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

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

Ciprian Tudor, Nakahiro Yoshida

To cite this version:

Ciprian Tudor, Nakahiro Yoshida. Asymptotic expansion for vector-valued sequences of random vari-

ables with focus on Wiener chaos. 2017. �hal-01659406�

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

Ciprian A. Tudor

Nakahiro Yoshida

1

Laboratoire Paul Painlev´e, Universit´e de Lille 1 F-59655 Villeneuve d’Ascq, France.

tudor@math.univ-lille1.fr

2

Graduate School of Mathematical Science, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153, Japan

3

Japan Science and Technology Agency CREST nakahiro@ms.utokyo.ac.jp

December 8, 2017

Abstract

We develop the asymptotic expansion theory for vector-valued sequences (F

)

of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F

. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of F

and we illustrate our results by several examples.

2010 AMS Classification Numbers: 62M09, 60F05, 62H12

Key Words and Phrases: 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 in probability and statistics. In particular, the asympotic expan-

∗

. The author was partially supported by Project ECOS NCRS-CONICYT C15E05,MEC PAI80160046 and MATHAMSUD 16-MATH-03 SIDRE Project.

†

The 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.

sion represents a technique that allows to give a precise approximation for the densities of probability distributions,

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 based on elements in Wiener chaos. The characterization of the convergence in distribution of se- quences of random variables belonging to a Wiener chaos of fixed order to the Gaussian law constitutes an important research direction in stochastic analysis in the last 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 [8] for the basics of this theory.

In this work, we are also concerned with some refinements of the Fourt Moment The- orem for sequences of random variables in a Wiener chaos of fixed order. More concretely, we aim to find 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 I

denote the qth multiple stochastic integral with respect to W . Let Z ∼ N (0, 1). Denote by d

(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 { F

= I

(f

), k ≥ 1 } of random variables in the q-th Wiener chaos. Assume that

k

lim

EF

= lim

k→∞

q! k f

k

= 1. (1) Then, the following statements are equivalent:

1. The sequence of random variables { F

= I

(f

), k ≥ 1 } converges in distribution as k → ∞ to the standard normal law N (0, 1).

2. lim

E[F

] = 3.

3. k DF

k

converges to q in L

(Ω) as k → ∞ . 4. d

(F

, Z) →

0.

A more recent point of interest in the analysis of the convergence to the normal dis-

tribution 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

the density of the random

variable F

(which exists due to a result in [14]). Since the total variation distance between

F and G (or between the distributions of the random variables F and G) can be written

as d

(F, G) =

R

R

| p

(x) − p

(x) | dx, we can notice that point 4. in the Fourth Moment Theorem implies that

k p

(x) − p(x) k

→

0 where p(x) =

2π

e

2

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

N

lim

E k DF

k

< ∞ (2) then the conditions 1.-4. in the Fourth Moment Theorem are equivalent to

sup

x∈R

| p

(x) − p(x) | ≤ c q

(EF

− 3). (3)

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

)

of the form p

(x) = p(x) + r

ϕ(x) such that r

→

0 which approximates the density p

in the following sense

sup

x∈R

| x

| | p

(x) − p

(x) | = o(r

) (4) for any α ∈ Z

. This improves the main result in [7] and it will also allow to generalize the Edgeworth -type expansion (see [9], [1]) for sequences on Wiener chaos to a larger class of measurable functions. The main idea to get the asymptotic expansion is to analyze the asymptotic behavior of the characteristic function of F

and then to obtain the approximate density p

by Fourier inversion of the dominant part of the characteristic function.

