Irène Gannaz
email: irene.gannaz@insa-lyon.fr

Asymptotic normality of wavelet covariances and multivariate wavelet Whittle estimators

Keywords: Multivariate processes, long-range dependence, covariance, wavelets, asymptotic normality, cerebral connectivity

Multivariate processes with long-range dependence properties can be encountered in many fields of application. Two fundamental characteristics in such frameworks are long-range dependence parameters and correlations between component time series. We consider multivariate long-range dependent linear processes, not necessarily Gaussian. We show that the covariances between the wavelet coefficients in this setting are asymptotically Gaussian. We also study the asymptotic distributions of the estimators of the long-range dependence parameter and the long-run covariance by a wavelet-based Whittle procedure. We prove the asymptotic normality of the estimators, and we provide an explicit expression for the asymptotic covariances. An empirical illustration of this result is proposed on a real dataset of rat brain connectivity.

Introduction

Univariate long-range dependent processes are processes with an autocovariance function with a power-law decay or equivalently a spectral density diverging at the zero frequency with a power-law rate. Univariate long-range dependence (LRD) has encountered much interest and is used widely in applications. See, for example, [START_REF] Percival | Wavelet methods for time series analysis[END_REF][START_REF] Beran | Long-Memory Processes[END_REF][START_REF] Pipiras | Long-range dependence and self-similarity[END_REF] and references therein.

Data is often recorded by multiple sensors where multivariate modeling brings better representation and can increase the consistency of inference. Multivariate processes with LRD properties are found in a wide range of applications, such as geoscience [START_REF] Whitcher | Wavelet estimation of a local long memory parameter[END_REF], finance [START_REF] Gençay | An introduction to wavelets and other filtering methods in finance and economics[END_REF], or neuroscience [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]. Extensions of univariate LRD models to multivariate frameworks were initiated by [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF], and this topic has met great interest over the last decades. Several models have been proposed, such as multivariate autoregressive fractionally integrated moving average (ARFIMA) models [START_REF] Lobato | Consistency of the averaged cross-periodogram in long memory series[END_REF][START_REF] Sela | Computationaly efficient methods for two multivariate fractionnaly integrated models[END_REF][START_REF] Kechagias | Definitions and representations of multivariate long-range dependent time series[END_REF]. In [START_REF] Kechagias | Definitions and representations of multivariate long-range dependent time series[END_REF], Kechagias and Pipiras provide properties in the time and spectral domains of linear representations of multivariate longrange dependent processes. A nonlinear example of multivariate long-range dependent processes was also proposed by Didier and Pipiras in [START_REF] Didier | Integral representations and properties of operator fractional Brownian motions[END_REF], where a multivariate Brownian motion was defined.

The specificity of the multivariate setting is that, in addition to LRD properties, it helps identify the correlation structure between the processes. The coupling between each component is characterized by the long-run covariance matrix [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]. A key point for real data application is the development of statistical tests on LRD parameters and on long-run covariance. For example, as illustrated here on a real data example, these characteristics are intrinsically related to the brain activity recordings in neuroscience. Some work has shown that their distributions can be modified by pathologies (see e.g. [START_REF] Maxim | Fractional Gaussian noise, functional MRI and Alzheimer's disease[END_REF] and [START_REF] Achard | Hubs of brain functional networks are radically reorganized in comatose patients[END_REF]). A statistical test may be useful to rigorously assess such observations. We focus on semiparametric estimators, which are more robust to model misspecification [START_REF] Robinson | Gaussian semiparametric estimation of long range dependence[END_REF]. A common estimation procedure in this framework is Whittle estimation, which is based on a Fourier decomposition of the processes [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF][START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF][START_REF] Nielsen | Local Whittle estimation of multi-variate fractionally integrated processes[END_REF]. The authors prove the consistency and the asymptotic distribution of their estimators. More recently, the asymptotic normality of estimators has been provided by Baek et al. [START_REF] Baek | Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity[END_REF], in prolongation of [START_REF] Robinson | Multiple local Whittle estimation in stationary systems[END_REF], in a multivariate framework where components can be co-integrated. An estimation with a Lasso penalty is also proposed in this setting by Pipiras et al. [START_REF] Baek | Semiparametric, parametric, and possibly sparse models for multivariate long-range dependence[END_REF], and Düker and Pipiras [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF] establish the asymptotic normality of this procedure.

As an alternative to Fourier, wavelet-based estimators can be used. Wavelet transforms are interesting especially because wavelet analysis performs an implicit differentiation, which offers the possibility to consider non-stationary processes. Wavelet-based Whittle estimation was introduced by Moulines et al. in [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF] for univariate long-range dependent time series. It was generalized to the multivariate setting by Achard and Gannaz [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]. Estimators are consistent and have theoretical rates comparable to Fourier-based estimators. The numerical performances of wavelet-based and Fourier-based estimators are also similar, as illustrated in [START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF].

The semiparametric multivariate long-range dependence framework

Let X = {X a (k), k ∈ Z, a = 1, . . . , p} be a multivariate stochastic process. We consider a process X with long-range dependence parameters d = (d 1 , d 2 , . . . , d p ). The stationary framework corresponds to LRD parameters d i ∈ (-1.2, 1/2). In this case, following [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF][START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF], we suppose that the cross-spectral density satisfies: for all λ ∈ [-π, π],

for all (a, b) , f a,b (λ) = 1 2π Ω a,b (1 -e -iλ ) -da (1 -e iλ ) -d b f S a,b (λ).
The functions f S a,b (•) correspond to the short-range dependence behavior of the process. This modelling is semiparametric since, if it imposes the LRD behavior, short-range dependence is left nonparametric through functions f S (•). Some assumptions on f S (•) are needed, which will be detailed below.

The LRD parameters, d, model the long-run dynamics of the process. This model is a multivariate extension of a scalar fractionally integrated process (the so-called I(d) process), and for any a ∈ {1, . . . , p}, the time series X a exhibits long-range dependence whenever 0 < d a < 1/2. The case -1/2 < d a < 0 corresponds to antipersistence, where the spectral density f a (•) tends toward 0 at the origin. The case d a = 0 is the weakdependence case, where the spectral density f a (•) tends toward a positive constant at the origin. See [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF][START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF]. For simplicity, the term LRD is used throughout the paper, regardless of the values of d.

Wavelet analysis performs an implicit differentiation, which offers the possibility to consider non stationary processes, that is, LRD parameters d i possibly higher than 1/2. Let L denote the difference operator, LX(t) = X(t + 1) -X(t). The kth difference operator, L k , k ∈ N, is defined by k recursive applications of L. Introduce D ∈ N p . We suppose that the multivariate process Z = L Da X a (k), k ∈ Z, a = 1, . . . , p is covariance stationary with a spectral density matrix given by, for all λ ∈ [-π, π]: for all (a, b) ∈ {1, . . . , p} 2 , f

(Da,D b ) a,b (λ) = 1 2π Ω a,b (1 -e -iλ ) -d * a (1 -e iλ ) -d * b f S a,b (λ),
where the long-range dependence parameters of Z are given by d * a ∈ (-1/2, 1/2) for all a = 1, . . . , p.

Let the overline be the conjugate operator and • be the Hadamard product. For any vector υ ∈ R p , diag(υ) stands for the p × p matrix with entries υ in the diagonal and 0 elsewhere.

The LRD assumption can be expressed as follows:

(M1) The generalized spectral density of the multivariate process X is, for all λ ∈ [-π, π],

f (λ) = Ω • (Λ 0 (d)f S (λ)Λ 0 (d)), with Λ 0 (d) = diag((1 -e -iλ ) -d ), (1)
where

d = D + d * , D ∈ N p , d * ∈ (-1/2, 1/2) p .
The matrix Ω is called fractal connectivity by [START_REF] Achard | Fractal connectivity of long-memory networks[END_REF] or long-run covariance matrix by [START_REF] Robinson | Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction[END_REF]. Similar to [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF][START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] we introduce some regularity assumptions on the short-range dependence, modeled by function f S (•).

The space H p (β, L) is defined as the class of non-negative symmetric functions g(•) : [π, π] → C p×p such that g(0) = 1 p×p and such that sup λ∈(-π,π)

|g(λ) -1 p×p | L|λ| β ,
with 1 p×p the p × p matrix with all entries equal to 1. We suppose that the following assumption is fulfilled:

(M2) f S (•) ∈ H p (β, L) with 0 < β 2 and 0 < L.

Assumption (M2) imposes that f S (0) has constant entries equal to 1. This assumption is necessary to make Ω identifiable in (M1).

When λ tends toward 0, the spectral density matrix can be approximated at the first order by

f (λ) ∼ Λ(d)Ω Λ(d), with Λ(d) = diag(|λ| -d e -iπd/2 ), (2)
where ∼ means that the ratio of the left-and right-hand sides converges to one.

Lobato [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF] uses Λ(d) = diag(λ -d ) as an approximation of f (•) whereas Shimotsu [START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF] chooses to approximate f (•) using Λ(d) = diag(λ -d e -i(π-λ)d/2 ), which corresponds to a second-order approximation due to the remaining term λ in the exponential. We refer to [6, Section 2.1] and references therein for examples of processes satisfying approximation (2).

Linear decomposition

We suppose hereafter that the multivariate process admits a linear representation.

(M3) There exists a sequence

{A (D) (u)} u∈Z in R p×p such that u∈Z max a,b=1,...,p |A (D) a,b (u)| 2 < ∞ and ∀t ∈ Z, L Da X a (t) a=1,...,p = u∈Z A (D) (t + u)ε(u)
with ε(t) weak white noise process, in R p . Let F t-1 denote the σ-field of events generated by {ε(s), s t -1}. Assume that

ε satisfies E[ε(t)|F t-1 ] = 0, E[ε a (t)ε b (t)|F t-1 ] = 1 1 a=b and E[ε a (t)ε b (t)ε c (t)ε d (t)|F t-1 ] = µ a,b,c,d with |µ a,b,c,d | µ ∞ < ∞, for all a, b, c, d = 1, . . . , p. Define for all λ ∈ R, A (D) * (λ) = t∈Z A (D) (t)e iλ t the Fourier series associated to {A (D) (u)} u∈Z . That is, A (D) * (λ) = A (D) * a,b (λ) a,b=1,...,p with A (D) * a,b (λ) = (2π) -1/2 t∈Z A (D) a,b (t)e -iλt , λ ∈ R .
We add the following assumption: (M4) For all (a, b) ∈ {1, . . . , p} 2 , for all λ ∈ R, the sequence (2

-j da |A (D) * a,b (2 -j λ)|) j 0 is convergent as j goes to infinity.
This assumption is necessary for technical reasons. It does not seem restrictive.

An example of a process that satisfies these assumptions is the causal multivariate linear representations with trigonometric power law coefficients proposed in [START_REF] Kechagias | Definitions and representations of multivariate long-range dependent time series[END_REF].

Wavelet representation

We introduce a discrete wavelet transform. Write L 2 (R) the set of square-integrable functions with respect to the Lebesgue measure. Let φ(•) and ψ(•) be two functions of L 2 (R). Their Fourier transforms are given by φ(λ) = (W5) For all i = 1, . . . , p,

(1 + β)/2 -α < d i M .
Recall that β in (W5) is the regularity of the short-range dependence behavior introduced in (M2).

These assumptions are the same as the ones considered in [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF][START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF][START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]. Assumptions (W1)-(W4) are usual when considering that φ(•) and ψ(•) are respectively the scalingfunction and the wavelet-function associated with a multiresolution analysis [START_REF] Cohen | Numerical analysis of wavelet methods[END_REF]. They are satisfied, for example, by Daubechies wavelets. These wavelets are parametrized by the number of vanishing moments M . Assumption (W2) holds with α an increasing function of M going to infinity (see [START_REF] Daubechies | Ten lectures on wavelets[END_REF]). Assumptions (W1)-(W5) are fulfilled by Daubechies wavelet basis with sufficiently large M .

Assumption (W3) implies that the wavelet transform performs an implicit differentiation of order M and makes it possible to consider nonstationary processes. In Fourier analysis, tapering procedures are necessary to consider nonstationary frameworks, see e.g. [START_REF] Velasco | Whittle pseudo-maximum likelihood estimation for nonstationary time series[END_REF][START_REF] Hurvich | An efficient taper for potentially overdifferenced longmemory time series[END_REF], and references therein.

At a given resolution j 0, for k ∈ Z, we define the dilated and translated functions

φ j,k (•) = 2 -j/2 φ(2 -j • -k) and ψ j,k (•) = 2 -j/2 ψ(2 -j • -k).
The wavelet coefficients of the process X are defined by

W(j, k) = R X(t)ψ j,k (t)dt j 0, k ∈ Z, where X(t) = k∈Z X(k)φ(t -k). For given j 0 and k ∈ Z, W(j, k) is a p- dimensional vector W(j, k) = W 1 (j, k) W 2 (j, k) . . . W p (j, k) T where W a (j, k) = R X a (t)ψ j,k (t)dt, a = 1, . . . , p.
Throughout the paper, we adopt the same convention as in [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF] and [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF]; that is, large values of the scale index j correspond to coarse scales (low frequencies). The index k is a location parameter, and W(j, k) captures information at scale j and location k on the behavior of the process X.

In practice, let X(1), . . . X(N X ) denote the observations of the process X. Since the wavelets have a compact support, only a finite number n j of coefficients are non-null at each scale j. Suppose without loss of generality that the support of ψ(•) is included in [0, T ψ ] with T ψ 1. For every j 0, define

n j := max (0, 2 -j (N X -T ψ + 1) -T ψ + 1).
At each scale j, the non-zero coefficients belong to {W(j, k), k = 0, . . . , n j }.

Let j 0 be the minimal scale and j 1 = j 0 + ∆ the maximal scale which are considered in the estimation procedure. Following [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF][START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF], the asymptotic behavior is given for N X and j 0 going to infinity. Results obtained in [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF][START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF][START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF] state that optimal rates in estimation are obtained when j 0 is high enough to remove the scales affected by lowrange dependence. In practice, the number of scales ∆ is finite. Yet, considering the asymptotic behavior, two cases may be distinguished: either the number of scales ∆ is finite and fixed when j 0 goes to infinity, or ∆ = j 1 -j 0 goes to infinity. The latter case seems natural, for example, when one takes all available scales above j 0 in estimation.

In the following, n will denote the number of wavelet coefficients used for estimation and < J > the mean of scales, that is,

n = j 1 j=j 0 n j and < J >= 1 n j 1 j=j 0 n j j .
Define also

η ∆ := ∆ u=0 u 2 -u 2 -2 -∆ and κ ∆ := ∆ u=0 (u -η ∆ ) 2 2 -u 2 -2 -∆ .
These sequences converge respectively to 1 and to 2 when ∆ goes to infinity [21, Lemma 13].

Moulines et al. state that under assumptions (W1)-(W5), the wavelet coefficient process {W(j, k), k ∈ Z} is covariance stationary for any given j 0 [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF]. Let

D u,τ (λ; δ) = t∈Z |λ + 2tπ| -δ ψ(λ + 2tπ) 2 u/2 ψ(2 u (λ + 2tπ)) e -i2 u τ (λ+2tπ) , D u,∞ (λ; δ) = 2 -u -1 τ =0 D u,τ (λ; δ) . (3)
Moulines, Roueff and Taqqu [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF] establish that D u,τ (λ; δ) is an approximation of the cross-spectral density between wavelet coefficients {W(j, k), k ∈ Z} and {W(j + u, 2 u k + τ ), τ = 0, . . . , 2 u -1, k ∈ Z}. The parameter δ captures the long-range dependence of the processes. Indeed, the cross-spectral density of (W a (j, k), W b (j + u,

2 j k + τ )) is approximated by D u,τ (λ; d a + d b ). Function D u,∞ (λ; δ)
allows us to consider between-scales dependence.

