, j )i, j=1,...,d the Stein–Malliavin matrix with components ( j)

19 20 21

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Λi, j = ⟨D FN(i) , D(−L)−1 FN ⟩ H for every i, j = 1, . . . , d. We consider the following assumptions: [C1] There exists a symmetric (strictly) positive definite d ×( d matrix C and a deterministic ) sequence (γ N ) N ≥1 converging to zero as N → ∞ such that FN , γ N−1 (Λ(FN ) − C) converges in distribution to a d +d ×d Gaussian random vector (Z 1 , Z 2 ) such that Z 1 = (Z 1(1) , . . . , Z 1(d) ), Z 2 = (Z 2(k,l) )k,l=1,...,d with ( j)

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EZ 1(i) Z 1 = Ci, j , for every i, j = 1, . . . , d, where C = (Ci, j )i, j=1,..d . [C2] For every i = 1, . . . , d, the sequence (FN(i) ) N ≥1 is bounded in Dℓ+1,∞ = ∩ p>1 Dℓ+1, p for some l ≥ d + 3. ( ( )) [C3] With γ N , C from [C1], for every i, j = 1, d the sequence γ N−1 Λi, j − Ci, j N ∈N is bounded in Dℓ,∞ for some l ≥ d + 3. Notice that the matrix Λ(FN ) is not necessarily symmetric (except when all the components belong to the same Wiener chaos) since in general Λi, j ̸= Λ j,i . Nevertheless, we )assumed in ( [C1] that it converges to the symmetric matrix C, in the sense that γ N−1 Λ(FN ) − C converges in law to normal random variable. In the case of random variables in Wiener chaos, from the main result in [12] (see Theorem 4 later in the paper), we can prove that, under the convergence in law of FN(i) , i = 1, . . . , d to the Gaussian law, the quantities Λi, j − Ci, j , Λ j,i − Ci, j converge to zero in L 2 (Ω ). Thus condition [C1] intuitively means that these two terms have the same rate of convergence to zero. Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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2.2. The truncation

1

Let us introduce the truncation sequence (Ψ N ) N ≥1 which will be widely used throughout our work. Its use allows one to avoid the condition on the existence of negative moments for the Malliavin derivative of FN stated in (2). We consider the function Ψ N ∈ Dℓ, p for p > 1 and ℓ ≥ d + 3 such that (( ) ) |Λ(FN ) − C|d×d 2 (7) ΨN = ψ K γ Nδ with some K , 0 < δ < 1, where ψ ∈ C ∞ (R; [0, 1]) is such that ψ(x) = 1 is |x| ≤ 12 and ψ(x) = 0 if |x| ≥ 1. The square in the argument of ψ in order to have Ψ N differentiable in the Malliavin sense. From this definition, clearly we have 1. 0 ≤ Ψ N ≤ 1 for every N ≥ 1. 2. Ψ N → N →∞ 1 in probability.

2 3 4 5

6

7 8 9 10

11 12

A crucial fact in our computations is that on the set {Ψ N > 0} we have 1 |Λ(FN ) − C|d×d ≤ γ Nδ . K This also implies (see e.g. [17]) that there exist two positive constants c1 , c2 such that for all N , c1 ≤ det Λ(FN ) ≤ c2

(8)

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if we chose a large K .

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2.3. Notation

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For every x = (x1 , . . . , xd ) ∈ Rd and α = (α1 , . . . , αd ) ∈ Z+ we use the notation ∂α ∂ α1 ∂ αd α α xα = x1 1 ....xd d and α = α1 .... αd ∂x ∂ x1 ∂ xd

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If x, y ∈ Rd , we denote by ⟨x, y⟩d := ⟨x, y⟩ their scalar product in Rd . By |x|d := |x| we will denote the Euclidean norm of x. The following will be fixed in the sequel: • The sequence (FN ) N ≥1 defined in Section 2.1. • The truncation sequence (Ψ N ) N ≥1 is as in Section 2.2. • oλ (γ N ) denotes a deterministic sequence that depends on λ such that N) o(γ N ) we refer as usual to a sequence such that o(γ → N 0. γN

21 22 23

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oλ (γ N ) → N →∞ 0. γN

By

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2

• By Φ(λ) = e

− λ2

we denote the characteristic function of Z ∼ N (0, 1) and by p(x) =

28

2

x √1 e− 2 2π

its density.

• By c, C, C1 , . . . we denote generic strictly positive constants that may vary from line to line and by ⟨·, ·⟩ we denote the Euclidean scalar product Rd . On the other hand, we will keep the subscript for the inner product in H , denoted by ⟨·, ·⟩ H . • We will use bold letters to indicate vectors in Rd when we need to distinguish them from their one-dimensional components.

Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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3. Asymptotic behavior of the characteristic function The first step in finding the asymptotic expansion for the sequence (FN ) N ≥1 which satisfies [C1]–[C3] is to analyze the asymptotic behavior as N → ∞ of the characteristic function of FN . In order to avoid troubles related to the integrability over the whole real line, we will work under truncation, in the sense that we will always keep the multiplicative factor Ψ N given by (7). Let λ = (λ1 , . . . , λd ) ∈ Rd and θ ∈ [0, 1] and let us consider the truncated interpolation ) ( 2 λT Cλ . (9) ϕ NΨ (θ, λ) = E Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 Notice that ϕ N (1, λ) = E(Ψ N ei⟨λ,FN ⟩ ), the “truncated” characteristic function of FN , while λT Cλ ϕ N (0, λ) = (EΨ N )e− 2 , the “truncated” characteristic function of the limit in law of FN . Therefore, it is useful to analyze the behavior of the derivative of ϕ NΨ (θ, λ) with respect to the variable θ . We have the following result. Proposition 1. Assume [C1]–[C3] and suppose that d ( ) ∑ a E Z 2(k,l) |Z 1 = ρk,l Z 1(a)

(10)

a=1 15 16

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for every k, l = 1, . . . , d, where (Z 1 , Z 2 ) is the Gaussian random vector from [C1] and the a coefficients ρk,l are real numbers for a, k, l = 1, . . . , d. Then we have the convergence ⎛ ⎞ d ∑ ∂ λT Cλ λ j λk λi ρ aj,k Cia ⎠ e− 2 . (11) γ N−1 ϕ NΨ (θ, λ) → N →∞ −iθ 2 ⎝ ∂θ i, j,k,a=1 Proof. By differentiating (9) with respect to θ and using the formula FN(i) = δ D(−L)−1 FN(i) for every i = 1, . . . , d, we can write ( ) ) ∂ Ψ 2 λT Cλ ( ϕ N (θ, λ) = E Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 i⟨λ, FN ⟩ + θ λT Cλ ∂θ ⎞ ⎛ d ∑ T 2 λ Cλ ( j) = iE ⎝Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 λ j δ D(−L)−1 FN ⎠ j=1

22

+θ

d ∑

(

λ j λk C j,k E Ψ N eiθ ⟨λ,FN ⟩−(1−θ

2 ) λT Cλ 2

)

j,k=1 23

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and by the duality relationship (65) ( ) d ∑ T ∂ Ψ iθ ⟨λ,FN ⟩−(1−θ 2 ) λ 2Cλ −1 ( j) ϕ (θ, λ) = i λ j E ΨN e iθ ⟨D(−L) FN , D⟨λ, FN ⟩⟩ H ∂θ N j=1 ( ) d ∑ T iθ⟨λ,FN ⟩−(1−θ 2 ) λ 2Cλ −1 ( j) +i λjE e ⟨DΨ N , D(−L) FN ⟩ H j=1

26

+θ

d ∑

(

λ j λk C j,k E Ψ N e

T iθ ⟨λ,FN ⟩−(1−θ 2 ) λ 2Cλ

)

.

j,k=1

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We obtain, since D⟨λ, FN ⟩ =

7

(k) k=1 λk D FN ,

∑d

1

( d ∑ ∂ Ψ 2 λT Cλ ϕ N (θ, λ) = −θ λ j λk E Ψ N eiθ⟨λ,FN ⟩−(1−θ ) 2 ∂θ j,k=1 )) ( (k) −1 ( j) × ⟨D FN , D(−L) FN ⟩ H − C j,k +i

d ∑

2

3

( ) 2 λT Cλ ( j) λ j E eiθ ⟨λ,FN ⟩−(1−θ ) 2 ⟨DΨ N , D(−L)−1 FN ⟩ H

4

j=1

:= A N (θ, λ) + B N (θ, λ).

(12)

Using the definition of the truncation function Ψ N , we can prove that γ N−1 B N (θ, λ)

=

γ N−1

d ∑

5

6

( ) 2 λT Cλ ( j) iλ j E eiθ ⟨λ,FN ⟩−(1−θ ) 2 ⟨DΨ N , D(−L)−1 FN ⟩ H

7

j=1

→ N →∞ 0.

