Jérémie Bigot

Rolando J Biscay Lirio

Jean-Michel Loubes

Lilian Muñiz Alvarez

Adaptive estimation of spectral densities via wavelet thresholding and information projection

Keywords: Spectral density estimation, adaptive estimation, wavelet thresholding, sequences of exponential families, Besov spaces. AMS classifications: Primary 62G07; secondary 42C40, 41A29

In this paper, we study the problem of adaptive estimation of the spectral density of a stationary Gaussian process. For this purpose, we consider a wavelet-based method which combines the ideas of wavelet approximation and estimation by information projection in order to warrants that the solution is a non-negative function. The spectral density of the process is estimated by projecting the wavelet thresholding expansion of the periodogram onto a family of exponential functions. This ensures that the spectral density estimator is a strictly positive function. The theoretical behavior of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. We also show the excellent practical performance of the estimator in some numerical experiments.

Introduction

The estimation of spectral densities is a fundamental problem in inference for stationary stochastic processes. Many applications in several fields such as weather forecast and financial series are deeply related to this issue, see for instance Priestley [START_REF] Priestley | Spectral Analysis and Time Series[END_REF]. It is known that the estimation of the covariance function of a stationary process is strongly related to the estimation of the corresponding spectral density. By Bochner's theorem the covariance function is non-negative definite if and only if the corresponding spectral density is a non-negative function. Hence in order to preserve the property of non-negative definiteness of a covariance function, the estimation of the corresponding spectral density must be a non-negative function. The purpose of this work is to provide a non-negative estimator of the spectral density.

Inference in the spectral domain uses the periodogram of the data, providing an inconsistent estimator which must be smoothed in order to achieve consistency. For highly regular spectral densities, linear smoothing techniques such as kernel smoothing are appropriate (see Brillinger [START_REF] Brillinger | Time Series: Data Analysis and Theory[END_REF]). However, these methods are not able to achieve the optimal mean-square rate of convergence for spectra whose smoothness is distributed inhomogeneously over the domain of interest. For this nonlinear methods are needed. One nonlinear method for adaptive spectral density estimation of a Assumption 1 The sequence (X 1 , ...X n ) is an n-sample drawn from a stationary sequence of Gaussian random variables.

Let ρ be the covariance function of the process, i.e. ρ (h) = cov (X t , X t+h ) with h ∈ Z. The spectral density f is defined as:

f (ω) = 1 2π h∈Z ρ (h) e i2πωh , ω ∈ [0, 1] .
We need the following standard assumption on ρ:

Assumption 2
The covariance function ρ is non-negative definite, such that there exists two constants

0 < C 1 , C 2 < +∞ such that h∈Z |ρ (h)| = C 1 and h∈Z hρ 2 (h) = C 2 .
Assumption 2 implies in particular that the spectral density f is bounded by the constant C 1 . As a consequence, it is also square integrable. As in Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], the data consist on a number of observations X 1 , ..., X n at regularly spaced points. We want to obtain a positive estimator for the spectral density function f without parametric assumptions on the basis of these observations. For this, we combine the ideas of wavelet thresholding and estimation by information projection.

Estimation by information projection

To ensure nonnegativity of the estimator, we will look for approximations over an exponential family. For this, we construct a sieve of exponential functions defined in a wavelet basis.

Let φ (ω) and ψ (ω), respectively, be the scaling and the wavelet functions generated by an orthonormal multiresolution decomposition of L 2 ([0, 1]), see Mallat [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] for a detailed exposition on wavelet analysis. Throughout the paper, the functions φ and ψ are supposed to be compactly supported and such that φ ∞ < +∞, ψ ∞ < +∞. Then, for any integer j 0 ≥ 0, any function g ∈ L 2 ([0, 1]) has the following representation: g (ω) = 2 j 0 -1 k=0 g, φ j 0 ,k φ j 0 ,k (ω) + +∞ j=j 0 2 j -1 k=0 g, ψ j,k ψ j,k (ω) , where φ j 0 ,k (ω) = 2 j 0 2 φ 2 j 0 ω -k and ψ j,k (ω) = 2 j 2 ψ 2 j ω -k . The main idea of this paper is to expand the spectral density f onto this wavelet basis and to find an estimator of this expansion that is then modified to impose the positivity property. The scaling and wavelet coefficients of the spectral density function f are denoted by a j 0 ,k = f, φ j 0 ,k and b j,k = f, ψ j,k .

To simplify the notations, we write (ψ j,k ) j=j 0 -1 for the scaling functions (φ j,k ) j=j 0 . Let j 1 ≥ j 0 and define the set

Λ j 1 = (j, k) : j 0 -1 ≤ j < j 1 , 0 ≤ k ≤ 2 j -1 .
Note that #Λ j 1 = 2 j 1 , where #Λ j 1 denotes the cardinal of Λ j 1 . Let θ denotes a vector in R #Λ j 1 , the wavelet-based exponential family E j 1 at scale j 1 is defined as the set of functions:

E j 1 =    f j 1 ,θ (.) = exp   (j,k)∈Λ j 1 θ j,k ψ j,k (.)   , θ = (θ j,k ) (j,k)∈Λ j 1 ∈ R #Λ j 1    . (2.1)
It is well known that Besov spaces for periodic functions in L 2 ([0, 1]) can be characterized in terms of wavelet coefficients (see e.g. Mallat [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]). Assume that ψ has m vanishing moments, and let 0 < s < m denote the usual smoothness parameter. Then, for a Besov ball B s p,q (A) of radius A > 0 with 1 ≤ p, q ≤ ∞, one has that for s * = s + 1/2 -1/p ≥ 0:

B s p,q (A) :=        g ∈ L 2 ([0, 1] : g s,p,q :=   2 j 0 -1 k=0 |a j 0 ,k | p   1 p +    ∞ j=j 0 2 js * q   2 j -1 k=0 |b j,k | p   q p    1 q ≤ A        ,
with the respective above sums replaced by maximum if p = ∞ or q = ∞ and where a j 0 ,k = g, φ j 0 ,k and b j,k = g, ψ j,k . The condition that s + 1/2 -1/p ≥ 0 is imposed to ensure that B s p,q (A) is a subspace of L 2 ([0, 1]), and we shall restrict ourselves to this case in this paper (although not always stated, it is clear that all our results hold for s < m).

Let M > 0 and denote by F s p,q (M ) the set of functions such that

F s p,q (M ) = {f = exp (g) : g s,p,q ≤ M } ,
where g s,p,q denotes the norm in the Besov space B s p,q . Note that assuming that f ∈ F s p,q (M ) implies that f is strictly positive. The following results hold.

Lemma 2.1 Suppose that f ∈ F s p,q (M ) with s > 1 p and 1 ≤ p ≤ 2. Then, there exists a constant M 1 such that for all f ∈ F s p,q (M ), 0 < M -1 1 ≤ f ≤ M 1 < +∞.
Let V j denote the usual multiresolution space at scale j spanned by the scaling functions (φ j,k ) 0≤k≤2 j -1 , and define A j < +∞ as the constant such that υ ∞ ≤ A j υ L 2 for all υ ∈ V j . For f ∈ F s p,q (M ), let g = log (f ). Then for j ≥ j 0 -1, define D j = g -g j L 2 and γ j = g -g j ∞ , where g j =

2 j -1 k=0 θ j,k ψ j,k , with θ j,k = g, ψ j,k .
The proof of the following lemma immediately follows from the arguments in the proof of Lemma A.5 in Antoniadis and Bigot [START_REF] Antoniadis | Poisson inverse problems[END_REF].

Lemma 2.2 Let j ∈ N. Then A j ≤ C2 j/2 . Suppose that f ∈ F s p,q (M ) with 1 ≤ p ≤ 2 and s > 1 p . Then, uniformly over F s p,q (M ), D j ≤ C2 -j(s+1/2-1/p) and γ j ≤ C2 -j(s-1/p) where C denotes constants depending only on M , s, p and q.

To assess the quality of the estimators, we will measure the discrepancy between an estimator f and the true function f in the sense of relative entropy (Kullback-Leibler divergence) defined by:

∆ f ; f = 1 0 f log f f -f + f dµ,
where µ denotes the Lebesgue measure on [0, 1]. It can be shown that ∆ f ; f is non-negative and equals zero if and only if f = f . We will enforce our estimator of the spectral density to belong to the family E j 1 of exponential functions, which are positive by definition. For this we will consider a notion of projection using information projection.

The estimation of density function based on information projection has been introduced by Barron and Sheu [START_REF] Barron | Approximation of density functions by sequences of exponential families[END_REF]. To apply this method in our context, we recall for completeness a set of results that are useful to prove the existence of our estimators. The proofs of the following lemmas immediately follow from results in Barron and Sheu [START_REF] Barron | Approximation of density functions by sequences of exponential families[END_REF] and Antoniadis and Bigot [START_REF] Antoniadis | Poisson inverse problems[END_REF].