For u 0, (δ 1 , δ 2 ) ∈ (-α, M ) 2 , define

I u (δ 1 , δ 2 ) = π -π D u,∞ (λ; δ 1 ) D u,∞ (λ; δ 2 ) dλ , (4)
where D u,∞ (λ; δ 2 ) is defined in (3). I u (δ 1 , δ 2 ) will naturally appear when studying the covariance between sample wavelet covariances.

Asymptotic normality of sample wavelet covariances and correlations

Define σ a,b (j) as the empirical covariance of the wavelet coefficients at a given scale j 0, between components a and b, and let σ a,b (j) denote the theoretical covariance,

σ a,b (j) = 1 n j n j -1 k=0 W a (j, k)W b (j, k), σ a,b (j) = E[W a (j, k)W b (j, k)].
Let Σ(j) = ( σ a,b (j)) a,b=1,...,p and Σ(j) = (σ a,b (j)) a,b=1,...,p be the two associated matrices in R p×p . In the following, for any matrix M ∈ C p×p , the maximal entry will be denoted by M ∞ = max a,b=1,...,p |M a,b |.

Proposition 2 in [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] proposes an approximation of the wavelet covariance at a given scale. It is recalled below.

Proposition 1 ([6]

). Suppose assumptions (M1)-(M2) and (W1)-(W5) hold. For all j 0, for all λ ∈ (-π, π),

Λ(j)(d) -1 Σ(j)Λ(j)(d) -1 -G ∞ CL2 -β j . ( 5
)
with constant C depending on β, min d , max d , max ,m |Ω ,m |, φ and ψ and

Λ j (d) = diag 2 jd , (6) G a,b = Ω a,b cos(π(d a -d b )/2)K(d a + d b ), a, b = 1, . . . , p, (7)
K(δ) = ∞ -∞ |λ| -δ | ψ(λ)| 2 dλ, δ ∈ (-α, M ).
For u 0, (δ 1 , δ 2 ) ∈ (-α, M ) 2 , let us introduce I u (δ 1 , δ 2 ) as

I u (δ 1 , δ 2 ) = 2π I u (δ 1 , δ 2 ) K(δ 1 )K(δ 2 )
, with I u (δ 1 , δ 2 ) defined in [START_REF] Whitcher | Wavelet estimation of a local long memory parameter[END_REF]. We also define G I G(u) ∈ R p 2 ×p 2 as:

G I G(u, u ) = diag vec Λ u∧u (d) -1 GΛ u∧u (d) -1 ( I |u-u | (d a + d b , d a + d b ) (a,b),(a ,b ))∈{1,...,p 2 } diag vec Λ u∧u (d) -1 GΛ u∧u (d) -1 . (8)
Remark 1. Observe that

I 0 (δ 1 , δ 2 ) = 2π π -π g ψ (λ; δ 1 )g ψ (λ; δ 2 )dλ π -π g ψ (λ; δ 1 )dλ π -π g ψ (λ; δ 2 )dλ where g ψ (λ; δ) = t∈Z |λ+2tπ| -δ | ψ(λ+2tπ)| 2 . It is straightforward that I 0 (δ 1 , δ 2 ) 2π.
Cauchy-Schwarz's inequality on the denominator also provides I 0 (δ 1 , δ 1 ) 1.

Here and subsequently, L -→ denotes a convergence in distribution. The asymptotic distribution of the sample wavelet covariance process is given in the following theorem.

Theorem 2. For all j 0 0, u 0, define

T(j 0 + u) = vec 2 -(j 0 +u)(da+d b ) σ a,b (j 0 + u), a, b = 1, . . . , p G = vec(G a,b , a, b = 1, . . . , p)
where vec(M) denotes the operation which transforms a matrix M ∈ R p 1 ×p 2 in a vector of R p 1 p 2 . Suppose assumptions (M1)-(M4) and (W1)-(W5) hold. Let 2 -j 0 β → 0 and

N -1 X 2 j 0 → 0. Then for all ∆ ∈ N, √ n j 0 +u T(j 0 + u) -G , u = 0, . . . , ∆ L -→ j 0 →∞ {Q(u), u = 0, . . . , ∆},
where

Q(•) is a centered Gaussian process with covariance function Cov (Q a,b (u), Q a ,b (u )) = V (a,b),(a ,b ) (u, u ) where V (a,b),(a ,b ) (u, u ) = 2 -|u-u |/2 G I G (a,a ),(b,b ) (u, u ) + G I G (a,b ),(a ,b) (u, u ) , (9)
and G I G(u, u ) is defined in [START_REF] Lobato | Consistency of the averaged cross-periodogram in long memory series[END_REF].

The proof is given in Appendix C. It is similar to the one of the univariate setting given in [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]Theorem 2]. It relies on decimated processes and limit theorems developed in [START_REF] Roueff | Central limit theorems for arrays of decimated linear processes[END_REF].

Remark 2. In the univariate setting, we obtain the same result as [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]Theorem 2]. The authors use a different normalization, by √ N X 2 -j 0 rather than by √ n j 0 +u . The correspondence between the results follows first from the equivalence √ n j 0 +u ∼ √ N X 2 -j 0 -u and second from approximation [START_REF] Gençay | An introduction to wavelets and other filtering methods in finance and economics[END_REF] for the expectancy term.

The main difference in the multivariate case is that the LRD properties in two processes can be different. It introduces a bias term through the presence of the cosinus term in G (7) and slightly modifies the variance through terms I u (•, •) (8).

Remark 3. In [START_REF] Whitcher | Wavelet analysis of covariance with application to atmospheric time series[END_REF], Whitcher et al. establish the asymptotic normality for wavelet correlations of bivariate multivariate time series with long-range dependence. The advantage of Theorem 2 is to provide an explicit form of the asymptotic variance.

Remark 4. As already pointed out by Roueff and Taqqu [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF], the covariance of the wavelet coefficients involves between-scales correlations which do not vanish when the sample size goes to infinity. This fact contrasts with the behavior of Fourier periodogram or Fourier-based Whittle estimation. In the variance formulation (9), these correlations appear through quantities { I u (δ 1 , δ 2 ), u 0, (a, b) ∈ {1, . . . , p} 2 }.

We can deduce the asymptotic normality for sample wavelet correlations by means of delta method. We do not present here the multivariate result for the sake of brevity, except when the matrix G is diagonal since formulas are more simple. We focus on the pointwise result to highlight the specificity of our setting. Then, under conditions of Theorem 2,

√ n j ( ρ a,b (j) -r a,b ) L -→ j→∞ N 0, V (ρ) a,b with V (ρ) a,b = I 0 (2d a , 2d b ) + I 0 (d a + d b , d a + d b )(r 2 a,b + r 4 a,b ) -( I 0 (2d a , 2d b ) + I 0 (2d b , d a + d b )) 2 r 2 a,b -( I 0 (2d a , 2d a ) + I 0 (2d b , 2d b )) r 2 a,b /2 . ( 10
)
When all off-diagonal entries of G are equal to 0, We recover a similar form of the asymptotic distribution, up to a normalization constant.

√ n j vec( ρ a,b (j), 1 a < b p) L -→ j→∞ N p(p-1)/2 0, diag vec 2 -j(2da+2d b ) I 0 (2d a , 2d b ), 1 a < b p .
Remark 6. In [32, Section 4.2.] Whitcher et al. use the convergence of the Fisher transform of ρ a,b (j) to a standard Gaussian distribution at a rate √ n j , when the correlation r a,b (j) is equal to zero. The result is true if we suppose that betweenscale wavelet coefficients are independent, which is asymptotically satisfied when the regularity of the wavelet goes to infinity [START_REF] Craigmile | Asymptotic decorrelation of between-scale wavelet coefficients[END_REF]. Corollary 3 illustrates that an additional normalization by

I 0 (2d a , 2d b ) -1/2 of ρ a,b (j) is necessary.
Some computed values of I 0 (2d a , 2d b ) are displayed in Table 1. It shows that I 0 (2d a , 2d b ) indeed decreases when the regularity increases. But between-scale wavelet coefficients dependence may not be negligible if the regularity is not high enough. For example, in the absence of long-range dependence, when d a = d b = 0, I 0 (0, 0) = 1.62 for Daubechies wavelets with M = 4 vanishing moments. Hence, in real data application, the approximation in [START_REF] Whitcher | Wavelet analysis of covariance with application to atmospheric time series[END_REF] may lead to false positives. Remark 7. The wavelet correlation at a given scale is also known as wavelet coherence.

d = (d 1 , d 2 ) (0, 0) (0, 0.1) (0, 0.2) (0, 0.3) (0, 0.4) M= 1 α= 1.
It is used in some applications, as in environmental studies by [START_REF] Whitcher | Wavelet analysis of covariance with application to atmospheric time series[END_REF], or in neurosciences in [START_REF] Achard | Wavelet-based graph inference using multiple testing[END_REF]. In such real data applications, the crucial point is the use of test procedures.

In particular, the test of the nullity of the correlations is essential. Corollary 3 shows that the asymptotic distribution depends on the long-range dependence parameters d.

Plugging in a consistent estimator of parameter d in [START_REF] Kechagias | Definitions and representations of multivariate long-range dependent time series[END_REF] allows for a test procedure to be built. For instance, one can use the wavelet Whitlle estimator described in Section 4 below.

Asymptotic normality of the parameters estimates

For clarity, the true parameters are denoted with an exponent 0 in this part.

The wavelet-based local Whittle procedure proposes to estimate the parameters by maximizing a pseudo-likelihood given by a Gaussian approximation of the wavelet coefficients {W(j, k), j 0, k = 0, . . . , n j }. Moulines et al. [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF] and Achard and Gannaz [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] prove that the wavelet-based Whittle approximation provides consistent estimators even for non-Gaussian processes. The Whittle procedure can also be applied in multivariate cases, which is not possible for example with the regression of the wavelet log-scalogram [START_REF] Abry | Wavelet analysis of long-range-dependent traffic, Information Theory[END_REF][START_REF] Achard | Fractal connectivity of long-memory networks[END_REF].

Let d and Ω be the wavelet Whittle estimators as defined in [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]Section 3.3]. They maximize the objective function

L(G(d), d) = 1 n j 1 j=j 0 n j log det (Λ j (d)G(d)Λ j (d)) + n j -1 k=0 W T j,k (Λ j (d)G(d)Λ j (d)) -1 W j,k ,
where the superscript T denotes the transpose operator and Λ j (d) and the matrix G(d) are defined respectively in ( 6) and ( 7).

The function L(•, •) corresponds to the negative log-likelihood of {W(j, k), j 0, k = 0, . . . , n j } under a Gaussian assumption, where Proposition 1 is used for a parametrization of the variance at each scale. The estimation of the vector of longrange dependence parameters d satisfies d = argmin

d∈R p R(d), with R(d) = log det( G(d)) + 2 log(2) 1 n j 1 j=j 0 j n j p =1 d . (11)
The covariance matrix Ω is estimated by

Ω a,b = G a,b ( d)/(cos(π( d a -d b )/2)K( d a + d b )), a, b = 1, . . . , p,
where

G(d) = 1 n j 1 j=j 0 n j Λ j (d) -1 Σ(j)Λ j (d) -1 . (12)
We introduce

I d ∆ (δ 1 , δ 2 ) = 2 κ ∆ I 0 (δ 1 , δ 2 ) (13)
+ 2 κ 2 ∆ ∆ u=1 (2 uδ 1 + 2 uδ 2 ) 2 -u 2 -2 -∆+u 2 -2 -∆ ((u + η ∆-u -η ∆ )(η ∆-u -η ∆ ) + κ ∆-u ) I u (δ 1 , δ 2 ) if ∆ < ∞, I d ∞ (δ 1 , δ 2 ) = I 0 (δ 1 , δ 2 ) + ∞ u=1 (2 uδ 1 + 2 uδ 2 ) 2 -u I u (δ 1 , δ 2 ) , if ∆ = ∞. (14)
Define also

G I d G(∆) = diag vec G 0 I d ∆ (d 0 a +d 0 b , d 0 a +d 0 b ) (a,b),(a ,b )∈{1,...,p 2 } diag vec G 0 . ( 15
)
The asymptotic normality of the estimator of the long-range dependence parameters is established by our next theorem.

Theorem 4. Suppose assumptions (M1)-( M4) and (W1)-(W5) hold. Let j 0 < j 1 j N with j N = max{j, n j 1} such that

j 1 -j 0 → ∆ ∈ {1, . . . , ∞}, log(N X ) 2 (N X 2 -j 0 (1+2β) + N -1/2 X 2 j 0 /2 ) → 0. Then √ n( d -d 0 )
converges in distribution to a centered Gaussian distribution with a variance equal to

V (d) (∆) = 1 2 log(2) 2 (G 0-1 • G 0 + I p ) -1 Υ(∆) (G 0-1 • G 0 + I p ) -1 , (16)
where I p is the identity matrix in R p×p and with entry (a, a ) of Υ (∆) , for (a, a ) ∈ {1, . . . , p} 2 , given by

Υ a,a (∆) = b,b =1,...,p (G 0-1 ) a,b (G 0-1 ) a ,b G I d G (a,a ),(b,b ) (∆) + G I d G (a,b ),(a ,b) (∆) (17)
where quantities G I d G(∆) are defined by [START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF].

The proof is given in Appendix F. Remark 8. In the univariate setting, we recover [23, Theorem 5], using the equality

∆-u v=0 2 -v 2 -2 -∆ (v -η ∆ )(u + v -η ∆ ) = 2 -2 -∆+u 2 -2 -∆ ((u + η ∆-u -η ∆ )(η ∆-u -η ∆ ) + κ ∆-u ),
in [START_REF] Achard | Hubs of brain functional networks are radically reorganized in comatose patients[END_REF]. Observe that the result is also normalized by

√ n rather than √ N X 2 -j 0 .
Remark 9. The condition on j 0 and j

1 , that is, log(N X ) 2 (N X 2 -j 0 (1+2β) +N -1/2 X 2 j 0 /2
) → 0 is more restrictive than the condition required for the consistency of the estimators given in [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]Theorem 6]. Roueff and Taqqu [23, Theorem 5] obtain a similar result in the univariate setting. As illustrated in [START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF], the condition log(N X ) 2 N -1/2 X 2 j 0 /2 → 0 means that the highest frequencies, which are affected by the short-range dependence, should be removed from the estimation. The additional condition log(N X ) 2 N X 2 -j 0 (1+2β) → 0 prevents us from choosing the scale

j 0 = N 1/(1+2β) X
giving the minimax rate [6, Corollary 7]. Yet a near minimax rate is possible, with only a logarithmic lost, choosing, for example,

j 0 = log(N X ) 3 N 1/(1+2β) X .
Remark 10. If the vector d 0 has all entries equal to d 0 , the resulting covariance is

1 4 log(2) 2 I ∆ (2d 0 , 2d 0 )(G 0-1 • G 0 + I p ) -1 .
We recognize a form of asymptotic variance similar to the ones given by [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF], [START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF] and [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF] with Fourier-based Whittle estimators. Note that they use a different approximation of spectral density at zero frequency. Lobato [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF] and Shimotsu [START_REF] Shimotsu | Gaussian semiparametric estimation of multivariate fractionally integrated processes[END_REF] consider respectively G 0 a,b = Ω a,b e iπ(da-d b )/2 and G 0 = Ω. Düker and Pipiras's modelling in [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF] is more general and does not suppose a linear representation of the time series. Their result is valid for a general form of matrix Ω. Additionally, Baek et al. [START_REF] Baek | Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity[END_REF] establish asymptotic normality of estimators in a bivariate model with possible co-integration.

