High order asymptotic expansion for Wiener functionals

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High order asymptotic expansion for Wiener functionals

Ciprian Tudor, Nakahiro Yoshida

To cite this version:

Ciprian Tudor, Nakahiro Yoshida. High order asymptotic expansion for Wiener functionals. 2019.

�hal-02291565�

High order asymptotic expansion for Wiener functionals ^∗

Ciprian A. Tudor

and Nakahiro Yoshida

1

Universit´e de Lille 1

3

Graduate School of Mathematical Sciences, University of Tokyo

4

CREST, Japan Science and Technology Agency

5

The Institute of Statistical Mathematics September 19, 2019

Abstract

By combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for a sequence of vector-valued random variables. Our asymptotic expansion formulas give the development of the characteristic functional and of the local density of the random vectors up to an arbitrary order. We analyzed in details an example related to the wave equation with space-time white noise which also provides interesting facts on the correlation structure of the solution to this equation.

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

Key Words and Phrases: Asymptotic expansion, Stein-Malliavin calculus, Central limit theorem, cumulants, wave equation.

∗This work was in part supported by Japan Science and Technology Agency CREST JP- MJCR14D7; 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 Statis- tical Mathematics.

†Universit´e de Lille 1: 59655 Villeneuve d’Ascq, France

‡Graduate School of Mathematical Sciences, University of Tokyo: 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. e-mail: [email protected]

1 Introduction

The asymptotic expansion of probability distributions for random variables and vec- tors represents a fundamental topic in probability theory and mathematical statistics.

This theory has been widely applied to several fields, including the efficiency of es- timators, hypothesis testing, information criterion for model selection, prediction theory, bootstrap methods and resampling plans, and information geometry. There exists now huge literature on asymptotic expansion. We refer, among many others, to [2] and [1] for the case of sequences of i.i.d. random variables and applications, to [3] for sequences of weakly dependent variables, to [9], [25], [26], [7], [27], [16], [28], [15] and [14] for asymptotic expansion of martingales and of classical diffusions, and to [12] or [20] for general sequences of random variables. The reader may consult the monographs [2], [4] and [8] for complete expositions on these topics.

Generally speaking, the asymptotic expansion theory aims at finding the ex- pansion of the density functions for a sequence of random variables that converges in law to a target distributions (usually, the Gaussian distribution, but other target distributions, such as mixed normal, are possible). The theory usually provides the leading term and second order term in the asymptotic development of the density function. For estimators and test statistics, while the leading term is used for confi- dence limits and testing, the higher order terms provide a more accurate inference.

Some higher order-type asymptotic expansions for particular sequences of diffusion- type can be found in [22], [24] with applications to statistics, and in [23], [6], [21], [17], [18], [19] with applications to finance.

Our purpose is to provide a general method to obtain the asymptotic expan-

sions of density functions up to an arbitrary order, i.e. to find the further terms that

appear in the asymptotic behavior of the density. Our approach combines the so-

called Fourier approach and the recent Stein-Malliavin theory (see [10]) and applies

to general sequences of random vectors. Our main finding is that the asymptotic ex-

pansion up to any order of the family of densities of a vector-valued random sequence

(F

)

is completely characterized by the expectation of the so-called Gamma fac-

tors associated to the sequence F

, or equivalently, by the joint cumulants of the

components of this sequence. These Gamma factors (defined in Section 2.1) are de-

fined in terms of the Malliavin operators of F

. Consequently, the knowledge of the

Taylor expansions of the cumulants, together with some regularity in the Malliavin

sense of F

, gives the higher order asymptotic expansion of the density. We also

mention that, in contrast with the classical assumptions in the martingale case (see

e.g. [26], [28]) or in the Malliavin calculus case (see [20]), the joint convergence in

distribution of F

together with its ”bracket” (which is the usual martingale bracket

when we deal with sequences of martingales and it is defined in terms of the Malliavin

derivative in the non-martingale case) is not assumed in our work.