Lemma 2.3 Let β ∈ R #Λ j 1 . Assume that there exists some θ (β) ∈ R #Λ j 1 such that, for all (j, k) ∈ Λ j 1 , θ (β) is a solution of f j,θ(β) , ψ j,k = β j,k .
Then for any function f such that f, ψ j,k = β j,k for all (j, k) ∈ Λ j 1 , and for all θ ∈ R #Λ j 1 , the following Pythagorian-like identity holds:

∆ (f ; f j,θ ) = ∆ f ; f j,θ(β) + ∆ f j,θ(β) ; f j,θ .
(2.

2)

The next lemma is a key result which gives sufficient conditions for the existence of the vector θ (β) as defined in Lemma 2.3. This lemma also relates distances between the functions in the exponential family to distances between the corresponding wavelet coefficients. Its proof relies upon a series of lemmas on bounds within exponential families for the Kullback-Leibler divergence and can be found in Barron and Sheu [START_REF] Barron | Approximation of density functions by sequences of exponential families[END_REF] and Antoniadis and Bigot [START_REF] Antoniadis | Poisson inverse problems[END_REF].

Lemma 2.4 Let θ 0 ∈ R #Λ j 1 , β 0 = β 0,(j,k) (j,k)∈Λ j 1 ∈ R #Λ j 1 such that β 0,(j,k) = f j,θ 0 , ψ j,k for all (j, k) ∈ Λ j 1 , and β ∈ R #Λ j 1 a given vector. Let b = exp log (f j,θ 0 ) ∞ and e = exp(1). If β -β 0 2 ≤ 1 2ebA j 1
then the solution θ β of

f j 1 ,θ , ψ j,k = β j,k for all (j, k) ∈ Λ j 1
exists and satisfies

θ β -θ 0 2 ≤ 2eb β -β 0 2 log f j 1 ,θ(β 0 ) f j 1 ,θ( β) ∞ ≤ 2ebA j 1 β -β 0 2 ∆ f j 1 ,θ(β 0 ) ; f j 1 ,θ( β) ≤ 2eb β -β 0 2 2 ,
where β 2 denotes the standard Euclidean norm for β ∈ R #Λ j 1 .

Following Csiszár [START_REF] Csiszár | I-divergence geometry of probability distributions and minimization problems[END_REF], it is possible to define the projection of a function f onto E j 1 . If this projection exists, it is defined as the function f j 1 ,θ * j 1 in the exponential family E j 1 that is the closest to the true function f in the Kullback-Leibler sense, and is characterized as the unique function in the family E j 1 for which

f j 1 ,θ * j 1 , ψ j,k = f, ψ j,k := β j,k for all (j, k) ∈ Λ j 1 .
Note that the notation β j,k is used to denote both the the scaling coefficients a j 0 ,k and the wavelet coefficients b j,k .

Let

I n (ω) = 1 2πn n t=1 n t ′ =1 X t -X X t ′ -X * e i2πω(t-t ′ ) ,
be the classical periodogram, where X t -X * denotes the conjugate transpose of X t -X and

X = 1 n n t=1
X t . The expansion of I n (ω) onto the wavelet basis allows to obtain estimators of a j 0 ,k and b j,k given by

a j 0 ,k = 1 0 I n (ω) φ j 0 ,k (ω) dω and b j,k = 1 0 I n (ω) ψ j,k (ω) dω. (2.
3)

It seems therefore natural to estimate the function f by searching for some θ n ∈ R #Λ j 1 such that

f j 1 , θn , ψ j,k = 1 0 I n (ω) ψ j,k (ω) dω := β j,k for all (j, k) ∈ Λ j 1 , (2.4)
where β j,k denotes both the estimation of the scaling coefficients a j 0 ,k and the wavelet coefficients b j,k . The function f j 1 , θn is the spectral density positive linear estimator.

Similarly, the positive nonlinear estimator with hard thresholding is defined as the function f HT j 1 , θn,ξ

(with θ n ∈ R #Λ j 1 ) such that f HT j 1 , θn,ξ , ψ j,k = δ ξ β j,k for all (j, k) ∈ Λ j 1 , (2.5)
where δ ξ denotes the hard thresholding rule defined by

δ ξ (x) = xI (|x| ≥ ξ) for x ∈ R,
where ξ > 0 is an appropriate threshold whose choice is discussed later on. The existence of these estimators is questionable. Thus, in the next sections, some sufficient conditions are given for the existence of f j 1 , θn and f HT j 1 , θn,ξ with probability tending to one as n → +∞.

Even if an explicit expression for θ n is not available, we use a numerical approximation of θ n , obtained via a gradient-descent algorithm with an adaptive step.

In this section we establish the rate of convergence of our estimators in terms of the Kullback-Leibler discrepancy over Besov classes.

We make the following assumption on the wavelet basis that guarantees that Assumption 2 holds uniformly over F s p,q (M ).

Assumption 3 Let M > 0, 1 ≤ p ≤ 2 and s > 1/p. For f ∈ F s p,q (M ) and h ∈ Z, let ρ(h) = 1 0 f (ω)e -i2πωh dω, C 1 (f ) := h∈Z |ρ (h)| and C 2 (f ) := h∈Z hρ 2 (h) .
Then, the wavelet basis is such that there exists a constant M * such that for all f ∈ F s p,q (M ),

C 1 (f ) ≤ M * and C 2 (f ) ≤ M * .

Linear estimation

The following theorem is the general result on the linear information projection estimator of the spectral density function. Note that the choice of the coarse level resolution level j 0 is of minor importance, and without loss of generality we take j 0 = 0 for the linear estimator f j 1 , θn .

Theorem 2.5 Assume that f ∈ F s 2,2 (M ) with s > 1 2 and suppose that Assumptions 1, 2 and 3 are satisfied. Define j 1 = j 1 (n) as the largest integer such that 2 j 1 ≤ n 1 2s+1 . Then, with probability tending to one as n → +∞, the information projection estimator (2.4) exists and satisfies:

∆ f ; f j 1 (n), θn = O p n -2s 2s+1 .
Moreover, the convergence is uniform over the class F s 2,2 (M ) in the sense that

lim K→+∞ lim n→+∞ sup f ∈F s 2,2 (M ) P n 2s 2s+1 ∆ f ; f j 1 (n), θn > K = 0.
This theorem provides the existence with probability tending to one of a linear estimator for the spectral density f given by f j 1 (n), θ j 1 (n) . This estimator is strictly positive by construction. Therefore the corresponding estimator of the covariance function ρ L (which is obtained as the inverse Fourier transform of f j 1 (n), θn ) is a positive definite function by Bochner's theorem. Hence ρ L is a covariance function.

In the related problem of density estimation from an i.d.d. sample, Koo [START_REF] Koo | Logspline Deconvolution in Besov Space[END_REF] has shown that, for the Kullback-Leibler divergence, n -2s 2s+1 is the fastest rate of convergence for the problem of estimating a density f such that log(f ) belongs to the space B s 2,2 (M ). For spectral densities belonging to a general Besov ball B s p,q (M ), Newman [START_REF] Neumann | Spectral density estimation via nonlinear wavelet methods for stationary non-Gaussian time series[END_REF] has also shown that n -2s 2s+1 is an optimal rate of convergence for the L 2 risk. For the Kullback-Leibler divergence, we conjecture that n -2s 2s+1 is the minimax rate of convergence for spectral densities belonging to F s 2,2 (M ). However, the result obtained in the above theorem is nonadaptive because the selection of j 1 (n) depends on the unknown smoothness s of f . Moreover, the result is only suited for smooth functions (as F s 2,2 (M ) corresponds to a Sobolev space of order s) and does not attain an optimal rate of convergence when for example g = log(f ) has singularities. We therefore propose in the next section an adaptive estimator derived by applying an appropriate nonlinear thresholding procedure.

Adaptive estimation

The bound on f is known

In adaptive estimation, we need to define an appropriate thresholding rule for the wavelet coefficients of the periodogram. This threshold is level-dependent and in this paper will take the form

ξ = ξ j,n = 2 2 f ∞ δ log n n + 2 j 2 ψ ∞ δ log n n + C * √ n , (2.6)
where δ ≥ 0 is a tuning parameter whose choice will be discussed later on and

C * = C 2 +39C 2 1 4π 2
. The following theorem states that the relative entropy between the true f and its nonlinear estimator achieves in probability the conjectured optimal rate of convergence up to a logarithmic factor over a wide range of Besov balls.

Theorem 2.6 Assume that f ∈ F s p,q (M ) with s > 1 2 + 1 p and 1 ≤ p ≤ 2. Suppose also that Assumptions 1, 2, 3 hold. For any n > 1, define j 0 = j 0 (n) to be the integer such that 2 j 0 ≥ log n ≥ 2 j 0 -1 , and j 1 = j 1 (n) to be the integer such that 2 j 1 ≥ n log n ≥ 2 j 1 -1 . For δ ≥ 6, take the threshold ξ j,n as in (2.6). Then, the thresholding estimator (2.5) exists with probability tending to one when n → +∞ and satisfies:

∆ f ; f HT j 0 (n),j 1 (n), θn,ξ j,n = O p n log n -2s 2s+1 .
Note that the choices of j 0 , j 1 and ξ j,n are independent of the parameter s; hence the estimator f HT j 0 (n),j 1 (n), θn,ξ j,n is an adaptive estimator which attains in probability what we claim is the optimal rate of convergence, up to a logarithmic factor. In particular, f HT j 0 (n),j 1 (n), θn,ξ j,n is adaptive on F s 2,2 (M ). This theorem provides the existence with probability tending to one of a nonlinear estimator for the spectral density. This estimator is strictly positive by construction. Therefore the corresponding estimator of the covariance function ρ N L (which is obtained as the inverse Fourier transform of

f HT j 0 (n),j 1 (n), θn,ξ j,n
) is a positive definite function by Bochner theorem. Hence ρ N L is a covariance function.