(13)

Indeed, by the chain rule (66) (( ) ) ( ) |Λ(FN ) − C|d×d 2 |Λ(FN ) − C|d×d 2 ′ K D K DΨ N = ψ γ Nδ γ Nδ

9

10

and by using [C2]–[C3] and (7), we get for every p > 1 ⏐ ⏐ ⏐ ⏐ ( j) γ N−1 |B N (θ, λ)| ≤ cγ N−1 E ⏐⟨DΨ N , D(−L)−1 FN ⟩ H ⏐ ( )1 1 δ 2 ≤ cγ N−1 P |Λ(FN ) − C|d×d ≥ γN 2K ( ( )2 p ) 12 −1−δp p(1−δ)−1 ≤ c E |Λ(FN ) − C|d×d γN ≤ cγ N

11

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where we used again [C2] and [C3]. Since δ < 1 and p is arbitrarily large, we obtain the convergence (13). On the other hand, by assumption [C1], we have the convergence γ N−1 A N (θ, λ) → N →∞ −θ

d ∑

( ) 2 λT Cλ ( j,k) λ j λk E eiθ⟨λ,Z 1 ⟩−(1−θ ) 2 Z 2 .

8

(14)

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j,k=1

Using our assumption (10), we can express the above limit in a more explicit way. We have ( ) ( ) 2 λT Cλ 2 λT Cλ ( j,k) ( j,k) E eiθ ⟨λ,Z 1 ⟩−(1−θ ) 2 Z 2 = E eiθ⟨λ,Z 1 ⟩−(1−θ ) 2 E(Z 2 |Z 1 ) ( ) d ∑ T (a) iθ ⟨λ,Z 1 ⟩−(1−θ 2 ) λ 2Cλ a =E e ρ j,k Z 1 (15) a=1 Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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and, moreover, for every a = 1, . . . , d, from the differential equation verified by the characteristic function of the normal distribution, ( ) ( ) 2 λT Cλ 2 λT Cλ E eiθ ⟨λ,Z 1 ⟩−(1−θ ) 2 Z 1(a) = e−(1−θ ) 2 E eiθ ⟨λ,Z 1 ⟩ Z 1(a)

4

= e−(1−θ

5

= e−

6

7

2 ) λT Cλ 2

λT Cλ 2

d −1 ∑ 2 λT Cλ ( λi Cia )θ 2 e−θ 2 iθ i=1

d d −θ ∑ λT Cλ ∑ ( λi Cia ) = iθ e− 2 λi Cia . i i=1 i=1

Thus, by replacing (15) and (16) in (14), we obtain ⎛ ⎞ d ∑ λT Cλ γ N−1 A N (θ, λ) → N →∞ −iθ 2 ⎝ λ j λk λi ρ aj,k Cia ⎠ e− 2 .

(16)

(17)

i, j,k,a=1 8

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Relation (17), together with (13) and (12) will imply the conclusion. ■ We will also need to analyze the asymptotic behavior of the partial derivatives with respect to the variable λ of (9). Let us first introduce some notation borrowed from [16]. For every u, z ∈ Rd , α ∈ Zd+ and for every symmetric d × d matrix C, we will write 1 T Cu

Pα (u, z, C) = e−i⟨u,z⟩d − 2 u

12

13

14

15 16

17

18

(−i)|α|

∂ α i⟨u,z⟩d + 1 u T Cu 2 e ∂u α

(18)

where |α| := α1 + · · · + αd if α = (α1 , . . . , αd ) ∈ Zd+ . Then clearly, ∂ α i⟨u,z⟩d + 1 u T Cu 1 T 2 = ei⟨u,z⟩d + 2 u Cu Pα (u, z, C). (19) e α ∂u In the next lemma we recall the explicit expression of the polynomial Pα together with some estimates on this polynomial. We refer to Lemma 1 in [16] for the proof. (−i)|α|

Lemma 1. If Pα is given by (18), then for every u, z ∈ Rd and every d × d matrix C, we have ∑ Pα (u, z, C) = Cβα (−i)β z α−β Sβ (u, C) 0≤β≤α

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with 1 T Cu

Sβ (u, C) = e− 2 u

∂ β 1 u T Cu e2 . ∂u β |β|

21

22

Moreover, there exist constants c j (independent of u, z) such that |Sβ (u, C)| ≤

|xβ| ∑

|β|

c j |u| j |C|( j+|β|)/2

j=0 23

24 25

d

for every u ∈ R and every d × d matrix C. Let us now state and prove the asymptotic behavior of the partial derivatives of the truncated interpolation ϕ NΨ . Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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Proposition 2. Assume [C1]–[C3] and (10). Let ϕ NΨ be given by (9). Then for every α ∈ Zd+ , we have ⎤ ⎡ ⎛ ⎞ d α α ∑ T Cλ ∂ ∂ ∂ λ ϕ Ψ (θ, λ)→ N →∞ α ⎣−iθ 2 ⎝ λ j λk λi ρ aj,k Cia ⎠ e− 2 ⎦ . γ N−1 α ∂λ ∂θ N ∂λ i, j,k,a=1 Proof. If α = (α1 , . . . , αd ) ∈ Zd+ , then by (12) we have

Cβα

j=1 β≤α

6

7

8

9

10

11

( ∂ α−β 2 λT Cλ λ · E Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 α−β j ∂λ

12

( j)

×i|β| Pβ (λ, θ FN , −(1 − θ 2 )C)⟨DΨ N , D(−L)−1 FN ⟩ H

) (20)

As in the proof of (13), the second summand above multiplied by γ N−1 converges to zero as N → ∞. Concerning the first summand in (20), the hypothesis [C1] implies that it converges as N → ∞, to ( d ∑ ∑ ) ∂ α−β ( −θ Cβα α−β λ j λk · E i|β| Pβ (λ, θ Z 1 , −(1 − θ 2 )C) ∂λ j,k=1 β≤α ) 2 λT Cλ ( j,k) eiθ ⟨λ,Z 1 ⟩−(1−θ ) 2 Z 2 = −θ

d ∑ ∑

Cβα

j,k=1 β≤α α

=

⎡

) ∂ α−β ( λ j λk · E α−β ∂λ

⎛

∂ β iθ ⟨λ,Z 1 ⟩−(1−θ 2 ) λT Cλ 2 e ∂λβ ⎞ ⎤

((

3

5

×i|β| Pβ (λ, θ FN , −(1 − θ 2 )C) ( )) (k) −1 ( j) × ⟨D FN , D(−L) FN ⟩ H − C j,k d ∑ ∑

2

4

∂α ∂α ∂ Ψ ∂α ϕ N (θ, λ) = α A N (θ, λ) + α B N (θ, λ) α ∂λ ∂θ ∂λ ∂λ with A N , B N defined in (12). Now, since by (19) ∂ α iθ ⟨λ,FN ⟩−(1−θ 2 ) λT Cλ 2 λT Cλ 2 = i|α| Pα (λ, θ FN , −(1 − θ 2 )C)eiθ ⟨λ,FN ⟩−(1−θ ) 2 αe ∂λ we can write ( d ∑ α−β ( ∑ ) ∂α ∂ Ψ 2 λT Cλ α ∂ ϕ N (θ, λ) = −θ Cβ α−β λ j λk · E Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 α ∂λ ∂θ ∂λ j,k=1 β≤α

+i

1

)

( j,k) Z2

13

14 15 16

17

18

)

d ∑ ∂ ⎣ λT Cλ 2⎝ −iθ λ j λk λi ρ aj,k Cia ⎠ e− 2 ⎦ . α ∂λ i, j,k,a=1

Then the conclusion is obtained. ■ Remark 1. If d = 1 and C = 1, then relation (11) becomes ∂ λ2 γ N−1 ϕ NΨ (θ, λ) → N →∞ −iθ 2 λ3 ρe− 2 ∂θ where ρ = E(Z 2 /Z 1 ). Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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The following integration by parts formula plays an important role in the sequel. Its proof is a slightly adaptation of the proof of Proposition 1 in [16]. For its proof the nondegeneracy (8) is needed. Lemma 2. Assume [C2]–[C3]. Consider a function f ∈ C lp (Rd ) with l ≥ 0 and let G be a random variable in Dl,∞ . Then it holds ) ( α ∂ f (FN )GΨ N = E ( f (FN )Hα (GΨ N )) (21) E ∂xα for α ∈ Zd+ , |α| ≤ l where the family of random variables Hα (GΨ N ) is bounded in L p (Ω ; R⊗|α| ) for every p ≥ 1. α

9 10 11

12

13

14

∂ ∂ Ψ We will use the integration by parts formula in Lemma 2 to show that ∂λ α ∂θ ϕ N (θ, λ) is dominated by an integrable function uniformly with respect to (θ, λ) ∈ [0, 1] × Rd . The ∂α α derivative ∂λ α will simply be denoted by ∂λ .