As mentioned above, our strategy is based on the Stein-Malliavin calculus combined with the so-called Fourier approach. We start by analyzing the behavior of the (truncated) characteristic function of the sequence (F

)

via an interpolation method and the Malliavin-type integration by parts. We notice the appearance of the Gamma factors in the principal part of asymptotic expansion of the characteristic function. The Fourier inversion, together with some regularity of the distribution expressed in term of the Malliavin calculus, allows to develop asymptotically the sequence of (local) densities of F

. Some regular ordering of the cumulants is assumed and this is checked in examples. Usually, the second order term in the asymptotic expansion comes from leading term in the expansion of third cumulant, while the third order term is due to the second and the fourth cumulant. A general formula is obtained.

As an example, we analyze the behavior of the spatial quadratic variation for the solution to the wave equation driven by a space-time white noise. We treat both the one dimensional case (i.e we fix the time t and we study the quadratic variation in space for the solution), as well as a two-dimensional case (i.e. we consider a the two- dimensional random sequence whose components are the spatial quadratic variations of the solution at two different times). In both cases, based on a sharp analysis of the correlation structure of the solution, we are able to find the asymptotic expansion up to at least the third order term. Let us emphasize that, besides being a toy example to apply our asymptotic expansion theory, this last part of our work shows some interesting facts related to the solution to the wave equation driven by a space-time white noise. We obtain the precise correlation of the increments of the solution, at fixed time and when the time is moving and it appears that the dependence structure of these increments depends in a non-trivial way on the spatial and temporal lags.

We organized our paper as follows. Section 2 presents, after the definition

and some basic properties of the Gamma-factors, the asymptotic expansion up to

an arbitrary order of the (truncated) characteristic function of a sequence of vector-

valued random variables (F

)

. This expansion depends on the Gamma-factors (or

equivalently, the cumulants) of the vector F

. Based on this expansion, we obtain

in Section 3, by inverting in Fourier sense the principal part of the characteristic

function, the approximate density for our sequence. This will approximate the local

(truncated) density of F

. The asymptotic expansion is further explicited in Section

4 where we show that if the cumulants of F

admit a specific Taylor expansion, then

a more precise expansion of the local density can be derived. In Section 5 we treat

in details a concrete example related to the solution to the wave equation driven by

an additive space-time white noise. The last section is the Appendix which contains

the basic tools of the Malliavin calculus.

2 Expansion of the characteristic functional

In this section, we analyze the asymptotic behavior of a general sequence of random vectors, by using an interpolation method and the Malliavin integration by parts. We will distinguish a principal part of the characteristic function, written in terms of the Gamma factors and a neglijible part. These two parts are then estimated separately.

2.1 The Gamma factors

Let (W (h), h ∈ H) be an isonormal process on a standard probability space (Ω, F , P ).

For the definition of the Malliavin operators with respect to W , see Section 6. The pseudo-inverse L

of L is defined by L

F = P

q=1

q

J

F for F = P

q=0

J

F ∈ L

(Ω), where J

is the orthogonal projection to the q-th chaos. For p ∈ N and F ∈ D

, we define the Gamma- factors Γ

(F ) is a reccursive way, see e.g. [10].

Γ

(F ) = F

Γ

(F ) = h DF, D( − L)

F i

, ....

Γ

(F ) = h DF, D( − L)

Γ

(F ) i

.

These variables are well defined by Lemma 1 below, if F is regular enough in the sense of the Malliavin calculus.

We have the following formula that links the Gamma-factors and the cumu- lants: for every m ≥ 1

k

(F ) = (m − 1)!E

Γ

(F )

. (1)

Recall that the mth cumulant of a random variable F ∈ L

(Ω) is given by k

(F ) = ( − i)

∂

∂t

ln E e

t=0

.

We also introduce the multidimensional Gamma factors of a random vector F = (F

, .., F

) ∈ D

( R

) are defined in the following way. For i = 1, .., d,

Γ

(F ) = F

and for i

, i

= 1, .., d,

Γ

(F ) = h DF

, D( − L)

F

i

while for i

, .., i

= 1, .., d

Γ

(F ) = h DF

, D( − L)

Γ

(F ) i

. (2)

The muldimensional Gamma factors Γ

(F ) are also related to the joint cumulants of the random vector F . Recall that if m = (m

, .., m

) ∈ N

, then the mth cumulant of the random vector F = (F

, .., F

) is

k

(F ) = k

(F

, .., F

) = ( − i)

∂

∂t

log E e

|

where | m | = m

+ ... + m

. See [10] for the precise link between the multidimensional Gamma factors and the cumulants.