(k), k = 1, . . . , N X } and {X 2 (k), k = 1, . . . , N X } in. That is, for i = 1, 2, d U i = argmin d i ∈R R i (d i ) with R i (d i ) = log 1 n j 1 j=j 0 n j 2 -2jd i σ ii (j) + 2 log(2) 1 n j 1 j=j 0 j n j d i .
According to Theorem 4 d U i , i = 1, 2, are asymptotically normal, with the same asymptotic variance σ 2 (d, j 1 -j 0 ) = V (d) (j 1 -j 0 ), given by [START_REF] Nielsen | Local Whittle estimation of multi-variate fractionally integrated processes[END_REF]. Let now d be the bivariate wavelet Whittle estimator defined in [START_REF] Didier | Integral representations and properties of operator fractional Brownian motions[END_REF]. Theorem 4 provides the asymptotic normality of d with the asymptotic variance given by ( 16), which is equal to

V (d) (j 1 -j 0 ) = (G 0-1 • G 0 + I p ) -1 2 σ 2 (d, j 1 -j 0 ) = 1 -ρ 2 /2 ρ 2 /2 ρ 2 /2 1 -ρ 2 /2 σ 2 (d, j 1 -j 0 ).
This result proves that we reduce the entrywise variance when we perform multivariate estimation instead of univariate estimation. A similar conclusion was obtained for Fourier-based estimation by [START_REF] Lobato | A semiparametric two-step estimator in a multivariate long memory model[END_REF] and [START_REF] Nielsen | Local Whittle estimation of multi-variate fractionally integrated processes[END_REF]. Achard and Gannaz [START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF] support this assertion on simulated data. In real data application, [START_REF] Achard | Wavelet Whittle estimation in multivariate time series models: application to fMRI data[END_REF] also establishes that the multivariate approach performs better than the univariate one, comparing their application on fMRI data where subjects were scanned twice.

Remark 12. Quantities I ∆ (δ 1 , δ 2 ) are computable for given δ 1 , δ 2 , ∆. Hence, plugging in ( 16)-( 17) consistent estimators of d and G, for example d and G( d) [6, Theorem 6], a test procedure on parameters d can be built.

We now study the asymptotic behavior of the estimation of long-run covariance. We show the asymptotic normality of G( d), defined in [START_REF] Maxim | Fractional Gaussian noise, functional MRI and Alzheimer's disease[END_REF].

Write

I G ∆ (δ 1 , δ 2 ) = I 0 (δ 1 , δ 2 ) + ∆ u=1 (2 uδ 1 + 2 uδ 2 )2 -u 2 -2 -∆+u 2 -2 -∆ I u (δ 1 , δ 2 ) if ∆ < ∞, I G ∞ (δ 1 , δ 2 ) = I 0 (δ 1 , δ 2 ) + ∞ u=1 (2 uδ 1 + 2 uδ 2 )2 -u I u (δ 1 , δ 2 ) if ∆ = ∞.
Let us also define

G I G G(∆) = diag vec G 0 I G ∆ (d 0 a +d 0 b , d 0 a +d 0 b ) (a,b),(a ,b )∈{1,...,p 2 } diag vec G 0 . ( 18
)
We are now in a position to formulate the asymptotic distribution of G( d).

Theorem 5. Suppose Assumptions (M1)-(M4) and (W1)-(W5) hold. Let

j 1 -j 0 → ∆ ∈ {1, . . . , ∞}, log(N X ) 2 (N X 2 -j 0 (1+2β) + N -1/2 X 2 j 0 /2 ) → 0. Then vec √ n G( d) -G 0 converges in distribution to a centered Gaussian distribution with a variance equal to V G(∆) , with V (G) (a,b),(a ,b ) (∆) = G I G G (a,a ),(b,b ) (∆) + G I G G (a,b ),(a ,b) (∆) (19)
where quantities G I G G(∆) are defined by [START_REF] Robinson | Multiple local Whittle estimation in stationary systems[END_REF].

The proof is given in Appendix G.

We can deduce a convergence result for correlations.

Corollary 6. Let (a, b) ∈ {1, . . . , p} 2 , a = b. Define r a,b = G a,b ( d) G a,a ( d) G b,b ( d) and r a,b = G 0 a,b G 0 a,a G 0 b,b
.

Then, under conditions of Theorem 5,

√ n ( r a,b -r a,b ) L -→ j→∞ N 0, V (r) a,b (∆) with V (r) a,b (∆) = I G ∆ (2d a , 2d b ) + I G ∆ (d a + d b , d a + d b )(r 2 a,b + r 4 a,b ) -(I G ∆ (2d a , 2d b ) + I G ∆ (2d b , d a + d b )) 2 r 2 a,b -(I G ∆ (2d a , 2d a ) + I G ∆ (2d b , 2d b )) r 2 a,b /2.
When all off-diagonal entries of G are equal to 0,

√ n vec( r a,b , 1 a < b p) L -→ j→∞ N p(p-1)/2 0, diag vec I G ∆ (2d a , 2d b ), 1 a < b p . ( 20
)
The proof is based on delta method, and it is similar to the proof of Corollary 3. It is thus omitted. The covariance structure of vec( r a,b , a, b = 1, . . . , p) can also be deduced from Theorem 5, but it is not displayed here.

Remark 13. The result is very similar to the one presented in Corollary 3. For all (a, b) ∈ {1, . . . , p} 2 , the sequence ( √ n j (ρ a,b (j) -r a,b )) j 0 converges in distribution as j goes to infinity. The strength of Corollary 6 is that all the scales are used to estimate r a,b , which reduces the variance. Indeed, ( √ n( r a,b -r a,b )) j 0 converges in distribution as j goes to infinity, with n = j 1 j=j 0 n j . Remark 14. When the LRD parameters are equal, i.e. d a = d b , Corollary 6 provides a more simple form, which is

√ n( r a,b -r a,b ) L -→ j→∞ N 0, I G ∆ (2d a , 2d a )(1 -r 2 a,b ) 2 .
Remark 15. The asymptotic variances V (G) (∆) and V (r) (∆), given respectively in Theorem 5 and Corollary 6, depend on parameters d. Test procedures can be built by plugging in V (G) (∆) and V (r) (∆) the estimator d, which is consistent [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF]Theorem 6].

Remark 16. In [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF], Düker and Pipiras propose a global test for non-connectivity. That is, a test of (H 0 ) ∀a = b, r a,b = 0 against (H 1 ) ∃a = b, r a,b = 0. A similar test can be developed in our setting, based on [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF]. Another possibility is to perform the

p(p -1)/2 tests of (H 0 a,b ) r a,b = 0 against (H 1 a,b ) r a,b = 0, for 1 a < b
p and to apply a multiple testing correction on the p-values, for instance, Bonferroni's or Sidak's [START_REF] Achard | Asymptotic control of FWER under Gaussian assumption: application to correlation tests[END_REF]. This approach may be less powerful than the previous one if we are interested in the global test, but it provides information on which correlations are significant.

We can go further than Theorem 4 and Theorem 5 by giving the joint distribution of estimators d and G( d).

Proposition 7. Suppose assumptions of Theorem 5 hold.

Let T = d -d 0 , vec G( d) -G 0 .
Then √ n T converges in distribution to a centered Gaussian distribution.

A proof is given in Appendix H. An explicit form of the asymptotic covariance term is given in (H.1)-(H.2). It is not displayed here to gain in clarity.

Remark 17. Baek et al. [START_REF] Baek | Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity[END_REF] and Düker and Pipiras [START_REF] Düker | Asymptotic results for multivariate local Whittle estimation with applications[END_REF] also find that the estimates of long-range dependence parameters and long-run covariance converge jointly to a Gaussian distribution in a Fourier-based Whittle estimation framework. As stated before, they consider a more general model, allowing for a complex-valued matrix Ω.

Illustration on real data

We illustrate here the asymptotically Gaussian behavior on real data rather than on simulations. We consider fMRI recordings on dead and live rats. The dataset is freely available at https://zenodo.org/record/2452871 [START_REF] Becq | Functional connectivity is preserved but reorganized across several anesthetic regimes[END_REF][START_REF] Becq | Brain networks of rats under anesthesia using resting-state fMRI: comparison with dead rats, random noise and generative models of networks[END_REF]. The duration of scanning is 30 minutes with a time repetition of 0.5 second so that N X = 3, 600 time points are available at the end of experience. After preprocessing as described in [START_REF] Pawela | Resting-state functional connectivity of the rat brain[END_REF], we extracted p = 51 time series, each one being associated with a brain region of the rat. fMRI recordings of brain activity are based on the hemodynamic response to a magnetic field, which may create some temporal and spatial dependence. They suffer from different sources of noise, including system-related instabilities, subject motion, or physiological fluctuations [START_REF] Liu | Noise contributions to the fMRI signal: An overview[END_REF]. Additionally, during the preprocessing step, we aggregate the time series of each voxel to obtain a unique time series for each brain region. This aggregation step may create LRD properties [START_REF] Leipus | Aggregation and long memory: recent developments[END_REF]. Our claim is that long-range dependence and long-run covariance are closely related to brain activity and not to recording artifacts or preprocessing. We would like to check this assertion on the dataset. This means that we expect d 0 = 0 and a diagonal matrix G 0 for a dead rat but not for a live one.

We estimate d and G by wavelet-based Whittle estimation, using multiwave package [START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF]. We follow the procedure described in [22, Section 5.2] to choose the scales. Estimation is performed taking j 0 = 4 and j 1 = 9, which is the maximal scale; that is, we remove the frequencies above 0.12 Hz.

Based on Theorem 4, for each rat, we can test if the LRD parameters are significant for each brain region. That is, for all a = 1, . . . , p, we test

(H (d) 0 a ) d a = 0 against (H (d)
1 a ) d a = 0 , replacing d and G respectively by d and G( d) in V (d) . We consider a level α = 5% and apply Bonferroni's multiple testing correction, i.e. each test is applied with a level α /p to ensure that the probability to have a false positive on the p tests is equal to α .

Next Corollary 6 allows to test the significance of the long-run correlation between each pair of brain regions. For all 1 a < b p, we test

(H (r) 0 a,b ) r a,b = 0 against (H (r) 1 a,b ) r a,b = 0 .
Similarly, we apply Bonferroni's multiple testing correction and we consider a level α /(p(p -1)/2) for each test.

The tests have been applied on one dead rat and one live rat. The results are displayed in Figure 1 as graphs. Figure 1 shows that, indeed, we can conclude that d 0 = 0 and that off-diagonal entries of G 0 are equal to zero for the dead rat. For the live rat, six brain regions (over 51) have a significant LRD parameter, and 483 correlations (over 1275 of {r a,b , 1 a < b p}) are significant. These observations tend to confirm that long-range dependence and long-run covariance result from brain activity.

Conclusion

In this paper, we consider a multivariate process with long-range dependence properties, with a linear representation. We first establish that the covariance between wavelet coefficients is asymptotically Gaussian. The variance is explicitely given, and the convergence is established under mild assumptions on the wavelet transform and on the process. The asymptotic normality for the wavelet-based Whittle estimators defined in [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] is also established.

These results allow to perform statistical tests on the LRD parameters and on the longrun covariance. We propose an application on fMRI data, where we have recordings on a dead rat and alive one. The tests of significance on the LRD parameters and on the long-run correlations highlight that these characteristics are intrinsically linked to brain activity.

The function A * (λ) satisfies

A * (λ)A * (λ) T = f (λ), (A.2)
where f (•) is defined in (M1).

For all λ ∈ (-π, π), let

A * (j; λ) = H j (λ)A * (λ),
with A * (λ) defined in (A.1). Let us also define {A(j; t), t ∈ Z} ∈ 2 (Z) such that

A * (j; λ) = (2π) -1/2 t∈Z A(j; t)e iλt .
Then, the wavelet coefficients can be written as

W(j, k) = l∈Z A(j; 2 j k -l)ε(l).
Implicit differentiation by wavelet representation. As the wavelet ψ admits M vanishing moments under (W3), H j can be factorized as H j (λ) = (1 -e iλ ) M H j (λ), with H j trigonometric polynomial, H j (λ) = t∈Z h j,t e itλ . It results that

W a (j, k) = l∈Z h j,2 j k-l (L M X a )(l).
Appendix A.2. New indexing of wavelet coefficients Let j 0 and k ∈ {0, . . . , n j -1}. We introduce the new indexing proposed by [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]. Let u = j -j 0 , u ∈ {0, . . . , ∆}, and define (i, s) such that k = 2 ∆-u (s -1) + i, with i ∈ {2 ∆-u , . . . , 2 ∆-u+1 -1} and s ∈ Z. We have 2 j k = 2 j 1 (s -1 + 2 u-∆ i). Index i varies from 1 to N = 2 ∆+1 -1 and each couple (j, k) corresponds to a unique couple (i, s). We can rewrite wavelet coefficients as

Λ j 0 (d) -1 W(j, k) = t∈Z V (i,j 0 ) (2 j 1 s -t)ε(t)
with Λ j 0 (d) is defined in (1) and

V (i,j 0 ) (t) = Λ j 0 (d) -1 A(j; 2 j (i -2 j 1 -j ) + t), j = j 1 -log 2 (i) ,
where log 2 (i) = ∆ -u is the integer part of log 2 (i). Write

Z (i,s,j 0 ) = t∈Z V (i,j 0 ) (2 j 1 s -t)ε(t) = Λ j 0 (d) -1 W(j, 2 -[log 2 (i)] (s -1) + i) .
W(j, k), V (i,j 0 ) , Z (i,s,j 0 ) belong respectively to R p , R p×p and R p .

For all u = 0, . . . , ∆, denoting j = j 0 + u, the empirical variance satisfies

Λ j 0 (d) -1 Σ(j)Λ j 0 (d) -1 = 1 n j n j -1 k=0 Λ j 0 (d) -1 W(j, k) W(j, k) T Λ j 0 (d) -1 = √ n j 1 n j 2 ∆-u+1 -1 i=2 ∆-u n -1/2 j 1 n j 1 -1 s=0 Y (i,s,j 0 ) S (i,j 0 ) + n -1 j T ψ (2 ∆-u -1) i=2 ∆-u Y (i,n j 1 ,j 0 ) R(j) , (A.3) with Y (i,s,j 0 ) = Z (i,s,j 0 ) Z (i,s,j 0 )T = (Z (i,s,j 0 ) a Z (i,s,j 0 ) b
) a,b=1,...,p . Indeed, when s ∈ {0, . . . , n j 1 -1} and i ∈ {2 ∆-u , . . . , 2 ∆-u+1 -1}, index k = 2 ∆-u (s -1)+i varies in {0, . . . , 2 ∆-u n j 1 -1}, and when s = n j 1 and i ∈ {2 ∆-u , . . . , T ψ (2

∆-u -1)}, index k varies from 2 ∆-u n j 1 to 2 ∆-u (n j 1 -1) + T ψ (2 ∆-u -1) = 2 -j (N X -T ψ + 1) -T ψ = n j -1. That is, {k = 0, . . . , n j -1} = {k = 2 ∆-u (s-1)+i, s = 0, . . . , n j 1 -1, i = 2 ∆-u , . . . , 2 ∆-u+1 -1} ∪ {k = 2 ∆-u (s -1) + i, s = n j 1 , i = 2 ∆-u , . . . , T ψ (2 ∆-u -1)}.
The proof of Theorem 2 consists in establishing first the asymptotic normality of {S (i,j 0 ) , i = 1, . . . , N } j 0 0 when j 0 goes to infinity, and second that (R(j)) j 0 is negligible. To prove the asymptotic normality of {S (i,j 0 ) , i = 1, . . . , N } j 0 0 , we will need to approximate the variables {Y (i,j 0 ) , i = 1, . . . , N } j 0 0 by m-dependent variables.