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. Our main assumption is that the vector-valued random sequence F

, γ

(Λ(F

) − C)

converges to a d+d × d-dimensional Gaussian vec- tor, where (γ

)

is a deterministic sequence converging to zero as N → ∞ , Λ(F

) is the so-called Stein-Malliavin matrix of F

whose components are Λ

= h DF

, D( − L)

F

i

for i, j = 1, d (L is the Ornstein-Uhlenbeck operator) and C

= lim

EF

F

. In the one-dimensional case the assumption becomes

F

, γ

( h DF

, D( − L)

F

i

− 1)

(5) converges in distribution, as N → ∞ , to a two-dimensional centered Gaussian vector (Z

, Z

) with EZ

= EZ

= 1 and EZ

Z

= 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

)

denotes a sequence of random variables converging to Z ∼ N (0, 1) such that M

is the terminal value of a continuous martingale and if h M i

is the martingale’s bracket, then one assumes in e.g. [17]), [15] that

h M i

→

1 in probability and the following joint convergence holds true

M

, h M i

− 1 γ

→

(Z

, Z

) (6) where (Z

, Z

) is a centered Gaussian vector as above. The notation ” →

” stands for the convergence in distribution.

In our assumption (5), the role of the martingale bracket is played by

h DF

, D( − L)

F

i

which becomes

k DF

k

when F

is an element of the qth Wiener chaos. The choice is natural, suggested by the identity EF

= E h F i

which playes the role of the Itˆo isometry and by the fact that

− 1 converges to zero in L

(Ω), due to the Fourth Moment Theorem.

We organized our paper as follows. In Section 2, we fix the general context of our work, the main asumptions 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 demsity and the asymptotic behavior via Fourier inversion. In Section 5, we illustrate our theory by several examples related to Wiener chaos.

2 Asssumptions and notation

We will here present the basic assumptions and the notation utilised throughout our work.

Let H be a real and separable Hilbert space endowed with the inner product h· , ·i

and consider (W (h), h ∈ H) an isonormal 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.

2.1 The main assumptions

Consider a sequence of centered random variables (F

)

in R

of the form F

=

F

, . . . , F

.

Denote by Λ(F

) = (Λ

)

the Stein-Malliavin matrix with components Λ

= h DF

, D( − L)

F

i

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

)

converging to zero as N → ∞ such that F

, γ

(Λ(F

) − C) converges in distribution to a d + d × d Gaussian random vector (Z

, Z

) such that Z

= (Z

, . . . , Z

), Z

= (Z

)

with

EZ

Z

= C

, for every i, j = 1, .., d.

[C2 ] For every i = 1, .., d, the sequence (F

)

is bounded in D

= ∩

D

for some l ≥ d + 3.

[C3 ] With γ

, C from [C1 ], for every i, j = 1, d the sequence γ

Λ

− C

N∈N

is bounded in D

for some l ≥ d + 3.

Notice that the matrix Λ(F

) is not necessarily symmetric (except when all the components belong to the same Wiener chaos) since in general Λ

6 = Λ

. Nevertheless, we assumed in [C1] that it converges to the symmetric matrix C, in the sense that γ

(Λ(F

) − 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 F

, i = 1, .., d to the Gaussian law, the quantities Λ

− C

, Λ

− C

converges to zero in L

(Ω). Thus condition [C1] intuitively means that these two terms have the same rate of convergence to zero.

2.2 The truncation

Let us introduce the truncation sequence (Ψ

)

which will be widely used throughout our work. Its use allows to avoid the condition on the existence of negative moments for the Malliavin derivative of F

(2).

We consider the function Ψ

∈ D

for p > 1 and ℓ ≥ d + 3 such that Ψ

= ψ

K | Λ(F

) − C |

γ

!

(7) with some K, 0 < δ < 1, where ψ ∈ C

( R ; [0, 1]) is such that ψ(x) = 1 is | x | ≤

and ψ(x) = 0 if | x | ≥ 1. The square in the argument pf ψ in order to have Ψ

differentiable in the Mlliavin sense.

From this definition, clearly we have

1. 0 ≤ Ψ

≤ 1 for every N ≥ 1.

2. Ψ

→

1 in probability.

A crucial fact in our computations is that on the set { Ψ

> 0 } we have

| Λ(F

) − C |

≤ 1 K γ

.