Lemma 3. Assume [C2]–[C3]. For every α ∈ Zd+ , there exists a positive constant Cα such that ⏐ ⏐ ⏐ ⏐ −1 ⏐ α ∂ Ψ (22) γ N ⏐∂λ ϕ N (θ, λ)⏐⏐ ≤ Cα (1 + |λ|)−ℓ+2 ∂θ for all λ ∈ Rd , θ ∈ [0, 1] and N ≥ 1. 2 λT Cλ

15 16 17 18 19

Proof. First we consider the case α = 0 ∈ Zd+ . Consider the function f (x) = eiθ ⟨λ,x⟩−(1−θ ) 2 β for x ∈ Rd with ∂∂x β f (x) = f (x)(iθ λ)β for β ∈ Zd+ . Suppose that β ∈ Zd+ satisfies |β| ≤ ℓ. For θ ∈ (0, 1], by using (12) and the integration by parts formula (21) and a similar formula for ⟨u, DΨ N ⟩ H with u ∈ Dℓ,∞ (H ) in place of GΨ N , to the function f ℓ-times, we obtain ∂ Ψ ϕ (θ, λ) ∂θ N

20

(iθλ)β

21

= −θ(iθ λ)β

d ∑

( ( )) 2 λT Cλ ( j) λ j λk E Ψ N eiθ ⟨λ,FN ⟩−(1−θ ) 2 ⟨D FN(k) , D(−L)−1 FN ⟩ H − C j,k

j,k=1

22

+ i(iθ λ)β

d ∑

( ) 2 λT Cλ ( j) λ j E eiθ ⟨λ,FN ⟩−(1−θ ) 2 ⟨DΨ N , D(−L)−1 FN ⟩ H

j=1

23

= −θ

d ∑

( )) ( T ( ) ( j) (k) iθ ⟨λ,FN ⟩−(1−θ 2 ) λ 2Cλ Hβ ⟨D FN , D FN ⟩ H − 1 Ψ N λ j λk E e

j,k=1

24

+

d ∑

( ( )) 2 λT Cλ ( j) λ j E eiθ ⟨λ,FN ⟩−(1−θ ) 2 Hβ ⟨iλD FN , DΨ N ⟩ H

j=1 25 26 27

We use this estimate for θ ≥ 1/2. For θ ∈ [0, 1/2], the inequality is trivial thanks to T the exponential function exp(−(1 − θ 2 ) λ 2Cλ ). Thus we obtained the desired estimate when α = 0. Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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11

For α ∈ Zd+ , α ̸= (0, . . . , 0), we can follow a similar argument with the estimate ( )m ( λT Cλ ) <∞ sup (1 − θ 2 )(1 + |λ|2 ) exp −(1 − θ 2 ) 2 θ,λ for every m ∈ Z+ . Thus we obtained the result. ■

1

2

3

Let us denote by ϕ N the sum of the characteristic function of the law N (0, C) and of the integral over [0, 1] with respect to θ of the limit in (11), i.e. ⎛ ⎞ d ∑ T 1 λT Cλ λ Cλ λ j λk λi ρ aj,k Cia ⎠ e− 2 . (23) ϕ N (λ) = e− 2 − iγ N ⎝ 3 i, j,k,a=1 The next result shows that the difference between the truncated characteristic function of FN and the characteristic function of the weak limit of the sequence FN can be approximated by ϕ N . In what follows, we will assume (10). Proposition 3. Suppose that [C1]–[C3] are fulfilled for ℓ = d + 3. For every N ≥ 1, let ϕ N be given by (23). Then )⏐ ∫ ⏐ ( ⏐ ⏐ α [ i⟨λ,F ⟩ ] −1 N Ψ ⏐dλ→ N →∞ 0 ⏐∂ E e − E[Ψ ]ϕ (λ) γN N N N ⏐ λ ⏐

4 5

6

7 8 9

10 11

12

Rd

for every α ∈ Zd+ .

13

Proof. If α ∈ Zd+ , using

14

[ ] λT Cλ ϕ NΨ (1, λ) = E ei⟨λ,FN ⟩ Ψ N and ϕ NΨ (0, λ) = E[Ψ N ]e− 2 we can write ( )⏐ ∫ ⏐ ⏐ −1 α [ i⟨λ,F ⟩ ] ⏐ N Ψ ⏐γ ∂ E e ⏐ N − E[Ψ N ]ϕ N (λ) ⏐dλ N λ ⏐ Rd ∫ ⏐ ∫ 1( ⏐ α ⏐∂ = γ N−1 ∂θ ϕ NΨ (θ, λ) − (−i)θ 2 E[Ψ N ] ⏐ λ R 0 ⎞ ⎛ ) ⏐ d ∑ ⏐ λT Cλ ×⎝ λ j λk λi ρ aj,k Cia ⎠ e− 2 dθ ⏐⏐dλ

15

16

17

18

19

i, j,k,a=1

∫

∫

≤ Rd

0

1

⎞ ⎛ ⏐ ⏐ d ∑ T ⏐ −1 α ( )⏐ a − λ 2Cλ ⏐ ⏐γ ∂ ∂θ ϕ Ψ (θ, λ) − ∂ α (−i)θ 2 ⎝ ⎠ λ j λk λi ρ j,k Cia e N λ ⏐dθ dλ ⏐ N λ

20

i, j,k,a=1

( )1 + C ∥1 − Ψ N ∥22 2 . The last quantity converges to zero as N → ∞ by the choice of the truncation function and also Lemma 3 and the dominated convergence theorem since the integrand is bounded uniformly in (θ, N ). ■ Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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Let ϕ FN be the characteristic function of the random vector FN , i.e. ϕ FN (λ) = Eei⟨λ,FN ⟩ for λ ∈ Rd . If ϕ N is given by (23), we have ϕ FN (λ) ( ) ( ) = ϕ N (λ)EΨ N + E Ψ N ei⟨λ,FN ⟩ − E(Ψ N )ϕ N (λ) + E (1 − Ψ N )ei⟨λ,FN ⟩ ( ) = ϕ N (λ) + ϕ N (λ)(EΨ N − 1) + E Ψ N ei⟨λ,FN ⟩ − E(Ψ N )ϕ N (λ) ( ) +E (1 − Ψ N )ei⟨λ,FN ⟩ ) ( Proposition 3 shows that the difference E Ψ N ei⟨λ,FN ⟩ −E(Ψ N )ϕ N (λ) is small while from the definition of the truncation function Ψ N we see that the term E(1−Ψ N )ei⟨λ,FN ⟩ +ϕ N (λ)(EΨ N −1) is also small. Consequently, ϕ FN (λ) can be approximated by ϕ N . Actually, we have ϕ FN (λ) = ϕ N (λ) + s N (λ) where s N (λ) is a “small term” such that C(1 + |λ|)−ℓ+2 and it satisfies

(24) γ N−1 s N (λ)

and its derivatives are dominated by

s N (λ) ≤ C∥1 − Ψ N ∥ p + oλ (γ N ) = oλ (γ N )

(25)

17

for every p ≥ 1, where oλ (γ N ) denotes a deterministic sequence that depends on λ such that oλ (γ N ) → N →∞ 0. γN Therefore ϕ N is the dominant part in the asymptotic expansion of ϕ FN . By inverting ϕ N , we get the approximate density.

18

4. The approximate density

14 15 16

19 20 21

22

23 24 25

26

27

28 29 30

31

In this paragraph, our purpose is to find the first and the second-order term in the asymptotic expansion of the density of FN . We will assume throughout this section [C1]–[C3] are fulfilled for ℓ = d + 3. Let us denote by ϕ FΨN the truncated characteristic function of FN , i.e. ( ) ϕ FΨN (λ) = E Ψ N ei⟨λ,FN ⟩ = ϕ NΨ (1, λ) (26) with ϕ NΨ (1, λ) from (9). We define the approximate density of FN via the Fourier inversion of the dominant part of pΨ FN defined by (26). Definition 1. For every N ≥ 1, we define the approximate density p N by ∫ 1 e−i⟨λ,x⟩ ϕ N (λ)dλ, x ∈ Rd p N (x) = (2π )d Rd where ϕ N is given by (23).

(27)

Let us first notice that in the case d = 1, we can obtain an explicit expression for p N in the next result. Note that for d = 1, assuming C1,1 = 1 and EZ 1 Z 2 = ρ, we have from (23) 1 λ2 λ2 (28) ϕ N (λ) = e− 2 − i λ3 ρe− 2 γ N , for every λ ∈ R. 3 2

32 33

34

Recall that p(x) =

x √1 e− 2 2π

denotes the density of the standard normal law. By Hk we denote

the Hermite polynomial of degree k ≥ 0 given by H0 (x) = 1 and for k ≥ 1 ) k ( 2 2 k x2 d − x2 Hk (x) = (−1) e e . dxk Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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13

Proposition 4. If p N is given by (27) then for every x ∈ R, ρ p N (x) = p(x) − γ N H3 (x) p(x). 3

1

2

Proof. This follows from (28) and the formula ∫ 1 e−iλx (iλ)k Φ(λ)dλ = Hk (x) p(x) 2π R

3

(29)

4

λ2

for every k ≥ 0 integer. Recall the notation Φ(λ) := e− 2 . ■

5

We prefer to work with the truncated density of FN , which can be easily controlled.

