Antoine Godichon-Baggioni
email: godichon@insa-toulouse.fr

Online estimation of the asymptotic variance for averaged stochastic gradient algorithms

Keywords: Stochastic Gradient Algorithm, Averaging, Central Limit Theorem, Asymptotic Variance

Stochastic gradient algorithms are more and more studied since they can deal efficiently and online with large samples in high dimensional spaces. In this paper, we first establish a Central Limit Theorem for these estimates as well as for their averaged version in general Hilbert spaces. Moreover, since having the asymptotic normality of estimates is often unusable without an estimation of the asymptotic variance, we introduce a new recursive algorithm for estimating this last one, and we establish its almost sure rate of convergence as well as its rate of convergence in quadratic mean. Finally, two examples consisting in estimating the parameters of the logistic regression and estimating geometric quantiles are given.

Introduction

High Dimensional and Functional Data Analysis are interesting domains which do not have stopped growing for many years. To consider these kinds of data, it is more and more important to think about methods which take into account the high dimension as well as the possibility of having large samples. In this paper, we focus on an usual stochastic optimization problem which consists in estimating

m := arg min h∈H E [g (X, h)] ,
where X is a random variable taking values in a space X and g : X × H -→ R, where H is a separable Hilbert space. In order to build an estimator of m, an usual method was to consider the solver of the problem generated by the sample, i.e to consider M-estimates (see [START_REF] Huber | Robust Statistics[END_REF] and [START_REF] Maronna | Robust statistics[END_REF] among others). In order to build these estimates, deterministic convex optimization algorithms (see [START_REF] Boyd | Convex optimization[END_REF]) are often used (see [START_REF] Vardi | The multivariate L 1 -median and associated data depth[END_REF], [START_REF] Oja | Asymptotic properties of the generalized median in the case of multivariate normality[END_REF] in the case of the median), and these methods are really efficient in small dimensional spaces.

Nevertheless, in a context of high dimensional spaces, this kind of method can encounter many computational problems. The main ones are that it needs to store all the data, which can be expensive in term of memory and that they cannot deal online with the data. In order to overcome this, stochastic gradient algorithms [START_REF] Robbins | A stochastic approximation method[END_REF] are efficient candidates since they do not need to store the data into memory, and they can be easily updated, which is crucial if the data arrive sequentially (see [START_REF] Duflo | Algorithmes stochastiques[END_REF], [START_REF] Duflo | of Applications of Mathematics[END_REF], Kushner and Yin (2003a) or [START_REF] Nemirovski | Robust stochastic approximation approach to stochastic programming[END_REF] among others). In order to improve the convergence, [START_REF] Ruppert | Efficient estimations from a slowly convergent robbins-monro process[END_REF] and [START_REF] Polyak | Acceleration of stochastic approximation[END_REF] introduced its averaged version (see also [START_REF] Dippon | Weighted means in stochastic approximation of minima[END_REF] for a weighted version). These algorithms have become crucial to statistics and modern machine learning [START_REF] Bach | Non-strongly-convex smooth stochastic approximation with convergence rate O (1/n)[END_REF], [START_REF] Bach | Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression[END_REF], [START_REF] Juditsky | Deterministic and stochastic primal-dual subgradient algorithms for uniformly convex minimization[END_REF]). There are already many results on these algorithms in the literature, that we can split into two parts: asymptotic results, such as almost sure rates of convergence [START_REF] Schwabe | On a stochastic approximation procedure based on averaging[END_REF][START_REF] Duflo | of Applications of Mathematics[END_REF][START_REF] Walk | Foundations of stochastic approximation[END_REF][START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF][START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF], and non asymptotic ones, such as rates of convergence in quadratic mean (Cardot et al., 2017;Godichon-Baggioni, 2016a;[START_REF] Bach | Non-strongly-convex smooth stochastic approximation with convergence rate O (1/n)[END_REF][START_REF] Bach | Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression[END_REF][START_REF] Nemirovski | Robust stochastic approximation approach to stochastic programming[END_REF].

In a recent work, [START_REF] Godichon-Baggioni | Lp and almost sure rates of convergence of averaged stochastic gradient algorithms with applications to online robust estimation[END_REF] introduces a new framework, with only locally strongly convexity assumptions, in general Hilbert spaces, which allows to obtain almost sure and L p rates of convergence. In keeping with it, and in order to have a deeper study of the stochastic gradient algorithm as well as of its averaged version (up to a new assumption), we first give the asymptotic normality of the estimates. In a second time, since a Central Limit Theorem is often unusable without an estimation of the variance, we introduce a recursive algorithm, inspired by [START_REF] Gahbiche | On the estimation of the asymptotic covariance matrix for the averaged robbins-monro algorithm[END_REF], to estimate the asymptotic variance of the averaged estimator and we establish its rates of convergence.

As far as we know, there was not yet an efficient and recursive estimate of the asymptotic variance in the literature. Finally, two examples of application are given. The first usual one consists in estimating the parameters of the logistic regression [START_REF] Bach | Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression[END_REF] while the second one consists in estimating geometric quantiles (see [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF] and [START_REF] Chakraborty | The spatial distribution in infinite dimensional spaces and related quantiles and depths[END_REF]), which are useful robust indicators in statistics. Indeed, they are often used in data depth and outliers detection [START_REF] Serfling | Depth functions in nonparametric multivariate inference[END_REF], [START_REF] Hallin | Semiparametrically efficient rank-based inference for shape. i. optimal rank-based tests for sphericity[END_REF]), as well as for robust estimation of the mean and variance (see [START_REF] Minsker | Scalable and robust bayesian inference via the median posterior[END_REF]), or for Robust Principal Component Analysis [START_REF] Gervini | Robust functional estimation using the median and spherical principal components[END_REF], [START_REF] Kraus | Dispersion operators and resistant second-order functional data analysis[END_REF]Panaretos (2012), Cardot andGodichon-Baggioni (2017)).

The paper is organized as follows: Section 2 recalls the framework introduced by Godichon-Baggioni (2016b) before giving two new assumptions which allow to get the rate of convergence of the estimators of the asymptotic variance. In section 3, the stochastic gradient algorithm as well as its averaged version are introduced and their asymptotic normality are given. The recursive estimator of the asymptotic variance is given in Section 4 and its almost sure as well as its quadratic mean rates of convergence are established. Applications, consisting in estimating the logistic regression parameters and in the recursive estimation of geometric quantiles, are given in Section 5 as well as a short simulation study. Finally, the proofs are postponed in Section 6 and in a Supplementary file.

Assumptions

Let H be a separable Hilbert space such as R d or L 2 (I) (for some closed interval I ⊂ R), we denote by ., . its inner product and by . the associated norm. Let X be a random variable taking values in a space X , and let G : H -→ R be the function we would like to minimize, defined for all h ∈ H by

G(h) := E [g(X, h)] ,
(1)

where g : X × H -→ R. Moreover, let us suppose that the functional G is convex. Finally, let us introduce the space of linear operators on H, denoted by S(H), equipped with the Frobenius (or Hilbert-Schmidt) inner product, which is defined by

A, B F := ∑ j∈J A(e j ), B(e j ) , ∀A, B ∈ S(H),
where e j j∈J is an orthonormal basis of H. We denote by . F the associated norm, and S(H) is then a separable Hilbert space. Let us recall the framework introduced by Godichon-Baggioni (2016b):

(A1) The functional g is Frechet-differentiable for the second variable almost everywhere. Moreover, G is differentiable and there exists m ∈ H such that ∇G(m) = 0.

(A2)

The functional G is twice continuously differentiable almost everywhere and for all positive constant A, there is a positive constant C A such that for all h ∈ B (m, A),

Γ h op ≤ C A ,
where Γ h is the Hessian of the functional G at h and . op is the usual spectral norm for linear operators.

(A3) There exists a positive constant such that for all h ∈ B (m, ), there is an orthonormal basis of H composed of eigenvectors of Γ h . Moreover, let us denote by λ min the limit inf of the eigenvalues of Γ m , then λ min is positive. Finally, for all h ∈ B (m, ), and for all eigenvalue λ h of Γ h , we have λ h ≥ λ min 2 > 0.