Appendix A.3. Approximation by a m-dependent process Following [START_REF] Roueff | Central limit theorems for arrays of decimated linear processes[END_REF], we introduce a non-negative infinitely differentiable function H(•) defined on R such that H(0) = 1 and

H(t) = 0 if |t| > 1/2. Write H(•) its Fourier transform, H(λ) = ∞ -∞ H(t)e -iλ t dt.
Since H is supposed infinitely derivable, when |λ| tends to infinity, H(λ) decreases to 0 faster than any polynomial. Hence, there exists c H > 0 such that | H(λ)| c H |λ| -δv-1 for all |λ| 1, with δ v defined in Lemma 10. Additionally,

1 2π ∞ -∞ H(λ) dλ = H(0) = 1.
Let us define, for all t ∈ R, for all λ ∈ R,

V (i,j 0 )(m) (t) = H(2 -j 1 t/m) V (i,j 0 ) (t) , V (i,j 0 )(m) * (λ) = (2π) -1/2 t∈Z V (i,j 0 )(m) (t)e -iλt = m 2π ∞ -∞ H(mξ)V (i,j) * (λ -2 -j 1 ξ) dξ , Z (i,s,j 0 )(m) = t∈Z V (i,j)(m) (2 j 1 s -t)ε(t) . Then for all a, b = 1, . . . , p, vectors Z (s,j 0 )(m) a,b = Z (1,s,j 0 )(m) a , . . . , Z (N,s,j 0 )(m) a , Z (1,s,j 0 )(m) b , . . . , Z (N,s,j 0 )(m) b T
∈ R N are m-dependent relatively to index s. That is, for all q 1, for all (s 1 , . . . , s q ) such that s r+1 s r + m for r = 1, . . . , q, vectors Z

(s 1 ,j 0 )(m) a,b , . . . , Z (sq,j 0 )(m) a,b are independent.
We will next study sequences {S (i,j 0 )(m) , i = 1, . . . , N } j 0 0 which are defined as follows:

Y (i,s,j 0 )(m) = Z (i,s,j 0 )(m) Z (i,s,j 0 )(m)T , (A.4)

S (i,j 0 )(m) = n -1/2 j 1 n j 1 -1 s=0 Y (i,s,j 0 )(m) . (A.5)
The outline of the proof of Theorem 2 is first to prove the asymptotic normality of {S (i,j 0 )(m) , i = 1, . . . , N } j 0 0 with the use of the results on decimated m-dependent processes of [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]. Next a similar result for {S (i,j 0 ) , i = 1, . . . , N } j 0 0 is deduced by letting m go to infinity.

Appendix B. Notations and technical lemmas

This section provides some technical results on the quantities introduced in the wavelet representation and in the approximation by a m-dependent process, respectively in Appendix A.2 and Appendix A.3. These results will be used for the proof of Theorem 2.

where u = ∆ -log 2 (i) , u = ∆ -log 2 (i ) .

We begin by providing some results on the behavior of (V (i,j 0 ) * (•)) in a sequence of lemmas.

Lemma 10. Suppose assumptions of Theorem 20 hold. Suppose ∆ < ∞. Then there exists δ v > 1/2 such that for all j j 0 , j -j 0 ∆, we have

sup |λ|<π V (i,j 0 ) * (λ)V (i,j 0 ) * (λ) T ∞ C v 2 j (1 + 2 j |λ|) -2δv , with j = j 0 + ∆ -log 2 (i) and C v = C A C 2 H1 2 ∆(da+d b ) < ∞, depending on L, β, Ω, d, ∆, φ(•) and ψ(•).
Proof. Recall that the Fourier transform A * (•) was defined in (A.1). Observe that

V (i,j 0 ) * (λ)V (i,j 0 ) * (λ) T = (2π) -1 |H j (λ)| 2 Λ j 0 (d) -1 A * (λ)A * (λ) T Λ j 0 (d) -1 .

Lemma 9 yields

V (i,j 0 ) * (λ)V (i,j 0 ) * (λ) T a,b C A (2π) -1 |H j (λ)| 2 2 (j-j 0 )(da+d b ) 2 j λ -da-d b .
From (I.1) we get, for all a, b = 1, . . . , p,

V (i,j 0 ) * (λ)V (i,j 0 ) * (λ) T a,b C A C 2 H1 2 (j-j 0 )(da+d b ) 2 j 2 j |λ| 1 + 2 j |λ| 2M -da-d b (1 + 2 j |λ|) -2α-da-d b .
Therefore, since d a + d b < 2 M and 0

2 j |λ| 1+2 j |λ| 1, V (i,j 0 ) * (λ)V (i,j 0 ) * (λ) T a,b C A C 2 H1 2 (j-j 0 )(da+d b ) 2 j (1 + 2 j |λ|) -2α-da-d b .
Lemma 10, hence, holds with

C v = C A C 2 H1 2 (j-j 0 )M and δ v = α + max a=1,...,p d a . Assumption (W5) ensures that δ v > 1/2.
The following lemma provides some convergence results on (V (i,j 0 ) * (•)). Lemma 11. For all i = 1, . . . , N , for all λ ∈ R, there exist Φ (i,j 0 ) (λ) ∈ (-π, π) p×p and

V (i,∞) * (λ) ∈ C p×p , such that 2 -j 1 /2 V (i,j 0 ) * (2 -j 1 λ)e -iΦ (i,j 0 ) (2 -j 1 λ) -→ j 0 →∞ V (i,∞) * (λ) , (B.2) 2 -j 1 V (i,j 0 ) * (2 -j 1 λ)V (i ,j 0 ) * (2 -j 1 λ) T -→ j 0 →∞ W (i,i ) * (λ) , (B.3)
where W (i,i ) * (λ) is defined in (B.1).

Proof.

Proof of (B.2). Let Φ (i,j 0 ) (λ) be the arguments of V (i,j 0 ) * (λ). Let (a, b) ∈ {1, . . . , p} 2 . From (I.2), we have

2 -j 1 /2 V (i,j 0 ) * a,b (2 -j 1 λ)e -iΦ (i,j 0 ) a,b (2 -j 1 λ) -2 -∆/2+∆da φ(2 -j 1 λ) ψ(2 u-∆ λ)2 -j 1 da A * a,b (2 -j 1 λ) C H2 2 j(1/2-α-M )+(u-∆)M |λ| M 2 -∆/2+∆da 2 -j 1 da A * a,b (2 -j 1 λ) .
Lemma 9 gives the inequality

2 -j 1 da A * a,b (2 -j 1 λ) 2 2 -2 j 1 da p a =1 A * a,a (2 -j 1 λ) 2 C A |λ| -2da ,
for all a, b = 1, . . . , p. Hence,

2 -j 1 /2 V (i,j 0 ) * a,b (2 -j 1 λ)e -iΦ (i,j 0 ) a,b (2 -j 1 λ) -2 -∆/2+∆da φ(2 -j 1 λ) ψ(2 u-∆ λ)2 -j 1 da A * a,b (2 -j 1 λ) C H2 C A |λ| M -2da 2 -∆/2+∆da+(u-∆)M 2 j(1/2-α-M ) .
Since 1/2 -α -M < 0, we obtain that the right-hand side goes to 0 when j 0 goes to infinity. By continuity, φ(2 -j 1 λ) tends to φ(0) = 1 when j 0 goes to infinity. We conclude (B.2) by Assumption (M4), which supposes that 2 -j 1 da A * a,b (2 -j 1 λ) converges when j 1 goes to infinity.

Proof of (B.3). By equality (A.2), we get

2 -j 1 V (i,j 0 ) * (2 -j 1 λ)V (i ,j 0 ) * (2 -j 1 λ) T -2 -j 1 +j/2+j /2 | φ(2 -j 1 λ)| 2 ψ(2 j-j 1 λ) ψ(2 j -j 1 λ) Λ j 0 (d) -1 f (2 -j 1 λ)Λ j 0 (d) -1 e i( 2 u-∆ i-2 u -∆ i )λ ∞ 2 -j 1 H j (2 -j 1 λ)H j (2 -j 1 λ) -2 j/2+j /2 | φ(2 -j 1 λ)| 2 ψ(2 j-j 1 λ) ψ(2 j -j 1 λ) Λ j 0 (d) -1 f (2 -j 1 λ)Λ j 0 (d) -1 ∞ Inequality (I.
3) gives that the right-hand side can be bounded by

C H3 2 -j 1 2 (j+j )(1/2-α-M ) 2 (u+u -2∆)M |λ| 2M Λ j 0 (d) -1 f (2 -j 1 λ)Λ j 0 (d) -1 ∞ .
With Lemma 8, the bound becomes max a,b=1,...,p

C H3 C f 2 -(2j 0 +u+u )(α+M ) 2 (u+u -2∆)(M +1/2) 2 ∆(da+d b ) |λ| 2M -da-d b .
This term goes to 0 when j 0 goes to infinity uniformly for λ ∈ (-π, π).

As φ(2 -j 1 λ) -→ j 0 →∞ φ(0) = 1 and as f (2 -j 1 λ) satisfies approximation (2), we obtain convergence (B.3).

We introduce some useful notations.

For (i, i ) ∈ {1, . . . , N } 2 , t ∈ Z, λ ∈ R, let V (i,∞) (t) = 1 √ 2π V (i,∞) * (λ)e i λ t d λ, W (i,i ) * (λ) = t∈Z W (i,i ) * (λ + 2tπ), (B.4)
where V (i,∞) * (•) and W (i,i ) * (•) have been defined respectively in (B.2) and in (B.1). We also define, for (s, s ) ∈ {1, . . . , N } 2 , (a, b, a , b ) ∈ {1, . . . , p} 4 ,

Θ ((i,s),(i ,s )) = ∞ -∞ W (i,i ) * (λ)e -i(s-s )λ dλ, (B.5) Γ (i,i ) (a,b),(a ,b ) = 2π π -π W (i,i ) * a,a (λ) W (i,i ) * b,b (λ)dλ+2π π -π W (i,i ) * a,b (λ) W (i,i ) * b,a (λ)dλ. (B.6)
We can first state a result on Θ. This result is useful to show that Γ (i,i ) is the asymptotic covariance matrix of (S (i,j 0 ) , S (i ,j 0 ) ) as j 0 goes to infinity. This will be proved in Lemma 16.

Lemma 12. For all (i, i ) ∈ {1, . . . , N } 2 , ∀j 0 0, ∀a = 1, . . . , p, sup

s∈Z t∈Z p a =1 V (i,j 0 ) a,a (2 j 1 s -t) 2 < ∞ , (B.7) ∀(s, s ) ∈ Z 2 , t∈Z V (i,j 0 ) (2 j 1 s -t)V (i ,j 0 ) (2 j 1 s -t) T -→ j 0 →∞ Θ ((i,s),(i ,s )) , (B.8)
with Θ ((i,s),(i ,s )) defined in (B.5).

Moreover, for all (a, b) ∈ {1, . . . , p} 2 , for all i ∈ {1, . . . , N },

sup t∈Z s∈Z V (i,j 0 ) a,b (2 j 1 s -t) 2 -→ j 0 →∞ 0 . (B.9)
Proof.

Proof of (B.7). By Parseval's identity and a change of variable, for all (a, a ) ∈ {1, . . . , p} 2 , t∈Z

V (i,j 0 ) a,a (2 j 1 s -t) 2 = π -π |V (i,j 0 ) * a,a (λ)| 2 dλ C v ∞ -∞ (1 + |λ|) -2δv dλ ,
which implies (B.7).

Proof of (B.8). Applying Parseval's theorem and the change of variable λ → 2 j 1 λ, we get the equality t∈Z

V (i,j 0 ) (2 j 1 s -t)V (i ,j 0 ) (2 j 1 s -t) T = 2 j 1 π -2 j 1 π 2 -j 1 V (i,j 0 ) * (2 -j 1 λ)V (i ,j 0 ) * (2 -j 1 λ) T e i(s-s )λ dλ.
The function under the integral converges to W (i,i ) * (λ)e i(s-s )λ by (B.3). Convergence under the integral can be applied thanks to dominated convergence, by Lemma 10. This gives (B.8).

Proof of (B.9). Observe that

V (i,j 0 ) a,b (2 j 1 s -t) = 1 √ 2π π -π V (i,j 0 ) * a,b (
λ)e -i(2 j 1 s+t)λ dλ.

Since the function λ → V

(i,j 0 ) * a,b (λ) is 2π-periodic,[31, Lemma 4], leads to V (i,j 0 ) a,b (2 j 1 s -t) = 1 √ 2π π -π 2 -j 1 2 j -1 l=0 V (i,j 0 ) * a,b 2 -j 1 (λ + 2πl))e -i(2 j 1 s-t)(2 -j 1 (λ+2πl) dλ = 1 √ 2π π -π 2 -j 1 2 j 1 -1 l=0 V (i,j 0 ) * a,b 2 -j 1 (λ + 2πl
))e -i t 2 -j 1 (λ+2πl) e isλ dλ.

Parseval's identity entails that s∈Z

V (i,j 0 ) a,b (2 j 1 s -t) 2 = π -π 2 -j 1 2 j 1 -1 l=0 V (i,j 0 ) * a,b (2 -j 1 (λ + 2πl))e -i t 2 -j 1 (λ+2πl) 2 dλ. Hence, s∈Z V (i,j 0 ) a,b (2 j 1 s -t) 2 2 -j 1 π -π 2 j 1 -1 l=0 2 -j 1 /2 V (i,j 0 ) * a,b (2 -j 1 (λ + 2πl)) 2 dλ.
Lemma 10 implies that s∈Z

V (i,j 0 ) a,b (2 j 1 s -t) 2 C v 2 -j 1 π -π 2 j 1 -1 l=0 (1 + |λ + 2πl|) -δv 2 dλ.
We can deduce the following inequalities s∈Z

V (i,j 0 ) a,b (2 j 1 s -t) 2 C v 2 -j 1 π -π 1 + 2 j 1 -2 l=0 (1 + |2πl|) -δv 2 dλ C v 2 -j 1 2π 1 + 2 j 1 -2 0 (1 + |2πξ|) -δv dξ 2 C v 2 -j 1 2π 1 + (1 + 2π 2 j 1 ) 1-δv 2 C v 4π2 -j 1 1 + (2π) 2 2 j 1 (2-2δv) .
The right-hand side goes to 0 when j 1 goes to infinity since

δ v > 1/2.
for all (a, b) ∈ {1, . . . , p} 2 , for all i ∈ {1, . . . , N },

sup t∈Z s∈Z V (i,j 0 )(m) a,b (2 j 1 s -t) 2 -→ j 0 →∞ 0 .
Moreover for all m 1, for all (a, b) ∈ {1, . . . , p} 2 , for all (i, i ) ∈ {1, . . . , N } 2 and (s,

s ) ∈ {0, . . . , n j 1 -1} 2 , t∈Z V (i,j 0 )(m) (2 j 1 s -t)V (i ,j 0 )(m) (2 j 1 s -t) T -→ j 0 →∞ Θ ((i,s),(i ,s ))(m) , (B.13)
with Θ ((i,s),(i ,s ))(m) defined in (B.11).