6

Definition 2. The local (or truncated) density of the random variable FN is defined as the inverse Fourier transform of the truncated characteristic function ∫ 1 e−i⟨λ,x⟩ ϕ FΨN (λ)dλ pΨ (x) = FN (2π )d Rd for x ∈ Rd .

7 8

9

10

The local density is well-defined since obviously the truncated characteristic function ϕ FΨN is integrable over Rd . Our purpose is to show that the local density p Ψ FN is well-approximated by the approximate density p N given by (27) in the sense that for every α ∈ Zd+ ⏐ ( )⏐ ⏐ ⏐ sup ⏐x α p Ψ (x) − p (x) ⏐. N FN

11 12 13 14

15

x∈Rd

is of order o(γ N ). This will be shown in the next result, based on the asymptotic behavior of the characteristic function of FN . Theorem 2. For every p ≥ 1, and for every integer α = (α1 , .., αd ) ∈ Zd+ ⏐ ( )⏐ ⏐ ⏐ sup ⏐x α p Ψ FN (x) − p N (x) ⏐ ≤ c∥1 − Ψ N ∥ p + o(γ N ) = o(γ N ).

16 17

18

(30)

19

x∈Rd

Proof. By integrating by parts with the help of Lemma 3 and Proposition 3, we have the estimate ∫ ( ) ( ) 1 α x α pΨ (x) − p (x) = x e−i⟨λ,x⟩ ϕ FΨN λ − ϕ N (λ) dλ N FN d (2π ) Rd ∫ ) α ( 1 −i⟨λ,x⟩ ∂ Ψ = e ϕ (λ) − ϕ (λ) dλ. ■ N F N (2π )d (−i)|α| Rd ∂λα ⏐ ⏐ In particular, sup d |x α |⏐ p Ψ (x)⏐ < ∞ for every α ∈ Zd . Let us make some comments

24

on the above result.

25

Remark 2.

26

x∈R ,n∈N

FN

+

• Theorem 2 extends the inequality (3) when FN belongs to the Wiener chaos of order q ≥ for every N ≥ 1. It is known that for every N ≥ 1, we have that γ N2 = V ar ( q1 ∥D FN ∥2H ) Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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behaves as EFN4 − 3, more precisely (see Section 5.2.2 in [9]) q −1 (EFN4 − 3) ≤ (q − 1)γ N2 . 3q It also extends some results in [3] (in particular Theorem 6.2 and Corollary 6. 6). • The appearance of truncation function Ψ in the right-hand side of (30) replaces the condition (2) on the existence of the negative moments of the Malliavin derivative. γ N2 ≤

As a consequence of Theorem 2 we will obtain the asymptotic expansion for the expectation of measurable functionals of FN . Theorem 3. Let p N be given by (27) and fix M ≥ 0, γ > 0. Then ⏐ ⏐ ∫ ⏐ ⏐ ⏐ f (x) p N (x)d x ⏐⏐ = o(γ N ) sup ⏐E f (Fn ) −

10 11

12 13

14

15

16

(31)

Rd

f ∈E(M,γ )

where E(M, γ ) is the set of measurable functions f : Rd → R such that | f (x)| ≤ M(1 + |x|γ ) for every x ∈ Rd . Proof. For any measurable function f that satisfies the assumptions in the statement, we can write ∫ E( f (Fn )) − f (x) p N (x)d x = E( f (FN )) − E( f (FN )Ψ N ) + E( f (FN )Ψ N ) Rd ∫ f (x) p N (x)d x − Rd ∫ [ ] = E( f (FN )(1 − Ψ N )) + f (x) p Ψ FN (x) − p N (x) d x Rd

17 18

19

with given in Definition 2. Then for every p > 1 and for α ≥ 0 large enough (it may depend on M, γ ) with p ′ such that 1p + p1′ = 1, we have, by using H¨older’s inequality, ⏐ ⏐ ∫ ⏐ ⏐ ⏐E f (Fn ) − f (x) p N (x)d x ⏐⏐ ⏐ pΨ FN

Rd

20

21

( )α ∫ )1 ( 2 1 ′ p′ ≤ E| f (FN )| p ∥1 − Ψ N ∥ p + | f (x)| dx 2 1 + |x| Rd ] )α [ ψ ( × sup 1 + |x|2 2 p FN (x) − p N (x) x∈Rd

22

23

24

25

26 27 28 29 30

≤ C∥1 − Ψ N ∥ p + o(γ N ) = o(γ N ). where we used (30). So the desired conclusion is obtained. ■ We insert some remarks related to the above result. Remark 3. • Theorem 3 is related to Theorem 9.3.1 in [9] (which reproduces the main result of [8]). Notice that we do not assume the differentiability of f and our result includes the uniform control in the case of the Kolmogorov distance. • In particular, the asymptotic expansion (31) holds for any bounded function f and for polynomial functions, but the class of examples is obviously larger. Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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In the case of sequences in a Wiener chaos of fixed order, we have the following result, which follows from the above results and from the hypercontractivity property (64). Corollary 1. Assume (FN ) N ≥1 = (Iq ( f N )) N ≥1 (with f N ∈ H ⊗q for every N ≥ 1) be a random 2 sequence in the q th Wiener chaos (q ≥ 2). ( Assume)EFN = 1 for every N ≥ 1 and suppose that the condition [C1] holds with γ N = V ar (

FN , γ N−1

(

∥D FN ∥2H q

i.e.

1 2

3 4

5

)) ∥D FN ∥2H −1 →(d) N →∞ (Z 1 , Z 2 ) q

6

where (Z 1 , Z 2 ) is a centered Gaussian vector with EZ 12 = EZ 22 = 1 and EZ 1 Z 2 = ρ ∈ [−1, 1]. Then we have the asymptotic expansions (30) and (31).

7 8

Proof. The results follow from Theorems 2 and 3 by noticing that (64) ensures that the conditions [C2] and [C3] are satisfied. ■

10

5. Examples

11

In the last part of our work, we will present three examples in order to illustrate our theory. The first example concerns a one-dimensional sequence of random variables in the second Wiener chaos and it is inspired from [9], Chapter 9. The second example is a multidimensional one and it involves quadratic functions of two correlated fractional Brownian motions while the last example involves a sequence in a finite sum of Wiener chaoses. 5.1. A one-dimensional in a Wiener chaos of a fixed order: Quadratic variations of stationary Gaussian sequences We consider a centered stationary Gaussian sequence (X k )k∈Z with EX k2 = 1 for every k ∈ Z. We set ρ(l) := EX 0 X l for every l ∈ Z

N ≥ 1.

12 13 14 15 16

17 18

19 20

21

so that ρ(0) = 1 and |ρ(l)| ≤ 1 for every l ∈ Z. Without loss of generality and in order to fit to our context, we assume that X k = W (h k ) with ∥h k ∥ H = 1 for every k ∈ Z, where (W (h), h ∈ H ) is an isonormal process as defined in the Appendix. Define the quadratic variation sequence N ) 1 ∑( 2 VN = √ Xk − 1 , N k=0

9

(32)

and let v N := EVN2 . Assume ρ ∈ ℓ2 (Z) (the Banach space of 2-summable functions endowed ∑ with the norm ∥ρ∥2ℓ2 (Z ) = k∈Z ρ(k)2 ). In this case, we know from [9] that the deterministic ∑ sequence v N converges to the constant 2 k∈Z ρ(k)2 > 0 and consequently it plays no role in the sequel. Define the renormalized sequence ( ) N VN 1 ∑ ⊗2 FN = √ = I2 √ h for every N ≥ 1. (33) vN N v N k=0 k Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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Obviously EFN = 0 and EFN2 = 1 for every N ≥ 1. It is well known (see [2], see also Chapter 7 in [9]) that, as N → ∞ FN →(d) Z ∼ N (0, 1). Let us see that the sequence (FN ) N ≥1 satisfies the conditions [C1]–[C3] with ( ) 1 2 2 γ N = V ar (34) ∥D FN ∥ H . 2 Actually, this follows from the computations contained in Section 5 of [9], but we recall the main ideas because they are needed later. Notice that ⟨D FN , D(−L)−1 FN ⟩ H = 12 ∥D FN ∥2H since FN belongs to the second Wiener chaos. To prove [C1], we need to assume 4

10

11 12 13

14

15 16 17 18 19

20 21

22

23 24

25

26 27

28

29

30

31 32

33

ρ ∈ ℓ 3 (Z).