Estimating the bound on f

Although the results of Theorem 2.6 are certainly of some theoretical interest, they are not helpful for practical applications. The (deterministic) threshold ξ j,n depends on the unknown quantities f ∞ and C * := C (C 1 , C 2 ), where C 1 and C 2 are unknown constants. To make the method applicable, it is necessary to find some completely data-driven rule for the threshold, which works well over a range as wide as possible of smoothness classes. In this subsection, we give an extension that leads to consider a random threshold which no longer depends on the bound on f neither on C * . For this let us consider the dyadic partitions of [0, 1] given by I n = j/2 Jn , (j + 1)/2 Jn , j = 0, ..., 2 Jn -1 . Given some positive integer r, we define P n as the space of piecewise polynomials of degree r on the dyadic partition I n of step 2 -Jn . The dimension of P n depends on n and is denoted by N n . Note that N n = (r + 1) 2 Jn . This family is regular in the sense that the partition I n has equispaced knots.

An estimator of f ∞ is constructed as proposed by Birgé and Massart [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] in the following way. We take the infinite norm of f n , where f n denotes the (empirical) orthogonal projection of the periodogram I n on P n . We denote by f n the L 2 -orthogonal projection of f on the same space. Then the following theorem holds.

Theorem 2.7 Assume that f ∈ F s p,q (M ) with s > 1 2 + 1 p and 1 ≤ p ≤ 2. Suppose also that Assumptions 1, 2 and 3 hold. For any n > 1, let j 0 = j 0 (n) be the integer such that 2 j 0 ≥ log n ≥ 2 j 0 -1 , and let j 1 = j 1 (n) be the integer such that 2 j 1 ≥ n log n ≥ 2 j 1 -1 . Take the constants δ = 6 and b ∈ 3 4 , 1 , and define the threshold

ξ j,n = 2 2 f n ∞ δ (1 -b) 2 log n n + 2 j 2 ψ ∞ δ (1 -b) 2 log n n + log n n .
(2.7)

Then, if f -f n ∞ ≤ 1 4 f ∞ and N n ≤ κ (r+1) 2 n log n
, where κ is a numerical constant and r is the degree of the polynomials, the thresholding estimator (2.5) exists with probability tending to one as n → +∞ and satisfies

∆ f ; f HT j 0 (n),j 1 (n), θn, ξ j,n = O p n log n -2s 2s+1 .
Note that, we finally obtain a fully tractable estimator of f which reaches the optimal rate of convergence without prior knowledge of the regularity of the spectral density, but also which gets rise to a real covariance estimator.

Remark 2.8 We point out that, in Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF] the condition f -f n ∞ ≤ 1 4 f ∞ is assumed. Under some regularity conditions on f , results from approximation theory entails that this condition is met. Indeed for f ∈ B s p,∞ , with s > 1 p , we know from DeVore and Lorentz [START_REF] Devore | Constructive Approximation[END_REF] that

f -f n ∞ ≤ C (s) |f | s,p N -s-1 p n , with |f | s,p = sup y>0 y -s w d (f, y) p < +∞, where w d (f, y) p is the modulus of smoothness and d = [s] + 1. Therefore f -f n ∞ ≤ 1 4 f ∞ if N n ≥ 4C (s) |f | s,p f ∞ 1 s-1 p := C (f, s, p)
, where C (f, s, p) is a constant depending on f , s and p.

Numerical experiments

In this section we present some numerical experiments which support the claims made in the theoretical part of this paper. The programs for our simulations were implemented using the MATLAB programming environment. We simulate a time series which is a superposition of an ARMA(2,2) process and a Gaussian white noise:

X t = Y t + c o Z t , (3.1)
where

Y t + a 1 Y t-1 + a 2 Y t-2 = b 0 ε t + b 1 ε t-1 + b 2 ε t-2
, and {ε t }, {Z t } are independent Gaussian white noise processes with unit variance. The constants were chosen as a 1 = 0.2, a 2 = 0.9, b 0 = 1, b 1 = 0, b 2 = 1 and c 0 = 0.5. We generated a sample of size n = 1024 according to (3.1). The spectral density f of (X t ) is shown in Figure 1. It has two moderately sharp peaks and is smooth in the rest of the domain.

Starting from the periodogram we considered the Symmlet 8 basis, i.e. the least asymmetric, compactly supported wavelets which are described in Daubechies [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. We choose j 0 and j 1 as in the hypothesis of Theorem 2.7 and left the coefficients assigned to the father wavelets unthresholded. Hard thresholding is performed using the threshold ξ j,n as in (2.7) for the levels j = j 0 , ..., j 1 , and the empirical coefficients from the higher resolution scales j > j 1 are set to zero. This gives the estimate

f HT j 0 ,j 1 ,ξ j,n = 2 j 0 -1 k=0 a j 0 ,k φ j 0 ,k + j 1 j=j 0 2 j -1 k=0 b j,k I b j,k > ξ j,n ψ j,k , (3.2)
which is obtained by simply thresholding the wavelet coefficients (2.3) of the periodogram. Note that such an estimator is not guaranteed to be strictly positive in the interval [0, 1]. However, we use it to built our strictly positive estimator f HT j 0 ,j 1 , θn, ξ j,n (see (2.5) to recall its definition). We want to find

θ n such that f HT j 0 ,j 1 , θn, ξ j,n , ψ j,k = δ ξ j,n β j,k for all (j, k) ∈ Λ j 1
For this, we take

θ n = arg min θ∈R #Λ j 1 (j,k)∈Λ j 1 f j 0 ,j 1 ,θ , ψ j,k -δ ξ j,n β j,k 2
, where f j 0 ,j 1 ,θ (.) = exp (j,k)∈Λ j 1 θ j,k ψ j,k (.) ∈ E j 1 and E j 1 is the family (2.1). To solve this optimization problem we used a gradient descent method with an adaptive step, taking as initial value

θ 0 = log f HT j 0 ,j 1 , ξ j,n + , ψ j,k , where f HT j 0 ,j 1 , ξ j,n (ω)
+ := max f HT j 0 ,j 1 , ξ j,n
(ω) , η for all ω ∈ [0, 1] and η > 0 is a small constant.

In Figure 1 we display the unconstrained estimator f HT j 0 ,j 1 ,ξ j,n as in (3.2), obtained by thresholding of the wavelet coefficients of the periodogram, together with the estimator f HT j 0 ,j 1 , θn, ξ j,n , which is strictly positive by construction. Note that these wavelet estimators capture well the peaks and look fairly good on the smooth part too.

We compared our method with the spectral density estimator proposed by Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], which is based on a model selection procedure. As an example, in Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], the author study the behavior of such estimators using a collection of nested models (S m ), with m = 1, ..., 100, where S m is the space of piecewise constant functions, generated by a histogram basis on [0, 1] of dimension m with equispaced knots (see Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF] for further details). In Figure 2 we show the result of this comparison. Note that our method better captures the peaks of the true spectral density.

Appendix

Throughout all the proofs, C denotes a generic constant whose value may change from line to line.

Technical results for the empirical estimators of the wavelet coefficients

4.1 Let n ≥ 1, β j,k := f, ψ j,k and β j,k := I n , ψ j,k for j ≥ j 0 -1 and 0 ≤ k ≤ 2 j -1.
Suppose that Assumptions 1, 2 and 3 hold. Then, .

Bias 2 β j,k := E β j,k -β j,k 2 ≤ C 2
C * = C 2 +39C 2 1 4π 2
, and V ar β j,k :

= E β j,k -E β j,k 2 ≤ C
n for some constant C > 0. Moreover, there exists a constant M 2 > 0 such that for all f ∈ F s p,q (M ) with s > 1 p and 1 ≤ p ≤ 2,

E β j,k -β j,k 2 = Bias 2 β j,k + V ar β j,k ≤ M 2 n . Proof. Note that Bias 2 β j,k ≤ f -E (I n ) 2 L 2 .
Using Proposition 1 in Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF], Assumptions 1 and 2 imply that f -

E (I n ) 2 L 2 ≤ C 2 +39C 2 1 4π 2 n
, which gives the result for the bias term. To bound the variance term, remark that

V ar β j,k = E I n -E (I n ) , ψ j,k 2 ≤ E I n -E (I n ) 2 L 2 ψ j,k 2
L 2 = 1 0 E|I n (ω) -E (I n (ω)) | 2 dω.
Then, under Assumptions 1 and 2, it follows that there exists an absolute constant C > 0 such that for all ω ∈

[0, 1], E|I n (ω) -E (I n (ω)) | 2 ≤ C n .
To complete the proof it remains to remark that Assumption 3 implies that these bounds for the bias and the variance hold uniformly over F s p,q (M ).

