domain: stat math

text:
El Mehdi Issouani
email: elmehdi.issouani@gmail.com

Patrice Bertail

Emmanuelle Gautherat

Exponential bounds for regularized Hotelling's T 2 n statistic in high dimension

Keywords: Eigenvalues, exponential bounds, high dimension, Hotelling statistics, penalized covariance, regularized covariance matrix, self-normalization

We obtain exponential inequalities for regularized Hotelling's T 2 n statistics, that take into account the potential high dimensional aspects of the problem. We explore the finite sample properties of the tail of these statistics by deriving exponential bounds for symmetric distributions and also for general distributions under weak moment assumptions (we never assume exponential moments). For this, we use a penalized estimator of the covariance matrix and propose an optimal choice for the penalty coefficient.

Introduction

In many applications (for instance in genomics or natural language processing), the dimension of the parameter of interest q is large in comparison to the sample size n and sometimes is increasing with n. Consider for instance the problem of estimating or testing a mean of variables in R q , with q > n; in that case, the empirical covariance matrix is not full rank and does not even converge to the true one when n tends to infinity and is ill-conditioned (see [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF]). As a consequence, the usual Hotelling's T 2 n tests in a large dimension framework are no longer valid. It is thus important to construct estimators and testing procedures that take into account the high dimensional aspects of the problem (as done for instance in Ledoit andWolf (2000, 2022) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF][START_REF] Ledoit | Quadratic shrinkage for large covariance matrices[END_REF], see also the references therein). One relevant proposition which has been developed in the statistical literature is to use a penalized estimator of the covariance matrix which is non-singular and to use this matrix in tests. In that spirit, [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF] [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF] have obtained asymptotically valid regularized Hotelling's T 2 n tests for the mean in the Gaussian case in a high dimensional framework, when n and q ≡ q(n) tend to infinity at some specific rate. [START_REF] Li | An adaptable generalization of hotelling's t 2 test in high dimension[END_REF] [START_REF] Li | An adaptable generalization of hotelling's t 2 test in high dimension[END_REF] have extended these results to some specific sub-gaussian distribution. The purpose of this paper is to further explore the finite sample properties of such tests by deriving exponential bounds of some correctly regularized Hotelling's T 2 n under general distributions, including ones with very few moments.

Such bounds allow to build conservative confidence regions for the parameter of interest. They are also of interest in statistical learning to control risk even with unbounded loss functions. For this, we derive exponential bounds for some regularized Hotelling's T 2 n statistics in the spirit of [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF], who obtained bounds for selfnormalized quadratic forms or the Hotelling's T 2 n statistic when q < n. We show that for symmetric distributions, only moments of order 2 are needed and we only assume the existence of moments of order 8 for general distributions.

Let Z, Z 1 , . . . , Z n be i.i.d. centered random vectors with probability distribution P, defined on a probability space (Ω, A, P) with values in R q(n) , B, P endowed with the L 2 norm ∥.∥ 2 . We denote E the expectation under P. Put Z (n) = (Z i ) 1≤i≤n . As n and q(n) go to infinity, notice that actually (Z (n) ) n defines a triangular array of random variables with varying dimensions. However, since we are interested in finite sample properties, we will drop the dependence in n. In particular, we use q instead of q(n). But keep in mind that q is a function of n in an asymptotic framework.

The covariance matrix of the observation is given by S 2 = E (ZZ ′ ), where we denote by Z ′ the transpose of Z and S the square root of S 2 . Denote by Zn = n -1 n i=1 Z i the sample mean. The sample covariance matrix is defined here by

S 2 n Z (n) = 1 n n i=1 Z i Z ′ i .
To simplify notations, we denote the sample covariance matrix of Z i 's by S 2 n when there is no confusion. Notice that we do not center by the empirical mean.

We recall that Hotelling's T 2 n , which can be seen as a quadratic form of self-normalized sums, is given by

T 2 n = n Z′ n S -2
n Zn , when q < n and S -2 n = S 2 n -1

. For some nonnegative real numbers, ρ 1 and ρ 2 , define Σ 2 n the linear combination of the identity matrix with the sample covariance matrix

Σ 2 n ≡ Σ 2 n (ρ 1 , ρ 2 ) = ρ 1 I q + ρ 2 S 2 n ,
with I q the identity matrix of size q. For ρ 1 = 0 and ρ 2 = 1, Σ 2 n (0, 1) = S 2 n is the empirical covariance matrix, which is singular for q > n. When ρ 2 = 1 and ρ 1 > 0 (and small), Σ 2 n corresponds to a Tikhonov regularization of the sample covariance matrix: see Tikhonov (1963) [START_REF] Tikhonov | On the regularization of ill-posed problems[END_REF]. It is precisely this estimator which is used in the tests proposed by Chen et al (2011) [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF]. However, it is shown in [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] that if one chooses adequately ρ 1 and ρ 2 then one can obtain a well-conditioned estimator of the covariance matrix which is invertible and more accurate than the sample covariance for some L 2 -distance. We denote by Σ 2 the expectation of Σ 2 n , which is given by

Σ 2 ≡ Σ 2 (ρ 1 , ρ 2 ) = ρ 1 I q + ρ 2 S 2 .
Actually, such modification ensures that we can control the distance between Σ 2 n and S 2 : this will be fundamental to obtain exponential bounds.

In the following, we are interested in the Hotelling's T 2 n statistic with a linear combination of the sample covariance and the identity, that we now call the regularized Hotelling's T 2 n statistic defined by

T 2 n (ρ 1 , ρ 2 ) = n Z′ n Σ -2 n (ρ 1 , ρ 2 ) Zn
generalizing the proposal of Chen et al (2011) [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF].

In the framework of high dimension, such quantities also appear naturally when studying empirical likelihood under a lot of constraints, penalized in its dual form by an L2-norm: see for instance Newey and Smith (2004) [START_REF] Newey | Higher order properties of gmm and generalized empirical likelihood estimators[END_REF], [START_REF] Lahiri | A penalized empirical likelihood method in high dimensions[END_REF], [START_REF] Lahiri | A penalized empirical likelihood method in high dimensions[END_REF], Carrasco and Kontchoni (2017) [START_REF] Carrasco | Regularized generalized empirical likelihood estimators[END_REF] among others.

When q < n, exponential bounds for T 2 n (0, 1) (that is, with the empirical covariance matrix instead of a regularized one) have been obtained by [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF]. Their exponential bound is controlled by two terms: [START_REF] Abramovitch | Handbook of mathematical tables[END_REF] an exponential term corresponding to a "Hoeffding" or [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] type of inequality applied to a symmetrized version of the observations and (2) an exponential bound which essentially controls the minimum eigenvalue of the sample covariance matrix and the proximity of S 2 n to S 2 . However, for q ≥ n such inequality can not hold since in that case the minimum eigenvalues of S 2 n is always 0. Moreover, it can easily be seen from the results of [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] that the bound becomes very bad when q > n or/and when q and n are of the same order. We obtain in this paper general results with an adequate choice of ρ 1 and ρ 2 when q is bigger than n and when q and n are such that q n → l ∈ ]0, ∞[. The paper is divided into four parts including this introduction. In the second part, we recall some known exponential inequalities for q < n under weak moments assumptions. Then we obtain an oracle exponential inequality for the regularized Hotteling's T 2 n , assuming that the values ρ 1 and ρ 2 are fixed and known. Some interesting sharp bounds which may be useful in statistical learning assuming symmetry are obtained for any n and q large. We then establish a general inequality for q = O(n) for non-symmetric distributions under a few moments' assumptions. In the third part, we estimate the optimal values ρ * 1 and ρ * 2 and show that the inequality remains valid up to some additional small terms controlling the concentration of these estimators around their true value. We illustrate our results with some simulations in the supplementary material.

Oracle exponential bounds for regularized Hotelling's T 2 n

In the following, we define the penalized Hotelling's T 2 n as the particular regularized Hotelling's statistic T 2 n (ρ, 1) with ρ ≥ 0. The aim of this section is to establish an oracle exponential inequality of the distribution of the penalized Hotelling's T 2 n in the case q ≥ n and when the distribution of the data is symmetric (Theorem 1 and Theorem 2) as well as in the general case, that is when the distribution is not necessarily symmetric (Theorem 3).

2.1. Known bounds for Hotelling's T 2 n Some bounds for T 2 n or self-normalized sums may be quite easily obtained in the symmetric case (that is for random variables having a symmetric distribution see [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF]) and are well-known in the unidimensional case q = 1. In non-symmetric and/or multidimensional cases with q < n, these bounds are new and not trivial to prove. One of the main tools for obtaining exponential inequalities in various settings is the famous Hoeffding inequality (see [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF] [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]). For centered independent real random variables Y 1 , . . . , Y n , that are bounded, say |Y i | < 1, for all i ∈ {1, . . . , n}, we have, for

a i ∈ [-1, 1] such that a 2 i = 1, ∀t > 0, P                ≥ t           = E           P           n i=1 Z i ε i 2 n i=1 Z 2 i ≥ t (Z i ) i∈{1,...,n}                     ≤ 2 exp - t 2 .
Pinelis (1994) [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] has obtained with a different technique, a sharp χ 2 type of bounds which generalizes this kind of results for multivariate data when q < n. He proved that, if Z has a symmetric distribution, without any moment assumption on the variables Z i , then one has

∀t > 0, P T 2 n ≥ t ≤ 2e 3 9 F q (t), (1)
where F q (t) is the cumulative distribution function (cdf) of a χ 2 (q) distribution. The density is denoted by f q . A crude approximation yields that for t large enough,

P T 2 n ≥ t ≤ e 3 9 2 2-q 2 Γ( q 2 ) t q 2 -1 exp(-t/2),
where Γ (x) = +∞ 0 t x-1 e -t dt is the gamma function. Notice that, for q = 1 this bound (only valid for large t) is better than the crude Hoeffding bound since we recover the missing factor 1 √ t in front of the exponential (see [START_REF] Talagrand | The missing factor in hoeffding's inequalities[END_REF] [START_REF] Talagrand | The missing factor in hoeffding's inequalities[END_REF]). When q < n, using a multidimensional version of Panchenko's symmetrization lemma [START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF] [START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF]) [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] have obtained an exponential inequality for the general distribution of Z with finite kurtosis γ 4 = E S -1 Z 4 2 . More precisely, they establish that under 0 < γ 4 < ∞, (i) for t > nq, P T 2 n ≥ t = 0. (ii) for any a > 1, and any nonnegative t such that 2q(1 + a) ≤ t ≤ nq, the following bound holds for some is an explicit constant C(q),

P(T 2 n ≥ t) ≤ 2e 3 9Γ( q 2 + 1) t -q(1 + a) 2(1 + a) q 2 exp - t -q(1 + a) 2(1 + a) + C(q)n 3-6 q+1 exp           - n 1 -1 a 2 γ 4 (q + 1)           .
The first term is essentially equivalent to the tail of a χ 2 (q) distribution (up to an explicit constant), while the second term controls the speed of convergence of S 2 n to S 2 , when γ 4 < ∞. The constant a controls the balance between these two terms on the right-hand side of the inequality and may be optimized. Notice that this second exponential term is small when q << n but explodes in n 3 if q/n → l > 0 for large n, making this bound totally useless in that case.

In the general multidimensional framework considered in [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] and in this paper, the main difficulty is to keep the self-normalized structure when symmetrizing the initial sum. In the next sections, the results of [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] obtained for q < n are extended to the case q ≥ n by using a regularized version of S 2 n . This inequality is based on an appropriate diagonalization of the regularized sample covariance matrix which allows applying Pinelis (1994)'s inequality [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] (see section 2.2). This crude inequality is refined in section 2.3. When dealing with the general case (see section 2.4), we establish first a multivariate symmetrization lemma 3 in the spirit of [START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF] [START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF]. This symmetrization partially destroys the self-normalized structure (the normalization is then

Σ 2 n +Σ 2 = 2Σ 2 n +(Σ 2 -Σ 2 n ) instead of the expected normalization Σ 2 n
), but the right standardization can be recovered (up to the factor 2) by obtaining a lower tail control of the distance between Σ 2 n and Σ 2 . To control this distance and make it as small as possible we will use the results of [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF].

Bounds for regularized Hotelling's T 2

n in a symmetric framework We now obtain a simple inequality for the regularized Hotelling's T 2 n in the symmetric case, based on previous results by [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF]. It essentially shows that the tail of the regularized Hotelling's T 2 n is controlled by the tail of a χ 2 (n) distribution.