(35)

γ N2

In this case (which coincides with the fourth cumulant of FN , denoted k4 (FN )) behaves as c N1 for N large, see Theorem 7.3.3 in [9]. The proof of Theorem 9.5.1 in [9] implies that, as N → ∞, ( ( )) 2 ( ) −1 ∥D FN ∥ H FN , γ N −1 = FN , γ N−1 ⟨D FN , D(−L)−1 FN ⟩ H − 1 →(d) (Z 1 , Z 2 ) 2 where (Z 1 , Z 2 ) is a correlated centered Gaussian vector with EZ 12 = EZ 22 = 1. Therefore condition [C1] is fulfilled. Conditions [C2] and [C3] are satisfied since FN belongs to the second Wiener chaos and one can use the hypercontractivity property of multiple integrals (64). Consequently Theorems 3 and 2 can be then applied to the sequence (32). Remark 4. Let (BtH )t∈R be a (two-sided) fractional Brownian motion with Hurst parameter H ∈ (0, 1). We recall that B H is a centered Gaussian process with covariance function 1 EBtH BsH = (|t|2H + |s|2H − |t − s|2H ) 2 H for every s, t ∈ R. Let X k = Bk+1 − BkH for every k ∈ Z. In this case ρ H (k) := EX 0 X k is given by ) 1( ρ H (k) = |k + 1|2H + |k − 1|2H − 2|k|2H , ∀k ∈ Z. (36) 2 Since ρ H (k) behaves as H (2H − 1)|k|2H −2 for |k| large enough, condition (35) is fulfilled for 0 < H < 58 . Therefore Theorems 2 and 3 can be applied for H ∈ (0, 58 ). 5.2. A multidimensional: quadratic variations of correlated fractional Brownian motions For every H ∈ (0, 1), denote ( ) H− 1 H− 1 f tH (u) = d(H ) (t − u)+ 2 − (−u)+ 2 , with d(H ) a positive constant that ensures that k ∈ Z, let H L kH = f k+1 − f kH .

∫ R

t ∈R f tH (u)2 du = |t|2H for every t ∈ R. For every (37)

Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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Consider (Wt )t∈R a Wiener process on the whole real line and define the two fractional Brownian motions ∫ H H i = 1, 2. f t i (s)d Ws , Bt i =

1 2

3

R

The fBms B H1 and B H2 are correlated. We actually have (see [5], Section 4), for every k, l ∈ Z, H

H

H

H

H

H

H

H

H

4 5

H

1 2 1 E(Bk+1 − Bk 1 )(Bl+1 − Bl 2 ) = ⟨ f k+1 − f k 1 , fl+12 − fl 2 ⟩ H = ⟨L k 1 , L l 2 ⟩ H = D(H1 , H2 )ρ H1 +H2 (k − l)

6

(38)

7

2

where D(H1 , H2 ) is an explicit constant such that D(Hi , Hi ) = 1 for i = 1,2. ( ) Assume H1 , H2 ∈ 0, 34 . Define the quadratic variations, for i = 1,2 and for N ≥ 1 ] N −1 [ )2 1 ∑ ( Hi H VN(i) = √ Bk+1 − Bk i − 1 N k=0

8 9

10

and

11

VN(i)

FN(i) = √

(39)

12

vn(i)

∑ (i) 2 (i) 2 with v (i) N = E(VN ) , i = 1,2. Recall that v N → N →∞ cvi := 2 k∈Z ρ Hi (k) for i = 1,2. (FN(1) ,

13

FN(2) )

Let us show that the sequence FN = satisfies the conditions [C1]–[C3] with 1 γ N = N − 2 . We will focus on [C1] (the assumptions [C2] and [C3] can be easily checked since we deal with elements of the second Wiener chaos and we can apply the hypercontractivity (64)). From the previous paragraph we know that for i = 1,2, the random vectors ) ( √ (40) FN(i) , N ⟨D FN(i) , D(−L)−1 FN(i) ⟩ H − 1 converge in distribution, when N → ∞, to some centered Gaussian vectors (Z i(1) , Z i(2) ) with E(Z i(1) )2 = Ci,i , i = 1,2 and with nontrivial correlation. The main tool to prove the two-dimensional sequence (FN(1) , FN(2) ) satisfies [C1] is the multidimensional version of the Fourth Moment Theorem. Let us recall this result which says that for sequences of random vectors on Wiener chaos, the componentwise convergence to the Gaussian law implies the joint convergence (see [12] or Theorem 6.2.3 in [9]). Theorem 4. Let d ≥ 2 and q1 , . . . , qd ≥ 1 integers. Consider the random vector ( ) ( ) FN = FN(1) , . . . , FN(d) = Iq1 ( f N(1) ), . . . , Iqd ( f N(d) )

( j)

16 17

18

19 20 21 22 23 24

25

26

with f N(i) ∈ H ⊗qi , i = 1, . . . , d. Assume that EFN(i) FN → N →∞ C(i, j) for every i, j = 1, . . . , d.

14 15

27

(41)

Then the following two conditions are equivalent: • FN converges in law, as N → ∞, to a d dimensional centered Gaussian vector with covariance matrix C. • For every i = 1, . . . , d, FN(i) converges in law, as N → ∞, to N (0, C(i, i)). Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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30 31 32

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We will need the two auxiliary lemmas below. The first concerns the asymptotic covariance of FN(1) and FN(2) . ) ( Lemma 4. Let FN(1) , FN(2) be defined by (39). Assume H1 , H2 ∈ 0, 34 . Then for every H1 , H2 ∈ (0, 1), EFN(1) FN(2) → N →∞ C1,2

5 6

with C1,2 =

7

2D 2 (H1 , H2 ) ∑ ρ H1 +H2 (l)2 . √ cv1 cv2 2

(42)

l∈Z

8

9

10

Consequently, ( ) (1) (2) FN(1) , FN(2) →(d) N →∞ (Z 1 , Z 1 ) where (Z 1(1) , Z 1(2) ) is a Gaussian vector with EZ 1(1) = EZ 1(2) = 0 and E(Z 1(1) )2 = C1,1 = 1, E(Z 1(2) )2 = C2,2 = 1, E(Z 1(1) Z 1(2) ) = C1,2

11 12

13

14

with C1,2 given by (42). Recall that we denoted by “→(d) ” the convergence in distribution. Proof. By the chaos expression of FN(i) (33) and by using the isometry (58) ⎞ ) ⎛ N −1 ( N −1 ∑ ∑ 1 H H EFN(1) FN(2) = √ EI2 (L i 1 )⊗2 I2 ⎝ (L j 2 )⊗2 ⎠ (1) (2) j=0 i=0 N vN vN 1

=

15

N 16

with L

H

√

2

(2) v (1) N vN

=

18

→ N →∞

19

22 23

24 25

26

H

H

⟨L i 1 , L j 2 ⟩2

i, j=0

1

= N

21

N −1 ∑

from (37). From the identity (38)

EFN(1) FN(2)

17

20

(43)

√

(2) v (1) N vN

2

2D 2 (H1 , H2 )

N −1 ∑

√

(2) v (1) N v N l=0 2D 2 (H1 , H2 )

√

cv1 cv2

N −1 ∑

2

i, j=0

ρ H1 +H2 (l)2 (1 − 2

∑ l∈Z

l ) N

ρ H1 +H2 (l)2 . 2

The sum l∈Z ρ H1 +H2 (l)2 is finite since H1 + H2 < 2 Theorem 4. ■ ∑

ρ H1 +H2 (i − j)2

3 . 2

The limit (43) will then follow from

We will also need to control the speed of convergence of the covariance function of FN(1) and In this case we need to impose an additional restriction on the Hurst parameters.

FN(2) .

(1) (2) Lemma ( 5 ) 5. Let FN , FN be defined by (39) and let C1,2 be given by (42). Assume H1 , H2 ∈ 0, 8 . Then √ N (EFN(1) FN(2) − C1,2 ) → N →∞ 0.