P | b j,k -b j,k | > 2 f ∞ x n + 2 j/2 ψ ∞ x n + C * √ n ≤ 2e -x ,
where

C * = C 2 +39C 2 1 4π 2 . Proof. Note that b j,k = 1 2πn n t=1 n t ′ =1 X t -X X t ′ -X * 1 0 e i2πω(t-t ′ ) ψ j,k (ω) dω = 1 2πn X T T n (ψ j,k )X * ,
where X = X 1 -X, ..., X n -X T , X T denotes the transpose of X and T n (ψ j,k ) is the Toeplitz

matrix with entries [T n (ψ j,k )] t,t ′ = 1 0 e i2πω(t-t ′ ) ψ j,k (ω) dω, 1 ≤ t, t ′ ≤ n.
We can assume without loss of generality that E (X t ) = 0, and then under under Assumptions 1 and 2, X is a centered Gaussian vector in R n with covariance matrix Σ = T n (f ). Using the decomposition X = Σ

1 2 ε, where ε ∼ N (0, I n ), it follows that b j,k = 1 2πn ε T A j,k ε, with A j,k = Σ 1 2 T n (ψ j,k )Σ 1 2
. Note also that E b j,k = 1 2πn tr (A j,k ), where tr (A) denotes the trace of a matrix A. Now let s 1 , . . . , s n be the eigenvalues of the Hermitian matrix A j,k with |s

1 | ≥ |s 2 | ≥ . . . ≥ |s n | and let Z = 2πn b j,k -E b j,k = ε T A j,k ε -tr (A j,k ). Then, for 0 < λ < (2|s 1 |) -1 one has that log E e λZ = n i=1 -λs i - 1 2 log (1 -2λs i ) = n i=1 +∞ ℓ=2 1 2ℓ (2s i λ) ℓ ≤ n i=1 +∞ ℓ=2 1 2ℓ (2|s i |λ) ℓ ≤ n i=1 -λ|s i | - 1 2 log (1 -2λ|s i |) ,
where we have used the fact that -log(1 -x) = +∞ ℓ=1 x ℓ ℓ for x < 1. Then using the inequality -u - 1 2 log (1 -2u) ≤ u 2 1-2u that holds for all 0 < u < 1 2 , the above inequality implies that

log E e λZ ≤ n i=1 λ 2 |s i | 2 1 -2λ|s i | ≤ λ 2 s 2 1 -2λ|s 1 |
,

where s 2 = n i=1 |s i | 2 .
Arguing as in Birgé and Massart [START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF], the above inequality implies that for any x > 0, P(|Z| > 2|s 1 |x + 2 s √ x) ≤ 2e -x , which implies

P | b j,k -E b j,k | > 2|s 1 | x n + 2 s n √ x ≤ 2e -x . (4.1)
Let τ (A) denotes the spectral radius of a matrix A. For the Toeplitz matrices Σ = T n (f ) and T n (ψ j,k ) one has that τ (Σ) ≤ f ∞ and τ (T n (ψ j,k )) ≤ ψ j,k ∞ = 2 j/2 ψ ∞ . These inequalities imply that

|s 1 | = τ Σ 1 2 T n (ψ j,k )Σ 1 2 ≤ τ (Σ) τ (T n (ψ j,k )) ≤ f ∞ 2 j/2 ψ ∞ . (4.2)
Let λ i , i = 1., , , .n, be the eigenvalues of T n (ψ j,k ). From Lemma 3.1 in Davies [START_REF] Davies | Asymptotic inference in stationary Gaussian time-series[END_REF], we have that lim sup

n→+∞ 1 n tr T n (ψ j,k ) 2 = lim sup n→+∞ 1 n n i=1 λ 2 i = 1 0 ψ 2 j,k (ω) dω = 1,
which implies that

s 2 = n i=1 |s i | 2 = tr A 2 j,k = tr (ΣT n (ψ j,k )) 2 ≤ τ (Σ) 2 tr T n (ψ j,k ) 2 ≤ f 2 ∞ n, (4.3)
where we have used the inequality tr (AB) 2 ≤ τ (A) 2 tr B 2 that holds for any pair of Hermitian matrices A, B. Combining (4.1), (4.2) and (4.3), we finally obtain that for any x > 0

P | b j,k -E b j,k | > 2 f ∞ x n + 2 j/2 ψ ∞ x n ≤ 2e -x . (4.4) Now, let ξ j,n = 2 f ∞ x n + 2 j/2 ψ ∞ x n + C *
√ n , and note that

P b j,k -b j,k > ξ j,n ≤ P b j,k -E b j,k > ξ j,n -E b j,k -b j,k , By Lemma 4.1, one has that E b j,k -b j,k ≤ C * √ n , and thus ξ j,n -E b j,k -b j,k ≥ ξ j,n -C * √ n
which implies using (4.4) that

P b j,k -b j,k > ξ j,n ≤ P b j,k -E b j,k > ξ j,n - C * √ n ≤ 2e -x ,
which completes the proof of Lemma 4.2.

Lemma 4.3 Assume that f ∈ F s p,q (M ) with s > 1 2 + 1 p and 1 ≤ p ≤ 2. Suppose that Assumptions 1, 2 and 3 hold. For any n > 1, define j 0 = j 0 (n) to be the integer such that 2 j 0 > log n ≥ 2 j 0 -1 , and j 1 = j 1 (n) to be the integer such that 2 j 1 ≥ n log n ≥ 2 j 1 -1 . For δ ≥ 6, take the threshold

ξ j,n = 2 2 f ∞ δ log n n + 2 j 2 ψ ∞ δ log n n + C * √ n as in (2.6), where C * = C 2 +39C 2 1 4π 2
. Let β j,k := f, ψ j,k and β ξ j,n ,(j,k) := δ ξ j,n β j,k with (j, k) ∈ Λ j 1 as in (2.5). Take β = (β j,k ) (j,k)∈Λ j 1 and

β ξ j,n = β ξ j,n ,(j,k) (j,k)∈Λ j 1
. Then there exists a constant M 3 > 0 such that for all sufficiently large n:

E β -β ξ j,n 2 2 := E   (j,k)∈Λ j 1 β j,k -δ ξ j,n β j,k 2   ≤ M 3 n log n -2s 2s+1 uniformly over F s p,q (M ).
Proof. Taking into account that

E β -β ξ j,n 2 2 = 2 j 0 -1 k=0 E (a j 0 ,k -a j 0 ,k ) 2 + j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I b j,k > ξ j,n + j 1 j=j 0 2 j -1 k=0 b 2 j,k P b j,k ≤ ξ j,n := T 1 + T 2 + T 3 , (4.5)
we are interested in bounding these three terms. The bound for T 1 follows from Lemma 4.1 and the fact that j 0 = log 2 (log n) ≤ 1 2s+1 log 2 (n):

T 1 = 2 j 0 -1 k=0 E (a j 0 ,k -a j 0 ,k ) 2 = O 2 j 0 n ≤ O n -2s 2s+1 . (4.6)
To bound T 2 and T 3 we proceed as follows. Write

T 2 = j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I b j,k > ξ j,n , |b j,k | > ξ j,n 2 + I b j,k > ξ j,n , |b j,k | ≤ ξ j,n 2
and

T 3 = j 1 j=j 0 2 j -1 k=0 b 2 j,k P b j,k ≤ ξ j,n , |b j,k | ≤ 2ξ j,n + P b j,k ≤ ξ j,n , |b j,k | > 2ξ j,n .
From Hardle, Kerkyacharian, Picard and Tsybakov [START_REF] Hardle | Wavelets, Approximation, and Statistical Applications[END_REF] we get that

T 2 + T 3 ≤ j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I |b j,k | > ξ j,n 2 + j 1 j=j 0 2 j -1 k=0 b 2 j,k I (|b j,k | ≤ 2ξ j,n ) + 5 j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I b j,k -b j,k > ξ j,n 2
:= T ′ + T ′′ + T ′′′ .
Now we bound T ′′′ . Using Cauchy-Schwarz inequality, we obtain

T ′′′ ≤ 5 j 1 j=j 0 2 j -1 k=0 E 1 2 b j,k -b j,k 4 P 1 2 b j,k -b j,k > ξ j,n 2
.