The proof is similar to that of Lemma 12 and it is thus omitted.

Appendix B.4. Asymptotic variance of (S (i,j 0 ) , i = 1, . . . , N ) j 0 0

Lemma 15 below studies the behavior of {Θ ((i,s),(i ,s )) , i, i = 1, . . . , N, s, s 0} and {Θ ((i,s),(i ,s ))(m) , i, i = 1, . . . , N, s, s 0} when summing over the parameters (s, s ). It is used next to prove Lemma 16 which establishes that the asymptotic covariances of (S (i,j 0 ) , i = 1, . . . , N ) j 0 0 are equal to (Γ (i,i ) , i, i = 1, . . . , N ) when j 0 goes to infinity. Lemma 15. Suppose conditions of Theorem 2 hold. For all (a, b, a , b ) ∈ {1, . . . , p} 4 , for all (i, i ) ∈ {1, . . . , N }, Proof. We only prove (B.14), since the proof of (B.15) is similar. Quantity Θ ((i,s),(i ,s )) can be written as

lim →∞ -1 s,s =0,..., -1 Θ ((i,s),(i ,s )) a,b Θ ((i,s),(i ,s )) a ,b = 2π π -π W (i,i ) * a,b (λ) W (i,i ) * a ,b (λ)dλ, (B.14) lim →∞ -1 s,s =0,..., -1 Θ ((i,s),(i ,s ))(m) a,b Θ ((i,s),(i ,s ))(m) a ,b = 2π π -π W (i,i )(m) * a,b (λ) W (i,i )(m) * a ,b (
Θ ((i,s),(i ,s )) = π -π W (i,i) * (λ) e -i (s-s )λ dλ, Hence, setting v = s -s , -1 s,s =0,..., -1 Θ ((i,s),(i ,s )) a,b Θ ((i,s),(i ,s )) a ,b = v∈Z -1 ( -|v|) + π -π W (i,i ) * a,b (λ) e -i v λ dλ π -π W (i,i ) * a ,b (λ) e -i v λ dλ ,
with ( -|v|) + = -|v| if -|v| 0 and 0 otherwise. Lemma 33 entails that, when goes to infinity, the above term converges to

2π π -π W (i,i ) * a,b (λ) W (i,i ) * a ,b (λ)dλ .
This is precisely the assertion of the lemma.

We can deduce from Lemma 15 that for all i, i = 1, . . . , N , Γ (i,i ) is the asymptotic covariance between S (i,j 0 ) and S (i ,j 0 ) . Lemma 16. For all m 1, for all (a, b, a , b ) ∈ {1, . . . , p} 4 , for all (i, i ) ∈ {1, . . . , N } 2 , lim

j 0 →∞ Cov(S (i,j 0 ) a,b , S (i ,j 0 ) a ,b ) = Γ (i,i ) (a,b),(a ,b ) , with Γ (i,i ) defined in (B.6).
Proof. We first decompose Cov(S (i,j 0 ) a,b , S (i ,j 0 ) a ,b ) in two terms and next study separately the two terms.

Step 1. Decomposition of Cov(S

(i,j 0 ) a,b , S (i ,j 0 ) a ,b ). Easy calculation shows that Cov(Y (i,s,j 0 ) , Y (i ,s ,j 0 ) ) = T ((i,s),(i ,s ),j 0 ) (a,a ),(b,b ) + T ((i,s),(i ,s ),j 0 ) (a,b ),(b,a ) -R ((i,s),(i ,s ),j 0 ) (a,b),(a ,b ) , with T ((i,s),(i ,s ),j 0 ) (a,a ),(b,b ) = t 1 ∈Z V(2 j 1 s -t 1 )V (i ,j 0 ) (2 j 1 s -t 1 ) T a,a t 2 ∈Z V (i,j 0 ) (2 j 1 s -t 2 )V (i ,j 0 ) (2 j 1 s -t 2 ) T b,b
, a similar expression for T (i,i ,s,s ,j 0 ) (a,b ),(b,a ) , and

R ((i,s),(i ,s ),j 0 ) (a,b),(a ,b ) = a 1 ,a 2 ,a 3 ,a 4 =1,...,p µ a 1 ,a 2 ,a 3 ,a 4 t∈Z V (i,j 0 ) a,a 1 (2 j 1 s -t)V (i,j 0 ) b,a 2 (2 j 1 s -t)V (i ,j 0 ) a ,a 3 (2 j 1 s -t) V (i ,j 0 ) b ,a 4 (2 j 1 s -t) - a 1 ,a 2 =1,...,p t∈Z V (i,j 0 ) a,a 1 (2 j 1 s -t)V (i,j 0 ) b,a 1 (2 j 1 s -t)V (i ,j 0 ) a ,a 2 (2 j 1 s -t)V (i ,j 0 ) b ,a 2 (2 j 1 s -t) - a 1 ,a 2 =1,...,p t∈Z V (i,j 0 ) a,a 1 (2 j 1 s -t)V (i,j 0 ) b,a 2 (2 j 1 s -t)V (i ,j 0 ) a ,a 1 (2 j 1 s -t)V (i ,j 0 ) b ,a 2 (2 j 1 s -t) - a 1 ,a 2 =1,...,p t∈Z V (i,j 0 ) a,a 1 (2 j 1 s -t)V (i,j 0 ) b,a 2 (2 j 1 s -t)V (i ,j 0 ) a ,a 2 (2 j 1 s -t)V (i ,j 0 ) b ,a 1 (2 j 1 s -t) .
Hence,

Cov(S (i,j 0 ) , S (i ,j 0 ) ) = 1 n j 1 n j 1 -1 s=0 n j 1 -1 s =0 T ((i,s),(i ,s ),j 0 ) (a,a ),(b,b ) + 1 n j 1 n j 1 -1 s=0 n j 1 -1 s =0 T ((i,s),(i ,s ),j 0 ) (a,b ),(b,a ) - 1 n j 1 n j 1 -1 s=0 n j 1 -1 s =0 R ((i,s),(i ,s ),j 0 ) (a,b),(a ,b )
. (B.16)

We shall now study separately the terms in the right-hand side.

Step 2. Study of R

((i,s),(i ,s ),j 0 ) (a,b)
. Let us study first R

((i,s),(i ,s ),j 0 ) (a,b),(a ,b )
. Cauchy-Schwarz's inequality yields to:

1 -1 s=0 -1 s =0 |R ((i,s),(i ,s ),j 0 ) (a,b),(a ,b ) | (µ ∞ + 3) a 1 ,a 2 ,a 3 ,a 4 =1,...,p sup t∈Z s∈Z V (i,j 0 ) a,a 1 (2 j 1 s -t) 2 1/2 sup t∈Z s∈Z V (i,j 0 ) b,a 2 (2 j 1 s -t) 1/2 sup s ∈Z t∈Z V (i ,j 0 ) a ,a 3 (2 j 1 s -t) 2 1/2 sup s ∈Z t∈Z V (i ,j 0 ) b ,a 4 (2 j 1 s -t) 2 1/2 .
The right-hand side does not depend on . Results (B.7) and (B.9) in Lemma 12 imply that it converges to 0 when j 0 goes to infinity. Hence,

1 n j 1 n j 1 -1 s=0 n j 1 -1 s =0 R ((i,s),(i ,s ),j 0 ) (a,b),(a ,b ) -→ j 0 →∞ 0. (B.17)
Step 3. Study of T

((i,s),(i ,s ),j 0 ) (a,b)
. First observe that

t 1 ∈Z V(2 j 1 s -t 1 )V (i,j 0 ) (2 j 1 s -t 1 ) T = π -π 2 -j 1 V (i,j 0 ) * (λ)V (i ,j 0 ) * (λ) T e i2 j 1 (s-s )λ dλ = 1 √ 2π π -π (2π) 1/2 2 -j 1 2 j 1 -1 q=0 V (i,j 0 ) * (2 -j 1 (λ + 2πq))V (i ,j 0 ) * (2 -j 1 (λ + 2πq)) T e i2 j 1 (s-s )λ dλ.
Last equality was obtained by [START_REF] Roueff | Central limit theorems for arrays of decimated linear processes[END_REF]Lemma 4], since the function λ → V (i,j 0 ) * (2 -j 1 (λ + 2π q))V (i ,j 0 ) * (2 -j 1 (λ + 2π q)) T e i2 j 1 (s-s )λ is 2π-periodic.

For all functions g 1 , g 2 in L 2 (-π, π), Q ∈ N, set

L Q (g 1 , g 2 ) = q∈Z 1 - |q| Q + π -π g 1 (λ)dλ π -π g 1 (λ)dλ .
For all ∈ N, 1

-1 s=0 -1 s =0 T ((i,s),(i ,s ),j 0 ) (a,b),(a ,b ) = L (g (j 0 )
a,a , g

(j 0 ) b,b ),
where

g (j 0 ) a,a (λ) = (2π) 1/2 2 -j 1 2 j 1 -1 q=0 V (i,j 0 ) * (2 -j 1 (λ + 2π q))V (i ,j 0 ) * (2 -j 1 (λ + 2π q)) T a,a ,
and a similar definition of g

(j 0 )
b,b (λ). We omit the dependence on i, i temporally to simplify notations.

Introduce

g (∞) a,a (λ) = (2π) 1/2 W a,a (λ) = (2π) 1/2 q∈Z W a,a (λ + 2π q).
We have

L (g (j 0 ) a,a , g (j 0 ) b,b ) -L (g (∞) a,a , g (∞) b,b ) = L (g (j 0 ) a,a -g (∞)
a,a , g

(j 0 ) b,b -g (∞) b,b ) + L (g (∞)
a,a , g

(j 0 ) b,b -g (∞) b,b ) + L (g (j 0 ) a,a -g (∞) a,a , g (∞) b,b ) M (g (j 0 ) a,a -g (∞)
a,a )M (g

(j 0 ) b,b -g (∞) b,b ) + M (g (∞)
a,a )M (g

(j 0 ) b,b -g (∞) b,b ) +M (g (j 0 ) a,a -g (∞) a,a )M (g (∞) b,b ),
with M Q (g 1 ) defined in Lemma 32. Last inequality results from Cauchy-Schwarz's inequality, which entails that

L Q (g 1 , g 2 ) M Q (g 1 )M Q (g 2 ).
Applying Lemma 32,

M (g (∞) a,a ) π -π g (∞) a,a (λ) 2 dλ 1/2 , (B.18)
M (g

(j 0 ) a,a -g (∞) a,a ) π -π g (j 0 ) a,a (λ) -g (∞)
a,a (λ)

| = (2π) 1/2 W a,a (λ)
(2π) 1/2 C v (1 + |λ|) -2δv , with δ v > 1/2. Next, notice that π -π g (j 0 ) a,a (λ) -g (∞) (λ)
a,a (λ)

2 dλ = 2π ∞ -∞ 2 -j 1 V (i,j 0 ) * (2 -j 1 λ)V (i,j 0 ) * (2 -j 1 λ) T a,a -W (i,i ) * a,a (λ) 2 dλ.
Convergence (B.3) and inequality (B. [START_REF] Baek | Semiparametric, parametric, and possibly sparse models for multivariate long-range dependence[END_REF]) ensure that the integral goes to 0 when j 0 goes to infinity by dominated convergence. Therefore, the right-hand side of (B.19) goes to 0 when j 0 goes to infinity. It results that L (g

(j 0 )
a,a , g

(j 0 ) b,b ) -L (g (∞) a,a , g (∞)
b,b ) can be bounded by a quantity which is independent of and which goes to 0 when j 0 goes to infinity. Consequently, L n j 1 (g

(j 0 ) a,a , g (j 0 ) b,b ) -L n j 1 (g (∞) a,a , g (∞) b,b ) -→ j 0 →∞ 0.
Observe that

L n j 1 (g (j 0 ) a,a , g (j 0 ) b,b ) = n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 T ((i,s),(i ,s ),j 0 ) (a,a ),(b,b ) L n j 1 (g (∞) a,a , g (∞) b,b ) = n -1 j 1 n j 1 -1 s=0 n j 1 -1 s=0 Θ ((i,s),(i ,s )) a,a Θ ((i,s),(i ,s )) b,b -→ j 0 →∞ 2π π -π W (i,i ) * a,a (λ) W (i,i ) * b,b (λ)dλ,
where the last convergence is given by Lemma 15. Hence,

n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 T ((i,s),(i ,s ),j 0 ) (a,a ),(b,b ) -→ j 0 →∞ 2π π -π W (i,i ) * a,a (λ) W (i,i ) * b,b (λ)dλ. (B.20)
Similarly,

n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 T ( (i,s),(i ,s ),j 0 ) (a,b ),(b,a ) -→ j 0 →∞ 2π π -π W (i,i ) * a,b (λ) W (i,i ) * b,a (λ)dλ. (B.21)
Step 4. End of the proof. Lemma 16 follows from (B.16), (B.17 We proceed to show that Γ (i,i )(m) goes to Γ (i,i ) when m goes to infinity, for all i, i = 1, . . . , N . That is, the asymptotic variance of S (•,j 0 )(m) when j 0 goes to infinity converges to the asymptotic variance of S (•,j 0 ) . Lemma 17. For all i, i = 1, . . . , N ,

lim m→∞ Γ (i,i )(m) = Γ (i,i ) .
Proof. To study the limit of

Γ (i,i )(m) a,b
when m goes to infinity, we will first prove that

Γ (i,i ) satisfies Γ (i,i ) = lim j 0 →∞ lim m→∞ Γ by Γ (i,i ,j 0 )(m) (a,b),(a ,b ) = 2π π -π W (i,i ,j 0 )(m) * a,a (λ)
W (i,i ,j 0 )(m) * b,b (λ)dλ + 2π π -π W (i,i ,j 0 )(m) * a,b (λ)
W (i,i ,j 0 )(m) * b,a (λ)dλ, with W
(i,i ,j 0 )(m) * (λ) = t∈Z W (i,i ,j 0 )(m) * (λ + 2tπ), W (i,i ,j 0 )(m) * (λ) = 2 -j 1 V (i,j 0 )(m) * (2 -j 1 λ)V (i ,j 0 )(m) * (2 -j 1 λ) T . Notice that π -π W (i,i ,j 0 )(m) * a,a (λ) W (i,i ,j 0 )(m) * b,b (λ)dλ =
∞ -∞ W (i,i ,j 0 )(m) * a,a (λ)W (i,i ,j 0 )(m) * b,b (λ)dλ.
When j 0 goes to infinity, W (i,i ,j 0 )(m) * (λ) converges to W (i,i )(m) * (λ). The convergence under the integral is obtained by dominated convergence thanks to Lemma 10. It results

that lim j 0 →∞ Γ (i,i ,j 0 )(m) = Γ (i,i )(m) .
Let us now study the convergence of ( Γ

(i,i ,j 0 )(m)
) with respect to m. We introduce

W (i,i ,j 0 ) * (λ) = 2 -j 1 V (i,j 0 ) * (2 -j 1 λ)V (i ,j 0 ) * (2 -j 1 λ) T , W (i,i ,j 0 ) * (λ) = t∈Z W (i,i ,j 0 ) * (λ + 2tπ), and
Γ (i,i ,j 0 ) (a,b),(a ,b ) = 2π π -π W (i,i ,j 0 ) * a,a (λ) W (i,i ,j 0 ) * b,b (λ)dλ+2π π -π W (i,i ,j 0 ) * a,b (λ) W (i,i ,j 0 ) * b,a (λ)dλ. (B.22)
Since V (i,s,j 0 ) * (•) is continuous, we can apply a convergence under the integral. Hence, for all λ ∈ R,

V (i,j 0 )(m) * (λ) = 1 2π π -π H(u)V (i,j 0 ) * (λ -u/m) du -→ m→∞ V (i,j 0 ) * (λ) 1 2π π -π H(u)du = V (i,j 0 ) * (λ) .
Additionally Lemma 13 entails that

∀j 0 0, sup m 1 sup |λ|<π 2 -j/2 V (i,s,j 0 )(m) * (λ) ∞ (1 + 2 j |λ|) δv < ∞ . Consequently, Γ (i,i ,j 0 )(m)
converges uniformly in m to Γ (i,i ,j 0 )

. Moreover, Γ (i,i ,j 0 )

converges to Γ (i,i ) when j 0 goes to infinity, the convergence under the integral being obtained by continuity.