(A4) There are positive constants , C such that for all h ∈ B (m, ),

∇G(h) -Γ m (h -m) ≤ C h -m 2 .

(A5) (a)

There is a positive constant L 1 such that for all h ∈ H,

E ∇ h g (X, h) 2 ≤ L 1 1 + h -m 2 .
(a') There is a positive constant L 2 such that for all h ∈ H,

E ∇ h g (X, h) 4 ≤ L 2 1 + h -m 4 .
(b) For all integer q, there is a positive constant L q such that for all h ∈ H,

E ∇ h g (X, h) 2q ≤ L q 1 + h -m 2q .
Let us now make some comments on assumptions. First, Assumption (A1) ensures the existence of a solution and enables to use a stochastic gradient descent, while (A2) gives some smoothness properties on the objective function. Assumption (A3) ensures the uniqueness of the minimizer of G, and (A4),(A5) give bounds of the gradient and of the remainder term of its Taylor's expansion. The main difference between this framework and the usual one for strongly convex objective is that we just assume the local strong convexity of the objective function, and in return, p-th moments of the gradient of the functional g have to be bounded. Note also that the Hessian of the functional G is not supposed to be compact, so that its smallest eigenvalue does not necessarily converge to 0 when the dimension tends to infinity (a counter example is given in Section 5). Remark that assumptions (A1) to (A5b) are deeply discussed in Godichon-Baggioni (2016b). Let us now introduce two new assumptions.

(A6) Let ϕ : H -→ S(H) be the functional defined for all h ∈ H by

ϕ (h) := E [∇ h g (X, h) ⊗ ∇ h g (X, h)] .
(a) The functional ϕ is continuous at m with respect to the Frobenius norm:

lim h→m E [∇ h g (X, m) ⊗ ∇ h g (X, m)] -E [∇ h g (X, h) ⊗ ∇ h g (X, h)] F = 0. (b)
The functional ϕ is locally lipschitz on a neighborhood of m: there are positive constants , C , such that for all h ∈ B (m, ),

E [∇ h g (X, m) ⊗ ∇ h g (X, m) -∇ h g (X, h) ⊗ ∇ h g (X, h)] F ≤ C h -m .
Assumption (A6a) enables to establish the asymptotic normality of the stochastic gradient descent as well as of its averaged version. Note that under (A5a), the functional ϕ is bounded, and more precisely

E [∇ h g (X, h) ⊗ ∇ h g (X, h)] F ≤ E ∇ h g (X, h) 2 ≤ L 1 1 + h -m 2 .
Assumption (A6b) can be verified by giving a bound, on a neighborhood of m, of the derivative of the functional ϕ. This last assumption allows to give the rate of convergence of the estimators of the asymptotic variance. An example is given for the special case of the geometric median in a supplementary file.

Remark 2.1. For all h ∈ H and A > 0,

B (h, A) = h ∈ H, h -h < A . Remark 2.2. Let h, h ∈ H, the linear operator h ⊗ h : H -→ H is defined for all h ∈ H by h ⊗ h (h ) := h, h h . Moreover, h ⊗ h F = h h . ( 2
)
3 The stochastic gradient algorithm and its averaged version

The Robbins-Monro algorithm

In what follows, let X 1 , ..., X n be independent random variables with the same law as X.

The stochastic gradient algorithm is defined recursively for all n ≥ 1 by

m n+1 = m n -γ n ∇ h g (X n+1 , m n ) , (3)
with m 1 bounded and (γ n ) is a step sequence of the form γ n := c γ n -α , with c γ > 0 and α ∈ 1 2 , 1 . Moreover, let (F n ) n≥1 be the sequence of σ-algebras defined for all n ≥ 1 by F n := σ (X 1 , ..., X n ). Then, the algorithm can be considered as a noisy (or stochastic) gradient algorithm since it can be written as

m n+1 = m n -γ n Φ (m n ) + γ n ξ n+1 , (4)
where Φ (m n ) := ∇G (m n ), and (ξ n ), defined for all n ≥ 1 by

ξ n+1 := Φ (m n ) -∇ h g (X n+1 , m n ),
is a martingale differences sequence adapted to the filtration (F n ). Finally, note that under assumptions (A1) to (A5a), it was proven in Godichon-Baggioni (2016b) that for all positive constant δ,

m n -m 2 = o (ln n) δ n α a.s. (5)
Moreover, assuming that (A5b) is also fulfilled, for all positive integer p, there is a constant C p such that for all n ≥ 1,

E m n -m 2p ≤ C p n pα . ( 6
)
In order to get a deeper study of this estimate, we now give its asymptotic normality.

Theorem 3.1. Suppose assumptions (A1) to (A5a') and (A6a) hold. Then, we have the convergence in law

lim n→∞ 1 √ γ n (m n -m) ∼ N (0, Σ RM ) ,
with

Σ RM := +∞ 0
e -sΓ m Σ e -sΓ m ds, and

Σ := E [∇ h g (X, m) ⊗ ∇ h g (X, m)] .
The proof is given in a Supplementary file. Note that the variance Σ RM does not depend on the step sequence (γ n ), but Theorem 3.1 could be written as

lim n→∞ n α/2 (m n -m) ∼ N (0, c γ Σ RM ) ,
Remark 3.1. Let M be a squared matrix, e M is defined by (see [START_REF] Horn | Matrix analysis[END_REF] among others)

e M = ∞ ∑ k=0 1 k! M k .
Thanks to assumptions (A2),(A3), 0 < λ min (Γ m ) ≤ λ max (Γ m ) < ∞, while under (A5a) and by dominated convergence,

Σ RM F ≤ +∞ 0 e -sΓ m 2 op Σ F ds ≤ +∞ 0 e -2sλ min Σ F ds ≤ L 1 2λ min ,
and Σ RM is so well defined.

Remark 3.2. Note that analogous results are given by [START_REF] Fabian | On asymptotic normality in stochastic approximation[END_REF][START_REF] Pelletier | On the almost sure asymptotic behaviour of stochastic algorithms[END_REF] in the particular case of finite dimensional spaces while, for analogous results in Banach and Hilbert spaces, one can also see [START_REF] Walk | Foundations of stochastic approximation[END_REF], [START_REF] Ljung | Stochastic approximation and optimization of random systems[END_REF], [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF].

Remark 3.3. Note that taking a step sequence of the form γ n = c n with c > 2 λ min is possible, and one can obtain the following asymptotic normality (see [START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF] among others for the case of finite dimensional spaces)

lim n→∞ √ n (m n -m) ∼ N 0, cΣ .
Nevertheless, it does not only necessitate to have some information on the Hessian Γ m , but cΣ is also not the optimal variance (see [START_REF] Duflo | of Applications of Mathematics[END_REF] and [START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF] for instance).

The averaged algorithm

As mentioned in Remark 3.3, having the parametric rate of convergence (O 1 n ) with the Robbins-Monro algorithm is possible taking a good choice of step sequence (γ n ). Never- theless, this choice is often complicated and the asymptotic variance which is obtained is not optimal. Then, in order to improve the convergence, let us now introduce the averaged algorithm (see [START_REF] Ruppert | Efficient estimations from a slowly convergent robbins-monro process[END_REF] and [START_REF] Polyak | Acceleration of stochastic approximation[END_REF]) defined for all n ≥ 1 by

m n = 1 n n ∑ k=1 m k .
This can be written recursively for all n ≥ 1 as

m n+1 = m n + 1 n + 1 (m n+1 -m n ) . (7)
It was proven in Godichon-Baggioni (2016b) that under assumptions (A1) to (A5a), for all δ > 0,

m n -m 2 = o (ln n) 1+δ n a.s. (8)
Suppose assumption (A5b) is also fulfilled, for all positive integer p, there is a positive constant C p such that for all n ≥ 1,

E m n -m 2p ≤ C p n p . ( 9
)
Finally, in order to have a deeper study of this estimate, we now give its asymptotic normality.