Let Z

= { 0, 1, 2, ... } .

Lemma 1. (a) Let ℓ ∈ Z

and r > 1. Then D( − L)

F ∈ D

(H) if F ∈ D

, and there exists a constant C

such that

D( − L)

F

≤ C

F

for all F ∈ D

.

(b) Let ℓ ∈ Z

, r > 1 and p ∈ { 2, 3, ... } . Then Γ

(F ) ∈ D

if F = (F

, ..., F

) ∈ D

( R

), and there exists a constant C

such that

Γ

(F )

≤ C

F

ℓ+1,pr

(3)

for all F ∈ D

and i

, ..., i

∈ { 1, ...d } . In particular, Γ

(F ) ∈ D

if F ∈ D

∞−

= ∩

D

.

Proof. Let r > 1. Since D( − L)

F = (I − L)

DF for F ∈ P , the set of polynomial functionals, we have

D( − L)

F

<

∼

(I − L)

D( − L)

F

=

(I − L)

DF

< ∼ DF

<

∼ F

for all F ∈ P uniformly. Therefore, D( − L)

is extended as a continuous linear operator from D

to D

( H ). Thus we obtained (a).

Suppose that F = (F

, ..., F

) ∈ D

( R

). The property (b) follows from (a) by induction. Indeed, since Γ

(F ) = h DF

, D( − L)

Γ

k)

i

, if (3) holds for k ( ≤ p − 1) in place of p, then

Γ

(F )

<

∼

F

D( − L)

Γ

k)

(F )

< ∼

F

Γ

k)

(F )

< ∼

F

F

ℓ,(k+1)r

< ∼ F

ℓ+1,(k+1)r

.

Thus, (3) holds for p = k + 1. The inequality (3) is trivial when p = 1, which completes the proof.

2.2 Interpolation

Let d ≥ 1. Consider a sequence of centered random variables (F

)

in R

of the form

F

=

F

, . . . , F

.

Let C = (C

)

be a deterministic d × d positive definite symmetric matrix. For every N ≥ 1, we introduce a truncation functional Ψ

which is a smooth random variable. A more precise form of this functional will be chosen later in Section 2. We consider the following condition for the truncation functional Ψ

: if p is a positive integer

[Ψ ] For each N ∈ N , Ψ

: Ω → [0, 1] and Ψ

∈ D

. Let us define the interpolation functional

e(θ, λ , F

) = exp

i θ h λ , F

i − 1

2 (1 − θ

) λ

C λ

(4) for every θ ∈ [0, 1] and λ ∈ R

. Let λ = (λ

, ..., λ

) ∈ R

and θ ∈ [0, 1] and let us consider the truncated interpolation

ϕ

(θ, λ ) = E

Ψ

e(θ, λ , F

)

. (5)

Notice that ϕ

(1, λ ) = E

Ψ

e

represents the ”truncated” characteristic func- tion of F

, while ϕ

(0, λ ) = E[Ψ

]e

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

.

The first step is to get the expansion of the derivative with respect to the variable θ of the characteristic functional.

Lemma 2. Suppose that F

∈ D

( R

) for any N ≥ 1 and that [Ψ] holds. Then

the functional ϕ

(θ, λ ) is well defined for (θ, λ ) ∈ [0, 1] × R

, and it holds that

∂

∂θ ϕ

(θ, λ ) = i(iθ)

X

i1,...,ip+1=1

λ

....λ

E

Ψ

e(θ, λ , F

)Γ

(F

)

+ i ( i θ)

X

i1,...,ip=1

λ

....λ

E

Ψ

e(θ, λ , F

) E

Γ

(F

)

+ i ( i θ)

X

i1,...,ip−1=1

λ

....λ

E

Ψ

e(θ, λ , F

) E

Γ

(F

) + · · ·

+ i ( i θ)

X

i1,i2,i3=1

λ

λ

λ

E

Ψ

e(θ, λ , F

) E

Γ

(F

)

+i(iθ) X

i1,i2=1

λ

λ

E

Ψ

e(θ, λ , F

) E

Γ

(F

)

− C

(6)

+ X

j=1

R

(θ, λ )

for (θ, λ ) ∈ [0, 1] × R

, λ = (λ

, . . . , λ

), where R

(θ, λ ) = i ( i θ)

X

i1,...,ij=1

λ

....λ

E h

e(θ, λ , F

)

DΨ

, D( − L)

Γ

(F

)

H

i

(7) for j = 1, 2, .., p. In (6), Γ

(F

) can be replaced by the symmetrized version Γ

(F

) = 2

Γ

(F

) + Γ

(F

)

.