It results that

lim m→∞ lim j 0 →∞ Γ (i,i ,j 0 )(m) = lim j 0 →∞ lim m→∞ Γ (i,i ,j 0 )(m) = Γ (i,i ) .
Convergence of Lemma 17 follows.

Appendix B.6. Sums of Γ (i,i ) Due to decomposition (A.3), we will have to manipulate sums of covariances of {S (i,j 0 ) , i = 1, . . . , N }. By Lemma 16, the covariances are equal to {Γ (i,i ) , i, i = 1, . . . , N }. The objective of this section is to give some results on these sums. Rather than using expressions (B.6), we use convergence (B.14). We first need the following Lemma on quantities {Θ

((i,s),(i ,s )) a,b
, i, i = 1, . . . , N, s, s 0}.

Lemma 18. Suppose conditions of Theorem 2 hold. Let (a, b) ∈ {1, . . . , p} 2 , (i, .

i ) ∈ {1, . . . , N } 2 , (s, s ) ∈ N 2 . Introduce Ξ ((u,s),(u ,s )) a,b = 2 (∆-u)(1-da-d b ) ∞ -∞ g u -u (λ; d a + d b ) e -i (k-2 u -u k )λ dλ, with g u -u (ξ; δ) = ψ(λ) ψ(2 u -u λ)|λ| -δ , and k = i + 2 ∆-u (s -1), k = i + 2 ∆-u (s -
Hence,

Θ ((i,s),(i ,s )) = ∞ -∞ 1 2 W (i,i ) * (λ) + W (i ,i) * (λ)
T e -i(s-s )λ dλ.

Replacing W (i,i ) * (λ) and W (i ,i) * (λ) by their expression (B.1), we get

Θ ((i,s),(i ,s )) a,b = Ω a,b cos π(d a -d b )/2 2 (u-∆)/2+(u -∆)/2+∆ (da+d b ) Ξ ((u,s),(u ,s )) a,b with Ξ ((u,s),(u ,s )) a,b = ∞ -∞ ψ(2 u-∆ λ) ψ(2 u -∆ λ)|λ| -da-d b e -i( 2 u-∆ (i-1)-2 u -∆ (i -1))λ e -i(s-s )λ dλ, = 2 (∆-u)(1-da-d b ) ∞ -∞ ψ(λ) ψ(2 u -u λ)|λ| -da-d b e -i (i+2 ∆-u s)λ+i (2 ∆-u s +i )2 u -u λ dλ.
We are now in a position to give a useful result on Γ. Namely, we consider the sum of {Γ (i,i ) , i = 2 ∆-u , . . . 2 ∆-u+1 -1, i = 2 ∆-u , . . . , 2 ∆-u +1 -1}, which corresponds to the contribution of the scales (j, j ) = (j 0 + u, j 0 + u ) to the asymptotic variance V of the sample wavelet covariance. with Ξ

((u,s),(u ,s )) a,b = 2 (∆-u)(1-da-d b ) ∞ -∞ g u -u (λ; d a + d b ) e -i (k-2 u -u k )λ dλ,
and

g u -u (ξ; δ) = ψ(λ) ψ(2 u -u λ)|λ| -δ , k = i + 2 ∆-u (s -1), k = i + 2 ∆-u (s -1).
To get all values in Z from k -2 u -u k , we introduce τ ∈ {0, . . . , 2 -(u -u) -1}. Then, when i, i , s and s vary respectively in {2 ∆-u , . . .

2 ∆-u+1 -1}, {2 ∆-u , . . . 2 ∆-u +1 -1}, {0, . . . -1} ad {0, . . . -1}, quantity q = k -2 u -u k + 2 u -u τ takes all relative integers values in {-Q, . . . , Q}, with Q = 2 ∆-u ( -1).
We have

2 ∆-u+1 -1 i=2 ∆-u 2 ∆-u +1 -1 i =2 ∆-u -1 -1 s=0 -1 s =0 Ξ ((u,s),(u ,s )) a,b Ξ ((u,s),(u ,s )) a ,b = Q -Q q Q 1 - q Q + 2 (∆-u)(2-da-d b -d a -d b ) ∞ -∞ 2 -(u -u) -1 τ =0 g(λ; d a + d b )e -i (q-2 u -u τ )λ) dλ ∞ -∞ 2 -(u -u) -1 τ =0 g(λ; d a + d b ) e -i (q-2 u -u τ ) λ dλ = 2 ∆-u -Q q Q 1 - q Q + 2 (∆-u)(2-da-d b -d a -d b ) π -π 2 -(u -u)/2 D u -u,∞ (λ; d a + d b )e i qλ dλ π -π 2 -(u -u)/2 D u -u,∞ (λ; d a + d b ) e i qλ dλ since D u -u;∞ (λ; d a +d b ) = 2 u -u -1 v=0
2 (u -u)/2 t∈Z g u -u (λ+2tπ)e i 2 u -u τ (λ+2tπ) . Applying Lemma 33, we obtain Let us use the notations introduced in Appendix A.2. The sample wavelet covariances satisfy (A.3). It results that the vector of empirical covariances at different scales can be written as

lim →∞ 2 ∆-u+1 -1 i=2 ∆-u 2 ∆-u +1 -1 i =2 ∆-u -1 s=0 s =0 Ξ ((u,s),(u ,s )) a,b Ξ ((u,s),(u ,s )) a ,b = 2 ∆-u 2 (∆-u)(2-da-d b -d a -d b )+u-u (2π) π -π D u -u;∞ (λ; d a + d b ) D u -u;∞ (λ; d a + d b ) dλ = 2 ∆-u +(∆-u)(2-da-d b -d a -d b ) I u -u (d a + d b , d a + d b )K(d a + d b )K(d a + d b ). (B.
2 -j 0 (da+d b )    σ a,b (j 0 ) . . . σ a,b (j 1 )    = √ n j 1 B j 0    S (1,j 0 ) a,b . . . S (N,j 0 ) a,b    +    R a,b (j 0 ) . . . R a,b (j 1 )    (C.1)
with

B j 0 =           0 . . . . . . . . . 0 2 ∆ times n -1 j 0 . . . n -1 j 0 0 . . . . . . 0
Recall that S (i,j 0 ) = 1 n j 1 n j 1 -1 s=0 Y (i,s,j 0 ) . The steps of the proof of the asymptotic normality of vec S (i,j 0 ) are the following:

• We approximate {Y We study variables (Y (i,s,j 0 )(m) ) i=1,...,N defined in (A.4), in Appendix A.3. The objective of this step is to prove that variables (Y (i,s,j 0 )(m) ) i=1,...,N are asymptotically Gaussian. They are defined from variables (Z (i,s,j 0 )(m) a , a = 1, . . . , p) i=1,...,N by Y (i,s,j 0 )(m) = Z (i,s,j 0 )(m) Z (i,s,j 0 )(m)T . We will study first the behavior of (Z (i,s,j 0 )(m) ) i=1,...,N and next deduce that variables (Y (i,s,j 0 )(m) ) i=1,...,N are asymptotically Gaussian.

For all a = 1, . . . , p and s ∈ N, let

Z (s,j 0 )(m) a = Z (1,s,j 0 )(m) a , Z (2,s,j 0 )(m) a , . . . , Z (N,s,j 0 )(m)
Z (s,∞)(m) a = (Z (i,s,∞)(m) a ) i=1,...,N and Cov Z (i,s,∞)(m) a , Z (i ,s ,∞)(m) b = Θ ((i,s),(i ,s ))(m) a,b . For s ∈ N, ν ∈ R p×p , write Y (s,j 0 )(m) a,b = Y (1,s,j 0 )(m) a,b , . . . , Y (N,s,j 0 )(m) a,b T Y (s,j 0 )(m) (ν) = a,b=1,...,p ν a,b Y (s,j 0 )(m) a,b = Y (1,s,j 0 )(m) (ν), . . . , Y (N,s,j 0 )(m) (ν) T
The continuous mapping theorem implies that, when j 0 goes to infinity, (Y (s,j 0 )(m) (ν)) s=0,...,n j 1 -1 converges in distribution to Y (s,∞)(m) (ν) s=0,...,n j 1 -1

given by

Y (s,∞)(m) (ν) = a,b=1,...,p ν a,b Y (s,∞)(m) a,b = Y (1,s,∞)(m) (ν), . . . , Y (N,s,∞)(m) (ν) T , with Y (s,∞)(m) a,b = (Y (i,s,∞)(m) a,b ) i=1,...,N and Y (i,s,∞)(m) a,b = Z (i,s,∞)(m) a Z (i,s,∞)(m) b . Appendix C.1.2. Second step: asymptotic normality of S (i,s,j 0 )(m)
We first prove that conditions of [31, Proposition 3] are satisfied by {Y (s,j 0 )(m) , s ∈ N, j 0 0}.

Lemma 22. For all m 1, for all ν ∈ R p×p , sup i=1,...,N sup s 0 sup j 0 0 E[Y (i,s,j 0 )(m) (ν)] < ∞, (C.2)
∀s, s 0, lim

j 0 →∞ Cov(Y (s,j 0 )(m) (ν), Y (s ,j 0 )(m) (ν)) = Cov(Y (s,∞)(m) (ν), Y (s ,∞)(m) (ν)), (C.3) lim →∞ lim j 0 →∞ Cov( -1/2 -1 s=0 Y (s,j 0 )(m) (ν)) = Γ (m) (ν), (C.4) with Γ (m) (ν) = a,b=1,...,p a ,b =1,...,p ν a,b ν a ,b Γ (i,i )(m) (a,b),(a ,b ) i,i =1,...,N and Γ (m) defined in (B.12). Proof. Proof of (C.2). Assertion (C.2) follows from the fact that E[Y (s,∞)(m) (ν)] = a,b=1,...,p ν a,b Θ ((i,s),(i,s))(m) a,b i=1,...,N . Proof of (C.3). Vector Z (s,∞)(m) a , Z (s,∞)(m) b , Z (s ,∞)(m) a , Z (s ,∞)(m) b
T follows a centered Gaussian distribution. We can therefore use Isserlis's theorem. We get

E(Y (i,s,∞)(m) (ν)Y (i,s ,∞)(m) (ν)) = a,b,a ,b =1,...,p ν a,b ν a ,b E(Z (i,s,∞)(m) a Z (i,s,∞)(m) b Z (i,s ,∞)(m) a Z (i,s ,∞)(m) b ) = a,b,a ,b =1,...,p ν a,b ν a ,b Θ ((i,s),(i,s))(m) a,b Θ ((i ,s ),(i ,s ))(m) a ,b +Θ ((i,s),(i ,s ))(m) a,a Θ ((i,s),(i ,s ))(m) b,b + Θ ((i,s),(i ,s ))(m) a,b Θ ((i,s),(i ,s ))(m) b,a .
It results that:

Cov(Y (i,s,∞)(m) (ν), Y (i ,s ,∞)(m) (ν)) = a,b,a ,b =1,...,p ν a,b ν a ,b Θ ((i,s),(i ,s ))(m) a,a Θ ((i,s),(i ,s ))(m) b,b + Θ ((i,s),(i ,s ))(m) a,b Θ ((i,s),(i ,s ))(m) b,a .
We deduce that it is sufficient to prove that, when j 0 goes to infinity, Cov(Y (i,s,j 0 )(m) (ν), Y (i ,s ,j 0 )(m) (ν)) converges to Following the proof of Lemma 16, we can write Cov(Y (i,s,j 0 )(m) (ν), Y (i ,s ,j 0 )(m) (ν)) as Proof of (C.4). The proof is based on decomposition (C.5) of Cov(Y (i,s,j 0 )(m) , Y (i ,s ,j 0 )(m) ). Following the step 2 and the step 3 of the proof of Lemma 16, we can establish that

Cov(Y (i,s,j 0 )(m) (ν), Y (i ,s ,j 0 )(m) (ν)) = a,b,a ,b =1,...,p ν a,b ν a ,b T (i,i ,s,s ,j 0 )(m) (a,a ),(b,b ) + T (i,i ,s,s ,j 0 )(m) (a,b ),(b,a ) - n j 1 -1 s =0 ν a,b ν a ,b R (i,i ,s,s ,j 0 )(m) (a,b),(a ,b ) , (C.5) with T (i,i ,s,s ,j 0 )(m) (a,a ),(b,b ) (ν) = t 1 ∈Z V (i,j 0 )(m) (2 j 1 s -t 1 )V (i ,j 0 )(m) (2 j 1 s -t 1 ) T a,a t 2 ∈Z V (i,j 0 )(m) (2 j 1 s -t 2 )V (i ,j 0 )(m) (2 j 1 s -t 2 ) T b,b
n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 R (i,i ,s,s ,j 0 )(m) (a,a ),(b,b ) -→ j 0 →∞ 0, n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 T (i,i ,s,s ,j 0 )(m) (a,b),(a ,b ) -→ j 0 →∞ 2π π -π W (i,i )(m) * a,a (λ) W (i,i )(m) * b,b (λ)dλ, n -1 j 1 n j 1 -1 s=0 n j 1 -1 s =0 T (i,i ,s,s ,j 0 )(m) (a,b ),(b,a ) -→ j 0 →∞ 2π π -π W (i,i )(m) * a,b (λ) W (i,i )(m) * b,a (λ)dλ.
The proof is very similar and it is not detailed here for the sake of concision. It relies on Lemma 32, Lemma 13, Lemma 14 and Lemma 15.

We are now in a position to give the asymptotic normality of variables S (i,s,j 0 )(m) , defined in (A.5).