Theorem 3.2. Suppose assumptions (A1) to (A5a') and (A6a) are verified. Then, we have the convergence in law

lim n→∞ √ n (m n -m) ∼ N (0, Σ) , with Σ := Γ -1 m Σ Γ -1 m , and Σ := E [∇ h g (X, m) ⊗ ∇ h g (X, m)].
The proof is given in Section 6. For analogous results, one can also see [START_REF] Schwabe | On a stochastic approximation procedure based on averaging[END_REF], [START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF], [START_REF] Dippon | The averaged robbins-monro method for linear problems in a banach space[END_REF].

Recursive estimation of the asymptotic variance

Some existing estimators

A first naive method to estimate the asymptotic variance could be to estimate the Hessian Γ m and the variance Σ as follows

Γ (n+1) m = Γ (n) m + 1 n + 1 ∇ 2 h g (X n+1 , m n ) -Γ (n) m , Σ n+1 = Σ n + 1 n + 1 ∇ h g (X n+1 , m n ) ⊗ ∇ h g (X n+1 , m n ) -Σ n ,
but the main problem is that under assumptions (A2), (A3) and (A5a), if H is an infinite dimensional space, then

Γ m F = ∞, while Γ -1 m Σ Γ -1 m F ≤ L 1 λ 2 min .
Another problem is that, in order to get a recursive estimator of the asymptotic variance, it needs to invert a matrix at each iteration, which costs much calculus time in high dimensional spaces. A second estimator of the asymptotic variance was introduced in Pelletier ( 2000), defined for all n ≥ 1 by

Σ n = 1 ln n n ∑ k=1 (m k -m n ) ⊗ (m k -m n ) , (10)
and under (A1) to (A6b),

E Σ n -Σ 2 F = O 1 ln n .
Thus, this estimator faces two main problems: it is not recursive and it converges very slowly. Finally, in order to solve the second problem, a faster algorithm was introduced by [START_REF] Gahbiche | On the estimation of the asymptotic covariance matrix for the averaged robbins-monro algorithm[END_REF], defined for all n ≥ 1 by

Σn := 1 -δ n 1-δ n ∑ k=1 1 k δ+s+µ exp - k 1-s 1 -s k ∑ j=1 j µ/2 e j 1-s 2(1-s) m j -m n ⊗ k ∑ j=1 j µ/2 e j 1-s 2(1-s) m j -m n , (11) with (1 + α)/2 < s < 1, µ ≥ 0 and s/2 < δ < (1 + s)/2.
This algorithm is first based on an usual decomposition of the stochastic gradient algorithm (see equation ( 18)) which enables to make appear a martingale term which carries the convergence rate (see equation ( 27)). In a second time, the objective is to find step sequences which enable to improve the rate of convergence of the variance estimate (see [START_REF] Gahbiche | On the estimation of the asymptotic covariance matrix for the averaged robbins-monro algorithm[END_REF] for technical details on assumptions on the step sequences). In the case of finite dimensional spaces, the following convergence in probability is given (under some assumptions)

n 1/2-s/2 (ln ln n) c Σn -Σ op P ---→ n→∞ 0,
with c > 0. A first technical problem is that only the convergence in probability is given, in the case of finite dimensional spaces, and for the usual spectral norm. A second one is that it is not recursive and it cannot be easily updated.

A recursive and fast estimate

We now give a recursive version of the algorithm defined by (11) to estimate the asymptotic variance in separable Hilbert spaces, before establishing its rates of convergence (almost sure and in quadratic mean). This algorithm is defined by

Σ n := 1 -δ n 1-δ n ∑ k=1 1 k δ+s+µ exp - k 1-s 1 -s k ∑ j=1 j µ/2 e j 1-s 2(1-s) m j -m j ⊗ k ∑ j=1 j µ/2 e j 1-s 2(1-s) m j -m j , (12) with (1 + α)/2 < s < 1, µ ≥ 0, and s/2 < δ < (1 + s)/2. ( 13
)
The difference with previous algorithm is the replacement of m n by m j , which enables the estimates to be written recursively for all n ≥ 1 as

V n+1 = V n + (n + 1) µ/2 exp (n + 1) 1-s 2(1 -s) (m n+1 -m n+1 ) , Σ n+1 = n n + 1 1-δ Σ n + 1 -δ (n + 1) δ+s+µ exp - (n + 1) 1-s 1 -s V n+1 ⊗ V n+1 , with V 1 = Σ 1 = 0.
Then, contrary to previous algorithms, this one does not need to store all the estimations into memory and can be easily updated. Finally, the following theorem ensures that it is quite fast.

Theorem 4.1. Suppose assumptions (A1) to (A5a') and (A6b) hold. Then, the sequence (Σ n ) defined by ( 12) verifies for all positive constant γ,

Σ n -Σ 2 F = o (ln n) γ n 1-s a.s.
Moreover, suppose (A5b) holds too, there is a positive constant C such that for all n ≥ 1,

E Σ n -Σ 2 F ≤ C n 1-s
The proof is given in Section 6.

Corollary 4.1. Suppose assumptions (A1) to (A5a') and (A6b) hold. Then, for all positive constant γ, Σn -

Σ 2 F = o (ln n) γ n 1-s a.s.
Moreover, suppose (A5b) holds too, there is a positive constant C such that for all n ≥ 1,

E Σn -Σ 2 F ≤ C n 1-s
Remark 4.1. The constant C in Theorem 4.1 depends on the constants introduced in assumptions, on the initialization of the stochastic gradient descent, and on α, δ, µ, s, c γ . Remark 4.2. Estimating recursively the asymptotic variance coupled with Theorem 3.2 can be useful to build online asymptotic confidence balls. Moreover, in the recent literature, non asymptotic convergence rates are often given under the form

E m n -m 2 ≤ Σ F n + R n ,
where R n is a rest term. Then, using the recursive variance estimates could enable to have, in practice, a precise bound of the quadratic mean error, and in the short term, it could allow to get precise non asymptotic confidence balls.

Remark 4.3. In order to get a faster algorithm (in term of computational time), one can consider a parallelized version of previous estimates. This consists in splitting the sample into p parts, and to run the algorithm on each subsample to get p estimates Σ n/p,i , before taking the mean of these p last ones.

Applications

Application to the logistic regression

Let d be a positive integer, and let Y ∈ {-1, 1} and X ∈ R d be random variables. In order to get the parameter m l ∈ R d of the logistic regression, the aim is to minimize the functional G l defined for all h ∈ R d by

G l (h) := E [log (1 + exp (-Y X, h ))] . ( 14
)
Under usual assumptions (see [START_REF] Bach | Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression[END_REF] among others), the functional G l is locally strongly convex and twice Fréchet differentiable with for all h ∈ R d ,

∇G l (h) = -E exp (-Y X, h ) 1 + exp (-Y X, h ) YX , ∇ 2 G l (h) = E exp (-Y X, h ) (1 + exp (-Y X, h )) 2 X ⊗ X .
Then, the parameters of the logistic regression and the asymptotic variance can be estimated simultaneously as:

m l n+1 = m l n + γ n exp -Y n+1 X n+1 , m l n 1 + exp (-Y n+1 X n+1 , m l n ) Y n+1 X n+1 , m l n+1 = m l n + 1 n + 1 m l n+1 -m l n , V l n+1 = V l n + (n + 1) µ/2 exp (n + 1) 1-s 2(1 -s) m l n+1 -m l n+1 , Σ l n+1 = n n + 1 1-δ Σ l n + 1 -δ (n + 1) δ+s+µ exp - (n + 1) 1-s 1 -s V l n+1 ⊗ V l n+1 .