Proof. The Malliavin derivative of the random variable e(θ, λ , F

) given by (4) can be calculated as follows

De(θ, λ , F

) = i θ e(θ, λ , F

) X

k=1

λ

DF

. (8)

We differentiate ϕ

(θ, λ ) given by (5) with respect to θ, and we use the identity (99)

as F

= δD( − L)

F

for every j = 1, .., d and the duality relationship and (8) to

obtain

∂

∂θ ϕ

(θ, λ ) = X

j=1

iλ

E

Ψ

e(θ, λ , F

)F

+

X

j=1

θλ

λ

E

Ψ

e(θ, λ , F

)C

= − θ X

j,k=1

λ

λ

E

Ψ

e(θ, λ , F

)

h DF

, D( − L)

F

i

− C

+i X

j=1

λ

E

e(θ, λ , F

) h DΨ

, D( − L)

F

i

= i ( i θ) X

i1,i2=1

λ

λ

E

Ψ

e(θ, λ, F

)

Γ

(F

) − C

+R

(θ, λ ) (9)

with

R

(θ, λ ) = i X

j=1

λ

E

e(θ, λ , F

) h DΨ

, D( − L)

F

i

.

The formula (9) has been also obtain in [20] and it allows to obtain the second order terms in the asymptotic expansion of the sequence (F

)

. In order to get the higher order term in this asymptotic expansion when p ≥ 2, we refine the above formula (9). By (99), we write

Γ

(F

) − C

= Γ

(F

) − E

Γ

(F

) + E

Γ

(F

)

− C

= δD( − L)

Γ

(F

) + E

Γ

(F

)

− C

(10) for every i

, i

∈ { 1, .., d } , and we apply the duality and (8) once again to obtain

E

Ψ

e(θ, λ , F

)δD( − L)

Γ

(F

)

= iθE

Ψ

e(θ, λ , F

) X

i3=1

λ

h DF

, D( − L)

Γ

(F

) i

+E

e(θ, λ , F

) h DΨ

, D( − L)

Γ

(F

) i

. (11)

From (9), (10) and (11), we obtain

∂

∂θ ϕ

(θ, λ ) = i ( i θ)

X

i1,i2,i3=1

λ

λ

λ

E

Ψ

e(θ, λ , F

)Γ

(F

)

+i(iθ) X

i1,i2=1

λ

λ

E

Ψ

e(θ, λ , F

) E

Γ

(F

)

− C

+R

(θ, λ ) + R

(θ, λ ) for every θ ∈ [0, 1] and λ ∈ R

, where

R

(θ, λ ) = i(iθ) X

i1,i2=1

λ

λ

E

e(θ, λ , F

) h DΨ

, D( − L)

Γ

(F

) i

.

By iterating this procedure, we obtain the desired expansion of

ϕ

(θ, λ ).

2.3 Approximation to the characteristic functional and esti- mate of the error

The result in Lemma 2 shows that the behavior of the characteristic functional ϕ

(θ, λ ) is closely related to the behavior of the Gamma factors (or equivalently, the cumulants) of F

. Let us define, for (θ, λ ) ∈ [0, 1] × R

, the following quantity, which will be called in the sequel the principal part

P

(θ, λ ) = i ( i θ)

X

i1,...,ip+1=1

λ

....λ

E

Γ

(F

)

+i(iθ)

X

i1,...,ip=1

λ

....λ

E

Γ

(F

)

+ i ( i θ)

X

i1,...,ip−1=1

λ

....λ

E

Γ

(F

) + · · ·

+ i ( i θ)

X

i1,i2,i3=1

λ

λ

λ

E

Γ

(F

)

+i(iθ) X

i1,i2=1

λ

λ

E

Γ

(F

)

− C

(12)

and let

R

(θ, λ ) = i(iθ)

X

i1,...,ip+1=1

λ

....λ

E

Ψ

e(θ, λ , F

)

Γ

(F

) − E

Γ

(F

)

+ X

j=1

R

(θ, λ ).