Theorem 1. Assume that Z has a symmetric distribution with finite covariance matrix then, without any additional moment assumption, we have, for any n > 1, for t > n, for any ρ 1 , ρ 2 > 0,

P T 2 n ρ 1 ρ 2 , 1 ≥ t = P n Z′ n Σ -2 n (ρ 1 , ρ 2 ) Zn ≥ t ρ 2 ≤ 2e 3 9 Fn (t) ≤ 2e 3 9 exp - (t -n) 2 4t , (2)
where F n is the cdf of a χ 2 (n) distribution. Moreover, for any ρ > 0, we have

P T 2 n (ρ, 1) -n √ 2n ≥ t = P n Z′ n Σ -2 n (ρ, 1) Zn -n √ 2n ≥ t ≤ 2e 3 9 exp             -t 2 2 1 + √ 2 t √ n             . The inequality (2) yields a control of T 2 n (ρ 1 , ρ 2 ) = n Z′ n Σ -2 n (ρ 1 , ρ 2 )
Zn , when using a linear shrinkage estimator of the variance. This in turn can be simplified in (3), to a truly penalized Hotelling's T 2 n . Note that for any ρ 1 , ρ 2 > 0,

Σ 2 n (ρ 1 , ρ 2 ) ρ 2 = ρ 2 S 2 n + ρ 1 I q ρ 2 = S 2 n + ρ 1 ρ 2 I q
and for any ρ > 0, Σ 2 n (ρ, 1) = S 2 n + ρI q is a penalized estimator of the covariance matrix. Inequality (3) can be interpreted as a Bernstein-type inequality.

Remark: These inequalities hold for any choice of ρ 1 and ρ 2 . However for the inequalities to be sharp, ρ 1 and ρ 2 should be chosen adequately. First from the proof of Theorem 1, we see that the inequality is sharp only when ρ 1 is close to 0, which is in accordance with what we know about Tikhonov regularisation (1963) [START_REF] Tikhonov | On the regularization of ill-posed problems[END_REF]. Actually when ρ 1 tends to 0, Σ -2 n (ρ 1 , ρ 2 ) is going to be identical to 1 ρ 2 (S 2 n ) -where (A) -is the Moore-Penrose or generalized inverse of A (which is unique for symmetric matrices). Notice that the proof of the theorem and the inequality remain valid if we use n Z′ n S 2 n -Zn rather than n Z′ n Σ -2 n (ρ 1 , ρ 2 ) Zn . In the procedure of Chen et al. (2011) [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF] this means that asymptotically there is no difference between standardizing by the regularized variance or by the generalized inverse of the covariance matrix. The regularization just serves as a trick to approximate the generalized inverse. However, the finite sample properties of the regularized Hotelling's T 2 n will strongly depend on the choice of ρ 1 and ρ 2 .

2.

3. An improved bound for penalized Hotelling's T 2 n in the symmetric case It can be seen from the proof of Theorem 1 that the penalized Hotelling's T 2 n statistic essentially behaves like a weighted sum of asymptotically χ 2 random variables. This also explains the results of Chen et al. (2015) [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF]. Actually, we can obtain a bound for this quantity relying on the results of [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] and Laurent and Massart (2000) [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF] (see p.24 of their paper) who control the tail of the weighted sum of independent χ 2 (1) random variables. Let λ = (λ j ) j=1,...,q ∈ R q + be the eigenvalues of S 2 n (ordered in a increasing order). We define for any ρ 1 , ρ 2 > 0, the following effective dimensions (see [START_REF] Chen | A regularized hotelling's t 2 test for pathway analysis in proteomic studies[END_REF] for other expressions of these quantities):

Θ 1 (λ, ρ 1 , ρ 2 ) = inf(n,q) j=1 λ j ρ 1 + ρ 2 λ j ; Θ 2 (λ, ρ 1 , ρ 2 ) = inf(n,q) j=1 λ 2 j (ρ 1 + ρ 2 λ j ) 2 ; Θ ∞ (λ, ρ 1 , ρ 2 ) = sup 1≤ j≤inf(n,q) λ j ρ 1 + ρ 2 λ j .
In the next result, we obtain a sharp bound for regularized and penalized Hotelling's T 2 n . Notice that, in that case, the recentering factor depends on Θ 1 (λ, ρ 1 , ρ 2 ) and is random. In the proof of Theorem 2.1, this value is essentially bounded by n/ρ 2 , which is a very bad approximation when ρ 2 is small. Theorem 2 tells that, for q ≥ n, the tail of the regularized Hotelling's T 2 n statistic behaves as the weighted sum of n independent χ 2 (1) r.v.'s where the weights are given by the random factors λ j ρ 1 +ρ 2 λ j . We get some Bernstein bounds for this weighted sum by first randomizing by some independent Gaussian r.v.'s, then conditioning on the data and applying Laurent and Massart (2000)'s Bernstein inequality [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF]. This inequality in turn can be transformed into some exact bounds for the statistics of interest.

Theorem 2. Assume that Z has a symmetric distribution then, without any moment assumption, we have, for any n > 1 and q > 0, for any t > 0 and for any ρ 1 , ρ 2 > 0,

P        T 2 n (ρ 1 , ρ 2 ) -Θ 1 (λ, ρ 1 , ρ 2 ) 2Θ 2 (λ, ρ 1 , ρ 2 ) 2 ≥ √ 2 √ t + Θ ∞ (λ, ρ 1 , ρ 2 ) Θ 2 (λ, ρ 1 , ρ 2 ) t        ≤ C exp(-t), with C = 3824.
Or equivalently, we have for the penalized Hotelling's statistic, for n > 1 and q > 0, for any t > 0 and, for any ρ > 0,

P T 2 n (ρ, 1) -Θ 1 (λ, ρ, 1) Θ 2 (λ, ρ, 1) ≥ √ 2t + Θ ∞ (λ, ρ, 1) Θ 2 (λ, ρ, 1) t ≤ C exp - t 2 .
In the symmetric case, this theorem enables us to easily obtain confidence regions of level 1 -δ, for δ ∈ [0, 1] for the regularized Hotelling's statistic, as stated in the following corollary.

Corollary 1. Put c(δ) = log C
δ with C = 3824. Then, for any n > 1 and q ≥ 1, for any t > 0 and for any ρ 1 , ρ 2 > 0, with probability 1 -δ, we have

T 2 n (ρ 1 , ρ 2 ) ≤ Θ 1 (λ, ρ 1 , ρ 2 ) + 2Θ 2 (λ, ρ 1 , ρ 2 ) c(δ) + Θ ∞ (λ, ρ 1 , ρ 2 ) Θ 2 (λ, ρ 1 , ρ 2 ) c(δ) ,
The proof of this corollary is left to the reader. This result holds for any n and q. When q ≤ n is large, we can actually put ρ 1 = 0 and get some Pinelis' type bounds (when the χ 2 distribution tail is itself approximated by a Gaussian tail).

The constant C comes from a result of Chasapis and al (2022) [START_REF] Chasapis | Rademacher-gaussian tail comparison for complex coefficients and related problems[END_REF] who extended a result of Pinelis [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] (1994). Indeed they state that, when symmetrizing, for smooth functions of quadratic forms, Rademacher variables may be replaced by standard normal variables. However, their constant is clearly not optimal and we expect the optimal C to be 2e 3 /9 as in Pinelis [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] (1994).

The bounds in Theorem 2 and Corollary 1 can be used in practice for testing purposes in particular in anomaly detection in statistical learning. See for instance the literature on intrusion detection systems using multivariate control charts based on Hotelling T 2 n (for instance [START_REF] Tracy | Multivariate control charts for individual observations[END_REF] [START_REF] Tracy | Multivariate control charts for individual observations[END_REF] and further works by these authors).

Bounds for regularized Hotelling's T 2 n for non symmetric distribution

We now consider Z with a general (not necessarily symmetric) distribution. We will later prove a symmetrization lemma that generalizes the one obtained in [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF]. In the following, we also use the results of [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] to optimally control the distance between Σ 2 n (ρ 1 , ρ 2 ) and S 2 . For this, consider the modified Frobenius scalar product between matrices and the corresponding norm given by

⟨A, B⟩ = T r (AB ′ ) q and ∥A∥ 2 = ⟨A, A⟩ = T r (AA ′ ) q .
Note that dividing the standard Frobenius scalar product by q enables the norm of the identity I q to be equal to 1, which is more convenient. In the following, we extend this modified Frobenius norm to vectors by considering, for any vector Z ∈ R q , ∥Z∥ 2 = T r ZZ ′ /q.

Additional notations and hypotheses

Put S 2 = σ k j 1≤k, j≤q and consider Λ the diagonal matrix of the eigenvalues of S 2 and O the matrix of associated eigenvectors. The eigenvalues are denoted µ 1 , . . . , µ q with

µ 1 ≤ µ 2 ≤ • • • ≤ µ q . We have S 2 = O ′ Λ 2 O. Now, for i ∈ {1, . . . , n}, we define Y i = OZ i with Y i = Y i,1 , . . . , Y i,q ′ .
In order to provide a well-conditioned estimator for large dimensional covariance matrices, Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] have studied the minimum of E Σ 2 n (ρ 1 , ρ 2 ) -S 2 2 . This minimization can be seen as a projection problem in the Hilbert space of random matrices, equipped with the inner product ⟨A,

B⟩ H = E [⟨A, B⟩] with associated norm ∥.∥ 2 H = E ∥.∥ 2 .
We assume the four following assumptions:

(A 1 ) ∃K 0 , K 1 > 0 such that, for any n and any q ≥ n, K 0 ≤ q n ≤ K 1 . (A 2 ) ∃K 2 > 0 such that, for any n and any q ≥ n, 1 q q j=1 E Y 8 1, j ≤ K 2 . (A 3 ) ∃K 3 > 0 such that for any n and any q ≥ n, 1 K 3 < µ 1 ≤ µ q < K 3 . (A 4 ) ∃K 4 > 0 such that for any n and any q ≥ n,

ν = q 2 n 2 × (i, j,k,l)∈Q Cov Y 1,i Y 1, j , Y 1,k Y 1,l 2 Card (Q) ≤ K 4 n ,
where Q denotes the set of all the quadruples that are made of four distinct integers between 1 and q.

Remarks: (A 2 ) and (A 4 ) are already assumed in Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF]. First assumption (A 1 ) essentially means that q = q(n) is of the same order as n. (A 2 ) states that the moment of order 8 is bounded in average: this condition holds if the following moment of order 8, 1 q q j=1 E Z 8 1, j is finite (by sub-multiplicative inequality and the fact that ∥O∥ = 1). This is a weak condition: we do not require exponential moments and allow for fat tail behavior of the sample. (A 3 ) ensures that the largest and the smallest eigenvalue of the true covariance matrix are bounded. This rules out the case when the components of the vector Z are too correlated: consider for instance the degenerate case where S 2 is a matrix full of 1, then in that case the smallest eigenvalue is 0 and the largest is q. The case of a vector with long memory components is studied in [START_REF] Merlevède | Unbounded largest eigenvalue of large sample covariance matrices: Asymptotics, fluctuations and applications[END_REF] [START_REF] Merlevède | Unbounded largest eigenvalue of large sample covariance matrices: Asymptotics, fluctuations and applications[END_REF] : they show that the largest eigenvalue is unbounded. Thus this case does not enter our framework. Assumption (A 4 ) is immediate in the Gaussian case, since ν = 0 because of the rotation which makes the Y 1, j 's j ∈ {1, . . . , q} independent. Obviously, for (Z 1, j ) j independent, ν = 0 as well. More generally if the components of the vector satisfy some adequate α-mixing conditions, then the sum in the hypothesis (A 4 ) can be seen as a sum of cumulants and may also be controlled using the arguments of [START_REF] Doukhan | Cumulants for stationary mixing random sequences and applications to empirical spectral density[END_REF] [START_REF] Doukhan | Cumulants for stationary mixing random sequences and applications to empirical spectral density[END_REF].

Inequalities for random variables with a general distribution

The next Theorem 3 extends Theorem 1 to general distributions which are not necessarily symmetric. From now on, following [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF], we denote ρ * 1 and ρ * 2 the optimal values defined as the minimum arguments of E Σ 2 n (ρ 1 , ρ 2 ) -S 2 2 . Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] have obtained

ρ * 1 = β 2 δ 2 σ 2 and ρ * 2 = α 2 δ 2 , with σ 2 = S 2 , I q ; α 2 = S 2 -σ 2 I q 2 ; β 2 = E S 2 n -S 2 2 and δ 2 = α 2 + β 2 = E S 2 n -σ 2 I q 2 .
Now, we define, for α 2 0,

ρ * = ρ * 1 ρ * 2 = β 2 α 2 σ 2
, which yields the optimal penalized estimator of S 2 n :

Σ * 2 n = Σ 2 n ρ * 1 , ρ * 2 ρ * 2 = S 2 n + ρ * I q .
If α 2 = 0, take Σ * 2 n = σ 2 I q (in that case we will just need to estimate σ 2 ). In Figure 1, the scalar product is ⟨, ⟩ H with its associated norm. We represent Σ 2 n (ρ * 1 , ρ * 2 ), the optimal combination of S 2 n and I q defined by orthogonal projection of the true covariance matrix S 2 on the random vector-space generated by S 2 n and I q . Thus Σ

2 * n = Σ 2 n (ρ * , 1) is the penalization of S 2 n by I q with ρ * = ρ * 1 ρ * 2
. The green dashed line represents the set of penalized estimators Σ 2 n (ρ, 1) for which we obtain universal bounds in Theorem 3. Theorem 3. Assume that Z has a general distribution with finite variance S 2 . Assume in addition that assumptions (A 1 ) to (A 3 ) hold. Put a * = 1 + K 3 ρ * . Then we have, for any n > 1, for any q ≥ n, and for t > 2n,

P T 2 n (ρ * , 1) ≥ (1 + a * ) t = P n Z′ n Σ * -2 n Zn ≥ (1 + a * ) t ≤ 2e 3 9 t -n 2 n 2 exp -t-n 2 Γ n 2 + 1 ,
Remark: Here the bounding function for large t behaves like a centered χ 2 (n) distribution, up to the factor 2e 3 9 . The term (1 + a * ) ensures that the smallest eigenvalue of Σ * 2 n does not contribute to the inequality. Notice that the inequality is still valid when using Σ 2 n , the regularized version of S 2 n instead of the penalized version Σ * 2 n , up to a small modification of the bound (1 + a * )t by the factor 1/ρ * 2 : for n > 1, q ≥ n, for any t > 2n

P T 2 n ρ * 1 , ρ * 2 ≥ 1 ρ ⋆ 2 (1 + a * ) t ≤ 2e 3 9 t -n 2 n 2 exp -t-n 2 Γ n 2 + 1 .