By the same inequality we get

E b j,k -b j,k 4 ≤ E I n -f 4 L 2 ψ j,k 4
L 2 = O E I n -f 4 L 2 . It can be checked that E I n -f 4 L 2 ≤ 8E I n -E (I n ) 4 L 2 + E (I n ) -f 4 L 2 . According to Comte [7], E I n -E (I n ) 4 L 2 = O n 2 . From the proof of Lemma 4.1 we get that E (I n ) -f 4 L 2 = O 1 n 2 . Therefore E I n -f 4 L 2 ≤ O n 2 + 1 n 2 = O n 2 . Hence E b j,k -b j,k 4 = O E I n -f 4 L 2 = O n 2 . For the bound of P b j,k -b j,k > ξ j,n 2
we use the result of Lemma 4.2 with x = δ log n, where δ > 0 is a constant to be specified later. We obtain

P b j,k -b j,k > ξ j,n 2 = P b j,k -b j,k > 2 f ∞ δ log n n + 2 j 2 ψ ∞ δ log n n + C * √ n ≤ 2e -δ log n = 2n -δ .
Therefore, for δ ≥ 6, we get

T ′′′ ≤ 5 j 1 j=j 0 2 j -1 k=0 E 1 2 b j,k -b j,k 4 P 1 2 b j,k -b j,k > ξ j,n 2 ≤ O n -1 log n ≤ O n -2s 2s+1 .
Now we follow results found in Pensky and Sapatinas [START_REF] Pensky | Functional deconvolution in a periodic setting: uniform case[END_REF] to bound T ′ and T ′′ . Let j A be the integer such that 2 j A > n log n 1 2s+1 > 2 j A -1 (note that given our assumptions j 0 ≤ j A ≤ j 1 for all sufficiently large n), then T ′ can be partitioned as

T ′ = T ′ 1 + T ′ 2 ,
where the first component is calculated over the set of indices j 0 ≤ j ≤ j A and the second component over j A + 1 ≤ j ≤ j 1 . Hence, using Lemma 4.1 we obtain

T ′ 1 ≤ C j A j=j 0 2 j n = O 2 j A n -1 = O n log n 1 2s+1 n -1 ≤ O n -2s 2s+1 . (4.7)
To obtain a bound for T ′ 2 , we will use that if f ∈ F s p,q (A), then for some constant C, dependent on s, p, q and A > 0 only, we have that

2 j -1 k=0 b 2 j,k ≤ C2 -2js * , (4.8)
for 1 ≤ p ≤ 2, where

s * = s + 1 2 -1 p . Taking into account that I |b j,k | > ξ j,n 2 ≤ 4 ξ 2 j,n |b j,k | 2 , we get T ′ 2 ≤ C n j 1 j=j A 2 j -1 k=0 4 ξ 2 j,n |b j,k | 2 ≤ C ( f ∞ ) 2 -2s * j A √ δ log n + ψ ∞ δn -s 2s+1 (log n) 4s+1 4s+2 2 j 1 j=j A 2 j -1 k=0 2 2js * |b j,k | 2 ≤ O 2 -2s * j A = O   n log n -2s * 2s+1   ,
where we used the fact that

√ δ log n + ψ ∞ δn -s 2s+1 (log n) 4s+1 4s+2
→ +∞ when n → +∞. Now remark that if p = 2 then s * = s and thus

T ′ 2 = O   n log n -2s * 2s+1   = O n log n -2s 2s+1 .
(4.9)

For the case 1 ≤ p < 2, the repeated use of the fact that if B, D > 0 then

I (|b j,k | > B + D) ≤ I (|b j,k | > B), enables us to obtain that T ′ 2 ≤ C n j 1 j=j A 2 j -1 k=0 I |b j,k | > ξ j,n 2 ≤ C n j 1 j=j A 2 j -1 k=0 |b j,k | -p |b j,k | p I |b j,k | -p < 2 f ∞ √ δ log n n -p ≤ C j 1 j=j A 2 j -1 k=0 1 n 2 f ∞ √ δ log n n -p |b j,k | p .
Since f ∈ F s p,q (A) it follows that there exists a constant C depending only on p, q, s and A such that

2 j -1 k=0 |b j,k | p ≤ C2 -pjs * , (4.10)
where s * = s + 1 2 -1 p as before. By (4.10) we get . Now, set j A as before, then T ′′ can be split into

T ′ 2 ≤ (log n) C ( f ∞ , δ, p) j 1 j=j A 2 j -1 k=0 (log n) -p 2 n 1-p 2 |b j,k | p ≤ C ( f ∞ , δ, p) (log n) 1-p 2 n 1-p 2 j 1 j=j A C2 -pjs * = O (log n) 1-p 2 n 1-p 2 2 -pj A s * = O n log n -2s 2s+1 . ( 4
T ′′ = T ′′ 1 + T ′′ 2 ,
where the first component is calculated over the set of indices j 0 ≤ j ≤ j A and the second component over j A + 1 ≤ j ≤ j 1 . Then

T ′′ 1 ≤ j A j=j 0 2 j -1 k=0 b 2 j,k I   |b j,k | 2 ≤ 32   4 f 2 ∞ δ log n n + 2 j 2 ψ ∞ δ log n n 2 + C 2 * n     .
Using repeatedly that (B + D) 2 ≤ 2 B 2 + D 2 for B, D ∈ R, we obtain the desired bound for T ′′ 1 :

T ′′ 1 ≤ C ( f ∞ ) j A j=j 0 2 j -1 k=0 δ log n n + 2 j ψ 2 ∞ δ 2 (log n) 2 n 2 + C (C * ) j A j=j 0 2 j -1 k=0 1 n ≤ C ( f ∞ , δ, C * ) log n n 2 j A + C ( f ∞ , δ, ψ ∞ ) (log n) 2 n 2 2 2j A = O (log n) 2s 2s+1 n -2s 2s+1 + (log n) 4s 2s+1 n -4s 2s+1 ≤ O n log n -2s 2s+1 .
(4.12)

To bound T ′′ 2 , note that T ′′ 2 ≤ j 1 j=j A +1 2 j -1 k=0 b 2 j,k = O 2 -2j A s * = O n log n -2s * 2s+1
, where we have used the condition (4.8). Now remark that if p = 2 then s * = s and thus

T ′′ 2 = O   n log n -2s * 2s+1   = O n log n -2s 2s+1 . (4.13) If 1 ≤ p < 2, T ′′ 2 = j 1 j=j A +1 2 j -1 k=0 |b j,k | 2-p |b j,k | p I (|b j,k | ≤ 2ξ j,n ) ≤ j 1 j=j A +1 2 j -1 k=0 8 f ∞ δ log n n + 8 f ∞ 2 j 1 2 ψ ∞ δ log n n + 4 log n n 2-p |b j,k | p ≤ (C ( f ∞ , ψ ∞ , δ)) 2-p log n n 2-p j 1 j=j A +1 2 j -1 k=0 |b j,k | p = O log n n 2-p 2 2 -pj A s * = O    n log n p 2 -1- p ( s+ 1 2 -1 p ) 2s+1    = O n log n -2s 2s+1 , (4.14)
where we have used condition (4.10) and the fact that C * ≤ √ log n for n sufficiently large, taking into account that the constant C * := C (C 1 , C 2 ) does not depend on n. Hence, by (4.12), (4.13) and

(4.14), T ′′ = O n log n -2s 2s+1
. Combining all terms in (4.5), we conclude that:

E β -β ξ j,n 2 2 = O n log n -2s 2s+1 .
This completes the proof. Lemma 4.4 Assume that f ∈ F s p,q (M ) with s > 1 2 + 1 p and 1 ≤ p ≤ 2. Suppose that Assumptions 1, 2 and 3 hold. For any n > 1, define j 0 = j 0 (n) to be the integer such that 2 j 0 > log n ≥ 2 j 0 -1 , and j 1 = j 1 (n) to be the integer such that 2 j 1 ≥ n log n ≥ 2 j 1 -1 . Define the threshold ξ j,n as in (2.7) for some constants δ = 6 and b ∈ 3 4 , 1 . Let β j,k := f, ψ j,k and β ξ j,n ,(j,k) := δ ξ j,n β j,k with (j, k) ∈ Λ j 1 as in (2.5). Take β = (β j,k ) (j,k)∈Λ j 1

and

β ξ j,n = β ξ j,n ,(j,k) (j,k)∈Λ j 1 . Then, if f -f n ∞ ≤ 1 4 f ∞ and N n ≤ κ (r+1) 2 n log n
, where κ is a numerical constant and r is the degree of the polynomials, there exists a constant M 4 > 0 such that for all sufficiently large n:

E β -β ξ j,n 2 2 := E   (j,k)∈Λ j 1 δ ξ j,n β j,k -β j,k 2   ≤ M 4 n log n -2s 2s+1 uniformly over F s p,q (M ).
Proof. Recall that f n is the L 2 orthogonal projection of f on the space P n of piecewise polynomials of degree r on a dyadic partition with step 2 -Jn . The dimension of P n is N n = (r + 1) 2 Jn . Let f n similarly be the orthogonal projection of I n on P n . By doing analogous work as the one done to obtain (4.5), we get that

E β -β ξ j,n 2
2

:= T 1 + T 2 + T 3 , (4.15)
where T 1 =

2 j 0 -1 k=0
E (a j 0 ,k -a j 0 ,k ) 2 do not depend on ξ j,n . Therefore, by (4.6), T 1 = O n -2s 2s+1 . For T 2 and T 3 we have that

T 2 = j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I b j,k > ξ j,n , |b j,k | > ξ j,n 2 + I b j,k > ξ j,n , |b j,k | ≤ ξ j,n 2
and

T 3 = j 1 j=j 0 2 j -1 k=0 b 2 j,k P b j,k ≤ ξ j,n , |b j,k | ≤ 2 ξ j,n + P b j,k ≤ ξ j,n , |b j,k | > 2 ξ j,n .
Using the same decomposition as in the proof of Lemma (4.3) we get that