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Proof. For every N ≥ 1 we can write √ N (EFN(1) FN(2) − C1,2 ) ⎡ ⎤ N −1 ∑ ∑ √ 1 l 1 = 2D 2 (H1 , H2 ) N ⎣ √ ρ H1 +H2 (l)2 (1 − ) − √ ρ H1 +H2 (l)2 ⎦ . N c c 2 2 (1) (2) v v 1 2 l∈Z v N v N l=0

1

2

3

Thus, for N large enough, using that v (i) N converges to cvi for i = 1,2, we can write ⏐ ⏐ √ ⏐ (1) (2) ⏐ N ⏐EFN FN − C1,2 ⏐ ⏐ ⏐ ⏐ ⏐ N −1 ∑ √ ∑ √ ⏐ 1 1 1 ⏐⏐ ≤ C N ρ H1 +H2 (l)2 + C √ l × ρ H1 +H2 (l)2 + C N ⏐⏐ √ −√ cv1 cv2 ⏐⏐ 2 2 N l=0 ⏐ v (1) v (2) l≥N N N := T1,N + T2,N + T3,N . Since for l large, the function ρ H (l) behaves as C × |l|2H −2 , we get, since H1 , H2 < H1 + H2 < 54 ) √ 5 T1,N ≤ C N N 2H1 +2H2 −3 = C N 2H1 +2H2 − 2 → N →∞ 0

5 8

4

5

6

7

(so

and

8 9

10

11

1 5 T2,N ≤ C √ N 2H1 +2H2 −2 = C N 2H1 +2H2 − 2 → N →∞ 0. N Concerning the summand T3,N , we have for N large enough ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ √ ⏐ 1 √ ⏐ 1 ⏐ ⏐ 1 1 ⏐ ⏐ ⏐ T3,N ≤ C N ⏐ √ − √ ⏐ + C N ⏐√ − √ ⏐⏐ c c v1 ⏐ v2 ⏐ ⏐ v (1) ⏐ v (2) N N ) √ ( (1) ≤ C N |v N − cv1 | + |v (2) N − cv2 | ( ) ∑ ∑ √ 5 5 2 2 ρ H2 (l) ≤ C(N 4H1 − 2 + N 4H2 − 2 )→ N →∞ 0 ≤C N ρ H1 (l) + l≥N

12

13

14

15

16

l≥N

and the conclusion follows. ■ The following lemma is also important in order to check [C1]. Lemma 6. Assume H1 < 58 and H2 < 58 . Let FN(1) , FN(2) be defined by (39) and let C1,2 be given by (42). Then ( ) √ 1 (1) (2) N ⟨D FN , D FN ⟩ H − C1,2 2 converges in distribution, as N goes to infinity, to a Gaussian random variable. Proof. Since EFN(1) FN(2) = E 12 ⟨D FN(1) , D FN(2) ⟩ H and by Lemma 5 ) ( √ 1 (1) (2) N E ⟨D FN , D FN ⟩ H − C1,2 → N →∞ 0 2 Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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18

19 20

21

22

23

24

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2

3 4

5

6

7

8

9 10

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it suffices to show that the sequence (Y N ) N ≥1 with ) 1√ ( Y N := N ⟨D FN(1) , D FN(2) ⟩ H − E⟨D FN(1) , D FN(2) ⟩ H (44) 2 converges in distribution as N goes to infinity, to a Gaussian random variable. We will apply the Fourth Moment Theorem. In order to do it, we need to show that EY N2 → N →∞ C0 > 0 and V ar (∥DY N ∥2H ) → N →∞ 0. From (33) and the product formula for multiple integrals (60), we can write ⎛ ⎞ N −1 ∑ 2 H H H H I2 ⎝ (L k 1 ⊗ L l 2 )⟨L k 1 , L l 2 ⟩ H ⎠ YN = √ (1) (2) k,l=0 N vN vN ⎞ ⎛ N −1 ∑ 2 H H I2 ⎝ ρ H1 +H2 (k − l)(L k 1 ⊗ L l 2 )⎠ = D(H1 , H2 ) √ 2 (1) (2) k,l=0 N vN vN

EY N2 = D 2 (H1 , H2 )

14

15

= D 2 (H1 , H2 )

17

18

(46)

with L kH defined by (37). Consequently, since any four functions f, g, u, v of two variable we have 1 ˜ u ⊗v⟩ ˜ = (⟨ f, u⟩⟨g, v⟩ + ⟨ f, v⟩⟨g, u⟩) ⟨ f ⊗g, 2 we will obtain

H ˜ lH2 , ⟨L k 1 ⊗L

16

(45)

×

1[

4

N −1 ∑

(2) N v (1) N vN

k,l,i, j=0

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2

2

H ˜ Hj 2 ⟩ L i 1 ⊗L

4

N −1 ∑

(2) N v (1) N vN

k,l,i, j=0

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2

2

] ρ H1 (k − i)ρ H2 (l − j) + D(H1 , H2 )2 ρ H1 +H2 (i − l)ρ H1 +H2 (k − j) .

2 2 2 For two sequences (u(n), n ∈ Z) and (v(n), n ∈ Z), we define their convolution by ∑ (u ∗ v)( j) = u(n)v( j − n).

(47)

n∈Z 19

20

We will need the Young’s inequality: if s, p, q ≥ 1 with

1 s

+1=

1 p

+

1 q

∥u ∗ v∥ℓs (Z) ≤ ∥u∥ℓ p (Z) ∥v∥ℓq (Z) .

then (48)

As the proof of Proposition 7.3.3 in [9], we have N −1 1 ∑ ρ H +H (k − l)ρ H1 +H2 (i − j)ρ H1 +H2 (i − l)ρ H1 +H2 (k − j) N k,l,i, j= 1 2 2 2 2 2

→ N →∞ ⟨ρ ∗3 H1 +H2 , ρ H1 +H2 ⟩ℓ2 (Z) 2

2

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where ⟨ρ ∗3 H1 +H2 , ρ H1 +H2 ⟩ℓ2 (Z) < ∞ when H1 + H2 ≤ 2

2

5 4

4

which implies that ρ H1 +H2 ∈ ℓ 3 (Z). The

second summand in (47) can be handled as follows 1 N

N −1 ∑

2

ρ H1 +H2 (k4 − k3 )ρ H1 +H2 (k2 − k1 )ρ H1 (k1 − k4 )ρ H2 (k3 − k2 ) 2

k1 ,k2 ,k3 ,k4 =0

3

2

N −1 N −1−k ∑1 1 ∑ ρ H1 +H2 (k4 − k3 )ρ H1 +H2 (k2 )ρ H1 (k4 )ρ H2 (k3 − k2 ) N k =0 k ,k ,k =−k 2 2 2 3 4 1 1 ∑ = ρ H1 +H2 (k4 − k3 )ρ H1 +H2 (k2 )ρ H1 (k4 )ρ H2 (k3 − k2 )

=

k2 ,k3 ,k4 ∈Z

1

2

2

4

5

2

( ) ( )] max(k2 , k3 , k4 ) min(k2 , k3 , k4 ) 1∨ 1− −0∧ 1|k2 |
2

k2 ,k3 ,k4 ∈Z

6

7

2

= ⟨ρ H1 +H2 ∗ ρ H1 ∗ ρ H2 , ρ H1 +H2 ⟩ℓ2 (Z) .

(49)

8

Again notice that ⟨|ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 |, |ρ H1 +H2 |⟩ℓ2 (Z) < ∞ since by H¨older’s and Young’s

9

2

2

2

2

4 3

inequalities, by using ρ H1 +H2 ∈ ℓ (Z),

10

2

⟨|ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 |, |ρ H1 +H2 |⟩ℓ2 (Z) ≤ C∥|ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 |∥ℓ4 (Z) 2

2

11

2

≤ C∥|ρ H1 +H2 |∥ ≤ ∥ρ H1 +H2 ∥ 2

×∥|ρ H2 |∥

4

ℓ 3 (Z)

2

4

ℓ 3 (Z)

× ∥|ρ H1 | ∗ |ρ H2 |∥ℓ2 (Z)

× ∥|ρ H1 |∥

4

ℓ 3 (Z)

4

ℓ 3 (Z)

12

13

14

where we applied twice (48). Thus EY N2 → N →∞ C0 :=

2D 2 (H1 , H2 ) ( ⟨ρ H1 +H2 ∗ ρ H1 ∗ ρ H2 , ρ H1 +H2 ⟩ℓ2 (Z) cv1 cv2 2 2 ) +D 2 (H1 , H2 )⟨ρ ∗3 H1 +H2 , ρ H1 +H2 ⟩ℓ2 (Z) 2

2

and the first part of (45) is obtained. Next, we check the convergence of the Malliavin derivative in (45). From (46) and the product formula (60) ⎛ N −1 2 ∑ 4D(H1 , H2 ) ⎝ ∥DY N ∥2H = I ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2 (2) 2 2 N v (1) N vN i, j,k.l=0 ⎞ H ˜ lH2 )⊗1 (L iH1 ⊗L ˜ Hj 2 )⎠ × (L k 1 ⊗L

where the contraction ⊗1 is defined in (63). Since for every f, g, u, v ∈ L 2 (R2 ), we have ˜ ˜ = ( f ⊗g)⊗ 1 (u ⊗v)

1 (⟨ f, u⟩(g ⊗ v) + ⟨ f, v⟩(g ⊗ u) + ⟨g, u⟩( f ⊗ v) + ⟨g, v⟩( f ⊗ u)) 4

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we obtain, via (38) ∥DY N ∥2H

=

D(H1 , H2 )2 (2) N v (1) N vN

⎛

N −1 ∑

I2 ⎝

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2

i, j,k.l=0

2

[ H H H H ρ H1 (k − i)(L l 2 ⊗ L j 2 ) + D(H1 , H2 )ρ H1 +H2 (k − j)(L l 2 ⊗ L i 1 )