Inequality with estimated parameters

We have proved an exponential inequality for the penalized Hotelling's T 2 n with theoretical values a * and ρ * . In practice these values are unknown. In this section, we estimate these quantities and obtain an inequality for the penalized Hotelling's T 2 n with estimated parameters. We first recall several results of [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] on the asymptotic behavior of regularized empirical covariance estimator Σ 2 n . Lemma 1 and proposition 1 below summarize these results with our notations and are proved by [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] in different lemmas and a theorem of their paper.

We use the same assumptions as in [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF]:

L 4
-→ denotes the fourth-moment convergence as n goes to infinity, i.e.

U n L 4 -→ U ⇐⇒ E (U n -U) 4 -→ n→∞ 0.
Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] essentially have shown that L 4 -consistent estimators for σ 2 , α 2 , β 2 and δ 2 are simply given by their empirical counterparts that is

σ2 n = S 2 n , I q ; δ2 n = S 2 n -σ2 n I q 2 ; α2 n = δ2 n -β2 n with β2 n = 1 n 2 n i=1 Z i (Z i ) ′ -S 2 n 2 and β2 n = min β2 n , δ2 n Lemma 1.
[ [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [11] lemma 3.2, lemma 3.3, lemma 3.4, lemma 3.5] Under assumptions (A 1 ) to (A 4 ), we have 1. σ 2 , α 2 and β 2 remain bounded (as n and q tend to ∞).

2. For all n, E σ2 n = σ 2 , and

σ2 n -σ 2 L 4 -→ 0 and σ4 n -σ 4 L 4 -→ 0. 3. δ2 n -δ 2 L 4 -→ 0. 4. β2 n -β 2 L 4 -→ 0 and β2 n -β 2 L 4 -→ 0. 5. α2 n -α 2 L 4 -→ 0.
After replacing the unobservable scalars σ 2 , α 2 , β 2 and δ 2 by their sample counterparts in the formula of Σ 2 n , Ledoit and Wolf obtained an estimation of the regularized empirical covariance matrix say

Σ2 n = β2 n δ2 n σ2 n I q + α2 n δ2 n S 2 n .
Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] have shown that Σ2 n and Σ 2 n are asymptotically equivalent in the modified Frobenius norm. Proposition 1. [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF], Theorem 3.2] Under the assumptions (A 1 )-(A 4 ), we have

1. lim n→∞ E Σ2 n -Σ 2 n 2 = 0.

Moreover, Σ2

n has the same asymptotic expected loss (or risk) as Σ 2 n i.e.

lim n→∞ E Σ2 n -Σ 2 2 -E Σ 2 n -Σ 2 2 = 0.
In the same way as Ledoit and Wolf (2000) [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] we define the optimal coefficients ρ *

Σ * 2 n = Σ 2 n ρ * 1 ρ * 2 , 1 = S 2 n + ρ * n I q , where ρ * n = β2 n σ2 n α2 n .
Similarly the unobservable threshold constant a * introduced in theorem 3 is estimated by â * n = 1+ K 3 ρ * n . The principle in Figure 2 is similar to the one in Figure 1 except that Σ2

n is determined first so that the regularized estimator belongs to the yellow line and the optimal estimator Σ 2 n = Σ 2 n ρ * 1 , ρ * 2 is the closest value to S 2 on this line. This difference induces an additional error term in our inequalities. Theorem 4 establishes an exponential bound for the penalized self-normalized sums, when Σ * 2 n is replaced by the estimator Σ2 * n and a * by â * n , up to a small error term that we control explicitly. Theorem 4. Under the assumptions (A 1 ) to (A 4 ), we have, for any n > 1, for any q > n, for any t > 2n and for any small value of ϵ > 0,

P T 2 n ( ρ * n , 1) ≥ t 1 + â * n + 2ϵ = P n Z′ n Σ * -2 n Zn ≥ t 1 + â * n + 2ϵ ≤ 2e 3 9 t -n 2 n 2 e -t-n 2 Γ n 2 + 1 + C (ϵ) nϵ , (3)
where

â * n = 1 + K 3 ρ *
n , and C (.) is a real nonnegative function, independent of n, defined by

C (ϵ) = 4K 1 K 2 2 + 1 q + K 1 + 2K 1 G ϵ 2K 1 + 4K 2 1 σ 4 ϵ G ϵ 2σ 2 K 1 + K 2 3 ϵ G ϵ K 3 .
The function G is defined explicitly in lemma 7. Notice that C(ϵ)/ϵ explodes when ϵ goes to 0.

These results essentially show that we have a χ 2 (n) control in the tail of the distribution, for a threshold larger than 2n(1 + â * n ) (recall that 2n is the variance of a χ 2 (n) distribution). The loss (1 + â * n ) is essentially due to the correlation between the components of Z and the deviation from homoscedasticity. The value of ϵ can not be too small but can be optimized by balancing the two terms in the inequality. For a given ϵ and a given level δ it is possible to solve numerically the second term of the inequality (3) equal to delta to get a valid bound for the Hotelling's T 2 for any n and q.

Proofs of the theorems

We set some notations that we will consider in the following proofs. S 2 n is a symmetric and diagonalizable matrix. Let's denote by O n an orthogonal matrix in M q (R) such that S 2 n = O ′ n Λ 2 n O n where Λ 2 n is a diagonal matrix and

Λ 2 n =                             λ 1 . . . λ n 0 ∖ 0                            
for any q > n.

Put Ŷi = O n Z i with Ŷi = Ŷi,1 . . . , Ŷi,q ′ . Let λ 1 ≤ • • • ≤ λ q denote eigenvalues of S 2 n .

Proof of theorem 1 and 2

We first establish a simple inequality for the penalized Hotelling's T 2 n in the symmetric case, based on previous results by Pinelis [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF]. The idea of the theorem is to use a rotation trick of the Z i that allows us to return to the "small" dimension case given by Pinelis. This yields a bound given by the survival function of a χ 2 with n degrees of freedom.

Proof of theorem 1. Note that Vectors Ŷi remain symmetric in distribution and uncorrelated. It is easy to see that, by construction, the empirical covariance matrix of the Ŷ1 , . . . , Ŷn is

1 n n i=1 Ŷi Ŷ′ i = 1 n n i=1 O n Z i Z ′ i O ′ n = O n S 2 n O ′ n = Λ 2 n .
This implies that, for any vector Ŷi , their coordinates for j ≥ n + 1 are zero. Indeed, for j ≥ n + 1, n -1 n i=1 Ŷ2 i, j = 0, implies in turn that each Ŷi, j = 0, for j ∈ {n + 1, . . . , q} and i ∈ {1, . . . , n}. Define Ỹi the n-dimensional vector version of Ŷi with these non-zero components, that is to say ∀ j ≤ n, Ỹi, j = Ŷi, j and their corresponding empirical mean Ȳn on the collection Ỹ(n) = Ỹi 1≤i≤n

. Thus, for all ρ 2 > 0, we have

n Z′ n Σ -2 n (ρ 1 , ρ 2 ) Zn = n        1 n n i=1 Ŷ′ i        ρ 1 I q + ρ 2 Λ 2 n -1        1 n n i=1 Ŷi        = n n j=1 n -1 n i=1 Ŷi, j 2 ρ 1 + ρ 2 λ j ≤ n n j=1 n -1 n i=1 Ŷi, j 2 ρ 2 λ j ≤ 1 ρ 2 n j=1 n -1/2 n i=1 Ŷi, j 2 λ j .
As λ j = n -1 n i=1 Ŷ2 i, j , we have reduced the problem to the sum of n self normalized sums, which can be seen as Hotelling's T 2 n of symmetric random variables in R n . In other words, n

Z′ n Σ -2 n (ρ 1 , ρ 2 ) Zn ≤ 1 ρ 2 n Ȳ′ n S -2 n Ỹ(n)
Ȳn . Thus, by applying Pinelis' equation (1) [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF], we have

∀t > 0, P n Z′ n Σ -2 n Zn ≥ t ρ 2 ≤ 2e 3 9 Fn (t) .
Recall that, if N 1 , . . . , N n are independent N(0, 1) random variables, then by Lemma 1 of Laurent and Massart (2000) [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF], one has, for u > 0,

P       n i=1 N 2 i -n √ 2n ≥ √ 2( √ u + u √ n )       ≤ e -u .
By inverting the polynomial in √ u, this is a Bernstein type inequality for i.i.d random variables

P       n i=1 N 2 i -n √ 2n ≥ ν       ≤ exp               - 2ν 2 1 + 1 + 2 √ 2 ν √ n 2               ≤ exp          - ν 2 2(1 + √ 2 ν √ n )          .
It follows that, for t > n,

Fn (t) = P       n i=1 N 2 i -n √ 2n ≥ t -n √ 2n       ≤ exp - (t -n) 2 4t .
Proof of theorem 2. Recall that : Zn = 1 n n i=1 Z i with Z i ∈ R q . Introduce independent Rademacher r.v.'s ε i taking the values ±1 with probability 1/2. Define Zϵ n = 1 n n i=1 ϵ i Z i . Then, in the symmetric case considered here, Zn and Zϵ n have the same distribution. Now write

n Zϵ ′ n Σ -2 n (ρ 1 , ρ 2 ) Zϵ n = n        1 n n i=1 ϵ i Ŷ′ i        ρ 1 I q + ρ 2 Λ 2 n -1        1 n n i=1 ϵ i Ŷi        = ϵ ′ VV ′ ϵ (4)
where Ŷ = ( Ŷ1 , . . . , Ŷn )

′ , ϵ = (ϵ 1 , . . . , ϵ n ) ′ and V = 1 √ n Ŷ ρ 1 I q + ρ 2 Λ 2 n -1/2
. Chasapis and al (2022) [START_REF] Chasapis | Rademacher-gaussian tail comparison for complex coefficients and related problems[END_REF] obtain an extension of Pinelis' result [START_REF] Pinelis | Extremal probabilistic problems and hotelling's t2 test under a symmetry condition[END_REF] stating that for smooth functions of quadratic forms, Rademacher variables may be replaced by standard normal variables. More precisely, define the Euclidian norm ∥x∥ 2 = √ ⟨x, x⟩ and consider ξ 1 , . . . , ξ n independent standard Gaussian random variables. Then, for any t ≥ 0, for any vectors ν 1 , . . . , ν n in R q , we have

P ∥ϵ 1 ν 1 + • • • + ϵ n ν n ∥ 2 ≥ t ≤ CP ∥ξ 1 ν 1 + • • • + ξ n ν n ∥ 2 ≥ t with C = 3824.
Since we have

ϵ 1 ν 1 + • • • + ϵ n ν n = ϵ ′ V
where V is the matrix of vectors ν i = ν i1 , . . . , ν iq corresponding to the rows, we can rewrite

∥ϵ 1 ν 1 + • • • + ϵ n ν n ∥ 2 2 = ϵ ′ V 2 2 = ϵ ′ VV ′ ϵ.
It follows that, for any u > 0,

P ϵ ′ VV ′ ϵ ≥ u ≤ CP ξ ′ VV ′ ξ ≥ u
By conditioning according to Ŷi 's and using equation (4), we have, for any u > 0 and, for any ρ 1 , ρ 2 > 0,

P n Zϵ ′ n Σ -2 n (ρ 1 , ρ 2 ) Zϵ n ≥ u = E P ϵ ′ VV ′ ϵ ≥ u Ŷ1 , . . . , Ŷn ≤ CE P ξ ′ VV ′ ξ ≥ u Ŷ1 , . . . , Ŷn .
Moreover recall from the preceding proof that we have

n Zϵ ′ n Σ -2 n (ρ 1 , ρ 2 ) Zϵ n = n        1 n n i=1 ϵ i Ŷ′ i        ρ 1 I q + ρ 2 Λ 2 n -1        1 n n i=1 ϵ i Ŷi        = n inf(q,n) j=1 n -1 n i=1 ϵ i Ŷi, j 2 ρ 1 + ρ 2 λ j = n inf(q,n) j=1 n -1 n i=1 ϵ i Ŷi, j 2 λ j λ j ρ 1 + ρ 2 λ j We obtain P n Zϵ ′ n Σ 2 n (ρ 1 , ρ 2 ) -1 Zϵ n > u ≤ CE           P           n inf(q,n) j=1 n -1 n i=1 ξ i Ŷi, j 2 λ j λ j ρ 1 + ρ 2 λ j > u Ŷ1 , . . . , Ŷn                     . (5)
Let us work now conditionally to Ŷ1 , . . . , Ŷn . Put K j = √ n n -1 n i=1 ξ i Ŷi, j / λ j for j ∈ {1, . . . , inf(q, n)}. Thus for any j k Cov K j , K k Ŷ1 , . . . ,