Application to the geometric median and geometric quantiles

Let H be a separable Hilbert space and let X be a random variable taking values in H. Let v ∈ H such that v < 1, the geometric quantile m v corresponding to the direction v (see [START_REF] Chaudhuri | On a geometric notion of quantiles for multivariate data[END_REF]) is defined by

m v := arg min h∈H E [ X -h -X ] -h, v , (15)
and in a particular case, the geometric median m (see [START_REF] Haldane | Note on the median of a multivariate distribution[END_REF]) corresponds to the case where v = 0. Under usual assumptions (see [START_REF] Kemperman | The median of a finite measure on a Banach space[END_REF] and [START_REF] Cardot | Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm[END_REF] among others), the functional G v is locally strongly convex and twice Fréchet-differentiable with for all h ∈ H,

∇G v (h) = -E X -h X -h + v , ∇ 2 G v (h) = E 1 X -h I H - (X -h) ⊗ (X -h) X -h 2 .
Then, it is possible to estimate simultaneously and recursively the geometric quantile m v as well as the asymptotic variance of the averaged estimator as follows:

m v n+1 = m v n + γ n X n+1 -m v n X n+1 -m v n + v , m v n+1 = m v n + 1 n + 1 (m v n+1 -m v n ) , V v n+1 = V v n + (n + 1) µ/2 exp (n + 1) 1-s 2(1 -s) (m v n+1 -m v n+1 ) , Σ v n+1 = n n + 1 1-δ Σ n + 1 -δ (n + 1) δ+s+µ exp - (n + 1) 1-s 1 -s V v n+1 ⊗ V v n+1 .
Note that under usual assumptions, the asymptotic variance obtained is the same as the one obtained with non-recursive estimates [START_REF] Maronna | Robust statistics[END_REF][START_REF] Gervini | Robust functional estimation using the median and spherical principal components[END_REF] in the special case of the geometric median.

A short simulation study

We focus here on the estimation of the geometric median. We consider from now that X is a random variable taking values in R d , with d ≥ 3, and following a uniform law on the unit sphere S d . Then, the geometric median m is equal to 0 and the Hessian of the functional G 0 at m verifies

Γ m = E 1 X I d - X X ⊗ X X = I d -E [X ⊗ X] = d -1 d I d .
Note that assumptions (A1) and (A6b) are then verified (see Section 3 in Godichon-Baggioni (2016b), Lemma A.1 in Godichon-Baggioni et al. ( 2017) and the supplementary file to be convinced). Finally, the asymptotic variance of the stochastic gradient estimate and of its averaged version verify

Σ RM = ∞ 0 e -sΓ m E X X ⊗ X X e -sΓ m ds = 1 2(d -1) I d , Σ = Γ -1 m E X X ⊗ X X Γ -1 m = d (d -1) 2 I d .
First, let us consider a stepsequence γ n = n -2/3 and let us study the quality of the Gaussian approximation of Q n , Q n , where

Q n := 2(d -1)n 1/3 (m n -m) ,
and

Q n := √ n d -1 √ d (m n -m) .
Figure 1 (respectively Figure 2) seems to confirm Theorem 3.1 (respectively Theorem 3.2)

since we can see that the estimated density of a component of

Q n (respectively Q n ) is close
to the density of N (0, 1), and so, even for small sample sizes (n = 200), which is also confirmed by a Kolmogorov-Smirnov test. In Figure 3, we consider the evolution of the quadratic mean error, with respect to the Frobenius norm, of the estimates (Σ n ) of Σ defined by ( 12), with regard to the sample size.

For this, we generate 100 samples, and use the parallelized version of the algorithms. 6 Proofs

Some decompositions of the algorithms

In order to simplify the proofs, let us now give some decompositions of the algorithms.

The Robbins-Monro algorithm

Let us recall that the stochastic gradient algorithm can be written as

m n+1 -m = m n -m -γ n Φ (m n ) + γ n ξ n+1 .
Linearizing the gradient, it comes

m n+1 -m = (I H -γ n Γ m ) (m n -m) + γ n ξ n+1 -γ n δ n , ( 16
)
where

δ n := Γ m (m n -m) -Φ (m n )
is the remainder term in the Taylor's expansion of the gradient. Thanks to previous decomposition and with the help of an induction (see [START_REF] Duflo | Algorithmes stochastiques[END_REF] or [START_REF] Duflo | of Applications of Mathematics[END_REF] for instance), one can check that for all n ≥ 1,

m n -m = β n-1 (m 1 -m) -β n-1 n-1 ∑ k=1 γ k β -1 k δ k + β n-1 n-1 ∑ k=1 γ k β -1 k ξ k+1 , (17)
with β n := ∏ n k=1 (I Hγ k Γ m ) for all n ≥ 1 and β 0 := I H . Finally, the asymptotic variance can be seen as the almost sure limit of the sequence of random variables

Γ -1 m ξ n ⊗ Γ -1 m ξ n n
(see the proof of Theorem 3.2). Then, in order to prove the convergence of the estimates, we need to exhibit this sequence. In this aim, one can rewrite equation ( 16) as

m n -m = T n γ n - T n+1 γ n + Ξ n+1 -∆ n , (18)
with

T n := Γ -1 m (m n -m) , Ξ n+1 := Γ -1 m (ξ n+1 ) , ∆ n := Γ -1 m (δ n ) .

The averaged algorithm

Summing equalities (18) and dividing by n, we obtain the following decomposition of the averaged estimator

m n -m = 1 n n ∑ k=1 T k γ k - T k+1 γ k - 1 n n ∑ k=1 ∆ k + 1 n n ∑ k=1 Ξ k+1 . ( 19
)
Finally, by linearity and applying an Abel's transform to the first term on the right-hand side of previous equality (see [START_REF] Delyon | Stochastic optimization with averaging of trajectories[END_REF] or [START_REF] Delyon | Accelerated stochastic approximation[END_REF] for instance),

Γ m (m n -m) = m 1 -m nγ 1 - m n+1 -m nγ n + 1 n n ∑ k=2 1 γ k - 1 γ k-1 (m k -m) - 1 n n ∑ k=1 δ k + 1 n n ∑ k=1 ξ k+1 .
(20)

The recursive estimator of the asymptotic variance

In order to simplify the proof of Theorem 4.1, we will introduce a new estimator of the variance. In this aim, let us now introduce the sequences (a n ) n≥1 and (b n ) n≥1 defined for all n ≥ 1 by a n := exp n 1-s 2(1-s) and b n := ∑ n k=1 a 2 k . Then, thanks to decomposition (18), let

T n := 1 √ b n n ∑ k=1 a k (m k -m) = 1 √ b n n ∑ k=1 a k γ k (T k -T k+1 ) + n ∑ k=1 a k ∆ k + n ∑ k=1 a k Ξ k+1 =: 1 √ b n (A 1,n + A 2,n + M n+1 ) . (21)
In order to simplify several proofs, we now give L p upper bounds of the terms on the righthand side of previous equality.

Lemma 6.1. Suppose assumptions (A1) to (A5b) hold. Then, for all positive integer p,

E   n ∑ k=1 a k γ k (T k -T k+1 ) 2p   = O exp pn 1-s 1 -s n pα , E   n ∑ k=1 a k ∆ k 2p   = O exp pn 1-s 1 -s n p(s-α , E   n ∑ k=1 a k Ξ k+1 2p   = O exp pn 1-s 1 -s n ps
The proof of this lemma as well as an analogous lemma which gives the asymptotic almost sure behavior of these terms are given in a Supplementary file. We can now introduce the following estimator

Σ n = 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ T k ⊗ T k , ( 22
)
and one can decompose Σ n as follows:

Σ n -Σ = Σ n - 1 -δ n 1-δ n ∑ k=1 1 k δ+s exp - k 1-s 1 -s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m + 1 -δ n 1-δ n ∑ k=1 1 k δ+s exp - k 1-s 1 -s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m -Σ n + Σ n -Σ.