This also implies (see e.g. [17]) that there exists two positive constants c

, c

such that for N large enough

c

≤ det Λ(F

) ≤ c

. (8)

2.3 Notation

For every x = (x

, ..., x

) ∈ R

and α = (α

, ..., α

) ∈ Z

we use the notation x

= x

....x

and ∂

∂x

= ∂

∂x

.... ∂

∂x

If x, y ∈ R

, we denote by h x, y i

:= h x, y i their scalar product in R

. By | x |

:= | x | we will denote the Euclidean norm of x.

The following will be fixed in the sequel:

• The sequence (F

)

defined in Section 2.1.

• The truncation sequence (Ψ

)

is as in Section 2.2.

• o

(γ

) denotes a deteministic sequence that depends on λ such that

N

→

0.

By o(γ

) we refer as usual to a sequence such that

N

→

0.

• By Φ(λ) = e

2

2

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

√1

2π

e

its density.

• By c, C, C

, ... we denote generic strictly positive constants that may vary from line to line and by h· , ·i we denote the Euclidean scalar product R

. On the other hand, we will keep the subscript for the inner product in H, denoted by h· , ·i

.

• We will use bold letters to indicate vectors in R

when we need to differentiate them

from their one-dimensional components.

3 Asymptotic behavior of the characteristic function

The first step in finding the asymptotic expansion for the sequence (F

)

which satisfies [C1] - [C3] is to analyze the asymptotic behavior as N → ∞ of the characteristic function of F

. 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

given by (7).

Let λ = (λ

, ..., λ

) ∈ R

and θ ∈ [0, 1] and let us consider the truncated interpola- tion

ϕ

(θ, λ ) = E

Ψ

e

. (9)

Notice that ϕ

(1, λ ) = E(Ψ

e

), the ”truncated” characteristic function of F

, while ϕ

(0, λ ) = (EΨ

)e

, the ”truncated”characteristic function of the limit in law of F

. Therefore, it is useful to analyze the behavior of the derivative of ϕ

(θ, λ ) with respect to the variable θ.

We have the following result.

Proposition 1 Assume [C1] - [C3] and suppose that

E

Z

| Z

=

d

X

a=1

ρ

Z

(10)

for every k, l = 1, .., d, where (Z

, Z

) is the Gaussian random vector from [C1]. Then we have the convergence

γ

∂

∂θ ϕ

(θ, λ ) →

−i θ





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e

. (11) Proof: By differentiating (9) with respect to θ and using the formula F

= δD( − L)

F

for every i = 1, .., d, we can write

∂

∂θ ϕ

(θ, λ ) = E

Ψ

e

i h λ , F

i + θ λ

C λ

= i E



Ψ

e

d

X

j=1

λ

δD( − L)

F





+θ

d

X

j,k=1

λ

λ

C

E

Ψ

e

and by the duality relationship (65)

∂

∂θ ϕ

(θ, λ ) = i

d

X

j=1

λ

E

Ψ

e

i θ h D( − L)

F

, D h λ , F

ii

+ i

d

X

j=1

λ

E

e

h DΨ

, D( − L)

F

i

+θ

d

X

j,k=1

λ

λ

C

E

Ψ

e

.

We obtain, since D h λ , F

i = P

k=1

λ

DF

,

∂

∂θ ϕ

(θ, λ ) = − θ

d

X

j,k=1

λ

λ

E

Ψ

e

h DF

, D( − L)

F

i

− C

+ i

d

X

j=1

λ

E

e

h DΨ

, D( − L)

F

i

:= A

(θ, λ ) + B

(θ, λ ). (12)

Using the definition of the truncation function Ψ

, we can prove that γ

B

(θ, λ ) = γ

d

X

j=1

E

e

h DΨ

, D( − L)

F

i

→

0. (13) Indeed, by the chain rule (66)

DΨ

= ψ

K | Λ(F

) − C |

γ

! D

| Λ(F

) − C |

γ

and by using [C2]-[C3] and (7), we get for every p > 1 γ

| B

(θ, λ ) | ≤ cγ

E

h DΨ

, D( − L)

F

i

≤ cγ

P

| Λ(F

) − C |

≥ 1 2 γ

≤ cE | Λ(F

) − C |

γ

≤ cγ

where we used again [C2]. Since δ < 1 and p is arbitrarly large, we obtain the convergence (13).