Ŷn = Cov        √ n n -1 n i=1 ξ i Ŷi, j λ j , √ n n -1 n i=1 ξ i Ŷi,k √ λ k Ŷ1 , . . . , Ŷn        = 1 n n i=1
Ŷi, j Ŷi,k

λ j λ k = 0.
Since K = K 1 , . . . , K inf(q,n) is a Gaussian vector (as a linear combination of independent variables) it follows that K 2 1 , . . . , K 2 inf(q,n) are iid χ 2 (1). Now, consider the vector b = b 1 , . . . , b q with nonnegative components (conditionally to Ŷi, j 's) defined by

b j = λ j ρ 1 + ρ 2 λ j .
A direct application of Laurent and Massart, lemma [START_REF] Laurent | Adaptive estimation of a quadratic functional by model selection[END_REF] to inf(q,n) j=1 b j K 2 j -1 gives for any u > 0

P         inf(q,n) j=1 b j K 2 j -1 > 2∥b∥ 2 √ u + 2∥b∥ ∞ u         ≤ exp (-u) .
In other words, for any u > 0, we have

P             inf(q,n) j=1 b j K 2 j -∥b∥ 1 2∥b∥ 2 2 > √ 2 √ u + √ 2 ∥b∥ ∞ ∥b∥ 2 u             ≤ exp (-u) (6)
Now by combining ( 5) and ( 6) we obtain the following result for the recentered version of our quantity of interest,

P             n Zϵ ′ n Σ -2 n (ρ 1 , ρ 2 ) Zϵ n -∥b∥ 1 2∥b∥ 2 2 > √ 2 √ u + √ 2 ∥b∥ ∞ ∥b∥ 2 u             ≤ CE             P             inf(q,n) j=1 b j K 2 j -∥b∥ 1 2∥b∥ 2 2 > √ 2 √ u + √ 2 ∥b∥ ∞ ∥b∥ 2 u Ŷ1, j , . . . , Ŷn, j j∈{1,...,inf(q,n)}                         ≤ C exp(-u).
The result of the theorem follows by noticing that

∥b∥ k = Θ k (λ, ρ 1 , ρ 2 ), k ∈ {1, 2, ∞} 4.2. Proof of theorem 3 4.2.1. A symmetrization lemma adapted to χ 2 distribution
The following lemma ensures that, if we have a χ 2 (k) type of control for the tail of a random variable ν, which stochastically dominates some random variable ξ, then we are also able to control the tail of ξ. For large values, this tail is essentially the same as the one of a χ 2 (k) distribution. We use exactly the same ideas as in Panchenko's lemma 1 and corollary 1 (which assumes an exponential control of the tail of the distribution of the variable ν). Lemma 2. Let ν and ξ be two real r.v.'s. For a ∈ R, put Φ a (x) = max (xa; 0). Assume that:

(i) for any a ∈ R, EΦ a (ξ) ≤ EΦ a (ν) (ii) there exists k and constants C 1 > 0, c 1 > 0, such that for any t > 0

P (ν ≥ t) ≤ C 1 Fk (c 1 t)
then, for t > 2k/c 1 , we have

P (ξ ≥ t) ≤ C 1 c 1 t -k 2 k 2 e -c 1 t-k 2 Γ k 2 + 1 and, for t > k/c 1 , we also get P (ξ ≥ t) ≤ C 1 Fk+2 (c 1 t -k) .
Proof of lemma 2 . We follow the lines of the proof of Panchenko's lemma, with a function Φ a with a = t -k c 1 given by Φ (x) = max (xt + k/c 1 ; 0), for t > k/c 1 . Remark that Φ (x) is convex, nondecreasing and that Φ (0) = 0 and Φ (t) = k/c 1 . We thus have by Markov's inequality

P (ξ ≥ t) ≤ EΦ (ξ) Φ (t) ≤ EΦ (ν) Φ (t) ≤ 1 Φ (t) Φ (0) + +∞ t-k/c 1 Φ ′ (x) P (ν ≥ x) dx ≤ C 1 c 1 k +∞ t-k/c 1 F k (c 1 x) dx.
By integration by parts, we get

+∞ t-k/c 1 Fk (c 1 x) dx = +∞ t-k/c 1 c 1 x f k (c 1 x) dx -(t -k/c 1 ) +∞ t-k/c 1 c 1 f k (c 1 x) dx. Recall that f k (u) = 1 2 k/2 Γ( k 2 ) u k 2 -1 exp(- u 2 ),
we thus have

c 1 k +∞ t-k/c 1 c 1 x f k (c 1 x) dx = c 1 2 k/2+1 k 2 Γ( k 2 ) +∞ t-k/c 1 (c 1 x) k+2 2 -1 exp(- c 1 x 2 )dx = F k+2 (c 1 t -k) .
It follows by straightforward calculations that, for t > k/c 1 ,

P (ξ ≥ t) ≤ C 1 F k+2 (c 1 t -k) - c 1 t -k k F k (c 1 t -k) .
Using the recurrence relation 26.4.8 of Abramovitch and Stegun ([1], page 941), for u ≥ 2k,

C 1 F k+2 (u -k) -u-k k F k (u -k) ≤ C 1 F k+2 (u -k) -F k (u -k) ≤ (u-k) 2 k/2 C 1 e -(u-k) 2 Γ( k 2 +1
) . We get with u = c 1 t, for t ≥ 2k/c 1 ,

P (ξ ≥ t) ≤ (c 1 t -k) 2 k/2 C 1 e -(c 1 t-k) 2 Γ k 2 + 1 . Moreover, for t > k/c 1 we have P (ξ ≥ t) ≤ C 1 F k+2 (c 1 t -k) .
Notice that we only lose 2 degrees of freedom in this case. It will not be important if k is large, typically of the order of n in our case.

Extension of Panchenko symmetrization lemma (see [17] Corollary 1, p. 2069)

Let J q = {u ∈ R q , ∥u∥ 2 = 1} be the unit circle of R q . Let X (n) = (X i ) 1≤i≤n be an independent copy of

Z (n) = (Z i ) 1≤i≤n . Since q > n, the matrix S 2 n Z (n) -X (n) = 1 n n i=1 (Z i -X i ) (Z i -X i ) ′ is not invertible. We derive from S 2 n Z (n) -X (n) the corresponding penalized empirical covariance matrix Σ2 n = 2ρ 1 I q + ρ 2 S 2 n Z (n) -X (n)
It is easy to see that

E S 2 n Z (n) -X (n) = 2S 2 and E S 2 n Z (n) -X (n) | Z (n) = S 2 n + S 2 . Since Σ2 n = ρ1 I q + ρ2 S 2 n Z (n) -X (n) , we get that E Σ2 n | Z (n) = ρ1 I q + ρ2 S 2 n + S 2 = 2ρ 1 I q + ρ 2 S 2 n + S 2 .
As a consequence, define

β2 = E S 2 n Z (n) -X (n) -2S 2 2 = E S 2 n Z (n) -S 2 2 + E S 2 n X (n) -S 2 2 = 2β 2 .
Similarly, put α2 = 2α 2 ; δ = 2δ 2 and σ2 = 2S 2 , I n = 2σ 2 then we have

ρ1 = α2 δ2 σ2 = 2 α 2 δ 2 σ 2 = 2ρ 1 and ρ2 = β2 δ2 = β 2 δ 2 = ρ 2 .
It thus follows with this natural choice of ρ 1 and ρ 2 that we have

E Σ2 n | Z (n) = Σ 2 n + Σ 2 and E Σ2 n = 2(ρ 1 I q + ρ 2 S 2 ) = 2Σ 2
The following lemma and its proof is an extension of corollary 1 of Panchenko (2003) (see [START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF]) with some adaptations to the multidimensional χ 2 case. See also [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] [START_REF] Bertail | Exponential bounds for multivariate self-normalized sums[END_REF] for the non penalized version of this result for q < n. Lemma 3. If there exists k ∈ N * , C 2 > 0 and c 2 > 0 such that, for all t ≥ 0,

P             sup u∈J q             √ nu ′ Zn -Xn u ′ Σ2 n u             ≥ √ t             ≤ C 2 Fk (c 2 t),
then, for all t ≥ 2k/c 2 ,

P        sup u∈J q        √ nu ′ Zn u ′ Σ 2 n + Σ 2 u        ≥ √ t        ≤ C 2 (c 2 t -k) 2 k/2 e -(c 2 t-k) 2 Γ k 2 + 1 and, for all t ≥ k/c 2 , P        sup u∈J q        √ nu ′ Zn u ′ Σ 2 n + Σ 2 u        ≥ √ t        ≤ C 2 F k+2 (c 2 t -k) Proof of Lemma 3 . Denote A n Z (n) = n sup u∈J q sup b>0 E 4b u ′ Zn -Xn -bu ′ Σ2 n u | Z (n) and C n Z (n) , X (n) = n sup u∈J q sup b>0 4b u ′ Zn -Xn -bu ′ Σ2 n u
We have by Jensen's inequality, that for any convex function ϕ

ϕ A n Z (n) ≤ E ϕ C n Z (n) , X (n) | Z (n) (7)
Finally, we can rewrite A n Z (n) and C n Z (n) , X (n) in an explicit form of self-normalized sums by maximizing according to b, the two expressions above, which leads to

A n Z (n) = sup u∈J q               √ nu ′ Zn ρ1 + ρ2 u ′ S 2 n + S 2 u        2        = sup u∈J q               √ nu ′ Zn u ′ Σ 2 n u + u ′ Σ 2 u        2       
Similarly, we have

C n Z (n) , X (n) = sup u∈J q                          √ nu ′ Zn -Xn u ′ Σ2 n u             2             
Now we conclude by applying lemma 1 to the inequality ( 7) with these expressions of A n Z (n) and

C n Z (n) , X (n) with C 2 = C 1 and c 2 = c 1 .
Proof of theorem 3 . We now control the Hotelling's T 2 n in the general case, by cutting its distribution tail into two parts. The first part allows us to get back to the expression above sup

u∈J q        √ nu ′ Zn √ u ′ Σ 2 n u+u ′ Σ 2 u 2       
controlled by Lemma 2. The second term is controlled by the largest eigenvalue of S 2 . Let

B n = sup u∈J q        u ′ Zn u ′ Σ 2 n u       
.

Notice that by construction we have, for any t > 0, (and particularly for any t > 2n)

n Z′ n Σ -2 n Zn ≥ t = n 1/2 B n ≥ √ t .
To transform the penalized self-normalised sum from the expression n Z′ n Σ 2 n -1 Zn to its "pseudo" version with the wrong normalization, sup

u∈J q        √ nu ′ Zn √ u ′ Σ 2 n u+u ′ Σ 2 u 2       
, let us introduce D n defined by

D n = sup u∈J q          1 + u ′ Σ 2 u u ′ Σ 2 n u          = sup u∈J q            1 + u ′ ρ 1 I q + ρ 2 S 2 u u ′ ρ 1 I q + ρ 2 S 2 n u            .
First, notice that we have

√ n B n D n = sup u∈J q        u ′ Zn u ′ Σ 2 n u        inf u∈J q                   1 + u ′ Σ 2 u u ′ Σ 2 n u          -1          ≤ sup u∈J q u ′ Zn √ u ′ Σ 2 n u 1 + u ′ Σ 2 u u ′ Σ 2 n u -1 ≤ sup u∈J q        √ nu ′ Zn √ u ′ Σ 2 n u+u ′ Σ 2 u 2        , (8)
for which we have an exponential bound by Lemma 3 and theorem 1. Thus by splitting the probability according to the event {D 2 n ≥ 1 + a}, for a > 1 and, for any t > 2n, we have

P n Z′ n Σ -2 n Zn ≥ t ≤ P       B n ≥ t n , D n ≤ √ 1 + a       + P D n ≥ √ 1 + a ≤ P B n D n ≥ t n (1 + a) + P D n ≥ √ 1 + a . ( 9
)
So now, it remains to treat the second term in the right-hand side of inequality [START_REF] Lahiri | A penalized empirical likelihood method in high dimensions[END_REF]. Notice that we have, for a > 1,

D n ≥ √ 1 + a =        sup u∈J q u ′ Σ 2 u u ′ Σ 2 n u ≥ a        = inf u∈J q u ′ Σ 2 n u u ′ Σ 2 u ≤ 1 a .
First, if S 2 = σ 2 I q is diagonal, then we have

u ′ Σ 2 u = u ′ (ρ 1 I q + ρ 2 σ 2 I q )u = ρ 1 + ρ 2 σ 2 .
Since inf

u∈J q u ′ Σ 2 n u = inf u∈J q u ′ (ρ 1 I q + ρ 2 S 2 n u) = ρ 1 ,
if we choose a such that a > (ρ 1 + ρ 2 σ 2 )/ρ 1 , then we have

P D n ≥ √ 1 + a ≤ P             inf u∈J q u ′ Σ 2 n u ρ 1 + ρ 2 σ 2 ≤ 1 a             = 0.
Remark that, in this case, we have ρ * 1 = σ 2 and ρ * 2 = 0 and it follows that the inequality is true for any a > 1. Notice that the proximity between S 2 and σ 2 I q is precisely controlled by the term α 2 = S 2 -σ 2 I q . Now consider the general case. First, notice that inf

u∈J q u ′ Σ 2 n u u ′ Σ 2 u = inf u∈J q u ′ Σ -1 Σ 2 n Σ -1 u = inf u∈J q        u ′ Σ -1 Σ -1 u 2 Σ 2 n Σ -1 u Σ -1 u 2 Σ -1 u 2 2        ≥ inf v∈J q v ′ Σ 2 n v × inf u∈J q u ′ Σ -2 u , with v = Σ -1 u Σ -1 u 2 ≥ ρ 1 µ 1 (Σ -2 ) = ρ 1 µ q (Σ 2 )
.