Proposition 23. Under conditions of Theorem 20, For i = 1, . . . , N , when j 0 goes to infinity, vec S (i,j 0 )(m) = vec n -1/2 j n j -1 s=0 Y (i,s,j 0 )(m) converges to a N p 2 (0, Γ (m) ) distribution by Proposition 23. We want to deduce a similar result for variables (S (i,j 0 ) ). Lemma 17 establishes that lim m→∞ Γ (m) = Γ. Hence Lemma 24 below entails that we can apply [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 3.2]. Proposition 21 follows.

vec S (i,j 0 )(m) , i = 1, . . . , N L -→ j 0 →∞ {Q (m) (i), i = 1, . . . , N }, where Q (m) (•) is a centered Gaussian process with covariance function Cov (Q (m) a,b (i), Q (m) a ,b (i )) = Γ (i,i )(m)
It remains to prove the following lemma. Lemma 24. For all i, i = 1, . . . , N , lim m→∞ lim j 0 →∞ Var S (i,j 0 )(m) -S (i,j 0 ) = 0.

Proof. Lemma 16 and Proposition 23 state respectively that for all (i, i ) ∈ {1, . . . , N } 2 , for all (a, b, a , b ) ∈ {1, . . . , p} 4 , we have lim j 0 →∞ Cov(S

(i,j 0 ) a,b , S (i ,j 0 ) a ,b ) = Γ (i,i ) (a,b),(a ,b ) and lim j 0 →∞ Cov(S (i,j 0 )(m) a,b S (i ,j 0 )(m) a ,b ) = Γ (i,i )(m) (a,b),(a ,b ) . Additionnally, by Lemma 17, lim m→∞ Γ (i,i )(m) = Γ (i,i ) . Consequently, lim m→∞ lim j 0 →∞ Cov(S (i,j 0 )(m) a,b , S (i,j 0 )(m) a ,b ) = lim j 0 →∞ Cov(S (i,j 0 ) a,b , S (i,j 0 ) a ,b ) = Γ (i,i ) (a,b),(a ,b ) .
Hence it is sufficient to prove that lim m→∞ lim j 0 →∞ Cov(S

(i,j 0 )(m) a,b , S (i ,j 0 ) a ,b ) = Γ (i,i ) (a,b),(a ,b ) .
To this aim, we will prove that limits can be inverted, that is, lim m→∞ lim j 0 →∞ Cov(S By Theorem 2, for all 0 < ∞, S a,b ( , j 0 ) is asymptotically Gaussian, with distribution N p (0, V (S) ( ω, )).

We will first establish the result when ∆ = j 1 -j 0 is finite and next when it is infinite.

• ∆ finite.
Since S a,b (∆, j 0 ) is asymptotically Gaussian, it is sufficient to prove that The convergence is obtained using (E.1).

The proof of Proposition 26 above is based on the following lemma.

Using Proposition 1,

E| S j 0 | = O √ nj 0 2 -j 0 β .
Thus E( S j 0 ) converges to zero when j 2 0 N X 2 -j 0 (1+2β) → 0.

Expression of

I (S) (δ 1 , δ 2 ) = 1 2 -2 -∆ ∆ u=0 ∆ u =0 2 -u/2-u /2 (u -η ∆ )(u -η ∆ ) 2 (δ 1 +δ 2 )u∨u -|u-u |/2 2 -uδ 1 -u δ 2 I |u-u | (δ 1 , δ 2 ) .
We can formulate this expression to recover a similar form to that of [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF]Theorem 5] and [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]Theorem 5]. The arguments are the same than those used in the proof of [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF]Proposition 10], but are recalled here to explicit the form of the variance. We can express I (S) (δ 1 , δ 2 ) as:

I (S) (δ 1 , δ 2 ) = 1 2 -2 ∆ ∆ γ =0 (γ -η ∆ ) 2 2 -γ I 0 (δ 1 , δ 2 ) + 1 2 -2 ∆ ∆ γ=1 ∆-γ γ =0 (γ + γ -η ∆ )(γ -η ∆ )2 -γ-γ (2 γδ 1 + 2 γδ 2 ) I γ (δ 1 , δ 2 ) ,
where we set γ = |u -u | and γ = u∧u . We can use the equalities ∆-γ γ =0 (γ -η ∆ ) 2 2 -γ = (2 -2 -∆+γ )(κ ∆-γ + (η ∆-γ -η ∆ ) 2 ) and ∆-γ γ =0 (γ -η ∆ )2 -γ = (2 -2 -∆+γ )(η ∆-γ -η ∆ ). We obtain

I (S) (δ 1 , δ 2 ) = κ ∆ I 0 (δ 1 , δ 2 ) + 1 2 -2 ∆ ∆ γ=1 (2-2 -∆+γ )((γ +η ∆-γ -η ∆ )(η ∆-γ -η ∆ )+κ ∆-γ ) 2 -γ (2 γδ 1 + 2 γδ 2 ) I γ (δ 1 , δ 2 ) .
We see that when ∆ < ∞, I (S) (δ 1 , δ 2 ) = When ∆ goes to infinity, the sequence κ ∆ converges to 2 and the sequence η ∆ converges to 1. We deduce the asymptotic form ( 14) when ∆ → ∞ by dominated convergence. Proof. Inequalities (I.1) and (I.2) are proved in [START_REF] Moulines | On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter[END_REF]Proposition 3]. Next, H j (λ)H j (λ) -2 j/2+j /2 | φ(λ)| 2 ψ(2 j λ) ψ(2 j λ) |H j (λ)| H j (λ) -2 j /2 φ(λ) ψ(2 j λ) + |H j (λ)| H j (λ) -2 j/2 φ(λ) ψ(2 j λ) + H j (λ) -2 j /2 φ(λ) ψ(2 j λ) H j (λ) -2 j/2 φ(λ) ψ(2 j λ) .

∞-

  ∞ φ(t)e -iλt dt and ψ(λ) = ∞ -∞ ψ(t)e -iλt dt, for all λ ∈ R. We suppose that φ(•) and ψ(•) satisfy the following assumptions: (W1) The functions φ(•) and ψ(•) are integrable, have compact supports, R φ(t)dt = 1 and R ψ 2 (t)dt = 1. (W2) There exists α > 1 such that sup λ∈R | ψ(λ)|(1 + |λ|) α < ∞. (W3) The mother wavelet ψ(•) has M > 1 vanishing moments. (W4) The function k∈Z k φ(• -k) is polynomial with degree for all = 1, . . . , M -1.

Corollary 3 .

 3 Let (a, b) ∈ {1, . . . , p} 2 , a = b, and j j 0 0. Define ρ a,b (j) = σ a,b (j) σ a,a (j) σ b,b (j) and r a,b = G a,b G a,a G b,b .

Remark 5 . 1 -ρ 2 ) 2 ;

 5122 Let ρ denote the sample correlation of a bivariate-Gaussian-distributed nsample with correlation ρ. Then √ n( ρ -ρ) see e.g. [33, Theorem 4.2.4]. When parameters d a and d b are equal, Corollary 3 entails that √ n j ( ρ a,b (j) -r a,b ) I 0 (2d a , 2d a )(1 -r 2 a,b ) 2 .

Remark 11 .ρ ρ 1 and d 1 =

 111 Consider the bivariate setting with Ω = 1 d 2 = d. Let d U 1 and d U 2 be the wavelet Whittle estimators obtained by separately considering the components {X 1

Figure 1 :

 1 Figure 1: Inferred graphs of cerebral activity for a dead rat (left) and a live rat (right). Each vertex of the graph corresponds to a brain region. Colored vertices are regions where the LRD parameter d a is significant, i.e. where the null hypothesis (H (r) 0 a ) is rejected. Two vertices a, b are connected by an edge if the long-run correlation is significant, i.e. if the null hypothesis (H (r) 0 a,b ) is rejected.

  λ)dλ, (B.15) where W (i,i ) * a ,b (•) and W (i,i )(m) * a ,b (•) are defined respectively in (B.4) and in (B.10).

  ), (B.20) and (B.21). Appendix B.5. Convergence of Γ (m) to Γ

  1).Then, under assumptions of Theorem 20,Θ ((i,s),(i ,s )) a,b = Ω a,b cos π(d a -d b )/2 2 (u-∆)/2+(u -∆)/2+∆ (da+d b ) Ξ ((u,s),(u ,s )) a,b . Proof. Recall that {Θ ((i,s),(i ,s )) a,b, a, b = 1, . . . , p, i = 1, . . . , N, s = 0, . . . , n j 1 -1} are defined in (B.5). For all (a, b) ∈ {1, . . . , p} 2 , Θ ((i,s),(i ,s ))

Lemma 19 .

 19 Suppose conditions of Theorem 2 hold. For all (a, b, a , b ) ∈ {1, . . . , p} 4 , for all ∆ ∈ N, (u, u ) ∈ {0, . . . , ∆} 2 , Proof. Quantities Γ (i,i ) (a,b),(a ,b ) can be expressed as: s),(i ,s )) a,b , a, b = 1, . . . , p, i = 1, . . . , N, s = 0, . . . , n j 1 -1} are noted in (B.5). Lemma 18 yields Θ ((i,s),(i ,s )) a,b = Ω a,b cos π(d a -d b )/2 2 (u-∆)/2+(u -∆)/2+∆ (da+d b ) Ξ ((u,s),(u ,s ))

25 )

 25 Lemma 19 results from (B.23), (B.24) and (B.25).

•

  (i,•,j) a,b , i = 1, . . . N } by the m-dependent process {Y (i,•,j)(m) a,b , i = 1, . . . N } defined in Appendix A.3. We establish that {Y (i,•,j)(m) a,b , i = 1, . . . N } is asymptotically normal when j goes to infinity, using [31, Proposition 2]. The asymptotic normality for a,b=1,...,p ν a,b S (i,j 0 ) a,b is obtained by letting m goes to infinity, using [44, Theorem 3.2]. Appendix C.1.1. First step, approximation by a m-dependent process

  i,s,j 0 )(m) a , a = 1, . . . , p) i=1,...,N are m-dependent decimated linear processes in R N . By [31, Proposition 2], we get that vec Z (s,j 0 )(m) a , a = 1, . . . , p converges in distribution to vec Z (s,∞)(m) a , a = 1, . . . , p , which follows a centered Gaussian distribution with

  a,b,a ,b =1,...,p ν a,b ν a ,b Θ ((i,s),(i ,s ))(m) a,a Θ ((i,s),(i ,s ))(m) b,b + Θ ((i,s),(i ,s ))(m) a,b Θ ((i,s),(i ,s ))(m) b,a to obtain equality (C.3).

,

  and lim j 0 →∞ R (i,i ,s,s ,j 0 )(m) (a,b),(a ,b ) = 0. Conclusion follows from (B.13).

  (a,b),(a ,b ) defined in (B.12). Proof. With results of Lemma 22, we can apply [31, Proposition 3], which gives the proposition. Appendix C.1.3. Third step: proof of Proposition 21

  j 0 ) a ,b ) = lim j 0 →∞ lim m→∞ Cov(S (i,j 0 )(m) a,b , S (i ,j 0 ) a ,b ).We can establish that Cov(S(i,j 0 )(m) a,b , S (i ,j 0 )a ,b ) converges as j 0 goes to infinity to2π π -π W (i,i ,j 0 )(m,∞) * a,a (λ) W (i,i ,j 0 )(m,∞) * b,b (λ)dλ + 2π π -π W (i,i ,j 0 )(m,∞) * a,b (λ) W (i,i ,j 0 )(m,∞) * b,a (λ)dλ,

  The proof is very similar to the one carried out for proving (C.4) and it is thus omitted. Additionally (C.6) converges to Γ (i,i ) (a,b),(a ,b ) when j 0 goes to infinity. Proof. For all 0, introduce S a,b ( , j 0 ) = u=0 ω a,b (u) √ n j 0 +u 2 -(j 0 +u)(da+d b ) ( σ a,b (j 0 + u) -σ a,b (j 0 + u)) .

E

  S a,b (∆, j 0 ) -S a,b (∆, j 0 ) goes to 0 when j 0 goes to infinity. Using Lemma 27, we haveE S a,b (∆, j 0 ) -S a,b (∆, j 0 ) C σ ∆ u=0 |ω a,b (u, j 0 ) -ω a,b (u)| . (E.2)We conclude with (E.1).• ∆ infinite.The convergence of S a,b ( , j 0 ) has been established when is finite and (E.1)-(E.2) imply that lim→∞ lim j 0 →∞ E |S a,b ( , j 0 ) -S a,b ( , j 0 )| = 0.Hence, it is sufficient to prove that lim →∞ limj 0 →∞ E |S a,b (∞, j 0 ) -S a,b ( , j 0 )| = 0. Lemma 27 gives E |S a,b (∞, j 0 ) -S a,b ( , j 0 )| ∞ u= +1|ω a,b (u, j 0 )| .

  a,b,a ,b =1,...,p V (S) (a,b),(a ,b ) ( ω (S) , ∆). It remains to prove that a,b,a ,b =1,...,p V (S) (a,b),(a ,b )( ω (S) , ∆) = 4 log(2) 2 κ 2 ∆ υ T V d(∆) υ. By expanding the expression, a,b,a ,b =1,...,p V (S) (a,b),(a ,b ) ( ω (S) , ∆)) = 4 log(2) 2 2 -2 -∆ a,b,a ,b =1,...,p υ a υ a (G 0-1 ) a,b (G 0-1 ) a ,b ∆ u=0 ∆ u =0 2 -u/2-u /2 (u -η ∆ )(u -η ∆ ) 2 -u(d 0 a +d 0 b )-u (d 0 a +d 0 b ) V (a,b),(a ,b ) (u, u ) = 4 log(2) 2 a,b,a ,b =1,...,p υ a υ a (G 0-1 ) a,b (G 0-1 ) a ,b G 0 aa G 0 bb I (S) (d 0 a + d 0 a , d 0 b + d 0 b ) +G 0 ab G0 a b I (S) (d 0 a + d 0 b , d 0 a + d 0 b ) , where

κ 2 ∆ 2 I

 22 ∆ (δ 1 , δ 2 ), with I ∆ (δ 1 , δ 2 ) defined in (13). Hence, a,b,a ,b =1,...,p V (S) (a,b),(a ,b ) ( ω (S) , ∆) = 4 log(2) 2 κ 2 ∆ υ T V d(∆) υ.

Appendix F. 3 .P-→ κ j 1

 31 Proof of Theorem 4The Taylor expansion of ∂ R(d)∂d at d at the neighborhood of d 0 gives √ n( dd 0 ) = d is such that dd 0 dd 0 .It has already been established in [6, Equation (E10)] that, under assumptions of Theorem 4,∂ 2 R(d) ∂d∂d T d -j 0 log(2) 2 2(G 0-1 • G 0 + I p ) . (F.8)This justifies also that the matrix∂ 2 R ∂d∂d T d in (F.7) is indeed invertible for sufficiently high N X when 2 -j 0 β + N -1/2 X 2 j 0 /2 → 0.Next Proposition 28 establishes that ∂ R ∂d d 0 converges to a centered Gaussian distribution with variance (2 log(2) 2 κ 2 j 1 -j 0 V d(∆) ). Theorem 4 then follows with (F.7) and (F.8).

1 / 2 . 2 .