3

2

⎞ +D(H1 , H2 )ρ H1 +H2 (l −

4

2

5

6

7

8

+ ρ H2 (l −

= C1

1

N −1 ∑

2 (2) 2 N 2 (v (1) N ) (v N )

i, j,k.l,i ′ , j ′ ,k ′ ,l ′ =0

2

ρ H1 +H2 (k ′ − l ′ )ρ H1 +H2 (i ′ − j ′ ) 2

× ρ H1 +H2 (k − i)ρ H1 +H2 (k ′ − i ′ )ρ H1 +H2 (l − l ′ )ρ H1 +H2 ( j − j ′ ) + C2

2

2

1

N −1 ∑

2 (2) 2 N 2 (v (1) N ) (v N )

i, j,k.l,i ′ , j ′ ,k ′ ,l ′ =0

2

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2

2

12

× ρ H1 (k − i)ρ H1 (k ′ − i ′ )ρ H2 (l − l ′ )ρ H2 ( j − j ′ )

13

+ C3

2

2

1

N −1 ∑

2 (2) 2 N 2 (v (1) N ) (v N )

i, j,k.l,i ′ , j ′ ,k ′ ,l ′ =0

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j) 2

2

14

ρ H1 +H2 (k ′ − l ′ )ρ H1 +H2 (i ′ − j ′ )

15

× ρ H1 (k − i)ρ H1 +H2 (k ′ − j ′ )ρ H2 (l − l ′ )ρ H1 +H2 ( j − i ′ ).

2

2

2

18

19

20

21

2

We can write, using the convolution symbol V ar (∥DY N ∥2H ) N −1 ) 1 ∑ [( |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | (l − j ′ )2 2 N 2 2 2 2 l, j ′ =0 ( ) + |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 | (l − j ′ )2 2 2 ( ) + |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | (l − j ′ ) 2 2 2 2 ( ) ] × |ρ H1 +H2 ||ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 | (l − j ′ )

≤C

2

22

⎠

2

ρ H1 +H2 (k ′ − l ′ )ρ H1 +H2 (i ′ − j ′ )

17

⊗

]

H Li 1 )

ρ H1 +H2 (k − l)ρ H1 +H2 (i − j)

11

16

H j)(L k 1

V ar (∥DY N ∥2H )

2

10

⊗

H L j 2)

and by the isometry formula (58) and commutativity of convolution if the reader would need,

2

9

H i)(L k 1

≤C

1 N

N −1 ∑

2

g(l)

(50)

l=−(N −1)

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23

with ( ) g(l) = |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | (l)2 2 2 2 2 ) ( + |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 | (l)2 2 2 ) ( + |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | ∗ |ρ H1 +H2 | (l) 2 2 2 2 ( ) × |ρ H1 +H2 ||ρ H1 +H2 | ∗ |ρ H1 | ∗ |ρ H2 | (l). 2

2

Since for any two sequences u, v ∈ ℓ2 (Z) we have √∑ √∑ 2 u( j) v( j)2 lim (|u| ∗ |v|)(n) ≤ n→∞

j≥M

1

2

j≥M

for every M ≥ 1 (see [9], page 168), we obtain as in [9], by using again Young’s inequality

3

g(N )→ N →∞ 0.

4

The convergence of V ar (∥DY N ∥2H ) to zero follows from (50) and C´esaro theorem. ■

5

Let us go back to the sequence FN = (FN(1) , FN(2) ) from (39). Proposition 5. The two-dimensional random sequence FN = satisfies condition [C1] with γ N = N

− 21

6

(

FN(1) , FN(2)

)

defined by (39)

.

8

Proof. Since for every N ≥ 1 ⟨D FN(1) , D(−L)−1 FN(2) ⟩ H = ⟨D FN(2) , D(−L)−1 FN(1) ⟩ H =

7

9

1 ⟨D FN(1) , D FN(2) ⟩ H 2

we need to show that the random vector ) ) ( ( ( √ √ 1 1 (1) (2) (1) 2 (2) 2 FN , FN , N ∥D FN ∥ H − 1 , N ∥D FN ∥ H − 1 , 2 2 ) √ 1 N ( ⟨D FN(1) , D FN(2) ⟩ H − C1,2 ) 2

(51)

10

11

12

(52)

converges to a Gaussian vector. All the components of the above vector are multiple integrals of order 2 or sum of a multiple integral of order 2 and a deterministic term tending to zero (the case of the last component). We will use the multidimensional Fourth Moment Theorem to prove the convergence of (52). From the convergence of the vectors (40), by using Lemmas 4 and 6, it suffices to check the condition (41) from the multidimensional Fourth Moment Theorem (Theorem 4), i.e. the covariances of the components of the vector (52) converge to constants as N → ∞. The covariance of FN(1) and FN(2) converges to C1,2 (42) due to Lemma 4. Concerning the other limits, we have from (33) (recall the symbol ∼ means that the sides have the same limit as Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

13

14 15 16 17 18 19 20 21 22

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N → ∞) EFN(1)

(

√ N

) ∥D FN(2) ∥2H −1 2

∼

3

=

4

5

→ N →∞

6

( N −1 ) ∑ H 1 1 ⊗2 CE I2 I2 (L i ) N i=0 ⎞ ⎛ N −1 ∑ H H ×⎝ ρ H2 ( j − k)(L j 2 ⊗ L k 2 )⎠ C

1 N

j,k=0 N −1 ∑

ρ H2 ( j − k)ρ H1 +H2 (i − j) 2

i, j,k=0

ρ H1 +H2 (i − k) 2 C⟨ρ H1 +H2 ∗ ρ H2 , ρ H1 +H2 ⟩ℓ2 (Z) < ∞ 2

7 8

9

10

11

12

13

14

Next, denote by Y N′ the last component of the vector (52), i.e. √ 1 Y N′ := N ( ⟨D FN(1) , D FN(2) ⟩ H − C1,2 ). 2 By Lemma 5 ( N −1 ) ∑ H 1 (1) ′ (1) ⊗2 EFN Y N ∼ EFN Y N ∼ C EI2 (L k 1 ) I2 N i=0 ⎞ ⎛ N −1 ∑ H H ×⎝ ρ H1 +H2 ( j − k)(L j 1 ⊗ L k 2 )⎠ =

15

=

16

C N

j,k=0 N −1 ∑

21

H

i, j,k=0

2

and similarly EFN(2) Y N′ → N →∞ C⟨ρ H1 +H2 ∗ ρ H2 , ρ H1 +H2 ⟩ℓ2 (Z) . 2

Finally, we can also prove that ( )( ) ∥D FN(1) ∥2H ∥D FN(2) ∥2H E −1 −1 2 2

∼

C ∑ ρ H (i − j)ρ H2 (k − l) N i, j,k,l 1

=

˜ j 1 ), (L k 2 ⊗L ˜ l 2 )⟩ × ⟨(L i 1 ⊗L ∑ C ρ H (i − j)ρ H2 (k − l) N i, j,k.l 1

H

22

23

H

2

C ∑ ρ H +H ( j − k)ρ H1 (i − j)ρ H1 +H2 (i − k) N i, j,k 1 2 2 2

2

20

H

˜ k 2 )⟩ ρ H1 +H2 ( j − k)⟨(L k 1 )⊗2 , (L j 1 ⊗L

2

19

2

→ N →∞ C⟨ρ H1 +H2 ∗ ρ H1 , ρ H1 +H2 ⟩ℓ2 (Z)

17

18

2

where the limit is obtained similarly as (49) and the last series is finite by using Young inequality 4 and ρ H1 +H2 , ρ H1 , ρ H2 ∈ ℓ 3 (Z). Also, by symmetry 2 ( ) (1) 2 (2) ∥D FN ∥ H EFN − 1 → N →∞ C⟨ρ H1 +H2 ∗ ρ H2 , ρ H1 +H2 ⟩ℓ2 (Z) . 2 2 2

H

H

H

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× ρ H1 +H2 (k − i)ρ H1 +H2 (l − j) 2 2 → N →∞ C⟨ρ H1 ∗ ρ H1 +H2 , ρ H2 ∗ ρ H1 +H2 ⟩ℓ2 (Z) , 2

1 2

2

(

) ( ) (1) 2 ∥D FN(1) ∥2H ∥D F ∥ N H E − 1 Y N′ ∼ E − 1 YN 2 2 → N →∞ C⟨ρ H1 ∗ ρ H1 +H2 , ρ H1 ∗ ρ H1 +H2 ⟩ℓ2 (Z) 2

2

and ) ( ) ∥D FN(2) ∥2H ∥D FN(2) ∥2H ′ − 1 YN ∼ E − 1 YN E 2 2 → N →∞ C⟨ρ H2 ∗ ρ H1 +H2 , ρ H2 ∗ ρ H1 +H2 ⟩ℓ2 (Z) . (

2

2

So, condition (41) is fulfilled and the conclusion follows from Theorem 4. ■

3

5.3. An example in a sum of finite Wiener chaoses

4

Consider a sequence I2 ( f N ) in a second Wiener chaos that satisfies condition [C1]–[C3] (take for example the sequence (33)). Consider a “perturbation” G N = Iq (g N ) with q ≥ 3 and such that for N ≥ 1 1 ∥g N ∥2H ⊗q ≤ C 1+β (53) N with some β > 0. In particular, (53) implies that G N → N →∞ 0 in L 2 (Ω ). Define, FN = I2 ( f N ) + G N ,

N ≥ 1.