Now, using the optimal values ρ * 1 and ρ * 2 , we have the decomposition

Σ 2 ρ * 1 , ρ * 2 = ρ * 1 I q + ρ * 2 S 2 ,
it follows that we get

µ q (Σ 2 ρ * 1 , ρ * 2 ) = ρ * 1 + ρ * 2 µ q (S 2 ) and inf u∈J q         u ′ Σ 2 n ρ * 1 , ρ * 2 u u ′ Σ 2 ρ * 1 , ρ * 2 u         ≥ ρ * 1 ρ * 1 + ρ * 2 µ q (S 2 )
.

It follows that if we choose a such that

1 a < 1 1 + µ q (S 2 ) ρ *
and, since a * = 1

+ K 3 ρ * > 1 + µ q (S 2 ) ρ *
by the assumption (A 3 ), then, if a ≥ a * , we get

P D n ≥ √ 1 + a = 0. ( 10
)
As a consequence, we obtain an exponential inequality for any value a ≥ a * . Combining ( 9) and (10), we get, for any a ≥ a * ,

∀t > 2n, P n Z′ n Σ -2 n Zn ≥ t (1 + a) ≤ P √ n B n D n ≥ √ t . (11)
Let X (n) = (X i ) 1≤i≤n be an independent copy of Z (n) = (Z i ) 1≤i≤n . Applying theorem 1 to (Z i -X i ) 1≤i≤n which is symmetric, we obtain

P             sup u∈J q             √ nu ′ Zn -Xn u ′ Σ2 n u             ≥ √ t             ≤ 2e 3 9 Fn (t) ,
Thus, applying the lemma 3 to the inequality above implies that, for all t ≥ 2n,

P        sup u∈J q        √ nu ′ Zn u ′ Σ 2 n + Σ 2 u        ≥ √ t        ≤ 2e 3 9 (t -n) 2 n/2 e -(t-n) 2 Γ n 2 + 1 . (12)
Finally by combining expressions ( 8), ( 11) and ( 12), the result holds.

Proof of Theorem 4

The following lemmas will allow us to control explicitly the deviation P 1 ρ * n -1 ρ * > ϵ for small positive values of ϵ. Lemma 4. (Inversion) Let w > 0, and consider (W n ) n≥1 a sequence of positive random variables. Assume that there exists a nonnegative constant C 3 , such that ∀ϵ > 0, ∃N > 0, ∀n > N,

P (|W n -w| > ϵ) ≤ C 3 n 1 ϵ 2 .
Then there exists a function C 3;1/w nonnegative, such that ∀ϵ > 0, ∀n > N

P 1 W n - 1 w > ϵ ≤ C 3;1/w (ϵ) nϵ 2 ,
where C 3;1/w (ϵ) = C 3 w 4 1 + (wϵ) 2/5 5 .

Proof of Lemma 4 . Since w > 0, we have

P 1 W n - 1 w > ϵ w = P w W n -1 > ϵ Now, ∀η ∈ ]0, w[ we get P 1 W n - 1 w > ϵ w ≤ P w W n -1 > ϵ, |W n -w| ≤ η + P (|W n -w| > η) ≤ (I) + (II).
On the interval w -η;

w + η , f : x → w x is Lipschitz with | f ′ (x)| ≤ w (w-η) 2 , thus we obtain ∀W n ∈ w -η; w + η , w W n -1 ≤ w (w -η) 2 |W n -w| . ∀η ∈ ]0; w[ , (I) ≤ P w (w -η) 2 |W n -w| > ϵ ≤ C 3 n × w 2 ϵ 2 (w -η) 4 and since ∀η ∈ ]0; w[ , (II) ≤ C 3 n × 1 η 2 ,
it follows that

P 1 W n - 1 w > ϵ w ≤ C 3 n × w 2 ϵ 2 (w -η) 4 + C 3 n × 1 η 2 ≤ C 3 n min η∈]0;w[          w 2 ϵ 2 1 -η w 4 w 4 + 1 w 2 η w 2          α= η w ≤ C 3 nw 2 min α∈]0;1[ 1 ϵ 2 (1 -α) 4 + 1 α 2 ≤ C 3 nw 2 min α∈]0;1[ 1 ϵ 2 (1 -α) 4 + 1 α 4 ≤ C 3 nw 2 1 + ϵ -2/5 5 . Setting ϵ ′ = ϵ w and C 3;1/w (ϵ ′ ) = ϵ ′2 × C 3 w 2 1 + (wϵ ′ ) -2/5 5 = C 3 w 4 1 + (wϵ ′ ) 2/5 5
, the result holds.

Lemma 5. (Product) Consider u, v two positive scalars, and (U n ), (V n ) some random sequences. Assume that there exists nonnegative constants C4 and C4 such that ∀ϵ > 0, ∀n ≥ 1 :

P (|U n -u| > ϵ) ≤ C4 n 1 ϵ 2 and P (|V n -v| > ϵ) ≤ C4 n 1 ϵ 2 .
Then there exists a function C 4;uv such that ∀ϵ > 0,

P (|U n V n -uv| > ϵ) ≤ C 4;uv (ϵ) n 1 ϵ 2 ,
where 2 is a positive function of ϵ depending on u, v, C4 and C4 .

C 4;uv (ϵ) = C4 2uv+ϵ u 2 + C4 (2u)
Proof of Lemma 5. By straightforward inequalities, we get

P (|U n V n -uv| > ϵ) = P (|U n V n -uV n + uV n -uv| > ϵ) ≤ P V n |U n -u| > ϵ 2 , u |V n -v| ≤ ϵ 2 + P u |V n -v| > ϵ 2 ≤ P v + ϵ 2u |U n -u| > ϵ 2 + P |V n -v| > ϵ 2u ≤ P |U n -u| > ϵu 2uv + ϵ + P |V n -v| > ϵ 2u ≤ C4 n 2uv + ϵ ϵu 2 + C4 n 2u ϵ 2 ≤ C 4;uv (ϵ) n 1 ϵ 2 .
Lemma 6. Proximity between σ 2 , α 2 , β 2 , δ 2 and their estimators Let u 2 ∈ σ 2 , α 2 , β 2 , δ 2 be one of these quantities of interest and û2 n its corresponding estimator. Then ∀n ≥ 1 and ∀ϵ > 0, we have :

P û2 n -u 2 > ϵ ≤ C u 2 (ϵ) nϵ 2 , with • C σ 2 = √ K 2 for the case where u 2 = σ 2 and û2 n = σ 2 n , • C δ 2 = 2K 4 + (100 + K 2 1 )K 2 + 2 4 √ 6K 5/4 2 + 4K 3/2 2 + 2 2 3K 2 2 + 4K 1/2 2 K 1/4 2 + 2 √ 6 K 2 1 K 2 + 4K 2 (1 + 3K 2 ) + 2K 4 for the case where u 2 = δ 2 and û2 n = δ 2 n , • C β 2 (ϵ) = 4K 2 1 √ K 2 + C δ 2 + 2K 1 √ K 2 ϵ for the case where u 2 = β 2 and û2 n = β 2 n , • C α 2 (ϵ) = 2 3 C δ 2 + 2 4 K 2 1 √ K 2 + 2 2 K 1 √ K 2 ϵ.
for the case where u 2 = α 2 and û2 n = α 2 n .

Proof of Lemma 6.

Consider σ2

n and σ 2 . Recall that σ2 n = 1 q q j=1 1 n n i=1 y 2 i j and σ 2 = 1 q q j=1 E y 2 1 j = 1 q q j=1 µ j . Following the ideas of Ledoit and Wolf [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] who obtain the convergence of the fourth order moment, we rather control the second order moment as follows :

E σ2 n -σ 2 2 = E                  1 q q j=1 1 n n i=1 y 2 i j -µ j         2          = E                  1 n n i=1 1 q q j=1 y 2 i j -µ j         2          = 1 n 2 n i 1 =1 n i 2 =1 E         1 q q j=1 y 2 i 1 j -µ j × 1 q q j=1 y 2 i 2 j -µ j         .
This last expression is equal to zero for any i 1 i 2 because of the independence between observations. Thus we get

E σ2 n -σ 2 2 = 1 n 2 n i=1 E                  1 q q j=1 y 2 1 j -µ j         2          = 1 n E                  1 q q j=1 y 2 1 j -µ j         2          = 1 n          E                  1 q q j=1 y 2 1 j         2          -         E         1 q q j=1 y 2 1 j                 2          ≤ 1 n E                  1 q q j=1 y 2 1 j         2          ≤ 1 n          E                  1 q q j=1 y 2 1 j         4                   1/2 ≤ 1 n         1 q q j=1 E y 8 1 j         1/2 .
Therefore, using the second assumption (A 2 ), one gets

E σ2 n -σ 2 2 ≤ √ K 2 n . (13)
Finally, we have by Markov inequality the bound

P σ2 n -σ 2 > ϵ ≤ E σ2 n -σ 2 2 ϵ 2 ≤ √ K 2 nϵ 2 .

Consider δ2

n and δ 2 . Combining the expressions (A.2) and (A.3) on page 394 in Ledoit and Wolf ([11]) we get

δ2 n -δ 2 = σ2 n -σ 2 2 -2σ 2 σ2 n -σ 2 + S 2 n 2 -E S 2 n 2 .
Similarly using their expressions, from page 394 (A.4) to page 399, and page 390 (A.1), we have respectively the inequalities

Var S 2 n 2 ≤ 1 n K 2 1 K 2 + 4K 2 (1 + 3K 2 ) + 2K 4 and σ 2 ≤ K 2 .
Combining these expressions with Bienaymé-Tchebychev, Markov and Cauchy-Schwartz inequalities, we obtain a control of P δ2 n -δ 2 > ϵ by a function of n, ϵ, A 2 , A 4 and Var(∥S 2 n ∥ 2 ) where A k = E σ2 n -σ 2 k . Indeed we have, by Markov inequality, for all ϵ > 0,

P δ2 n -δ 2 > ϵ ≤ 1 ϵ 2 E σ2 n -σ 2 4 + 4σ 4 E σ2 n -σ 2 2 + E S 2 n 2 -E S 2 n 2 2 + 4σ 2 E σ2 n -σ 2 3 +4σ 2 E σ2 n -σ 2 S 2 n 2 -E S 2 n 2 + 2E σ2 n -σ 2 2 S 2 n 2 -E S 2 n 2 ≤ 1 ϵ 2        A 4 + 4σ 4 A 2 + Var S 2 n 2 + 4σ 2 A 2 A 4 + 4σ 2 A 2 Var S 2 n 2 + 2 A 4 Var S 2 n 2       
. Now by some previous controls established by Ledoit and Wolf ([11], page 394) we have

A 4 ≤ 96K 2 n ; Var S 2 n 2 ≤ 1 n K 2 1 K 2 + 4K 2 (1 + 3K 2 ) + 2K 4 = 1 n K and σ 2 ≤ K 2 .
Using the control stated in (13), A 2 ≤ √ K 2 /n, we can easily get the explicit constant C δ 2 as a function of K 1 , K 2 , and K 4 . For all ϵ > 0, for all n ∈ N * , we have

P δ2 n -δ 2 > ϵ ≤ 1 nϵ 2 96K 2 + 4K 2 K 1/2 2 + K + 4K 1/2 2 96K 1/2 2 K 2 + 4K 1/2 2 K 1/2 2 K + 2 96K 2 K ≤ 1 nϵ 2 2K 4 + (100 + K 2 1 )K 2 + 2 4 √ 6K 5/4 2 + 4K 3/2 2 + 2 2 3K 2 2 +4K 1/2 2 K 1/4 2 + 2 √ 6 K 2 1 K 2 + 4K 2 (1 + 3K 2 ) + 2K 4 ≤ C δ 2 nϵ 2 .