The truncated characteristic function can be written in terms of the principal part P

(θ, λ ) and of the residual term R

(θ, λ ).

Lemma 3. Suppose that F

∈ D

( R

) and that [Ψ] holds. Then, for any positive number c, the functional ϕ

(θ, λ ) admits the expression

ϕ

(θ, λ ) = ϕ

(0, λ ) exp Z

0

P

(θ

, λ )dθ

+ Z

0

exp Z

θ1

P

(θ

, λ )dθ

R

(θ

, λ )dθ

(13) for (θ, λ ) ∈ [0, 1] × R

.

Proof. By Lemma 2, we have

∂

∂θ ϕ

(θ, λ ) = R

(θ, λ ) + ϕ

(θ, λ )P

(θ, λ ). (14) Solve (14) to obtain (13).

In order to obtain the high-order asymptotic expansion, we need to assume several conditions. The first assumptions stated below concern the Malliavin regular- ity of F

and of its associated Gamma factors. We fix a positive number q, that will be the order of the asymptotic expansion. Given a positive integer ℓ, we consider the following conditions.

[A1] (i) F

∈ D

∞

and

sup

N∈N

F

ℓ+1,r

< ∞ (15)

for every r > 1.

(ii) For some positive constant a and some d × d non-singular matrix C

satis- fying C = 2

C

+ C

, it holds that Γ

(F

) − C

ℓ,r

= O(N

)

as N → ∞ for every r > 1.

Let ℓ

∈ { 1, ..., ℓ } . Consider also the condition [A2 ] (i) For some r > 1,

Γ

(F

) − E

Γ

(F

)

ℓ1,r

= O (N

) (16) as n → ∞ for i

, ..., i

∈ { 1, ..., d } .

(ii) For some r > 1,

Γ

(F

) − E

Γ

(F

)

= o(N

) (17) as n → ∞ for i

, ..., i

∈ { 1, ..., d } .

Obviously, a sufficient condition for [A2] is [A2

] For some r > 1,

Γ

(F

) − E

Γ

(F

)

= o(N

) as n → ∞ for i

, ..., i

∈ { 1, ..., d } .

In what follows, we will work with Ψ

defined by Ψ

= ψ(Ξ

) for a function ψ ∈ C

( R ; [0, 1]) such that ψ(x) = 1 for x ∈ [ − 1/2, 1/2] and ψ(x) = 0 for x ∈ ( − 1, 1)

, and a functional Ξ

given by

Ξ

= K

N

Γ

(F

) − C

, (18) where K

is a positive number and Γ

(F

) = (Γ

(F

))

. By choosing a suffi- ciently large K

, we may assume, due to (18), that there exist positive constants c

and c

such that

c

≤ det Γ

(F

) ≤ c

(19) for all N ∈ N and a.s. ω ∈ Ω whenever | Ξ

| < 1.

Let Λ

= { λ ∈ R

; | λ | ≤ N

} , where ξ is a positive constant. We will put a condition ([A3]) concerning ξ later.

Lemma 4. Suppose that [A1] is fulfilled.

(a) Suppose that [A2] (i) is satisfied. Then, for any α ∈ Z

+

, there exists a constant C

such that

sup

N∈N

sup

θ∈[0,1]

1

N

∂

∂ λ

R

(θ, λ )

≤ C

(1 + | λ | )

( λ ∈ R

) (20)

(b) Suppose that [A2] (ii) is satisfied. Then sup

θ∈[0,1]

∂

∂ λ

R

(θ, λ )

= o(N

) as n → ∞ for every λ ∈ R

and α ∈ Z

+

. Remark 1. (i) Under the assumption that sup

F

ℓ+1,(p+1)r

< ∞ , we have sup

N∈N

k Γ

(F

) k

< ∞

thanks to Lemma 1. In this situation, Condition (16) is requesting the order N

to the norm. (ii) It is possible to show a similar result under the condition that

Γ

(F

)

= o(N

)

instead of [A2] though we will not pursue it here. (iii) Logically, ℓ in Lemma 4 can be ℓ

.