Proof of Theorem 3.2

Proof of Theorem 3.2. Let us recall that the averaged algorithm can be written as

Γ m (m n -m) = m 1 -m nγ 1 - m n+1 -m nγ n + 1 n n ∑ k=2 1 γ k - 1 γ k-1 (m k -m) - 1 n n ∑ k=1 δ k + 1 n n ∑ k=1 ξ k+1 . It is proven in Godichon-Baggioni (2016b) that m 1 -m √ nγ 1 = o (1) a.s, m n+1 -m √ nγ n = o(1) a.s, 1 √ n n ∑ k=2 1 γ k - 1 γ k-1 (m k -m) = o(1) a.s, 1 √ n n ∑ k=1 δ k = o(1) a.s.
In order get the asymptotic normality of the martingale term 1 n ∑ n k=1 ξ k+1 , let us check that assumptions of Theorem 5.1 in [START_REF] Jakubowski | Tightness criteria for random measures with application to the principle of conditioning in Hilbert spaces[END_REF] are fulfilled, i.e let (e i ) i∈I be an orthonor- mal basis of H and ψ i,j := Σ e i , e j for all i, j ∈ I, we have to verify ∀η > 0, lim

n→∞ P sup 1≤k≤n 1 √ n ξ k+1 > η = 0, ( 23
) lim n→∞ 1 n n ∑ k=1 ξ k+1 , e i ξ k+1 , e j = ψ i,j a.s, ∀i, j ∈ I, (24)
∀ > 0, lim

N→∞ lim sup n→∞ P 1 n n ∑ k=1 ∞ ∑ j=N ξ k+1 , e j 2 > = 0. ( 25
)
Proof of ( 23) Let η > 0, applying Markov's inequality,

P sup 1≤k≤n 1 √ n ξ k+1 > η ≤ n ∑ k=1 P 1 √ n ξ k+1 > η ≤ 1 n 2 η 4 n ∑ k=1 E ξ k+1 4 .
Then, applying Lemma H.1, there is a positive constant C such that

P sup 1≤k≤n 1 √ n ξ k+1 > η ≤ 1 n 2 η 4 n ∑ k=1 C = C nη 4 .
Proof of (24). First, note that

1 n n ∑ k=1 ξ k+1 ⊗ ξ k+1 = 1 n n ∑ k=1 E [ξ k+1 ⊗ ξ k+1 |F k ] + 1 n n ∑ k=1 k+1 , with k+1 := ξ k+1 ⊗ ξ k+1 -E [ξ k+1 ⊗ ξ k+1 |F k ].
Remark that ( n ) is a sequence of martingale differences adapted to the filtration (F n ), and one can check that

lim n→∞ 1 n n ∑ k=1 k+1 = 0 a.s.
Let us now prove that the sequence of operators (E [ξ k+1 ⊗ ξ k+1 |F k ]) converges almost surely to Σ , with respect to the Frobenius norm. Note that

E [ξ k+1 ⊗ ξ k+1 |F k ] -Σ = E [∇ h g (X k+1 , m k ) ⊗ ∇ h g (X k+1 , m k ) |F k ] -Σ -Φ (m k ) ⊗ Φ (m k ) F ≤ E [∇ h g (X k+1 , m k ) ⊗ ∇ h g (X k+1 , m k ) |F k ] -Σ F + Φ (m k ) ⊗ Φ (m k ) F .
Then, thanks to assumption (A6a), since

Φ(m k ) ≤ C m k -m and since (m k ) converges to m almost surely (see Godichon-Baggioni (2016b)), lim k→∞ E [∇ h g (X k+1 , m k ) ⊗ ∇ h g (X k+1 , m k ) |F k ] -Σ F = 0 a.s, lim k→∞ Φ (m k ) ⊗ Φ (m k ) F = lim n→∞ Φ(m k ) 2 = 0 a.s.
In a particular case, for all i, j ∈ I,

lim k→∞ E [ξ k+1 ⊗ ξ k+1 |F k ] (e i
), e j = ψ i,j := Σ (e i ), e j a.s.

Thus, applying Toeplitz's lemma,

lim n→∞ 1 n n ∑ k=1 E [ξ k+1 ⊗ ξ k+1 |F k ] (e i
), e j = ψ i,j a.s.

Finally, for all i, j ∈ I,

lim n→∞ 1 n n ∑ k=1 ξ k+1 , e i ξ k+1 , e j = lim n→∞ 1 n n ∑ k=1 ξ k+1 ⊗ ξ k+1 (e i ), e j = ψ i,j a.s.
Proof of (25). Let > 0, applying Markov's inequality,

P 1 n n ∑ k=1 ∞ ∑ j=N ξ k+1 , e j > ≤ 1 n 2 n ∑ k=1 ∞ ∑ j=N E ξ k+1 , e j 2 = 1 n 2 n ∑ k=1 ∞ ∑ j=N E E ξ k+1 , e j 2 |F k .
Since for all j ∈ I, ξ k+1 , e j 2 = ξ k+1 ⊗ ξ k+1 (e j , e j , and by linearity

P 1 n n ∑ k=1 ∞ ∑ j=N ξ k+1 , e j > ≤ 1 2 ∞ ∑ j=N 1 n n ∑ k=1 E E ξ k+1 ⊗ ξ k+1 (e j ), e j |F k = 1 2 ∞ ∑ j=N 1 n n ∑ k=1 E E [ξ k+1 ⊗ ξ k+1 |F k ] (e j ), e j .
Since E [ξ k+1 ⊗ ξ k+1 |F k ] converges almost surely to Σ and by dominated convergence, lim sup

n P 1 n n ∑ k=1 ∞ ∑ j=N ξ k+1 , e j > ≤ 1 ∞ ∑ j=N Σ (e j ), e j . Moreover, since Σ = E [∇ h g (X, m) ⊗ ∇ h g (X, m)], thanks to assumption (A5a), ∞ ∑ j=1 Σ (e j ), e j = E [∇ h g (X, m) ⊗ ∇ h g (X, m)] F ≤ E ∇ h g (X, m) 2 ≤ L 1 .
Thus, since for all j ∈ I, Σ (e j ), e j ≥ 0, lim N→∞ ∞ ∑ j=N Σ (e j ), e j = 0, which concludes the proof.

Proof of Theorem 4.1

For the sake of simplicity, the proof is given for mu = 0 (the case where µ > 0 is strictly analogous). Let us recall that equation ( 12) can be written as

Σ n -Σ = Σ n - 1 -δ n 1-δ n ∑ k=1 1 k δ+s exp - k 1-s 1 -s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m + 1 -δ n 1-δ n ∑ k=1 1 k δ+s exp - k 1-s 1 -s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m -Σ n + Σ n -Σ. (26)
In order to prove Theorem 4.1, we just have to give the rates of convergence of the terms on the right-hand side of previous equality. The following lemma gives the almost sure and the rate of convergence in quadratic mean of the first term on the right-hand side of previous equality.

Lemma 6.2. Suppose assumptions (A1) to (A5a') and (A6b) hold. Then, for all γ > 0,

Σ n - 1 -δ n 1-δ n ∑ k=1 1 k δ+s e -k 1-s 1-s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m 2 F = o (ln n) γ n 1-s a.s.
Moreover, suppose assumption (A5b) holds too. Then,

E   Σ n - 1 -δ n 1-δ n ∑ k=1 1 k δ+s e -k 1-s 1-s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m 2 F   = O 1 n 1-s .
The proof is given in a Supplementary file. The following lemma gives the almost sure and the rate of convergence in quadratic mean of the second term on the right-hand side of equality (26). Lemma 6.3. Suppose assumptions (A1) to (A5a') and (A6b) hold. Then, for all γ > 0,

1 -δ n 1-δ n ∑ k=1 1 k δ+s e -k 1-s 1-s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m -Σ n 2 F = o (ln n) γ n 2(1-s) a.s.
Moreover, suppose assumption (A5b) holds too. Then

E   1 -δ n 1-δ n ∑ k=1 1 k δ+s e -k 1-s 1-s k ∑ j=1 e j 1-s 2(1-s) m j -m ⊗ k ∑ j=1 e j 1-s 2(1-s) m j -m -Σ n 2 F   = O 1 n 2(1-s) .
The proof is given in a Supplementary file. Finally, the following Proposition gives the almost sure and the rate of convergence in quadratic mean of the last term on the right-hand side of equality (26). Proposition 6.1. Suppose assumptions (A1) to (A5a') and (A6b) hold. Then, there is a positive constant γ such that

Σ n -Σ 2 F = o (ln n) δ n 1-s a.s.
Suppose assumption (A5b) holds too. Then, there is a positive constant C such that for all n ≥ 1,

E Σ n -Σ 2 F ≤ C n 1-s .
Proof of Proposition 6.1. Applying equality (2), one can check that

Σ n -Σ F ≤ 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A 1,k 2 + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A 2,k 2 + 2 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A 1,k A 2,k + 2 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A 1,k M k+1 + 2 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A 2,k M k+1 + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k M k+1 ⊗ M k+1 -Σ F , (27)
where A 1,k , A 2,k , M k+1 are defined in (21). The following Lemma gives the rate of convergence in quadratic mean of the first terms on the right-hand side of previous inequality.