Consider β2

n and β 2 . Since δ 2 = α 2 + β 2 yielding δ 2 ≥ β 2 , Ledoit and Wolf showed ( [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF], proof of Lemma 3.4 page 401, lines from -12 to -6) that

-max | β2 n -β 2 |, | δ2 n -δ 2 | ≤ β2 n -β 2 ≤ | β2 n -β 2 |. From this we deduce | β2 n -β 2 | ≤ max max | β2 n -β 2 |, | δ2 n -δ 2 | , | β2 n -β 2 | ≤ max | β2 n -β 2 |, | δ2 n -δ 2 | . Controlling | β2 n -β 2 | leads to a control for | δ2 n -δ 2 | and | β2 n -β 2 |
. By the same arguments as in [START_REF] Ledoit | A well conditioned estimator for large dimensional covariance matrices[END_REF] (proof of Lemma 3.4, page 399, equation (A.7)), we have the following expression

β2 n -β 2 = 1 n ∥S 2 n -S 2 ∥ 2 +        1 n 2 n i=1 ∥Z i Z ′ i -S 2 ∥ 2 -E        1 n 2 n i=1 ∥Z i Z ′ i -S 2 ∥ 2               .
Now, splitting the probability into two terms, on the one hand, using Markov inequality on the first term and applying Bienaymé-Tchebychev inequality to the second term, we get

P | β2 n -β 2 | > ϵ ≤ 2 ϵ E 1 n ∥S 2 n -S 2 ∥ 2 + 4 ϵ 2 Var        1 n 2 n i=1 ∥Z i Z ′ i -S 2 ∥ 2        .
Following Ledoit and Wolf ([11], proof of Lemma 3.1 page 391 line +5), we have

E S 2 n -S 2 2 ≤ K 1 K 2 .
Moreover, we have (in the proof of Lemma 3.4, page 401 line +3)

Var        1 n 2 n i=1 Z i Z ′ i -S 2 2        ≤ K 2 1 K 2 /n.
We obtain

P β2 n -β 2 > ϵ ≤ 2 ϵ K 1 √ K 2 n + 4 ϵ 2 K 2 1 √ K 2 n .
Finally, with P δ2 n -δ 2 > ϵ ≤ C δ 2 nϵ 2 and

P β2 n -β 2 > ϵ ≤ P β2 n -β 2 > ϵ + P δ2 n -δ 2 > ϵ ,
we obtain

P β2 n -β 2 > ϵ ≤ 1 nϵ 2 4K 2 1 K 2 + C δ 2 + 2K 1 K 2 ϵ ≤ C β 2 (ϵ) nϵ 2 . 20
Remark that C β 2 (ϵ) tends to 4K

-α 2 = δ2 n -β2 n -δ 2 + β 2 .
For all ϵ > 0, we get

P α2 n -α 2 > ϵ ≤ P δ2 n -δ 2 > ϵ 2 + P β2 n -β 2 > ϵ 2 ≤ 2 2 C δ 2 nϵ 2 + 2 2 C β 2 (ϵ/2) nϵ 2 ≤ 1 nϵ 2 2 3 C δ 2 + 2 4 K 2 1 K 2 + 2 2 K 1 K 2 ϵ ≤ C α 2 (ϵ) nϵ 2 .
Remark that C α 2 (ϵ) tends to 2 3 C δ 2 + 2 4 K 2 1 √ K 2 when ϵ tends to 0.

In the next lemma 7, we control the proximity between 1/ ρ * n and 1/ρ * , that we denote g n (ϵ) and show that it is of order O (1/n). For this, we first apply product lemma 5 to β2 n and σ2 n . Then, we apply the inverse lemma 4 to β2 n σ2 n . Finally, we use another time product lemma 5 applied to α2 n and 1/ β2 n σ2 n . Lemma 7. Proximity between 1/ρ * and 1/ ρ * n For any ϵ > 0, have

g n (ϵ) = P 1 ρ * n - 1 ρ * > ϵ ≤ G(ϵ) nϵ 2
with C β 2 and C α 2 defined in lemma 6 and

G(ϵ) = C 3;1/β 2 σ 2 (ϵ) 2α 2 + ϵβ 2 σ 2 2 + 2 2 C α 2 (ϵ) β 4 σ 4 C 3;1/β 2 σ 2 (ϵ) =           K 1/2 2 2σ 2 β 2 + ϵ 2 β 8 σ 12 + 2 2 C β 2 (ϵ) β 8 σ 4           1 + β 2 σ 2 ϵ 2/5 5
.

Remark : the function C 3;1/β 2 σ 2 (ϵ) may be clearly bounded by a polynomial of degree 4 in ϵ. As a consequence, the function G(ϵ) may be bounded by a polynomial of degree 6.

Proof of Lemma 7. We apply the product lemma 5 to obtain a control for β2 n σ2 n thanks to lemma 6 which gives us some control of σ2 n and β2 n . For all ϵ > 0, one gets

P β2 n σ2 n -β 2 σ 2 > ϵ ≤ C 4;β 2 σ 2 (ϵ) nϵ 2 , (14)
with

C 4;σ 2 β 2 (ϵ) = K 1/2 2 2σ 2 β 2 + ϵ σ 2 2 + C β 2 (ϵ) 2σ 2 2 .
We now apply the inverse lemma 4 with inequality ( 14) and obtain a control of 1/ β2 n σ2 n . That is, for all ϵ > 0, we have

P 1 β2 n σ2 n - 1 β 2 σ 2 > ϵ ≤ C 3;1/β 2 σ 2 (ϵ) nϵ 2 , with C 3;1/β 2 σ 2 defined by C 3;1/β 2 σ 2 (ϵ) = C 4;β 2 σ 2 (ϵ) β 8 σ 8 1 + β 2 σ 2 ϵ 2/5 5
.

Applying the product lemma 5 with u = 1/(β 2 σ 2 ) and v = α 2 , we obtain

C 4;1/ρ * (ϵ) = C 3;1/β 2 σ 2 (ϵ) 2α 2 + ϵβ 2 σ 2 2 + C α 2 (ϵ) 2 2 β 4 σ 4 .
Lemma 7 gives a control of the first term on the right-hand side of this inequality so that it is sufficient to control the second term. Write

n 1 2 Ȳn 2 = 1 qn q j=1        n i=1 Y i, j        2 = 1 qn q j=1 n i=1 Y 2 i, j + 1 qn q j=1 n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j = I 1 + I 2 Since E ( I 1 ) = E 1 qn q j=1 n i=1 Y 2 i, j = σ 2
, use Bienaymé-Tchebychev inequality and the independence of the Y i 's to get

P         1 qn q j=1 n i=1 Y 2 i, j -σ 2 >         1 q q j=1 Y 4 1, j         + 1 nq 2 q j=1 q k=1 k j E Y 2 1, j Y 2 1,k ≤ 1 nq K 2 + 1 nq 2 q j=1 q k=1 k j E Y 4 1, j E Y 4 1,k ≤ 1 nq K 2 + 1 n         1 q q j=1 E Y 4 1, j         2 ≤ 1 nq K 2 + 1 n E         1 q q j=1 Y 4 1, j         ≤ 1 n K 2 1 q + 1 . (17)
Finally, combining inequalities [START_REF] Newey | Higher order properties of gmm and generalized empirical likelihood estimators[END_REF][START_REF] Panchenko | Symmetrization approach to concentration inequalities for empirical processes[END_REF], we get the following control for I 1 , for η > 0

P I 1 -E(I 1 ) > η 2 ≤ 4 η 2 1 n K 2 1 q + 1 . (18)
Now, we focus on I 2 . Using the independence between the observations Y i 's, we have

E (I 2 ) = E              1 qn q j=1 n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j              = 0.
By Bienaymé-Tchebychev inequality, we have, for η > 0

P I 2 > η 2 ≤ 4 η 2 E                           1 q q j=1 1 n n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j              2              . (19)
Furthermore, since 1 n = (n-1) 2 4 2 n(n-1)

2

, we can express the expectation above as the expectation of a U-statistic

E                           1 q q j=1 1 n n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j              2              = (n -1) 2 4 E                           2 n(n -1) n i=1 n i ′ =1 i ′ i 1 q q j=1 Y i, j Y i ′ , j              2              .
More precisely, this is a U-statistic of degree 2 with kernel w

(Y i , Y i ′ ) = 1 q q j=1 Y i, j Y i ′ , j , with E [w (Y i , Y i ′ )] = 0 and degenerated gradients E [ w (Y i , Y i ′ ) | Y i ] = 0 and E [ w (Y i , Y i ′ ) | Y i ′ ] = 0,
Using the expression of the variance of this U-statistic as given in Lee (2019) [START_REF] Lee | U-statistics: Theory and Practice[END_REF], it follows that

E                           1 q q j=1 1 n n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j              2              = (n -1) 2 4 1 n(n-1) 2 n -2 0 Var (w (Y i , Y i ′ )) = n -1 2n E                  1 q q j=1 Y 1, j Y 2, j         2          . (20)
Now, we have by independence

E                  1 q q j=1 Y 1, j Y 2, j         2          = E         1 q 2 q j=1 q k=1 Y 1, j Y 2, j Y 1,k Y 2,k         = 1 q 2 q j=1 q k=1 E Y 1, j Y 1,k 2 .
Recall that E Y 1, j Y 1,k = 0 if j k. By using Hölder inequalities repetitively and by hypothesis (A 2 ), we have

1 q q j=1 E Y 2 1, j 2 ≤ 1 q q j=1 E Y 8 1, j 1 2 ≤ K 1 2 2 , yielding E                  1 q q j=1 Y 1, j Y 2, j         2          ≤ 1 q K 2 . (21)
Finally, combining equations [START_REF] Talagrand | The missing factor in hoeffding's inequalities[END_REF][START_REF] Tikhonov | On the regularization of ill-posed problems[END_REF] and ( 21), we obtain a control for I 2 as follows

P I 2 > η 2 = P              1 qn q j=1 n i=1 n i ′ =1 i ′ i Y i, j Y i ′ , j > η 2              ≤ 1 η 2 2(n -1) qn K 2 .
Finally, assumption (A 1 ) implies

P (|∆ n | > ϵn) = P q 1 ρ * n - 1 ρ * n 1/2 Ȳn 2 > ϵn ≤ P n 1 2 Ȳn 2 1 ρ * n - 1 ρ * > ϵ K 1 ≤ P n 1 2 Ȳn 2 -σ 2 1 ρ * n - 1 ρ * > ϵ 2K 1 + P 1 ρ * n - 1 ρ * > ϵ 2σ 2 K 1 .
Using the fact that P(AB > ϵ) ≤ P(A > √ ϵ) + P(B > √ ϵ), and the definition of the function g n in lemma 7, we have

P (|∆ n | > ϵn) ≤ P n 1 2 Ȳn 2 -σ 2 > ϵ 2K 1 + g n ϵ 2K 1 + g n ϵ 2σ 2 K 1 ≤ P I 1 -σ 2 > 1 2 ϵ 2K 1 ) + P I 2 > 1 2 ϵ 2K 1 + g n ϵ 2K 1 + g n ϵ 2σ 2 K 1 .
Therefore, by inequalities ( 18) and ( 22), considering η = ϵ 2K 1 , we get

P (|∆ n | > ϵn) ≤ 4 ϵ 2K 1 2 × √ K 2 n 1 q + 1 + 1 2 n -1 n √ K 2 q + g n ϵ 2K 1 + g n ϵ 2σ 2 K 1 ≤ 4K 1 √ K 2 ϵn 2 + 1 q + K 1 + g n ϵ 2K 1 + g n ϵ 2σ 2 K 1 .
We now complete the proof of the theorem by handling the term (II). By lemma 7, we get

P (|â n -a * | > ϵ) = P 1 ρ * n - 1 ρ * > ϵ K 3 = g n ϵ K 3 . ( 22
)
With inequalities ( 15), ( 15), (22), and (22), and using the expression of G to bound g n given in lemma 7, we obtain

P n Z′ n Σ * -2 n Zn ≥ u 1 + â * n + 2ϵ ≤ P n Z′ n Σ * -2 n Zn ≥ u (1 + a * ) + 4K 1 √ K 2 ϵn 2 + 1 q + K 1 + g n ϵ 2K 1 + g n ϵ 2σ 2 K 1 + g n ϵ K 3 ≤ 2e 3 9 u -n 2 n 2 e -u-n 2 Γ n 2 + 1 + 1 n C (ϵ) ϵ ,
where C(ϵ) is independent of n such that In the independent case, it seems to be of the order 2q/n up to some factor probably depending on the variance of the eigenvalues of the matrix. Notice that when q = n the center of the distribution is rather stable but with a smaller variance as n grows. In the dependent case, the "optimal" penalization can decrease drastically even if the value of α is fixed but is even more stable (in mean). This can be explained by the fact that we have

C (ϵ) = 4K 1 K 2 2 + 1 q + K 1 + 2K 1 G ϵ 2K 1 + 4K 2 1 σ 4 ϵ G ϵ 2σ 2 K 1 + K 2 3 ϵ G ϵ K 3 .
α 2 = ∥S 2 -σ 2 I q ∥ 2 = 1 q q k=1 σ 2 k -σ 2 2 + 2 q q k, j<k
Cov 2 (Z 1,k , Z 1, j ).