Proof of Lemma 4. Let e Γ

(F

) = Γ

(F

) − E

Γ

(F

) . For α ∈ Z

+

, we have

∂

∂ λ

R

(θ, λ ) = i(iθ)

X

i1,...,ip+1=1

α γ

E

(θ, λ ) ∂

∂ λ

λ

....λ

+ X

j=1

∂

∂ λ

R

(θ, λ ) (21)

where

E

(θ, λ ) = E

Ψ

∂

∂ λ

e(θ, λ , F

) Γ e

(F

)

. By definition,

∂

∂ λ

e(θ, λ , F

) = p

θF

, (1 − θ

)C λ , (1 − θ

)C

e(θ, λ , F

) (22)

for a polynomial p

.

By Lemma 1 and [A1] (i), Γ

(F

) ∈ D

1

for any r

> 1. We apply the IBP formula (see Lemma 2 in [20]) to the interpolation functional e(θ, λ , F

) that is taking advantage of (19) and based on the chain rule

Df (F

), D( − L)

F

H

=

X

i2=1

∂f

∂x

(F

)Γ

(F

) for f ∈ C

( R

) of at most polynomial growth, in order to obtain

( i θ λ )

E

Ψ

e

p

θF

, (1 − θ

)C λ , (1 − θ

)C Γ e

(F

)

= E

e

Φ

Ψ

p

θF

, (1 − θ

)C λ , (1 − θ

)C Γ e

(F

)

for λ ∈ Λ

and β ∈ Z

+

with the sum of the components | β | = ℓ

. Here Φ

is a linear functional and we see that

sup

N∈N θ∈[0,1]

λ∈Rd

Φ

Ψ

p

θF

, (1 − θ

)C λ , (1 − θ

)C

N

e Γ

(F

)

e

1

< ∞ .

(23) Here we used (15), (16) for which r is strictly larger than 1, and the estimate

(1 − θ

)C λ ≤

C

1 + (1 − θ

) λ

C λ .

For each λ = (λ

) ∈ Λ ˇ

, we choose a component i that attains max {| λ

| ; j = 1, ...d } and use the above estimate for β = (i, i..., i). Then we obtain

sup

n∈N, θ∈[1/2,1],λ∈Rd

N

| λ |

E

(θ, λ )

< ∞ . (24)

Moreover, there exists a positive constant c

such that sup

N∈N, θ∈[0,1/2)

e(θ, λ , F

)

≤ c

exp − c

| λ |

(25) for all λ ∈ R

. From (24) and (25), we obtain

sup

N∈N

sup

θ∈[0,1]

E

(θ, λ ) ∂

∂ λ

λ

....λ

≤ K

(1 + | λ | )

(26)

for all λ ∈ R

, where K

is a constant depending on α − γ but independent of λ ∈ R

. Since k DΨ

k

= O(N

) for every m > 0, for Λ

= {| λ | ≤ N

} and for every k > 0 and s > 0, we see

N

lim

sup

λ∈ΛN

N

| λ |

∂

∂ λ

R

(θ, λ )

< ∼ lim

N→∞

N

= 0 (27) if we choose a sufficiently large m. Now, from (26) and (27), we obtain Inequality (20).

The property (b) is rather easy to prove since it only requires λ -wise estimate of an explicit expression of

R

(θ, λ ) by using [A2] (ii).

Estimate of ∂

/∂ λ

ϕ

(θ, λ ) will be necessary in the proof of Proposition 1 down below. The only possibility is to repeatedly apply the IBP formula. The estimate outside of Λ

cannot gain improvement by the decay of cumulants, and it is the worst one, in other words, it uses the highest order of repetition of the IBP formula among other terms.

Lemma 5. Suppose that [A1] (i) is fulfilled. Then, for every α ∈ Z

+

, there exists a constant C

such that

∂

∂ λ

ϕ

(θ, λ )

≤ C

(1 + | λ | )

(28) for all θ ∈ [0, 1], λ ∈ R

and N ∈ N ,with ℓ from condition [A1].