(54) ( ) Clearly FN verifies [C2]. Let us show that FN satisfies [C1] with γ N = V ar 12 ∥D I2 ( f N )∥2H (which behaves as C √1N for N large). We have, by the product formula (60) ) ( ( ) 1 −1 −1 1 −1 2 γ N ⟨D FN , D(−L) FN ⟩ − 1 = γ N ∥D I2 ( f N )∥ H − 1 + γ N−1 ∥DG N ∥2H 2 q ( + (2 + q)γ N−1 Iq ( f N ⊗1 g N ) ) +(q − 1)Iq−1 ( f ⊗2 g)

5 6 7

8

9

10

√

Notice that, by (53) ( ) 1 1 E γ N−1 ∥DG N ∥2H ≤ C N 2 ∥g N ∥2H ⊗q ≤ C N −β− 2 → N →∞ 0. Also, since from (63)∥ f N ⊗k g N ∥ H ⊗(2+q−2k) ≤ ∥ f N ∥ H ⊗2 ∥g N ∥ H ⊗q for k = 1,2 ( )2 E γ N−1 Iq ( f N ⊗1 g N ) ≤ C N ∥ f N ⊗1 g∥2H ⊗q ≤ C N ∥ f N ∥2H ⊗2 ∥g N ∥2H ⊗q ≤ C N −β

11 12

13

14 15

16

(55)

17

18

(56)

and ( )2 E γ N−1 Iq−2 ( f N ⊗2 g N ) ≤ C N ∥ f N ⊗2 g∥2H ⊗(q−2) ≤ C N ∥ f N ∥2H ⊗2 ∥g N ∥2H ⊗q ≤ C N −β . (57) ( ) Relations (55)–(57), together with the fact that γ N−1 21 ∥ D I2 ( f N )∥2 − 1 converges to a Gaussian law as N → ∞, implies [C1]. The same relations together with (64) will give [C3]. Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

19

20

21

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Acknowledgments

6

The first author was partially supported by Project ECOS CNRS-CONICYT C15E05, MEC PAI80160046 and MATHAMSUD 16-MATH-03 SIDRE Project. The second author was in part supported by Japan Science and Technology Agency CREST JPMJCR14D7; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (Scientific Research) and by a Cooperative Research Program of the Institute of Statistical Mathematics.

7

Appendix. Elements from Malliavin calculus

2 3 4 5

8 9 10 11 12 13 14 15 16

17

18

We briefly describe the tools from the analysis on Wiener space that we will need in our work. For complete presentations, we refer to [10] or [9]. Consider H a real separable Hilbert space and (W (h), h ∈ H ) an isonormal Gaussian process on a probability space (Ω , A, P), which is a centered Gaussian family of random variables such that E [W (ϕ)W (ψ)] = ⟨ϕ, ψ⟩ H . Denote by In the multiple stochastic integral with respect to B (see [10]). This mapping In is actually an isometry between the Hilbert space H ⊙n (symmetric tensor product) equipped with the scaled norm √1n! ∥ · ∥ H ⊗n and the Wiener chaos of order n which is defined as the closed linear span of the random variables h n (W (h)) where h ∈ H, ∥h∥ H = 1 and h n is the Hermite polynomial of degree n ∈ N ( 2) n ( ( 2 )) (−1)n x x d exp exp − , x ∈ R. h n (x) = n! 2 dxn 2 The isometry of multiple integrals can be written as follows: for m, n positive integers,

19

E (In ( f )Im (g)) = n!⟨ f˜, g⟩ ˜ H ⊗n

20

E (In ( f )Im (g)) = 0

21 22 23 24

25

if m = n,

if m ̸= n. (58) ( ) It also holds that In ( f ) = In f˜ where f˜ denotes the symmetrization of f . We recall that any square integrable random variable which is measurable with respect to the σ -algebra generated by W can be expanded into an orthogonal sum of multiple stochastic integrals F=

∞ ∑

In ( f n )

(59)

n=0 26 27

28

where f n ∈ H ⊙n are (uniquely determined) symmetric functions and I0 ( f 0 ) = E [F]. Let L be the Ornstein–Uhlenbeck operator ∑ LF = − n In ( f n ) n≥0

29 30 31

∑ 2 2 if F is given by (59) and it is such that ∞ n=1 n n!∥ f n ∥H⊗n < ∞. For p > 1 and α ∈ R we introduce the Sobolev–Watanabe space Dα, p as the closure of the set of polynomial random variables with respect to the norm α

32

33

∥F∥α, p = ∥(I − L) 2 F∥ L p (Ω) where I represents the identity. We denote by D the Malliavin derivative operator that acts on smooth functions of the form F = g(W (h 1 ), . . . , W (h n )) (g is a smooth function with compact Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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support and h i ∈ H )

1

n ∑ ∂g DF = (W (h 1 ), . . . , W (h n ))h i . ∂ xi i=1

2

The operator D is continuous from Dα, p into Dα−1, p (H ) . We recall the product formula for multiple integrals. It is well-known that for f ∈ H ⊙n and g ∈ H ⊙m n∧m ∑ (n ) (m ) In ( f )Im (g) = r! Im+n−2r ( f ⊗r g) (60) r r

3 4 5

6

r =0

where f ⊗r g means the r -contraction of f and g. This contraction is defined in the following way. Consider a complete orthonormal system (e j ) j≥1 in H and let f ∈ H ⊗n , g ∈ H ⊗m be two symmetric functions with n, m ≥ 1. Then ∑ f = λ j1 ,..., jn e j1 ⊗ · · · ⊗ e jn (61)

7 8 9

10

j1 ,.., jn ≥1

and

11

g=

∑

βk1 ,...,km ek1 ⊗ .. ⊗ ekm

(62)

12

k1 ,...,km ≥1

where the coefficients λi and β j satisfy λ jσ (1) ,..., jσ (n) = λ j1 ,..., jn and βkπ (1) ,...,kπ (m) = βk1 ,...,km for every permutation σ of the set {1, . . . , n} and for every permutation π of the set {1, . . . , m}. Actually λ j1 ,..., jn = ⟨ f, e j1 ⊗ · · · ⊗ e jn ⟩ and βk1 ,...,km = ⟨g, ek1 ⊗ .. ⊗ ekm ⟩ in (61) and (62). If f ∈ H ⊗n , g ∈ H ⊗m are symmetric given by (61), (62) respectively, then the contraction of order r of f and g is given by ∑ ∑ ∑ f ⊗r g = λi1 ,...,ir , j1 ,.., jn−r βi1 ,...,ir ,k1 ,..,km−r

13 14 15 16 17

18

i 1 ,...,ir ≥1 j1 ,..., jn−r ≥1 k1 ,...,km−r ≥1

( ) ( ) × e j1 ⊗ .. ⊗ e jn−r ⊗ ek1 ⊗ .. ⊗ ekm−r

(63)

for every r = 0, . . . , m ∧ n. In particular f ⊗0 g = f ⊗ g. Note that f ⊗r g belongs to H ˜ r g the for every r = 0, . . . , m ∧ n and it is not in general symmetric. We will denote by f ⊗ symmetrization of f ⊗r g. Another important property of finite sums of multiple integrals is the hypercontractivity. ∑ Namely, if F = nk=0 Ik ( f k ) with f k ∈ H ⊗k then ( )p (64) E|F| p ≤ C p EF 2 2 . ⊗(m+n−2r )

for every p ≥ 2. The adjoint of D is denoted by δ and is called the divergence (or Skorohod) integral. Its domain (Dom(δ)) coincides with the class of stochastic processes u ∈ L 2 (Ω × T ) such that |E⟨D F, u⟩| ≤ c∥F∥2

20 21 22 23 24

25

26 27 28

29

for all F ∈ D1,2 and δ(u) is the element of L 2 (Ω ) characterized by the duality relationship E(Fδ(u)) = E⟨D F, u⟩ H .

19

30

(65)

The chain rule for the Malliavin derivative (see Proposition 1.2.4 in [10]) will be used several times. If ϕ : R → R is a continuously differentiable function having bounded derivative and Please cite this article in press as: C.A. Tudor, N. Yoshida, Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications (2018), https://doi.org/10.1016/j.spa.2018.09.018.

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F ∈ D1,2 , then ϕ(F) ∈ D1,2 and Dϕ(F) = ϕ ′ (F)D F.

(66)

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