In the independent case, α 2 is essentially the empirical variance of the eigenvalues. But in the dependent case, the covariance terms clearly increase which induces a reduction of the penalizing term since ρ * = β 2 α 2 σ 2 . Now, we focus on the distribution of the optimal penalization when there is a strong dependent component. The graphics panel in Figure 4 compares the distribution of ρ * for case (iii) (independent case, first column) and case (iv) (dependent case, second column) respectively on the first row for fixed sample size n = 50 and varying q ′ s, on the second row for q = n varying in 50, 100, 200. Finally, the last row shows this distribution when q = 2n and n varies in 50, 100, 200. Figure 4 compares the distribution of the "optimal" estimated penalty for identical values of α 2 (depending on q) for the two scenarios, that is, the left (i.i.d.) and the right column (dependent case) and for different values of q. α 2 is equal respectively to 35.74, 55.63, 67.12, 74.19, 78.83, 82.04, 84.37, 86.13 for the values of q equal to 50, 100, 150, 200, . . . , 400. We see that, even for an identical value of α 2 , i.e. for a given distance between the true covariance matrix and the diagonal matrix, the distribution of optimal penalty term is systematically more concentrated around smaller values in the dependent case (second column). This conclusion is true for all values of q and n. In other words, the stronger the dependence, the smaller the optimal penalty term. Recall that, in Figure 3, we consider a fixed value α 2 = 1, 10 for all values of q. The comparison of Figures 3 and4 shows that when the α 2 term is big, this leads to a smaller penalization term. Furthermore, this penalization becomes smaller when q grows with n. This is quite in contradiction with the practice which suggests using a penalization of the order 2q/n as noticed in Figure 3. The distance to the homoscedastic framework has thus a very strong impact on the penalty. The following Figure 5 and Figure 6 give the histogram of the penalized Hotelling's statistic obtained by K = 999 Monte-Carlo simulations, respectively for the independent and dependent case but with the same α 2 . We present first the case for s = 0.6 (Figure 5) and then the case s = 0.99 (Figure 6). From this simulation study, we conclude that our bounds give some interesting information both on the optimal penalty that one may choose and on the order of the bounds. However, there is still room for improvement.

Fig. 1 :

 1 Fig. 1: True covariance S 2 , sample covariance S 2 n , and Σ 2 n (ρ * 1 , ρ * 2 ), Σ 2 * n respectively regularized and penalized sample covariance.

Fig. 2 :

 2 Fig. 2: True covariance S 2 , sample covariance S 2 n , regularized and penalized estimators of S 2 n , respectively Σ2 n and Σ * 2 n .

Fig. 3 :

 3 Fig. 3: Distributions of ρ * , independent (first column, case (i)), dependent with s=0.6 (second column, case (ii)). Vertical lines represent the empirical mean of the corresponding distribution.

Fig. 4 :

 4 Fig. 4: Distributions of ρ * , independent (first column, case (iii)), dependent with s=0.99 (second column, case (iv)) with the same α 2 . Vertical lines represent the empirical mean of the corresponding distribution.

Fig. 6 :

 6 Fig. 6: Distributions of T 2n ( ρ ⋆ n , 1), the penalized Hotelling's statistic, in independent (first column, case (iii)) and dependent with s=0.99 frameworks (second column, case (iv))

Fig. 7 :

 7 Fig. 7: Comparison of the true tail of the penalized Hotelling's statistic and the tail given by the bound for different values of n, q. s = 0.6 in the right column. The red dotted lines refer to the bounds for the ordered corresponding n.

Fig. 8 :

 8 Fig. 8: Comparison of the true tail of the penalized Hotelling's statistic and the tail given by the bound for different values of n, q. s = 0.99 in the right column. The red dotted lines refer to the bounds for the ordered corresponding n.

2 1 √

 1 K 2 + C δ 2 when ϵ tends to 0. and α 2 + β 2 = δ 2 , one can easily see that α2 n

	Consider α2 n and α 2 .
	Since we have α2 n = δ2 n -β2

n

Acknowledgment

This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

Credit authorship contribution statement

The three authors contributed equally. This paper is a part of the PhD thesis of El Mehdi ISSOUANI.

Appendix

In the first part of this section we provide all the proofs of Theorem 1, 2, 3 and 4 given in the sections 2, 3 and 4. In the second part of the appendix, we detail all the calculations to obtain an explicit constant C(ϵ) appearing in Theorem 4 when replacing the true quantities by their empirical estimators.

Remark that when ϵ tends to 0, C 4;1/ρ * tends to

n and a * = 1 + K 3 ρ * . For any u > 2n, we have for ϵ > 0

We start by establishing a control for (I). Define

Zn , then we have

Since u > 2n > n, we have

Theorem 3 gives us an exponential bound controlling the first term of the right hand of the inequality when a = a * and u > 2n. Now use the following matrix factorisation A -1 -B -1 = A -1 (B -A) B -1 to control the second term in the right hand with A = Σ * 2 n and B = Σ * 2 n . It is easy to see that B -A = ρ *ρ * n I q , then we obtain

Recall that

then using the same rotation matrix O n , we obtain Σ * -2 n Σ * -2

It follows that

Since, for any

, and because we have ∥x∥ 2 2 = q∥x∥ 2 , we get

Supplementary material -Simulations

In this supplementary material, we explore graphically for different distributions, how the dependence structure of the observations and the distance to homoscedasticity impact the penalization constants and the tail of the T 2 Hotelling distribution. We generate Gaussian random variables with a given covariance structure corresponding respectively to the following scenarios:

• scenario a) the components Z i, j , j ∈ {1, . . . , q} are independent with variance σ j, j , that is Z i are i.i.d N(0, S 2 ) with S 2 = diag(σ j, j ) 1≤ j≤q for i ∈ {1, . . . , n}. The σ j, j are themselves generated randomly in a LN(0, η 2 ). We actually expect the variance of the eigenvalues to have a strong influence on the penalized term. The variance η 2 is calibrated for comparison with the dependent case and chosen equal to log(1

to ensure that the distance between S 2 and σ 2 I q is indeed equal to α 2 (which is chosen the same in the dependent case Notice that in our framework the quantity α 2 is a measure of the complexity of the problem. Actually, if α 2 = 0, we can directly use the identity matrix instead of the empirical variance and there is no need for penalizing. For this reason, we are going to compare our simulation results for some given fixed values of α 2 respectively in the dependent and independent cases. For that, we now consider four simulation cases:

(i) scenario a) with α 2 close to 1.10 (note that actually the value of α 2 depends on q but is close to this value in all simulations) corresponding to a standard deviation η = 0.71;

(ii) scenario b) with the same values of α 2 as in (i) corresponding to a dependence parameter s = 0.6;

(iii) scenario a) with α 2 equal respectively to 35.74, 55.63, 67.12, 74.19, 78.83, 82.04, 84.37, 86.13 corresponding to η between 1.89 and 2.11 respectively for the value of q ∈ {50, 100, 150, 200, . . . , 400};

(iv) scenario b) with the same values of α 2 as in (iii) corresponding to a dependence parameter s = 0.99.

For each set of parameters, (i) to (iv), for n ∈ {50, 75, 100, . . . , 200}, we generate n r.v.'s of size q ∈ {50, 100, 150, . . . , 400} with q ≥ n. The procedure is repeated K = 999 times independently to obtain Monte-Carlo approximations respectively of the distributions of the penalized T 2 n -Hotelling's statistic (with estimated parameters) and the distribution of the penalizing parameter ρ * n . The graphics in Figure 3 compare the distribution of ρ * for case (i) (independent case, first column) and case (ii) (dependent case, second column of the panel) respectively.

-on the first row: for fixed sample size n = 50 and varying q ′ s equal 50, 200 and 400, -on the second row : for q = n equal to 50, 100, 200, -on the last row shows this distribution when q = 2n and n is equal respectively to 50, 100, 200.

The figures in panel 3 show that the dependence structure tends to lead to smaller penalization constants. By comparing the rows, it seems that there is a proportionality between the penalization parameter ρ * and q/n. n ( ρ ⋆ n , 1), the penalized Hotelling's statistic, in independent (first column, case (i)) and dependent with s=0.6 frameworks (second column, case (ii)).

Compare figures 5 and 6, focusing first on the first column corresponding to the independent case. We see the importance of the value α (the distance to homoscedasticity) in the distribution. Increasing α 2 tends to lead to a smaller penalization and to a less precise approximation of the covariance matrix yielding a shift of the distribution of the Hotelling's statistic on the right. Comparing the two columns (independent and dependent case), we see that the distributions are centered around quite similar values but tend to be more concentrated in the independent case. Increasing the value of α 2 in figure 6 tends to reverse this phenomenon. These figures also emphasize the role of the ratio q/n.


Online ACTRIS MOOC on observing technologies

Sustainability of ACTRIS beyond the term of the project requires actions towards both end-users and stakeholders that will address education, training, and dissemination of ACTRIS results to the community and the public. Training is key to this end. Ensuring proper training of scientists/engineer/students across the ACTRIS components and beyond is essential in the long-run to make sure of efficient use of ACTRIS Research Infrastructure platforms.

Developing a widespread knowledge of Earth Observation relevance and challenges also from the technical viewpoint will require to address the different strategies to transfer the technical information on the use of instruments and data sources within the experimental framework. The use of forefront methods and their fast evolvements imply the challenge to maintain a very high level of qualification of all the data producers. While ACTRIS-2 maintains its unique cross-border training courses that have made the reputation of the RI, it also needs to differentiate its training offers and open to new users by offering additional opportunities within and outside the ACTRIS consortium to train new users on the ACTRIS products and tools. A proposal was made to develop an online MOOC (Massive Open Online Course) -type ACTRIS course directed towards new operators of observing facilities (to be accessed through the ACTRIS data centre).

Among the main objectives of the MOOC was to develop a platform where harmonized practices for the best atmospheric composition observations could be found, and seen as a place for sharing scientific and technical experiences as well as training new operators. The potential services provided have to be explained and the whole data provision chain well described.

Definition and interest of online resources

An online open course is a sequence with a pedagogical objective, containing training sessions and skill evaluation tests. The sequence can have different formats such as video, quiz, serious games. Online courses are exclusively broadcasted on the web from a dedicated platform. The course is easily accessible and designed for a well-defined profile of attendees. An online course is not just a passive activity. Attendees are asked to evaluate their skills along the course. A certificate of qualification can be obtained at the end. Depending on the objectives we can distinguish:

-MOOC (Massive Online Open courses) designed to have the larger number of participants. They are well adapted to present general topics like "Air quality issues". -Small Online Open Courses (SOOC) dealing with more focused subjects and consequently expecting less participants. Example is an online training course presenting technical procedures on how to operate instruments or analyze data -Corporate Open Online Courses (COOC) on topics specific to a defined community. An example can be the description of an infrastructure organization or of internal processes Online courses can be structured and combined as a set to explain/present from the general framework to very technical procedures. In addition to online courses, other online resources are useful to both promote and present different types of platforms.

The interests of online courses are multiple: -The rise in competence is very flexible. Participants can learn at different speeds depending on their a-priori level and their availability -It is a very powerful tool when potential learners are spread geographically since it can be accessible from anywhere, providing Internet access is available. Resources can also be made available as transcripts (for videos) or PDF files in case of low-speed connections, videos can be subtitled in different languages. -An online course is most of the time very innovative and playful, using serious game, tutorials, auto-evaluation tests. It aims at favouring an active learning behaviour. Besides by monitoring/encouraging the involvement of participants, the courses can be adapted and optimized. -Beyond the training session, the online course will largely promote it producers and their framework.

Depending on the subject, although very efficient for theoretical aspects, the online courses may have to be completed by additional practical sessions in classroom or in situ.

How to create a MOOC

The time and the difficulty to create a MOOC is very often underestimated. The creation of MOOC has to be led as a large project involving a team combining several skills. A general schematic of the structuration of the different tasks and timeline are presented in Figure 1. The team should include experts in the concerned topics, experts in training engineering and communication specialists. A project leader has to be appointed in order to manage the entire process from designing to operation.

Fig. 1: Production steps of an online course

Regarding the production process and the corresponding needs, the production cost can be evaluated. By considering a MOOC (in mother language) of 4 to 5 weeks with a volume of 2 to 5 hours per week for the learners, the Institute-Mines-Telecom (IMT) has estimated the cost at 18 Pers.Months including half for the teaching team and half for the support (IMT Report "4 years of MOOC from 2013 to 2018"). The support mainly stands for the video production (3 PM), the management (2 PM), communication (2 PM), learning engineering (1 PM) and the broadcast platform (1 PM).

Existing online resources (not exhaustive)

Table 1 lists a few resources (in English 1 ) which have been (in grey) or are still available online at no cost for the users.