T 2 + T 3 ≤ j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I |b j,k | > ξ j,n 2 + j 1 j=j 0 2 j -1 k=0 E b 2 j,k I |b j,k | ≤ 2 ξ j,n + 5 j 1 j=j 0 2 j -1 k=0 E b j,k -b j,k 2 I b j,k -b j,k > ξ j,n 2 := T ′ + T ′′ + T ′′′ .
Now we bound T ′′′ . Using Cauchy-Schwarz inequality, one obtains

T ′′′ ≤ 5 j 1 j=j 0 2 j -1 k=0 E 1 2 b j,k -b j,k 4 P 1 2 b j,k -b j,k > ξ j,n 2 ,
From Lemma 4.2 we have that for any y > 0 the following exponential inequality holds:

P b j,k -b j,k > 2 f ∞ y n + 2 j 2 ψ ∞ y n + C * √ n ≤ 2e -y . (4.16) As in Comte [7], let Θ n,b = fn ∞ f ∞ -1 < b , with b ∈ (0, 1). Then, using that P b j,k -b j,k > B + D ≤ P b j,k -b j,k > B for B, D > 0, and taking x n = δ log n (1-b) 2 , one gets P b j,k -b j,k > ξ j,n 2 ≤ P b j,k -b j,k > 2 f n ∞ x n n + 2 j 2 ψ ∞ x n n ≤ P b j,k -b j,k > 2 f n ∞ x n n + 2 j 2 ψ ∞ x n n | Θ n,b P (Θ n,b ) + P b j,k -b j,k > 2 f n ∞ x n n + 2 j 2 ψ ∞ x n n | Θ c n,b P Θ c n,b := P 1 P (Θ n,b ) + P 2 P Θ c n,b .
In Comte [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]

is proved that if f -f n ∞ ≤ 1 4 f ∞ then P Θ c n,b ≤ O n -4 for the choices of 1 ≥ b ≈ 4 6 5 π = 0.841 ≥ 3 4 and N n ≤ 1 36(r+1) 2 n log n , where κ = 1
36 is the numerical constant in the hypothesis of our theorem. Following its proof it can be shown that this bound can be improved taking κ = 1 36( 75 ) and b as before (see the three last equations of page 290 in [START_REF] Comte | Adaptive estimation of the spectrum of a stationary Gaussian sequence[END_REF]). With this selection of κ we obtain that P Θ c n,b ≤ O n -6 . Using that P (Θ n,b ) = O (1) and P 2 = O (1), it only remains to bound the conditional probability P 1 . On Θ n,b the following inequalities hold:

(a) f n ∞ > (1 -b) f ∞ and (b) f n ∞ < (1 + b) f ∞ . (4.17)
Then, using (4.17a) we get

P 1 ≤ P b j,k -b j,k > 2 f ∞ δ log n n + 2 j 2 ψ ∞ δ log n n ≤ 2e -δ log n = 2n -δ ,
where the last inequality is obtained using (4.16) for y = δ log n > 0. Hence, using that δ = 6, we

get P b j,k -b j,k > ξ j,n 2 ≤ O n -6 . Therefore T ′′′ ≤ C j 1 j=j 0 2 j -1 k=0 n -2 ≤ O n -2 2 j 1 ≤ O n -2s 2s+1 . Now we bound T ′ . Let j A be the integer such that 2 j A > n log n 1 2s+1 > 2 j A -1 , then T ′ = T ′ 1 + T ′ 2 ,
where the first component is computed over the set of indices j 0 ≤ j ≤ j A and the second component over j A + 1 ≤ j ≤ j 1 . Hence, using Lemma 4.1 we obtain

T ′ 1 ≤ j A j=j 0 2 j -1 k=0 E b j,k -b j,k 2 ≤ O 2 j A n = O n log n 1 2s+1 n -1 ≤ O n -2s 2s+1 . To bound T ′ 2 , note that T ′ 2 := T ′ 2,1 + T ′ 2,2
, where

T ′ 2,1 = j 1 j=j A 2 j -1 k=0 E b j,k -b j,k 2 I |b j,k | > ξ j,n 2 , Θ n,b and
T ′ 2,2 = j 1 j=j A 2 j -1 k=0 E b j,k -b j,k 2 I |b j,k | > ξ j,n 2 , Θ c n,b
.

Using that on Θ n,b inequality (4.17a) holds and following the same procedures as in the proof of Theorem 2.6, we get the desired bound for T ′ 2,1 .

T ′ 2,1 ≤ C n j 1 j=j A 2 j -1 k=0 I |b j,k | > 2 (1 -b) f ∞ x n n + 2 j 2 ψ ∞ x n n (4.18) ≤ C 4 1 n j 1 j=j A 2 j -1 k=0 |b j,k | 2 f 2 ∞ δ log n n + 2 j 2 ψ ∞ δ log n (1-b)n 2 ≤ O 2 -2s * j A = O   n log n -2s * 2s+1   ,
where we have used that

√ δ log n + ψ ∞ (1 -b) -1 δn -s 2s+1 (log n) 4s+1 4s+2
→ +∞ when n → +∞ and that condition (4.8) is satisfied. Now remark that if p = 2 then s * = s and thus

T ′ 2,1 = O n log n -2s 2s+1 . (4.19)
For the case 1 ≤ p < 2, from (4.18) we have that

T ′ 2,1 ≤ C n j 1 j=j A 2 j -1 k=0 I |b j,k | > 2 f ∞ δ log n n + 2 j 2 ψ ∞ δ log n (1 -b) n ≤ C n j 1 j=j A 2 j -1 k=0 |b j,k | -p |b j,k | p I |b j,k | -p < 2 f ∞ δ log n n -p ≤ (log n) C ( f ∞ , δ, p) (log n) -p 2 n 1-p 2 j 1 j=j A 2 j -1 k=0 |b j,k | p = O (log n) 1-p 2 n 1-p 2 2 -pj A s * ≤ O n log n -2s 2s+1 , (4.20)
where we have used condition (4.10). Hence

T ′ 2,1 = O n log n -2s 2s+1 .
Now we bound T ′ 2,2 . Using Cauchy-Schwarz inequality, we have . Now we bound T ′′ . Set j A as before, then

T ′ 2,2 ≤ C j 1 j=j A 2 j -1 k=0 nP 1 2 |b j,k | > 2 f n ∞ x n n + 2 j 2 ψ ∞ x n n + log n n | Θ c n,b P 1 2 Θ c n,b ≤ C j 1 j=j A 2 j -1 k=0 nP 1 2 Θ c n,b ≤ C j 1 j=j A 2 j -1 k=0 n -2 ≤ O 2 j 1 n 2 ≤ O n -2s 2s+1 , ( 4
T ′′ = T ′′ 1 + T ′′ 2 ,
where the first component is calculated over the set of indices j 0 ≤ j ≤ j A and the second component over j A + 1 ≤ j ≤ j 1 . Recall that

x n = δ log n (1-b) 2 , then T ′′ 1 ≤ T ′′ 1,1 + T ′′ 1,2
, where

T ′′ 1,1 = j A j=j 0 2 j -1 k=0 b 2 j,k P |b j,k | ≤ 4 2 (1 + b) f ∞ x n n + 2 j 2 ψ ∞ x n n + log n n and T ′′ 1,2 = j A j=j 0 2 j -1 k=0 b 2 j,k P Θ c n,b ,
where we have used that given Θ n,b inequality (4.17b) holds. For T ′′ 1,1 we have

T ′′ 1,1 = j A j=j 0 2 j -1 k=0 b 2 j,k P   |b j,k | 2 ≤ 16 2 (1 + b) f ∞ x n n + 2 j 2 ψ ∞ x n n + log n n 2   ≤ 16 j A j=j 0 2 j -1 k=0 2 (1 + b) f ∞ δ log n (1 -b) 2 n + 2 j 2 ψ ∞ δ log n (1 -b) 2 n + log n n 2 ≤ C j A j=j 0 2 j -1 k=0 C ( f ∞ , b) δ log n (1 -b) 2 n + ψ 2 ∞ δ 2 log n (1 -b) 4 n + log n n = O n log n -2s 2s+1 , (4.22)
where we have used repeatedly that (B + D) 2 ≤ 2 B 2 + D 2 for all B, D ∈ R.

To bound T ′′ 1,2 we use again that P Θ c n,b ≤ O n -6 and that condition (4.8) is satisfied. Then . Now we bound T ′′ 2 .

T ′′ 1,2 ≤ j A j=j 0 2 j -1 k=0 b 2 j,k n -6 ≤ n -6 j A j=j 0 C2 -2js * = O n -6 2 -2j 0 s * ≤ O n -1 . ( 4
T ′′ 2 ≤ j 1 j=j A +1 2 j -1 k=0 b 2 j,k P |b j,k | ≤ 2 ξ j,n ≤ j 1 j=j A +1 2 j -1 k=0 b 2 j,k = O 2 -2j A s * = O   n log n -2s * 2s+1   ,
where we have used again the condition (4.8). Now remark that if p = 2 then s * = s and thus

T ′′ 2 = O n log n -2s * 2s+1 = O n log n -2s 2s+1
. For 1 ≤ p < 2, we proceed as follows.