 122 Applying inequalities (I.1) and (I.2) to the right-hand side gives (I.3).The following lemma is[26, Lemma 1]. It is used in the proofs of Lemma 16 and of Lemma 22.Lemma 32 ([26]). Let Q ∈ N. For all function g ∈ L 2 (-π, π), write M Q (g) = q∈Z (1 -|q|/Q) + π -πg(λ)e -i q λ dλ 2 Suppose g 1 and g 2 are C-valued functions of L 2 ((-π, π)). Then,|M Q (g 1 ) -M Q (g 2 )| 2 2π π -π |g 1 (λ) -g 2 (λ)| 2 dλ.Next, the following lemma states the convergence of a series of bivariate Fourier coefficients. It is used in the proofs of Lemma 15 and of Lemma 19.Lemma 33. Suppose {w * 1 (λ), λ ∈ (-π, π)} and {w * 2 (λ), λ ∈ (-π, π)} are C-valued functions of L 2 ((-π, π)). Then q∈Z (1 -|q|/Q) e -i q λ dλ = q∈Z (1 -|q|/Q) + c 1q c2q , 66 with c 1q and c 2q the q th Fourier coefficient respectively of functions w * 1 and w * The sequence (1 -|q|/Q) + c 1q c 2q converges to c 1q c 2q when Q goes to infinity. By Cauchy-Schwarz's inequality and Parseval's equality, Thus dominated convergence entails that the series q∈Z (1 -|q|/Q) + c 1q c 2q converges to q∈Z c 1q c 2q . By Parseval's theorem, q∈Z c 1q c 2q = 2π π -π w * 1 (λ)w * 2 (λ)dλ, which concludes the proof.

Table 1 :

 1

	00 5.43	5.56	5.67	5.77	5.86
	M= 2 α= 1.34 2.65	2.66	2.67	2.68	2.69
	M= 3 α= 1.64 1.85	1.86	1.86	1.86	1.87
	M= 4 α= 1.91 1.62	1.62	1.61	1.61	1.61
	M= 5 α= 2.18 2.05	2.04	2.04	2.03	2.02
	M= 6 α= 2.43 1.90	1.91	1.92	1.93	1.94
	M= 7 α= 2.68 1.22	1.23	1.24	1.25	1.26
	M= 8 α= 2.93 1.01	1.01	1.01	1.01	1.01

Values of I 0 (2d 1 , 2d 2 ) with respect to d = (d 1 , d 2 ) for Daubechies's wavelets with different values of vanishing moments M in (W3). Parameter α characterizes the regularity of the wavelets in (W2).

(i,i ,j 0 )(m), where Γ (i,i ,j 0 )(m) (a,b),(a ,b ) a,b,a ,b are defined

∆-u+1 -1 i=2 ∆-u 2 ∆-u +1 -1 i =2 ∆-u Γ (i,i ) (a,b),(a ,b ) = 2 ∆-u G I G (a,a ),(b,b ) (u) + G I G (a,b ),(a ,b) (u) ,where G I G(u) is defined in[START_REF] Lobato | Consistency of the averaged cross-periodogram in long memory series[END_REF].

∆-1 times n -1 j 0 +1 . . . n -1 j 0 +1

Acknowledgments

The author gratefully acknowledge to the two anonymous referees for their comments that led to substantial improvements in the article. The author also wishes to express her thanks to Sophie Achard for fruitful exchanges. Emmanuel Barbier and Guillaume Becq are also acknowledged for providing the data of the resting state fMRI on the rats.

Appendix A. Expression of wavelet coefficients

Let (a, b) ∈ {1, . . . , p} 2 . The objective is to study the asymptotic normality of the sample wavelet covariance { σ a,b (j 0 + u), u = 1, . . . , ∆} when j 0 goes to infinity. To this end, we introduce a new indexing of wavelet coefficients similar to that in [23, pages 543 and 544]. This new indexing enables to approximate the sequence of wavelet coefficients with m-dependent variables and to use the results on linear decimated processes of [START_REF] Roueff | Central limit theorems for arrays of decimated linear processes[END_REF]. The section is structured as follows. In Appendix A.1, we give a linear representation of wavelet coefficients, the new indexing of the coefficients is defined in Appendix A.2. Appendix A.3 finally introduces the approximation by a m-dependent process.

Appendix A.1. Linear representation of wavelet coefficients

Consider a scale j 0 and k ∈ Z. Define, for all l ∈ Z, h j,l = R φ(t + l)2 -j/2 ψ(2 -j t)dt, the discrete wavelet filter associated to (φ(•), ψ(•)). Then under (W1), the vector of wavelet coefficients W(j, k) defined in Section 2.2 can be written as W(j, k) = l∈Z h j,2 j k-l X(l), with W(j, k) ∈ R p . For all λ ∈ R let us denote H j (λ) = l∈Z h j,l e -iλ l = R l∈Z φ(t + l)2 -j/2 ψ(2 -j t)dt, the discrete Fourier transform of {h j,l , l ∈ Z}.

Suppose that the multivariate process X = {X a (k), k ∈ Z, a = 1, . . . , p} satisfies Assumption (M3). To express wavelet coefficients, we introduce, for all λ ∈ (-π, π),

and {A(t), t ∈ Z} ∈ 2 (Z) such that A * (λ) = (2π) -1/2 t∈Z A(t)e iλt .

Appendix B.1. Useful inequalities concerning the linear wavelet representation

We first give two lemmas, respectively on the behavior of the spectral density f (•) and of the function A * (•). Proof. Let (a, b) ∈ {1, . . . , p} 2 and λ ∈ (-π, π). By (M1),

Proof. The lemma is straightforward combining (A.2) and Lemma 8.

Appendix B.2. Preliminary results on

For all i, i = 1, . . . , N , for all λ ∈ R, let us define also

Appendix B.3. Preliminary results on the m-dependent processes

We define similar quantities than in Appendix B.2 in the m-dependent setting. That is,

We also denote

(B.12) We will prove in Appendix C.3 that Γ (i,i )(m) is the asymptotic covariance matrice of (S (i,j 0 )(m) , S (i ,j 0 )(m) ) as j 0 goes to infinity.

We now provide some general results on the behavior of (V (i,j 0 )(m) ) and (V (i,j 0 )(m) * ), in much the same way as in Lemma 10, Lemma 11 and Lemma 12 for (V (i,j 0 ) ) and (V (i,j 0 ) * ).

Lemma 13. Suppose assumptions of Theorem 20 hold. Suppose ∆ < ∞. Then there exists δ v > 1/2 such that for all j j 0 , j -j 0 ∆, we have

Proof. The lemma follows from Lemma 10 and [31, Lemma 5]. Lemma 14. Suppose assumptions of Theorem 20 hold. For all i = 1, . . . , N , for all m 1, sequences {V

, a, b = 1, . . . , p, j 0 0} verify the following properties:

The asymptotic normality is given by Theorem 20 below. Proposition 1 enables to approximate {2 -j(da+d b ) σ a,b (j), a, b = 1, . . . , p, j 0} by G and hence entails Theorem 2.

Theorem 20. Suppose Assumptions (M1)-(M4) and (W1) to (W5) hold. Let 2 -j 0 β → 0 and N -1 X 2 j 0 → 0. Then

where Q(•) is the centered Gaussian process defined in Theorem 2.

The construction of the proof is adapted from that of [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]Theorem 2]. The proof has been structured as follows. Appendix A proposes a writing of wavelet coefficients as decimated linear processes, and provides an approximation by a m-dependent decimated linear processes, m 0. Notations and technical results on the decimated decompositions are stated in Appendix B. They are useful for applying the propositions of [START_REF] Roueff | Central limit theorems for arrays of decimated linear processes[END_REF], which lead to the asymptotic normality described in Theorem 20. The current section deals with this step.

The objective is to show that vec( σ a,b (j), a, b = 1, . . . , p) is asymptotically Gaussian when j goes to infinity. The proof is divided into the following steps:

• Appendix C.1 establishes that the vector vec(S a,b (i, j 0 ), a, b = 1, . . . , p) i=1,...,N is asymptotically Gaussian when j 0 goes to infinity; the proof is base on the approximation by m-dependent processes introduced in Appendix A.3.

• Appendix C.2 proves that the terms vec(R a,b (j), a, b = 1, . . . , p) j 0 j j 1 are negligible.

Appendix C.3 compiles all elements above to prove Theorem 20.

Appendix C.1. Asymptotic normality of S (i,j)

The asymptotic normality of vec S (i,j 0 ) is given by the following proposition.

Proposition 21. Under conditions of Theorem 20,

Next we can study the limit of Cov(S

a ,b ) as m goes to infinity and state that it converges uniformly to Γ [START_REF] Achard | Wavelet-based and Fourier-based multivariate Whittle estimation: multiwave[END_REF]). In the proof of Lemma 17, we have proved that lim j 0 →∞ Γ

This concludes the proof.

Appendix C.2. Study of R(j)

The following lemma gives the convergence of {R a,b (j), a, b = 1, . . . , p, j 0} to zero when j goes to infinity.

goes to zero in probability as j 0 goes to infinity.

Proof. Hölder's inequality gives

. Using Parseval's equality and Lemma 10, we get Theorem 20 Equality (C.1) states that for all a, b = 1, . . . , p,

By Proposition 25, n 1/2 j R a,b (j) goes to 0 in probability when j goes to infinity.

Proposition 21 entails that the first term is asymptotically Gaussian and centered. We now explicit the asymptotic variance.

Replacing B j 0 by its expression, the equation above can be reformulated as

Using the fact that n j 1 n

, and using Proposition 21, it results that Gaussian distribution when j goes to infinity and that its asymptotic covariance satisfies:

We deduce first that

where we have used that V(u, u) = V(0, 0) for all u ∈ Z. Replacing also V (a 1 ,a 2 ),(a 3 ,a 4 ) (0, 0) for all a 1 , a 2 , a 3 , a 4 = 1, . . . , p by its expression given in [START_REF] Sela | Computationaly efficient methods for two multivariate fractionnaly integrated models[END_REF], we obtain the asymptotic distribution of Corrollary 3.

Second, suppose that the off-diagonal entries of G are equal to zero. Then,

Replacing V (a,b),(a ,b ) (u, u ) by its expression, it results that the right-hand side is equal to 0 for all (a, b) / ∈ {(a , b ), (b , a )}, and is equal to

Appendix E. Additional results on the sample wavelet covariance

The objective of this section is to prove that some linear combinations of sample wavelet covariances may be asymptotically Gaussian. Some conditions are given in the following proposition. It corresponds to [START_REF] Roueff | Asymptotic normality of wavelet estimators of the memory parameter for linear processes[END_REF]Theorem 3]. The arguments and the scheme of proof are the same. They are recalled here since the setting and the notations are slightly different.

Proposition 26. Suppose assumptions of Theorem 2 hold. Let ∆ ∈ N ∪ {∞}. Let {ω(u, j 0 ), u ∈ N, j 0 ∈ N} be a sequence of R p×p such that for all u ∈ N, ω(u, j 0 ) -→

Then vec(S(∆, j 0 )) converges in distribution to N p (0, V (S) ( ω, ∆)) when j 0 goes to infinity, with

Lemma 27. Suppose conditions of Theorem 20 hold. There exists C σ depending on Ω, d, φ(•), ψ(•), L and β such that for all (a, b) ∈ {1, . . . , p},

Proof. Since we only consider one scale, suppose momentarily that j 0 = j = j 1 . Based on notations of Appendix A.2, equation (A.3), and Proposition 25,

Lemma 22 and Lemma 24 imply that Var S a,b (1, j) is finite. Lemma 27 is then straightforward.

Appendix F. Proof of Theorem 4

In this section the true parameters are denoted with an exponent 0.

Observe that conditions of [6, Proposition 6] are satisfied since we suppose assumption (M3). Consequently, under assumptions of Theorem 4, conditions of Theorem 6 of [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] hold. It entails that

The proof of Theorem 4 is based on a Taylor expansion of the objective function. We first recall some useful results obtained [START_REF] Achard | Multivariate wavelet Whittle estimation in long-range dependence[END_REF] in Appendix F.1, next we give a normality result on the first derivative of the objective function in Appendix F.2. Appendix F.3 finally gives the proof of Theorem 4.

Appendix F.1. Some results about the objective function and its derivative

and

The derivatives of the criterion R(d) are equal to

when G(d) -1 exists.

For any a = 1, . . . , p, let i a be a p × p matrix whose a-th diagonal element is one and all other elements are zero. Let a and b be two indexes in 1, . . . , p. The first derivative of

. Asymptotic normality of the first derivative

The objective of this section is to prove that

is asymptotically Gaussian.

Proposition 28. Under assumptions of Theorem 4,

where V d(∆) is defined in equation [START_REF] Baek | Asymptotics of bivariate local Whittle estimators with applications to fractal connectivity[END_REF].

Proof. For any vector υ = (υ a ) a=1,...,p ∈ R p , we want to prove that υ T ∂ R

As expressed in [6, page 36], G(d) can be written as

Applying [6, Proposition 8], we get

We introduce

It is easily seen that

Reformulating (F.4), we get

Thus we can write S j 0 as

and ω

a,b (u, j 0 ) = -2 log(2)

Lemma 13 of [START_REF] Moulines | A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series[END_REF] states that

a,b (u, j 0 ) hence converges to ω

Applying Proposition 26, vec S (d) (∆, j 0 ) -E(S (d) (∆, j 0 )) is asymptotically Gaussian, with distribution N p (0, V (S) ( ω (S) , ∆)). Consequently, S j 0 -E( S j 0 ) follows asymptotically the Gaussian distribution N p (0, a,b,a ,b =1,...,p V (S) (a,b),(a ,b ) ( ω (S) , ∆)).

The end of the proof is divided into two steps. First we prove that E( S j 0 ) goes to 0 when j 0 goes to infinity, and next we establish that the asymptotic variance above, that is, a,b,a ,b =1,...,p V

Convergence of E( S j 0 ) toward 0.

Taking the expectancy of S j 0 ,

.

Appendix G. Proof of Theorem 5

In this section the true parameters are denoted with an exponent 0.

We have

, it is sufficient to establish the asymptotic distribution of G a,b ( d). More precisely, we want to prove that √ n vec G( d) -G 0 converges in distribution to a centered Gaussian distribution. We decompose

The first term converges to the desired distribution as established in Lemma 29, while the second one is negligible by Lemma 30.

Lemma 29 and Lemma 30 are given hereafter.

Lemma 29. Under assumptions of Theorem 5, √ n vec G(d 0 ) -G 0 converges as j 0 goes to infinity to a centered Gaussian distribution, with variance Θ G(∆) defined in [START_REF] Baek | Semiparametric, parametric, and possibly sparse models for multivariate long-range dependence[END_REF].

Using inequality [START_REF] Gençay | An introduction to wavelets and other filtering methods in finance and economics[END_REF],

We can write

when j 0 goes to infinity. Applying Proposition 26, we obtain that T 0 (j 0 ) -E(T 0 (j 0 )) converges as n goes to infinity to a centered Gaussian distribution, with variance

where

Quantity I G (δ 1 , δ 2 ) can be simplified as:

When ∆ goes to infinity,

Lemma 30. Under assumptions of Theorem 5, √ n G( d) -G(d 0 ) tends to 0 in probability when j 0 goes to infinity.

Proof. A Taylor expansion at order one at d 0 gives

Achard and Gannaz in [6, Section E.2.4.] state that, when

Next, using the derivative of G(d) given in (F.4), we have:

Similarly to what was done in Appendix F, we can establish that log(2)

) → 0 we conclude that this term goes to 0, which concludes the proof.

Appendix H. Proof of Proposition 7

In this section the true parameters are denoted with an exponent 0.

From the proof of Theorem 4 (using equations (F.7), (F.8), (F.5) and (F.6)) and Theorem 5 (using equations (G.1), (G.2), (G.3) and Lemma 30), we can extract the following results