1 Some potentially interesting resources exist already in French, for instance:  e-learning modules from the Airducation project (https://www.airducation.eu/parcoursenvironnement);  training schools for Bachelor students using integrated numerical tools (digital notebooks) and observation databases at SIRTA,  e-learning modules developed in the framework of Labex-IPSL (https://claroline.locean-ipsl.upmc.fr/) and at LACY (https://lacy.univ-reunion.fr/production/films-multimedia/mooc/)  videos presenting observatories (SIRTA: https://www.youtube.com/watch?v=4-fmTc2JYUc; Puy-de-Dôme: https://videotheque.cnrs.fr/index.php?urlaction=doc&id_doc=4056; MAIDO: https://lacy.univreunion.fr/production/films-multimedia/visite-virtuelle-maido)

Atmospheric physics and chemistry

This course provides an introduction to the physics and chemistry of the atmosphere including experience with computer codes, aerosols and theories of their formation, evolution, and removal. Topics covered include, gas and aerosol transport from urban to continental scales, coupled models of radiation, transport, and chemistry, solution of inverse problems to deduce emissions and removal rates, emissions control technology and costs, and applications to air pollution and climate.

Typical cost for the implementation of a MOOC and implications of an ACTRIS-MOOC

Implementing a MOOC within ACTRIS cannot be proposed without a careful cost-benefits analysis that should not only include cost of developing the course but also its maintenance in the short-to-medium terms. There is little publicly available information on MOOC costs and the original dedicated budget in ACTRIS-2 was close to 4 k€. Discussion with different parties (administrators, faculty members, researchers, and other actors in the MOOC space) clearly showed that institutional resources required for the development and delivery of MOOCs, is much higher, ranging from several Tens of K€ to several thousands of k€. The original budget planed was therefore clearly underestimated. Clearly, MOOCs are a one-time investment profitable to many users and should still be considered as a cost-effective way to transfer knowledge. In the case of ACTRIS however, to expected number of potential users following a MOOC on the technical implementation of ACTRIS techniques will remain limited to mostly self-motivated learners. As an indication, the whole series of ACTRIS courses offered with WP2 and WP3 in ACTRIS-2 are currently gathering less than 100 participants / year which could be a higher estimate of followers of an ACTRIS MOOC. Estimates of typical cost/students of current online courses is ranging anywhere from 50€ to 500€/course. Considering a 5-year MOOC lifetime, an initial investment of 25k€ and 5 k€/year maintenance, and 50 followers/session, cost/student will be on an acceptable range of 100€/course.

During ACTRIS-2, several institutions were contacted to evaluate their interest for funding (or contributing to funding) a MOOC on Earth Observation techniques from the ground. We received positive answers from:

 The World Meteorological Organisation that was seeing the MOOC as a additional resource to GAWTEC (https://www.gawtec.de/)  The IMT-Douai, which is a higher education institution already experienced in on-line courses  The space agencies (ESA, EUMETSAT) although it was only seen as a potential addition to already existing on-line resources. Recently (EGU2019) ACTRIS-2 attended a meeting with key stakeholders and science representatives to propose and identify the priorities for training and applications to be promoted in the next years in the field of atmospheric composition. Specific needs of ACTRIS for training dedicated to in-situ component of Earth Observation was mentioned by ACTRIS to the Space agencies organizing the session, which have in their agenda an obvious priority on satellite observations from past, current and planned missions.

Conclusion and suggestions for implementation

Training is a focal point of ACTRIS activities that will need to be further developed in the future. For this purpose, online resources constitute high potential tools in addition to face-to-face training sessions such as summer and winter schools. Several online resources related to ACTRIS scope already exist within the atmospheric science community and could be promoted through a dedicated platform. For instance, the ACTRIS website may include a section related to available "online resources" structured depending on the needs: general knowledge on atmospheric science, observing systems, up to technical procedures. That will simply be an inventory of existing resources in the laboratories involved within ACTRIS, and beyond, at the European level. Developing specific targeted MOOC-type on-line resources for ACTRIS operations is a more complex issue that will require a substantial amount of additional resources. At minima, the simplest approach could be to use current resources as as they are, simply providing subtitled/translated or easily modified version with the authorization of owners to meet the needs of the infrastructure at a reasonable cost. Ideally, the approach should be more ambitious.

Here are the following recommendations:  Focus of a Corporate Open Online Courses (COOC) rather than a MOOC. COOCs are more appropriate in terms of product to meet the internal needs of ACTRIS, i.e. training resources to develop technical skills. A MOOC will ideally have larger scope that ACTRIS and it is clear that an offer already exist on the market. Developing a MOOC, with specific ACTRIS sessions, is an option that could be under the responsibility of one or more higher education partner(s) in ACTRIS, already proposing online resources related to atmospheric sciences.  In that respect, a suggestion at this stage would be to identify ACTRIS partners willing to design and produce a core MOOC covering ACTRIS topics, which could be interlinked with existing online resources. The objectives of this ACTRIS MOOC would be for the followers: 1) to understand and be aware of the main scientific issues ACTRIS deals with, 2) to be convinced of the usefulness of atmospheric observations, 3) to know how these observation data are produced and 4) to have a presentation of the services ACTRIS can provide.  A wider ENVRI (Environmental Research Infrastructures) approach can be suggested, perhaps at this stage limited to the atmospheric domain, with other RIs such as ICOS, IAGOS participating.  A more targeted COOC focusing on the technical aspects of Earth Observation from the ground is clearly a better choice than a MOOC in the context of ACTRIS. In the organizational framework of the RI, it would clearly be under the responsibility of ACTRIS Topical centres that have the expertise to evaluate the need for such resources. It is a possibility in the future to include among their activities the development of operating procedures which will be made online accessible.  A 5-year workplan is needed to both design a dedicated COOC that would be accessible through the ACTRIS website. It is recommended that the COOC would address the full data chain, from provision to data transmission, and possibly including specific sessions dedicated to data usage. AT this stage, it is more reasonable to address the internal needs of ACTRIS community (also including the international community involved in Earth Observations from the ground) rather than focussing on a wider audience, without a clear concept for an added-value product.  It remains that online resources COOC/MOOC remain tools that could be more efficiently use in the future, for many applications in particular related to QA/QC in the RIs, that will prove to be a cost-efficient solution for the operational phase of the RI.

Table1: Non-exhaustive list of resources in English

	MOOC MOOC Open	Earth Observation Introduction to Atmospheric	Introduction to optical Earth observation -monitoring Introduction to GIS and the principles of spatial data Detailed overview of the chemical transformations	5 weeks 6 weeks 13 lectures,	ESA (FutureLearn) / University of West Florida (Canvas network) / MIT (MITOpenCourseWare) /
	Type b d course	Title from Space: the Geographical chemistry Optical View Information Systems (GIS)	Content / Objective our planet from satellites, using photography, imaging in personal life as well as applications of GIS across that control the abundances of key trace species in in various wavelengths, lidar and other optical sensing various industries. Major components of the course the Earth's atmosphere. Emphasizes the effects of technologies. include computer representation of geographic human activity on air quality and climate. Topics	Duration (4) (2-3) PDF files with (hours/wk) simplified	https://www.futurelearn.com/courses/optical-earth-https://www.canvas.net/browse/uwf-geodata-https://ocw.mit.edu/courses/civil-and-environmental-Provider (platform) / link observation center/courses/geographical-information-systems-gis-0 engineering/1-84j-atmospheric-chemistry-fall-2013/lecture-notes/
	MOOC a MOOC e c MOOC Open course	Echoes in space / Introduction to radar remote sensing Discovering Science: Atmospheric pollution causes and Air Quality / Air Chemistry	Basics that will help to understand where this technology is coming from, how the images are How satellite data is acquired and used, the range of information, the basics of GIS databases, spatial include photochemistry, kinetics, and data types available, and the terminology and analysis with GIS, and application areas of GIS. At the thermodynamics important to the chemistry of the techniques involved. end of the course, students will have an atmosphere; stratospheric ozone depletion; oxidation acquired and which manifold applications already use Detailed case studies of how this data is used in understanding of elementary GIS theory and chemistry of the troposphere; photochemical smog; Radar Remote Sensing to help protect our planet diverse fields, from climate science to humanitarian examples of GIS-based solutions in the world around aerosol chemistry; and sources and sinks of  History of Radar technology and the discovery of relief, monitoring of urban change to agriculture, and them. greenhouse gases and other climate forcers. electromagnetic waves many other areas. Learn about the role that radiation and greenhouse  Image acquisition Geometry of airborne and gases play in our understanding of atmospheric space borne Radar systems  Land applications of Radar remote sensing  Applications of radar remote sensing over Water pollution and its environmental, health, social and system are revealing extraordinarily diverse management Focus on issues related to air quality. Basics of air missions to other planets and moons in our solar  Application of Radar remote sensing for Hazard chemistry. As levels of CO2 in the atmosphere increase, you will discover the solutions scientists are exploring to reduce levels. And, discover how	5 weeks (3) content 2 weeks (5) (2-3) 3 weeks	ESA (EO College) / https://eo-college.org/courses/echoes-in-space/ University of Leeds (FutureLearn) / https://www.futurelearn.com/courses/discovering-science-https://www.edx.org/course/air-pollution-causes-and-IMT Lille Douai (EdX) / atmospheric-chemistry
	MOOC b MOOC a Open course	Monitoring Climate from Space impacts Monitoring Atmospheric Composition Atmosphere, Ocean and Environmental Change	What is Earth observation? How do we observe the Earth with satellites? And what role does Earth economic effects atmospheres. Explore how an exponential increase in the discovery of new exoplanets is helping us learn  Explore how we observe and measure the about the evolution of our own atmosphere. observation play in climate policy and planning? atmosphere with satellites, ground-based and How do we use different types of mission, other forms of in-situ measurements instrumentation and data to study changes to our  Understand the importance of satellite atmosphere, land, oceans and ice? How does Earth observation help us set policy; plan for climate risk, resilience and adaptation; and manage resources and biodiversity? How do we make sense of the large amount of data produced by Earth observation? Can crowdsourcing and citizen science play a role in developing climate change models? observations and other forms of measurements Explores the physical processes that control Earth's for atmospheric monitoring atmosphere, ocean, and climate. Quantitative  Investigate how atmospheric data is used in resources. pollutants in the atmosphere Earth's climate, global warming, energy, and water monitoring long-range transportation of and productivity, the seasons, El Niño, the history of  Recognise the importance of the data for climate, the ozone layer, air pollution, ocean currents in conjunction with models Topics include clouds, rain, severe storms, regional policy and decision-making, in a range of arenas, methods for constructing mass and energy budgets.	5 weeks (3) 5 weeks (3) 35 lectures, videos of 45 (self-paced) min each	impacts ESA (FutureLearn) / https://www.futurelearn.com/courses/climate-from-space EUMETSAT, CAMS, ECMWF (FutureLearn) / https://www.futurelearn.com/courses/atmospheric-composition Yale University (Open Yale courses) / https://oyc.yale.edu/geology-and-geophysics/gg-140?qt-course=2#qt-course

  Closed for now; b Not available anymore; c Archived (still open for registration but support is not guaranteed between sessions; first session in Oct. 2018 was followed by ~870 learners from 71 countries); d Closed for now, last session in 2015; e Running at the time of the report (started onMar. 25, 2019)

			modeling and approximation, and the observed large-		planetary-sciences/12-810-dynamics-of-the-atmosphere-
			scale phenomenology of the atmosphere. Only then		spring-2008/lecture-notes/
			are the basic equations derived in rigorous manner.
			The equations are then applied to important
			problems and methodologies in meteorology and
			climate, with discussions of the history of the topics
			where appropriate. Problems include the Hadley
			circulation and its role in the general circulation,
			atmospheric waves including gravity and Rossby
			waves and their interaction with the mean flow, with
			specific applications to the stratospheric quasi-
			biennial oscillation, tides, the super-rotation of Venus'
			atmosphere, the generation of atmospheric turbulence, and stationary waves among other problems. The quasi-geostrophic approximation is derived, and the resulting equations are used to examine the hydrodynamic stability of the circulation	27 lectures, PDF files for 13 of them	MIT (MITOpenCourseWare) / https://ocw.mit.edu/courses/chemical-engineering/10-571j-atmospheric-physics-and-chemistry-spring-2006/lecture-notes/
			with applications ranging from convective adjustment
			to climate.
	Open	The Atmosphere	The composition and circulation of the atmosphere	19 lectures,	UC Irvine (UCI Open) /
	course		with a focus on explaining the fundamentals of	videos of ~1	http://ocw.uci.edu/courses/ess_5_the_atmosphere.html
			weather and climate. Topics include solar and	hour each
			It is intended for undergraduates and first year terrestrial radiation, clouds, and weather patterns.
			graduate students.
	Open	Dynamics of the	This course begins with a study of the role of	13 chapters,	MIT (MITOpenCourseWare) /
	course	atmosphere	dynamics in the general physics of the atmosphere,	complete	https://ocw.mit.edu/courses/earth-atmospheric-and-
			the consideration of the differences between	PDF files

a