Lemma 6.4. Suppose Assumptions (A1) to (A6b) hold. Then, for all i, j ∈ {1, 2},

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A i,k A j,k 2   = o 1 n 1-s , E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k A i,k M k+1 2   = o 1 n 1-s .
The proof of this lemma as well as its "almost sure version" are given in a Supplementary file.

Then, we just have to bound the last term on the right-hand side of inequality (27). First let us decompose M k+1 ⊗ M k+1 as

M k+1 ⊗ M k+1 = k ∑ j=1 a 2 j Ξ j+1 ⊗ Ξ j+1 + k ∑ j=1 a j Ξ j+1 ⊗ M j + k ∑ j=1 a j Ξ j+1 ⊗ M k+1 -M j+1 + k ∑ j=1 a j M j ⊗ Ξ j+1 + k ∑ j=1 a j M k+1 -M j+1 ⊗ Ξ j+1 .
Note that for all j, M j is F j -measurable and E Ξ j+1 ⊗ M j |F j = 0. Moreover,

1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k M k+1 ⊗ M k+1 -Σ = 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ Ξ j+1 -Σ + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ M j + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j ξ j+1 ⊗ M k+1 -M j+1 + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j M j ⊗ Ξ j+1 + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j M k+1 -M j+1 ⊗ Ξ j+1 .
The end of the proof consists in giving a bound of the quadratic mean of each term on the right-hand side of previous equality. Note that the almost sure rates of convergence are not proven since it is quite analogous.

Bounding E 1 ∑ n k=1 k -δ ∑ n k=1 1 k δ 1 b k ∑ k j=1 a j Ξ j+1 ⊗ M j 2 F . First, note that 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ M j = 1 ∑ n k=1 k -δ n ∑ k=1 n ∑ j=k 1 k δ 1 b k a k Ξ k+1 ⊗ M k .
Moreover, with the help of an integral test for convergence, one can check that there is a positive constant C such that for all positive integers k ≤ n, n

∑ j=k 1 k δ 1 b k ≤ C k δ exp - k 1-s (1 -s) . ( 28
)
Furthermore, since Ξ j+1 ⊗ M j j is a sequence of martingale differences adapted to the fil- tration F j , let

( * ) := E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ M j 2 F   = E   1 ∑ n k=1 k -δ n ∑ k=1 n ∑ j=k 1 k δ 1 b k a k Ξ k+1 ⊗ M k 2 F   = 1 ∑ n k=1 k -δ 2 n ∑ k=1 n ∑ j=k 1 k δ 1 b k 2 a 2 k E Ξ k+1 ⊗ M k 2 F 20
Then, applying equality (2) and Cauchy-Schwarz's inequality,

( * ) ≤ 1 ∑ n k=1 k -δ 2 n ∑ k=1 n ∑ j=k 1 k δ 1 b k 2 a 2 k E Ξ k+1 2 M k 2 ≤ 1 ∑ n k=1 k -δ 2 n ∑ k=1 n ∑ j=k 1 k δ 1 b k 2 a 2 k E Ξ k+1 4 E M k 4 .
Finally, applying Lemmas 6.1 and H.1 as well as inequality ( 28),

( * ) = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ-s = O 1 n 1-s .
With analogous calculus, one can check

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j M j ⊗ Ξ j+1 2 F   = O 1 n 1-s . Bounding E 1 ∑ n k=1 k -δ ∑ n k=1 1 k δ 1 b k ∑ k j=1 a j Ξ j+1 ⊗ M k+1 -M j+1 2 F . First, note that k ∑ j=1 a j Ξ j+1 ⊗ M k+1 -M j = k ∑ j=1 k ∑ j =j+1 a j a j Ξ j+1 ⊗ Ξ j +1 = k ∑ j =2 j -1 ∑ j=1 a j a j Ξ j+1 ⊗ Ξ j +1 .
Note that ∑ j -1 j=1 a j a j Ξ j+1 ⊗ Ξ j +1 j is a sequence of martingale differences adapted to the filtration F j . Furthermore,

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ M k+1 -M j 2 F   = 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k E   k ∑ j =2 j -1 ∑ j=1 a j a j Ξ j+1 ⊗ Ξ j +1 2 F   + 1 ∑ n k=1 k -δ 2 E n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =2 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 , k ∑ i =2 i -1 ∑ i =1 a i a i Ξ i +1 ⊗ Ξ i +1 .
Then end of the proof consists in bounding the two terms on the right-hand side of previous equality. First, since ∑ j -1 j=1 a j a j Ξ j+1 ⊗ Ξ j +1 j is a sequence of martingale differences adapted to the filtration F j , let

( ) := 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k E   k ∑ j =2 j -1 ∑ j=1 a j a j Ξ j+1 ⊗ Ξ j +1 2 F   = 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k k ∑ j =2 E   j -1 ∑ j=1 a j a j Ξ j+1 ⊗ Ξ j +1 2 F   .
Then, applying equality (2) and Cauchy-Schwarz's inequality,

( ) = 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k k ∑ j =2 a 2 j E   j -1 ∑ j=1 a j Ξ j+1 2 Ξ j +1 2   ≤ 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k k ∑ j =2 a 2 j E Ξ j +1 4 E   j -1 ∑ j=1 a j Ξ j+1 4  
Finally, applying Lemma H.1, H.2 and 6.1,

( ) = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k k ∑ j =2 a 4 j j s = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k a 4 k k 2s = O 1 n min{2-2δ,1} . Then, since δ < (1 + s)/2, 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k 2δ 1 b 2 k E   k ∑ j =2 j -1 ∑ j=1 a j a j Ξ j+1 ⊗ Ξ j +1 2 F   = o 1 n 1-s .
In the same way, by linearity, let

( ) := 1 ∑ n k=1 k -δ 2 E n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =2 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 , k ∑ i =2 i -1 ∑ i =1 a i a i Ξ i +1 ⊗ Ξ i +1 = 1 ∑ n k=1 k -δ 2 n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ E   j ∑ j =2 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 , j ∑ i =2 i -1 ∑ i =1 a i a i Ξ i +1 ⊗ Ξ i +1 F   + 1 ∑ n k=1 k -δ 2 n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ E   j ∑ j =2 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 , k ∑ i =j+1 i -1 ∑ i =1 a i a i Ξ i +1 ⊗ Ξ i +1 F   .
Since (Ξ i ) is a sequence of martingale differences adapted to the filtration (F i ),

n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ E   j ∑ j =2 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 , k ∑ i =j+1 i -1 ∑ i =1 a i a i Ξ i +1 ⊗ Ξ i +1 F   = n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =2 j -1 ∑ j =1 k ∑ i =j+1 i -1 ∑ i =1 a i a i a j a j E Ξ j +1 ⊗ Ξ j +1 , Ξ i +1 ⊗ Ξ i +1 F = n ∑ k=2 k-1 ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =2 j -1 ∑ j =1 k ∑ i =j+1 i -1 ∑ i =1 a i a i a j a j E Ξ j +1 ⊗ Ξ j +1 , Ξ i +1 ⊗ E [Ξ i +1 |F i ] F = 0. Furthermore, since ∑ j j =2 ∑ j -1 j =1 a j a j Ξ j +1 ⊗ Ξ j +1 j
is a sequence of martingale differences adapted to the filtration F j and applying equality (2),

( ) = 1 ∑ n k=1 k -δ 2 n ∑ k=2 k ∑ j=1 b -1 k k -δ b -1 j j -δ E   j ∑ j =1 j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 2 F   = 1 ∑ n k=1 k -δ 2 n ∑ k=2 k ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =1 E   j -1 ∑ j =1 a j a j Ξ j +1 ⊗ Ξ j +1 2 F   = 1 ∑ n k=1 k -δ 2 n ∑ k=2 k ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =1 a 2 j E   j -1 ∑ j =1 a j Ξ j +1 2 Ξ j +1 2   .
Applying Cauchy-Schwarz's inequality as well as Lemmas H.1 and 6.1,

( ) ≤ 1 ∑ n k=1 k -δ 2 n ∑ k=1 k ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =1 a 2 j E   j -1 ∑ j =1 a j Ξ j +1 4 F   E Ξ j +1 4 F = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 k ∑ j=1 b -1 k k -δ b -1 j j -δ j ∑ j =1
a 4 j j s .