On the other hand, by assumption [C1], we have the convergence

γ

A

(θ, λ ) →

− θ

d

X

j,k=1

λ

λ

E

e

Z

. (14)

Using our assumption (10), we can express the above limit in a more explicit way. We have E

e

Z

= E

e

E(Z

| Z

)

= E e

d

X

a=1

ρ

Z

!

(15) and, moreover, for every a = 1, .., d, from the differential equation verified by the charac- teristic function of the normal distribution,

E

e

Z

= e

E

e

Z

= e

− 1 i θ (

d

X

i=1

λ

C

)θ

e

= e

− θ i (

d

X

i=1

λ

C

) = i θe

d

X

i=1

λ

C

. (16)

Thus, by replacing (15) and (16) in (14), we obtain γ

A

(θ, λ ) →

−i θ





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e

. (17) Relation (17), together with 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 ∈ R

, α ∈ Z

+

and for every symmetric d × d matrix C, we will denote by

P

(u, z, C) = e

( −i )

∂

∂u

e

(18) where | α | := α

+ ... + α

if α = (α

, .., α

) ∈ Z

. Then clearly,

( −i )

∂

∂u

e

= e

P

(u, z, C). (19)

In the next lemma we recall the explicit expression of the polynomial P

together

on some estimates on this polynomial. We refer to Lemma 1 in [16] for the proof.

Lemma 1 If P

is given by (18), then for every u, z ∈ R

and evey d × d matrix C, we have

P

(u, z, C) = X

0≤β≤α

C

( −i )

z

S

(u, C ) with

S

(u, C) = e

∂

∂u

e

. Moreover, we have the estimate

| S

(u, C ) | ≤

|β|

X

j=0

c

| u |

| C |

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

.

Proposition 2 Assume [C1] -[C3] and (10). Let ϕ

be given by (9). Then for every α ∈ Z

+

, we have

γ

∂

∂ λ

∂

∂θ ϕ

(θ, λ ) →

∂

∂ λ



 −i θ





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e



 .

Proof: If α = (α

, ..., α

) ∈ Z

, then by (12) we have

∂

∂ λ

∂

∂θ ϕ

(θ, λ ) = ∂

∂ λ

A

(θ, λ ) + ∂

∂ λ

B

(θ, λ ) with A

, B

defined in (12). Now, since by (19)

∂

∂ λ

e

= i

P

( λ , θF

, − (1 − θ

)C)e

we can write

∂

∂ λ

∂

∂θ ϕ

(θ, λ )

= − θ

d

X

j,k=1

λ

λ

E

Ψ

e

i

P

( λ , θF

, − (1 − θ

), C)

h DF

, D( − L)

F

i

− C

+ i

d

X

j=1

λ

E

Ψ

e

i

P

( λ , θF

, − (1 − θ

), C) h DΨ

, D( − L)

F

i

(20)

As in the proof of (13), the second summand above multiplied by γ

converges to zero as N → ∞ . Concerning the first summand in (20), the hypothesis [C1] implies that it converges as N → ∞ , to

− θ

d

X

j,k=1

λ

λ

E

i

P

( λ , θZ

, − (1 − θ

)C)e

Z

= − θ

d

X

j,k=1

λ

λ

E

∂

∂ λ

e

Z

= ∂

∂ λ



 −i θ





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e



 .

Then the conclusion is obtained.

Remark 1 If d = 1, relation (11) becomes γ

∂

∂θ ϕ

(θ, λ) →

−i θ

λ

ρe

2 2

where ρ = E(Z

/Z

).