T ′′ 2 = j 1 j=j A +1 2 j -1 k=0 E b 2 j,k I |b j,k | ≤ 2 ξ j,n , Θ n,b + b 2 j,k I |b j,k | ≤ 2 ξ j,n , Θ c n,b ≤ j 1 j=j A +1 2 j -1 k=0 E b 2 j,k I |b j,k | ≤ 4 2 (1 + b) f ∞ x n n + 2 j 2 ψ ∞ x n n + log n n + j 1 j=j A +1 2 j -1 k=0 b 2 j,k P |b j,k | ≤ 8 f n ∞ x n n + 2 j 2 ψ ∞ x n n + 4 log n n | Θ c n,b P Θ c n,b := T ′′ 2,1 + T ′′ 2,2
, where we have used that on Θ n,b inequality (4.17b) holds. Now we bound T ′′ 2,1 .

T ′′ 2,1 ≤ j 1 j=j A +1 2 j -1 k=0 |b j,k | 2-p |b j,k | p I |b j,k | ≤ 8 (1 + b) f ∞ x n n + 2 j 2 ψ ∞ x n n + 4 log n n ≤ j 1 j=j A +1 2 j -1 k=0 8 (1 + b) f ∞ δ log n (1 -b) 2 n + 2 j 1 2 ψ ∞ δ log n (1 -b) 2 n + 4 log n n 2-p |b j,k | p ≤ C ( f ∞ , b, δ, ψ ∞ ) 2-p log n n 2-p j 1 j=j A +1 2 j -1 k=0 |b j,k | p ≤ O log n n 2-p 2 2 -pj A s * = O    log n n 2-p 2 n log n - p ( s+ 1 2 -1 p ) 2s+1    = O n log n -2s 2s+1 . (4.24)
where we have used that condition (4.10) is satisfied. To bound T ′′ 2,2 we use again that P Θ c n,b ≤ O n -6 and that condition (4.8) also holds. Then, from (4.23) we get . Combining all terms in (4.15), we conclude that:

T ′′ 2,2 ≤ j 1 j=j A +1 2 j -1 k=0 b 2 j,k P Θ c n,b ≤ n -6 j 1 j=j A +1 C2 -2js * = O n -6 2 -2j A s * ≤ O n -1 . ( 4
E β -β ξ j,n 2 2 = O n log n -2s 2s+1 .
This completes the proof.

Proof of Theorem 2.5

First, one needs the following proposition. Proposition 4.5 Let β j,k = f, ψ j,k and β j,k = I n , ψ j,k with (j, k) ∈ Λ j 1 . Suppose that f ∈ F s p,q (M ) with s > 1/p and 1 ≤ p ≤ 2. Let M 1 > 0 be a constant such that M -1

1 ≤ f ≤ M 1 (see Lemma 2.1). Let ǫ j 1 = 2M 2
1 e 2γ j 1 +1 D j 1 A j 1 . If ǫ j 1 ≤ 1, then there exists θ * j 1 ∈ R #Λ j 1 such that:

f j 1 ,θ * j 1 , ψ j,k = f, ψ j,k = β j,k for all (j, k) ∈ Λ j 1
Moreover, the following inequality holds (approximation error)

∆ f ; f j 1 ,θ * j 1 ≤ M 1 2 e γ j 1 D 2 j 1 .
Suppose that Assumptions 1 and 2 hold. Let η j 1 ,n = 4M 2 1 e 2γ j 1 +2ǫ j 1 +2 A 2

j 1 #Λ j 1
n . Then, for every λ > 0 such that λ ≤ η -1 j 1 ,n there exists a set Ω n,1 of probability less than M 2 λ -1 , where M 2 is the constant defined in Lemma 4.1, such that outside the set Ω n,1 there exists some θ n ∈ R #Λ j 1 which satisfies:

f j 1 , θn , ψ j,k = I n , ψ j,k = β j,k for all (j, k) ∈ Λ j 1 .
Moreover, outside the set Ω n,1 , the following inequality holds (estimation error)

∆ f j 1 ,θ * j 1 ; f j 1 , θn ≤ 2M 1 e γ j 1 +ǫ j 1 +1 M 2 λ #Λ j 1 n .
Proof. Approximation error: Recall that β j,k = f, ψ j,k and let β = (β j,k ) (j,k)∈Λ j 1 . Define by g j 1 =

(j,k)∈Λ j 1 θ j,k ψ j,k an approximation of g = log (f ) and let β 0,(j,k) = f j 1 ,θ j 1 , ψ j,k = exp (g j 1 ) , ψ j,k with θ j 1 = (θ j,k ) (j,k)∈Λ j 1 and β 0 = β 0,(j,k) (j,k)∈Λ j 1 . Observe that the coefficients β j,k -β 0,(j,k) , (j, k) ∈ Λ j 1 , are the coefficients of the orthonormal projection of f -f j 1 ,θ j 1 onto V j .

Hence by Bessel's inequality,

β -β 0 2 2 ≤ f -f j 1 ,θ j 1 2 L 2
. Using Lemma 2.1 and Lemma 2 in Barron and Sheu [START_REF] Barron | Approximation of density functions by sequences of exponential families[END_REF], we get that:

β -β 0 2 2 ≤ f -f j 1 ,θ j 1 2 dµ ≤ M 1 f -f j 1 ,θ j 1 2 f dµ ≤ M 1 e 2 log f f j 1 ,θ j 1 ∞ f log f f j 1 ,θ j 1 2 dµ ≤ M 2 1 e 2 g-g j 1 ∞ g -g j 1 2 L 2 = M 2 1 e 2γ j 1 D 2 j 1 .
Then, one can easily check that b = e log f j 1 ,θ j 1 ∞ ≤ M 1 e γ j 1 . Thus the assumption that ǫ j 1 ≤ 1 implies that the inequality β -β 0 2 ≤ M 1 e γ j 1 D j 1 ≤ 1 2beA j 1 is satisfied. Hence, Lemma 2.4 can be applied with θ 0 = θ j 1 , β = β and b = exp log f j 1 ,θ j 1 ∞ , which implies that there exists

θ * j 1 = θ (β) such that f j 1 ,θ * j 1 , ψ j,k = β j,k for all (j, k) ∈ Λ j 1 .
By the Pythagorian-like relationship (2.2), we obtain that ∆ f ; f j 1 ,θ * j 1 ≤ ∆ f ; f j 1 ,θ j 1 . Now we use a result wich states that if f and g are two functions in L 2 ([0, 1]) such that log f g is bounded.

Then

∆ (f ; g) ≤ 1 2 e log f g ∞ 1 0 f log f g 2
dµ, where µ denotes the Lebesgue measure on [0, 1].(see Lemma A.1 in Antoniadis and Bigot [START_REF] Antoniadis | Poisson inverse problems[END_REF]). Hence, it follows that

∆ f ; f j 1 ,θ * j 1 ≤ 1 2 e log f f j 1 ,θ j 1 ∞ f log f f j 1 ,θ j 1 2 dµ = M 1 2 e g-g j 1 ∞ g -g j 1 2 L 2 = M 1 2 e γ j 1 D 2 j 1 .
which completes the proof for the approximation error.

Estimation error: Applying again Lemma 2.4 with

θ 0 = θ * j 1 , β 0,(j,k) = f j 1 ,θ 0 , ψ j,k = β j,k , β = β, where β = β j,k (j,k)∈Λ j 1
, and b = exp log

f j 1 ,θ * j 1 ∞ we obtain that if β -β 2 ≤ 1 2ebA j 1 with β = (β j,k ) (j,k)∈Λ j 1 then there exists θ n = θ β such that f j 1 , θn , ψ j,k = β j,k for all (j, k) ∈ Λ j 1 .
Hence, it remains to prove that our assumptions imply that the event β -

β 2 ≤ 1 2ebA j 1 holds with probability 1 -M 2 λ -1 .
First remark that b ≤ M 1 e γ j 1 +ǫ j 1 and that by Markov's inequality and Lemma 4.1 we obtain that for any λ > 0, P β -

β 2 2 ≥ λ #Λ j 1 n ≤ 1 λ n #Λ j 1 E β -β 2 2 ≤ M 2 λ -1 .
Hence, outside a set Ω n,1 of probability less than

M 2 λ -1 then β -β 2 2 ≤ λ #Λ j 1 n . Therefore, the condition β -β 2 ≤ 1 2ebA j 1 holds if λ #Λ j 1 n 1 2 ≤ 1 2ebA j 1 , which is equivalent to 4e 2 b 2 A 2 j 1 λ #Λ j 1 n ≤ 1. This last inequality is true if η j 1 ,n = 4M 2 1 e 2γ j 1 +2ǫ j 1 +2 A 2 j 1 #Λ j 1 n
≤ 1 λ , using that b 2 ≤ M 2 1 e 2γ j 1 +2ǫ j 1 . Hence, outside the set Ω n,1 , our assumptions imply that there exists θ n = θ β such that f j 1 , θn , ψ j,k = β j,k for all (j, k) ∈ Λ j 1 . Finally, outside the set Ω n,1 , by using the bound given in Lemma 2.4, one obtains the following inequality for the estimation error

∆ f j 1 ,θ * j 1 ; f j 1 , θn ≤ 2M 1 e γ j 1 +ǫ j 1 +1 λ #Λ j 1 n .
which completes the proof of Proposition 4.5.