Finally, applying Lemma H.2,

( ) = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 k ∑ j=1 b -1 k k -δ b -1 j j -δ a 4 j j 2s = O 1 ∑ n k=1 k -δ 2 n ∑ k=1 b -1 k k -2δ k 2s a 2 k = O 1 n 1-s .
Thus,

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j Ξ j+1 ⊗ M k+1 -M j+1 2 F   = O 1 n 1-s .
Moreover, with analogous calculus, one can check

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ 1 b k k ∑ j=1 a j M k+1 -M j+1 ⊗ Ξ j+1 2 F   = O 1 n 1-s . Bounding 1 ∑ n k=1 k -δ ∑ n k=1 1 k δ b k ∑ k j=1 a 2 k (Ξ k+1 ⊗ Ξ k+1 -Σ). First , note that 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 k (Ξ k+1 ⊗ Ξ k+1 -Σ) = 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 k (E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Σ) + 1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 k (Ξ k+1 ⊗ Ξ k+1 -E [Ξ k+1 ⊗ Ξ k+1 |F k ])
The end of the proof consists in bounding the quadratic mean of the terms on the right-hand side of previous equality. First, applying Lemma H.4, let

( ) := E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j (E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Σ) 2 F   ≤ 1 ∑ n k=1 k -δ 2    n ∑ k=1 1 k δ b k E   k ∑ j=1 a 2 j (E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Σ) 2 F      2 ≤ 1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j E E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Σ 2 F 2
Then, applying inequality (6) and Corollary H.1,

( ) = O   1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j E m n -m 2 2   = O   1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j j -α/2 2   .
Furthermore, thanks to Lemma H.2,

( ) = O   1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k δ b k a 2 k k s-α/2 2   = O 1 ∑ n k=1 k -δ 2 n 2-2δ-α = O 1 n α .
Thus, since α > 1/2,

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j (E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Σ) 2 F   = o 1 n 1-s .
Moreover, applying Lemma H.4, let

( ) := E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j (Ξ k+1 ⊗ Ξ k+1 -E [Ξ k+1 ⊗ Ξ k+1 |F k ]) 2 F   ≤ 1 ∑ n k=1 k -δ 2    n ∑ k=1 1 k δ b k E   k ∑ j=1 a 2 j (Ξ k+1 ⊗ Ξ k+1 -E [Ξ k+1 ⊗ Ξ k+1 |F k ]) 2 F      2 . Furthermore, since (E [Ξ k+1 ⊗ Ξ k+1 |F k ] -Ξ k+1 ⊗ Ξ k+1
) is a sequence of martingale differ- ences adapted to the filtration (F k ) and applying Lemma H.1,

( ) ≤ 1 ∑ n k=1 k -δ 2   n ∑ k=1 1 k δ b k k ∑ j=1 a 4 j E (Ξ k+1 ⊗ Ξ k+1 -E [Ξ k+1 ⊗ Ξ k+1 |F k ]) 2 F   2 = O    1 ∑ n k=1 k -δ 2   n ∑ k=1 1 k δ b k k ∑ j=1 a 4 j   2    .
Then, applying Lemma H.2,

( ) = O   1 ∑ n k=1 k -δ 2 n ∑ k=1 1 k δ b k a 2 k k s/2 2   = O   1 ∑ n k=1 k -δ 2 n ∑ k=1 k -δ-s/2 2   = O 1 n 2-s .
Finally,

E   1 ∑ n k=1 k -δ n ∑ k=1 1 k δ b k k ∑ j=1 a 2 j (Ξ k+1 ⊗ Ξ k+1 -E [Ξ k+1 ⊗ Ξ k+1 |F k ]) 2 F   = o 1 n 1-s ,
which concludes the proof.

Figure 1 :

 1 Figure 1: Estimated density of a component of Q n (in blue) compared to the standard gaussian density (in red), with n = 200 (on the left) and n = 5000 (on the right).

Figure 2 :

 2 Figure 2: Estimated density of a component of Q n (in blue) compared to the standard gaussian density (in red), with n = 200 (on the left) and n = 5000 (on the right).

Figure 3 :

 3 Figure 3: Evolution of the quadratic mean error of the estimation of the asymptotic variance Σ with respect to the Frobenius norm for d = 10 (on the left) and d = 5000 (on the right).


Soraya Mezouar

Nathalie Bardin

Didier Raoult

L Mège

Cléa Melenotte
email: melenotteclea@gmail.com

Laetitia Gay

Jean- Louis Mège

The sexual dimorphism of anticardiolipin autoantibodies in acute Q fever patients

Keywords: Q fever, Coxiella burnetii, anticardiolipin antibodies, sex, gender, autoimmunity Q fever, Coxiella burnetii, anticardiolipin antibodies, sex, gender, autoimmunity

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Viral and bacterial infections have been reported to be associated with autoimmune clinico-biological manifestations [START_REF] Konstantinov | Infections and antineutrophil cytoplasmic antibodies: Triggering mechanisms[END_REF]. The elevation of anticardiolipin (aCL) antibodies has been described in cases of human immunodeficiency virus (HIV), hepatitis B virus, hepatitis C virus, malaria, syphilis, leprosy, leptospirosis and Q fever [START_REF] Sène | Antiphospholipid antibodies, antiphospholipid syndrome and infections[END_REF]. In case of Q fever, the elevation of aCL antibodies has recently been reported as predictive for acute Q fever complications, such as acute Q fever endocarditis, haemophagocytic syndrome, alithiasic cholecystitis and thrombosis [START_REF] Million | Thrombosis and antiphospholipid antibody syndrome during acute Q fever[END_REF][START_REF] Melenotte | Clinical features and complications of Coxiella burnetii infections from the French National Reference Center for Q Fever[END_REF]. Interestingly, an unbalanced sex ratio has been described during C. burnetii infection, the disease affecting men more frequently than women (sex ratio men/women: 2.2). Here, we investigated the sexual dimorphism of autoantibodies during C. burnetii infection with a specific focus on aCL antibodies.

Methods

Between 1991 and 2016, 2,434 patients with C. burnetii infection were followed in the French national reference centre for Q fever. Acute Q fever was diagnosed by the association of acute clinical symptoms (fever, hepatitis and/or pneumonia) with the following serologic criteria: IgG titers representing phase II (200) and IgM titers representing phase II (50) or seroconversion within 3 months of the primary symptoms [START_REF] Eldin | From Q fever to Coxiella burnetii infection: A paradigm change[END_REF]. Persistent C. burnetii infection was defined by the persistence of clinical symptoms for more than 3 months in addition to a positive microbiological criteria and to the identification of an infectious focus [START_REF] Eldin | From Q fever to Coxiella burnetii infection: A paradigm change[END_REF].

1,323 patients were tested for IgG aCL antibodies assessed by enzyme-linked immunosorbent assay at first serological diagnosis. A cut-off threshold of 22 IgG anti-phospholipid-binding units (GPLU) was used to define positive samples.

The study was approved by the local ethics committee of the IHU (Institut Hospitalo-Universitaire)-Méditerranée Infection under the registration number 12-016 [START_REF] Melenotte | Clinical features and complications of Coxiella burnetii infections from the French National Reference Center for Q Fever[END_REF].

Statistical analyses were performed using Prism 6.0 (Graphpad Software Inc.) and SPSS 22 Statistics Software. Multivariate models were adjusted for sex and age at baseline [START_REF] Melenotte | Clinical features and complications of Coxiella burnetii infections from the French National Reference Center for Q Fever[END_REF]. P<0.05 was considered as significant.

Results

Among the 1,323 patients tested for aCL antibodies, 1,013 patients had acute Q fever and 310 had persistent C. burnetii focalized infection (Figure 1). Nine hundred and eighteen patients were men (sex ratio men/women: 2.26). The sexual dimorphism was more pronounced in cases of persistent focalized C. burnetii infection (sex ratio: 2.27) than in cases of acute Q fever (sex ratio: 2.16) (p=0.07). Eighty four patients with acute Q fever evolved towards a persistent C. burnetii infection, 70 were men (83.3%), 14 were women (16.7%), and the sex ratio was 5.