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

( R

) with l ≥ 0 and let G be a random variable in D

. Then it holds

E ∂

∂x

f (F

)GΨ

= E (f (F

)H

(GΨ

)) (21) where the random variable H

(GΨ

) belongs to L

(Ω) for every p ≥ 1.

We will use the integration by parts formula in the above Lemma 2 to show that

∂^α

∂λ^α ∂

∂θ

ϕ

(θ, λ ) is dominated by an integrable function uniformly with respect to (θ, λ ) ∈ [0, 1] × R

.

Lemma 3 Assume [C2]-[C3]. For every α ∈ Z

, there exists a positive constant C

such that

γ

∂

∂

∂θ ϕ

(θ, λ )

≤ C

(1 + | λ | )

(22)

for all λ ∈ R

, θ ∈ [0, 1] and N ≥ 1.

Proof: First we consider the case α = 0 ∈ Z

. Consider the function f (x) = e

for x ∈ R

with

f (x) = f (x)(i λ )

.

For θ ∈ (0, 1], by using (12) and the integration by parts formula (21) to the function f ℓ-times, we obtain

( i θ λ )

∂

∂θ ϕ

(θ, λ )

= − θ( i θ λ )

d

X

j,k=1

λ

λ

E

Ψ

e

h DF

, D( − L)

F

i

− C

+ i ( i θ λ )

d

X

j=1

λ

E

e

h DΨ

, D( − L)

F

i

= − θ

d

X

j,k=1

λ

λ

E

Ψ

e

H

q

h DF

, DF

i

− 1 Ψ

+

d

X

j=1

λ

E

e

H

h i λ ( − q)

DF

, DΨ

i

We use this estimate for θ ≥ 1/2. For θ ∈ [0, 1/2], the inequality is trivial thanks to the exponential function exp( − (1 − θ

)

). Thus we obtained the desired estimate when α = 0. For α ∈ Z

, α 6 = (0, ...0), we can follow a similar argument with the estimate

sup

θ,λ

(1 − θ

)(1 + λ

)

exp − (1 − θ

) λ

C λ 2

< ∞ for every m ∈ Z

. Thus we obtained the result.

Let us denote by ϕ

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.

ϕ

( λ ) = e

− i γ

1 3





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e

. (23) The next result shows that the difference between the truncated characteristic func- tion of F

and the characteristic function of the weak limit of the sequence F

can be approximated by ϕ

.

Proposition 3 Suppose that [C1]-[C3] are fulfilled for ℓ = d + 3. For every N ≥ 1, let ϕ

be given by (23). Then

γ

Z

Rd

∂

E

e

i Ψ

− E[Ψ

]ϕ

( λ )

d λ →

0

for every α ∈ Z

.

Proof: If α ∈ Z

, using

ϕ

(1, λ ) = E

e

Ψ

and ϕ

(0, λ ) = e

we can write

Z

Rd

γ

∂

E

e

Ψ

− E[Ψ

]ϕ

( λ )

d λ

= Z

R

∂

Z

0

γ

∂

ϕ

(θ, λ ) − ( −i )θ

E[Ψ

]





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e

dθ

d λ

≤ Z

R^d

Z

0

γ

∂

∂

ϕ

(θ, λ ) − ∂

( −i )θ





d

X

i,j,k,a=1

λ

λ

λ

ρ

C



 e

dθd λ +C k 1 − Ψ

k

.

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 ).