Our assumptions on j 1 (n) imply that 1 2 n

1 2s+1 ≤ 2 j 1 (n) ≤ n 1 2s+1
. Therefore, using Lemma 2.2, one has that for all f ∈ F s 2,2 (M ) with s > 1/2

γ j 1 (n) ≤ Cn 1-2s 2(2s+1) , A j 1 (n) ≤ Cn 1 2(2s+1) , D j 1 (n) ≤ Cn -s 2s+1 ,
where C denotes constants not depending on g = log(f ). Hence, lim

n→+∞ ǫ j 1 (n) = lim n→+∞ 2M 2 1 e 2γ j 1 (n) +1 A j 1 (n) D j 1 (n) = 0,uniformly over F s 2,2 (
M ) for s > 1/2. For all sufficiently large n, ǫ j 1 (n) ≤ 1 and thus, using Propo-sition 4.5, there exists θ

* j 1 (n) ∈ R #Λ j 1 (n) such that ∆ f ; f j,θ * j 1 (n) ≤ M 1 2 e γ j 1 (n) D 2 j 1 (n) ≤ Cn -2s 2s+1 for all f ∈ F s 2,2 (M ). ( 4

.26)

By the same arguments it follows that lim

n→+∞ η j 1 (n),n = lim n→+∞ 4M 2 1 e 2γ j 1 (n) +2ǫ j 1 (n) +2 A 2 j 1 (n) #Λ j 1 (n) n
= 0, uniformly over F s 2,2 (M ) for s > 1/2. Now let λ > 0. The above result shows that for sufficiently large n, λ ≤ η -1 j 1 (n),n , and thus using Proposition 4.5 it follows that there exists a set Ω n,1 of probability less than M 2 λ -1 such that outside this set there exists θ n ∈ R #Λ j 1 (n) which satisfies:

∆ f j 1 (n),θ * j 1 (n) ; f j 1 (n), θn ≤ 2M 1 e γ j 1 (n) +ǫ j 1 (n) +1 M 2 λ #Λ j 1 (n) n ≤ Cλn -2s 2s+1 , (4.27)
for all f ∈ F s 2,2 (M ). Then, by the Pythagorian-like identity (2.2) it follows that outside the set Ω n,1

∆ f ; f j 1 (n), θn = ∆ f ; f j 1 (n),θ * j 1 (n)

+ ∆ f j 1 (n),θ * j 1 (n) ; f j 1 (n), θn , and thus Theorem 2.5 follows from inequalities (4.26) and (4.27).

Proof of Theorem 2.6

First, one needs the following proposition.

Proposition 4.6 Let β j,k := f, ψ j,k and β ξ j,n ,(j,k) := δ ξ j,n β j,k with (j, k) ∈ Λ j 1 . Assume that f ∈ F s p,q (A) with s > 1/p and 1 ≤ p ≤ 2. Let M 1 > 0 be a constant such that M -1 1 ≤ f ≤ M 1 (see Lemma 2.1). Let ǫ j 1 = 2M 2 1 e 2γ j 1 +1 D j 1 A j 1 . If ǫ j 1 ≤ 1, then there exists θ * j 1 ∈ R #Λ j 1 such that:

f j 1 ,θ * j 1
, ψ j,k = f, ψ j,k = β j,k for all (j, k) ∈ Λ j 1

Moreover, the following inequality holds (approximation error)

∆ f ; f j 1 ,θ * j 1 ≤ M 1 2 e γ j 1 D 2 j 1 .
Suppose that Assumptions 1 and 2 hold. Let η j 1 ,n = 4M 2 1 e 2γ j 1 +2ǫ j 1 +2 A 2 j 1 n log n -2s 2s+1 . Then, for every λ > 0 such that λ ≤ η -1 j 1 ,n there exists a set Ω n,2 of probability less than M 3 λ -1 , where M 3 is the constant defined in Lemma 4.3, such that outside the set Ω n,2 there exists some θ n ∈ R #Λ j 1 which satisfies: f HT j 1 , θn,ξ j,n , ψ j,k = δ ξ j,n β j,k = β ξ j,n ,(j,k) for all (j, k) ∈ Λ j 1 .

Moreover, outside the set Ω n,2 , the following inequality holds (estimation error) ∆ f j 1 ,θ * j 1

; f HT j 1 , θn,ξ j,n ≤ 2M 1 e γ j 1 +ǫ j 1 +1 λ n log n -2s 2s+1 .

Proof. Approximation error: The proof is the same that the one of Proposition 4.5. Estimation error: Applying Lemma 2.4 with θ 0 = θ * j 1 , β 0,(j,k) = f j 1 ,θ 0 , ψ j,k = β j,k , β = β ξ j,n , where β ξ j,n = β ξ j,n ,(j,k) (j,k)∈Λ j 1

, and b = exp log f j 1 ,θ * j 1

∞

we obtain that if β ξ j,n -β 2 ≤ 1 2ebA j 1 with β = (β j,k ) (j,k)∈Λ j 1 then there exists θ n = θ β ξ j,n such that f HT j 1 , θ j 1 ,ξ j,n , ψ j,k = β ξ j,n ,(j,k) for all (j, k) ∈ Λ j 1 .

Hence, it remains to prove that our assumptions imply that the event β ξ j,n -β Hence, outside a set Ω n,2 of probability less than M 3 λ -1 , it holds that β ξ j,n -β Therefore, the condition β ξ j,n -

β 2 ≤ 1 2ebA j 1 holds if λ n log n -2s 2s+1 1 2 ≤ 1 2ebA j 1
, which is equivalent to 4e 2 b 2 A 2 j 1 λ n log n -2s 2s+1 ≤ 1. Using that b 2 ≤ M 2 1 e 2γ j 1 +2ǫ j 1 the last inequality is true if

η j 1 ,n = 4M 2 1 e 2γ j 1 +2ǫ j 1 +2 A 2 j 1 n log n -2s
2s+1 ≤ 1 λ . Hence, outside the set Ω n,2 , our assumptions imply that there exists θ n = θ β ξ j,n such that f HT j 1 , θn,ξ j,n , ψ j,k = β ξ j,n ,(j,k) for all (j, k) ∈ Λ j 1 . Finally, outside the set Ω n,2 , by using the bound given in Lemma 2.4, one obtains the following inequality for the estimation error ∆ f j 1 ,θ * j 1

; f HT j 1 , θn,ξ j,n ≤ 2M 1 e γ j 1 +ǫ j 1 +1 λ n log n -2s 2s+1 , which completes the proof of Proposition 4.6.

Our assumptions on j 1 (n) imply that 1 2 n log n ≤ 2 j 1 (n) ≤ n log n . Therefore, using Lemma 2.2, one has that for all f ∈ F s p,q (M ) with s > 1/p,

γ j 1 (n) ≤ C n log n -s-1 p , A j 1 (n) ≤ n log n 1 2 , D j 1 (n) ≤ C n log n -s *
, where C denotes constants not depending on g = log(f ). Hence,

lim n→+∞ ǫ j 1 (n) = lim n→+∞ 2M 2
1 e 2γ j 1 (n) +1 A j 1 (n) D j 1 (n) = 0, uniformly over F s p,q (M ) for s > 1/p. For all sufficiently large n, ǫ j 1 (n) ≤ 1 and thus, using Proposition 4.6, there exists θ * j 1 (n) ∈ R #Λ j 1 (n) such that

∆ f ; f j 1 (n),θ * j 1 (n) ≤ M 1 2 e γ j 1 (n) D 2 j 1 (n) ≤ C n log n -2s *
for all f ∈ F s p,q (M ).

Lemma

Figure 1 :

 1 Figure 1: True spectral density f , wavelet thresholding estimator f HT j 0 ,j 1 , ξ j,n and final positive estimator

Figure 2 :

 2 Figure 2: True spectral density f , final positive estimator f HT j 0 ,j 1 , θn, ξ j,n and estimator via model

2 ≤ 1 2ebA j 1 holds with probability 1 -≤

 2111 M 3 λ -1 . First remark that b ≤ M 1 e γ j 1 +ǫ j 1 and that by Markov's inequality and Lemma 4.3 we obtain that for any λ > 0, M 3 λ -1 .

Acknowledgments

This work was supported in part by Egide, under the Program of Eiffel excellency Phd grants, as well as by the BDI CNRS grant.

Now remark that if p = 2 then s * = s > 1 (by assumption), thus

(4.28)

By the same arguments it holds that

uniformly over F s p,q (M ) for s > 1/p. Now let λ > 0. The above result shows that for sufficiently large n, λ ≤ η -1 j 1 (n),n , and thus using Proposition 4.6 it follows that there exists a set Ω n,2 of probability less than M 3 λ -1 such that outside this set there exists θ n ∈ R #Λ j 1 (n) which satisfies:

for all f ∈ F s p,q (M ). Then, by the Pythagorian-like identity (2.2) it follows that outside the set Ω n,2 ,

, and thus Theorem 2.6 follows from inequalities (4.28) and (4.29).

Proof of Theorem 2.7

The proof is analogous to the one of Theorem 2.6. It follows from Lemma 4.4.