Proof. This lemma can be proved by estimatrion quite similar to that for E

(θ, λ ) in the proof of Lemma 4. We can repeat the IBP formula (i.e. Lemma 2 in [20]) ℓ-times in this case for θ ∈ [1/2, 1]. Estimation for θ ∈ [0, 1/2) is also similar.

Recall the expression (13) of the characteristic function ϕ

(θ, λ ) in Lemma 3. Based on the estimated in Lemma 4, we can show that the last summand in the right-hand side can be neglegted. Consequently, the dominant part of ϕ

(θ, λ ) will come from the first term in right-hand side of (13), which involves the exponential of the principal part. We will introduce a new random variable, in which we keep the first terms in the Taylor expansion of the exponential function. For k ∈ N , let

P

( λ ) = X

j=0

1 j!

Z

P

(θ, λ )dθ

and

R

( λ ) = ϕ

(0, λ ) X

j=k+1

1 j!

Z

P

(θ, λ )dθ

+ Z

0

exp Z

θ1

P

(θ

, λ )dθ

R

(θ

, λ )dθ

.

Then, if one has the expansion (13), then

ϕ

(1, λ ) = ϕ

(0, λ )P

( λ ) + R

( λ ). (29)

At this point, let us make a brief summary to comment on the role of the parameters that appear in our work. Recall that d is the dimension of the random vector, p + 1 is the maximum order of the cumulant that appear in the decomposition of the characteristic functional ϕ

(θ, λ ), k comes from the Taylor expansion of the exponential function while ℓ, ℓ

, q appear in assumptions [A1] − [A2]. N

will be the the size of the error term. Next, we need to assume some relation between all these parameters. Condition (30) will be needed to evaluate the residual terms.

[A3 ] The numbers q

∈ (0, ∞ ), ξ ∈ (0, ∞ ), ℓ ∈ N and ℓ

∈ N satisfy q

(k + 1) > q, ξ(ℓ − d) > q, ℓ

> p + 1 + d and

N

E

Γ

(F

)

− C +

X

j=3

N

E

Γ

(F

) = O(1) (30) as N → ∞ . The expectation of a matrix is understood componentwise and

· denotes the Euclidean norm.

Remark 2. (a) Condition (30) imposes a restriction on ξ from above. Besides, p will be asked to be large according to the value of q in order to satisfy [A2]. In this sense, p is a function of q: p = p(q).

(b) If we take ℓ

= ℓ, then Condition [A3] requires ℓ > d + max

q ξ , p + 1

to ℓ. If d = 1, p = 1, then ℓ ≥ 4 at least.

(c) Increasing k causes only increase of complexity of the formula and does not require F

to pay more, once we found p for [A2] and q

for (30).

(d) The index q

depends on the largest order of terms among the terms appearing

in (30). Often the term for j = 3 dominates other terms.

(e) In most common situations, the order p of the controllable cumulants is the essen- tial parameter. It determines the possible order q of the asymptotic expansion.

The parameters ℓ, k, ξ and q

are somewhat technical but automatically de- termined by p and the dimension d. See Section 4.2, that treats a regularly ordered expansion.

Lemma 6. (a) Suppose that Conditions [A1], [A2] (i) and [A3] are satisfied. Then there exists a constant K such that

N

1

∂

∂ λ

R

( λ )

≤ K(1 + | λ | )

for all (λ, N ) ∈ R

× N .

(b) Suppose that Conditions [A1], [A2] (ii) and [A3] are satisfied. Then

∂

∂ λ

R

( λ ) = o(N

) as n → ∞ for every λ ∈ R

.

Proof. It follows from [A3] that sup

θ∈[0,1],λ∈Rd

1

∂

∂ λ

P

(θ, λ )

= O(N

) (31) as N → ∞ for every α ∈ Z

+

. Therefore P

undertakes λ ’s by itself.

The property (a) follows from Lemma 4 (a), and the representations of R

( λ ) by R

(θ, λ ) and P

(θ, λ ) and q

(k + 1) > q. Similarly, the property (b) can be proved by using Lemma 4 (b).

3 Asymptotic expansion of the expectation and its error bound

The approximate density of the sequence (F

)

is defined as the Fourier inverse of the dominant part of the truncated characteristic function of F

as follows.

f

(x) = 1 (2π)

Z

Rd

e

ϕ

( λ )d λ , x ∈ R

(32)