Let ϕ

be the characteristic function of the random vector F

, i.e. ϕ

( λ ) = Ee

for λ ∈ R

. If ϕ

is given by (23), we have

ϕ

(λ)

= ϕ

( λ )EΨ

+ E

Ψ

e

− E(Ψ

)ϕ

( λ ) + E

(1 − Ψ

)e

= ϕ

( λ ) + ϕ

( λ )(EΨ

− 1) + E

Ψ

e

− E(Ψ

)ϕ

( λ ) + E

(1 − Ψ

)e

Proposition 3 shows that the difference E Ψ

e

− E(Ψ

)ϕ

( λ ) while from the definition of the truncation function Ψ

we see that the term E((1 − Ψ

)e

+ ϕ

( λ )(EΨ

− 1) is also small. Consequently, ϕ

( λ ) can be approximated by ϕ

. Actually, we have

ϕ

( λ ) = ϕ

( λ ) + s

( λ ) (24) where s

( λ ) is a ”small term’” such that γ

s

( λ ) and its derivatives are dominated by the right-hand side of (20) and it that satisfies

s

( λ ) ≤ C k 1 − Ψ

k

+ o

(γ

) (25) for every p ≥ 1, where o

(γ

) denotes a deteministic sequence that depends on λ such that

oλ(γN)

γN

→

0.

Therefore ϕ

is the dominant part in the asymptotic expansion of ϕ

. By inverting

ϕ

, we get the approximate density.

4 The approximate density

In this paragraph, our purpose is to find the first and second-order term in the asymptotic expansion of the density of F

. We will assume throughout this section [C1 ]-[C3 ] are fulfilled for ℓ = d + 3. Let us denote by ϕ

the truncated characteristic function of F

, i.e.

ϕ

( λ ) = E

Ψ

e

= ϕ

(1, λ ) (26)

with ϕ

(1, λ ) from (9).

We define the approximate density of F

via the Fourier inversion of the dominant part of p

N

defined by (26).

Definition 1 For every N ≥ 1, we define the approximate density p

by p

(x) = 1

(2π)

Z

R

e

ϕ

( λ )d λ , x ∈ R

(27) where ϕ

is given by (23).

Let us first notice that in the case d = 1, we can obtain an explicit expression for p

in the next result. Note that for d = 1, assuming C

= 1 and EZ

Z

= ρ, we have from (23)

ϕ

(λ) = e

2 2

− i 1

3 λ

ρe

2

2

, for every λ ∈ R . (28) Recall that p(x) =

2π

e

denotes the density of the standard normal law. By H

we denote the Hermite polynomial of degree k ≥ 0 given by H

(x) = 1 and for k ≥ 1

H

(x) = ( − 1)

e

d

dx

e

.

Proposition 4 If p

is given by (27) then for every x ∈ R , p

(x) = p(x) − ρ

3 γ

H

(x)p(x).

Proof: This follows from (28) and the formula 1

2π Z

R

e

( i λ)

Φ(λ)dλ = H

(x)p(x) (29) for every k ≥ 0 integer. Recall the notation Φ(λ) := e

2 2

.

We prefer to work with the truncated density of F

, which can be easier controlled.

Definition 2 The local (or truncated) density of the random variable F

is defined as the inverse Fourier transform of the truncated characteristic function

p

(x) = 1 (2π)

Z

Rd

e

ϕ

( λ )d λ for x ∈ R

.

The local density is well-defined since obviously the truncated characteristic function ϕ

N

is integrable over R

.

Our purpose is to show that the local density p

N

is well-approximated by the approximate density p

given by (27) in the sense that for every α ∈ Z

sup

x∈R^d

x

p

(x) − p

(x) .

is of order o(γ

). This will be showed in the next result, based on the asymptotic behavior of the characteristic function of F

.

Theorem 2 For every p ≥ 1, and for every integer α = (α

, .., α

) ∈ Z

+

sup

x∈Rd

x

p

(x) − p

(x)

≤ c k 1 − Ψ

k

+ o(γ

) = o(γ

). (30) Proof: We have, by integrating by parts

x

p

(x) − p

(x)

= x

1 (2π)

Z

R

e

ϕ

N

( λ − ϕ

(λ) d λ

= 1

(2π)

( −i )

Z

R^d

e

∂

∂ λ

ϕ

N

( λ ) − ϕ

( λ ) d λ and then, by using (24) and Lemma 3 we obtain

sup

x∈Rd

x

p

N

(x) − p

(x)

≤ 1 (2π)

Z

Rd

d

d λ

ϕ

N

(λ) − ϕ

( λ )

d λ

≤ c Z

R^d

(1 + | λ | )

d λ (o(γ

) + c k 1 − Ψ

k

)

≤ o(γ

) + c k 1 − Ψ

k

= o(γ

)

where we use ℓ − 2 > d since ℓ ≥ d + 3 in [C2]. The fact that c k 1 − Ψ

k

= o(γ

) follows from the choice of our truncation function Ψ

.

Let us make some comments on the above result.

Remark 2 • Theorem 2 extends the inequality (3) when F

belongs to the Wiener chaos of order q ≥ for every N ≥ 1. It is known that for every N ≥ 1, we have that γ

= V ar(

k DF

k

) behaves as EF

− 3, more precisely (see Section 5.2.2 in [8])

γ

≤ q − 1

3q (EF

− 3) ≤ (q − 1)γ

.

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.

As a consequence of Theorem 2 we will obtain the asymptotic expansion for the expectation of measurable functionals of F

.

Theorem 3 Let p

be given by (27) and fix M ≥ 0, γ > 0. Then sup

f∈E(M,γ)

Ef(F

) − Z

R^d

f (x)p

(x)dx

= o(γ

) (31)

where E (M, γ ) is the set of measurable functions f : R

→ R such that | f (x) | ≤ M (1 + | x |

) for every x ∈ R

.

Proof: For any measurable function f that satisfies the assumptions in the statement, we can write

Ef (F

) − Z

Rd

f (x)p

(x)dx = Ef (F

) − Ef(F

)Ψ

+ Ef (F

)Ψ

− Z

Rd

f(x)p

(x)dx

= Ef (F

)(1 − Ψ

) + Z

Rd

f(x) h p

N

(x) − p

(x) i dx with p

N

defined in (26). Then for every p > 1 and for α ≥ 0 large enough (it may depend on M, γ ) with p

such that

+

= 1, we have, by using H¨older’s inequality,

Ef (F

) − Z

R^d

f (x)p

(x)dx

≤

E | f(F

) |

p′

k 1 − Ψ

k

+ Z

Rd

| f (x) | 1

1 + | x |

dx × sup

x∈R^d

1 + | x |

h p

N

(x) − p

(x) i

≤

E | f(F

) |

p′

k 1 − Ψ

k

+ (c k 1 − Ψ

k

+ o(γ

))

≤ C k 1 − Ψ

k

+ o(γ

) = o(γ

).

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 [8] (which reproduces the main result of [9]). 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.

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 (F

)

= (I

(f

))

(with f

∈ H

for every N ≥ 1) be a random sequence in the q th Wiener chaos (q ≥ 2). Assume EF

= 1 for every N ≥ 1 and suppose that the condition [C1] holds with γ

= V ar

kDFNk²H

q

i.e.

F

, γ

k DF

k

q − 1

→

(Z

, Z

)

where (Z

, Z

) is a centered Gaussian vector with EZ

= EZ

= 1 and EZ

Z

= ρ ∈ (0, 1).

Then we have the asymptotic expansions (30) and (31).

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

5 Examples

In the last part of our work, we will present three examples in order two illustrate our theory. The first example concerns a one-dimensional sequence of random variables in the second Wiener chaos and it is inspired from [8], 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

)

with EX

= 1 for every k ∈ Z . We set

ρ(l) := EX

X

for every l ∈ Z so 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

= W (h

) with k h

k

= 1 for every k ∈ Z , where (W (h), h ∈ H) is an isonormal process as defined in Section 6. Define the quadratic variation sequence

V

= 1

√ N

N

X

k=0

X

− 1

, N ≥ 1. (32)

and let v

:= EV

. Assume ρ ∈ ℓ

( Z ) (the Banach space of 2-summable functions endowed with the norm k ρ k

= P

k∈Z

ρ(k)

). In this case, we know from [8] that the deterministic