The aCL antibodies levels were significantly higher in cases of acute Q fever (mean 88.2 ± 14 GPLU and median 18 GPLU) than in cases of persistent C. burnetii infection (mean 14.2 ± 2.4 GPLU and median 7 GPLU) (Figure 2A, p<0.0001). In addition, the proportion of positive aCL antibodies were significantly higher in cases of acute Q fever (464/1,013, 45.8%) than in cases of persistent C. burnetii infection (32/310, 10.3%) (p<0.05). In cases of acute Q fever, men had significantly higher aCL antibodies levels (mean 113.3 ± 20.3 GPLU and median 24 GPLU) than women (mean 50 ± 7 GPLU and median 11 GPLU) (Figure 2B, p<0.0001) and men presented a significantly higher proportion of positive aCL antibodies (351/692, 50.7%) compared to women (113/321, 35.2%) (p<0.05) for all age groups (Figure 2C). In men, higher levels of aCL antibodies were observed in almost all age groups (Figure 2D). Being a male was associated with a significant risk of positive aCL antibodies, regardless of age (OR=1.6;[95% CI,, p<0.001) [START_REF] Melenotte | Clinical features and complications of Coxiella burnetii infections from the French National Reference Center for Q Fever[END_REF].

Acute Q fever hepatitis affected men more than women (p=0.007) and was associated with a significant increase in aCL in men (p=0.047). In case of acute Q fever, being male was a risk factor for acute Q fever to progress to persistent cardiovascular C. burnetii infection (OR=2.14; [95% CI, 1.17-3.91], p=0.013).

Discussion

Here, we report that aCL antibody levels are higher in men than in women with acute C. burnetii infection. This is distinct from the sexual dimorphism observed in the antiphospholipid syndrome (APS), in which women are more affected by the disease than men (sex ratio men/women: 1:5) [START_REF] Cervera | Antiphospholipid syndrome: Clinical and immunologic manifestations and patterns of disease expression in a cohort of 1,000 patients[END_REF]. In addition, the aCL antibodies sexual dimorphism observed in association with C. burnetii infection is different from that observed in association with HIV infection, in which aCL antibodies are described as significantly higher in women compared to men [START_REF] Abdollahi | Serum concentrations of antiphospholipid and anticardiolipin antibodies are higher in HIV-infected women[END_REF].

Our data support the possibility of a hormonal influence on the expression of autoantibodies in case of C. burnetii infection. It is known that women are described to be more prone to autoimmune diseases than men, and this increased susceptibility disappears after menopause [START_REF] Klein | Sex differences in immune responses[END_REF]. A positive correlation has been shown between elevated aCL antibodies levels and high estradiol serum concentration in patients with systemic lupus erythematosus (SLE) and rheumatoid arthritis [START_REF] Kaliterna | Does estrogen stimulate the pathogenic sort of anticardiolipin antibodies?[END_REF][START_REF] Seriolo | Relationships between serum 17 betaoestradiol and anticardiolipin antibody concentrations in female patients with rheumatoid arthritis[END_REF]. In addition, hyper-oestrogenic levels in premenopausal SLE women are associated with an increased risk of APS and cardiovascular manifestations [START_REF] Kaliterna | Does estrogen stimulate the pathogenic sort of anticardiolipin antibodies?[END_REF]. In gonadoctemized or intact non-autoimmune male and female C57BL/6 mice, oestrogen treatment induces the expression of aCL autoantibodies [START_REF] Verthelyi | Characterization of Estrogen-Induced Autoantibodies to Cardiolipin in Non-Autoimmune Mice[END_REF]. It has been hypothetized that oestrogens act as enhancers of cell proliferation and stimulate humoral B cell response [START_REF] Straub | The complex role of estrogens in inflammation[END_REF].

In addition, the influence of testosterone should not be neglected. While oestrogen increases the autoantibody production, androgens decrease it [START_REF] Straub | The complex role of estrogens in inflammation[END_REF]. Men with hypogonadism are at risk of developing autoimmune diseases of which the Klinefelter syndrome is a glaring example. This syndrome has been associated with SLE, progressive systemic sclerosis, polymyositis, mixed connective tissue disease and with an increased frequency of aCL antibody positivity [START_REF] Oktenli | Study of Autoimmunity in Klinefelter's Syndrome and Idiopathic Hypogonadotropic Hypogonadism[END_REF]. Testosterone deficiency may promote autoantibody formation, however, in cases of C. burnetii infection, testosterone rather appears to favour aCL autoantibodies production [START_REF] Melenotte | Clinical features and complications of Coxiella burnetii infections from the French National Reference Center for Q Fever[END_REF].

Other mechanisms such as chromosomal influence could be involved. In transgenic SJL mice, the XX sex chromosome was compared with XY, and showed greater susceptibility to autoimmune encephalomyelitis and SLE [START_REF] Smith-Bouvier | A role for sex chromosome complement in the female bias in autoimmune disease[END_REF]. Women with Turner's syndrome were at significantly increased risk of autoimmune thyroid diseases and type 1 diabetes mellitus [START_REF] Jørgensen | Autoimmune diseases in women with Turner's Syndrome[END_REF].

As limitations of this study, we have no data on patient's hormonal serum levels, we have not studied the chromosomal influence on autoantibody secretion and we did not consider the gender impact, i.e. gender-related social habits that could influence the results.

In conclusion, we highlight a relationship between sex and markers of autoimmunity during Q fever. We noted that acute Q fever induces a higher autoimmune response in men compared to women. However, the mechanisms of sexual dimorphism are multifactorial and although age was taken into account, other factors such as hormone levels and chromosomal impact should be included in future analyses.

CALIBRATING AND MEASURING WAKES AND DRAG FORCES OF INLAND VESSELS IN CONFINED WATER IN A TOWING TANK.

Clément Caplier, Guillaume Gomit, Germain Rousseaux, Damien Calluaud, Ludovic Chatellier and Laurent David, Institut Pprime, CNRS -University of Poitiers -ISAE-ENSMA, France.

SUMMARY

This paper gives a review of the experimental methods developed in the towing tank of the Pprime Institute of the University of Poitiers, France, for the characterization of ship wakes and drag forces in confined waters. Different waterway configurations in calm water and in the presence of a current are reproduced in the experimental facility and small scale ship models of different block coefficients are considered. Stereoscopic optical methods have been developed in the laboratory for the measurement of the free surface deformation around the ship. The full wake generated by the ship is fully characterized and its hydraulic and undulatory properties are analyzed in both real space and spectral domain. In addition, a system for the measurement of the ship drag force has been set up and visualizations of the wakes have been performed in parallel with a highspeed camera, to relate the ship resistance crisis with its visual footprint in the wake.

NOMENCLATURE

INTRODUCTION

When a ship progresses in a confined waterway, such as a river, a canal or an estuary, it faces an increase in the advancing resistance and also experiences various phenomena in the waterway. There is a lowering of the water level, combined with the generation of a return current around the ship hull. The waves generated in its wake interact with the current of the waterway and reflect on the river banks, causing erosion and sediment transport issues. To prevent their appearance in the waterway, for both economical (reduce fuel consumption) and ecological (reduce bank erosion) reasons, many numerical studies are performed [START_REF] Linde | Three-dimensional numerical simulation of ship resistance in restricted waterways: effect of ship sinkage and channel restriction[END_REF]Pacuraru and Domnisoru, 2018;Razgallah et al., 2018;[START_REF] Terziev | Numerical investigation of the behaviour and performance of ships advancing through restricted shallow waters[END_REF]. However, the specific properties of the confined water wakes are not completely described numerically or theoretically. Experimental investigations are still necessary to understand the generation and the propagation of the waves and the flow with respect to the functional and geometrical parameters of the ship and the waterway. This path of research has been investigated in the past few years by the Hydrodynamic and Environmental Flows team of the Pprime Institute of the University of Poitiers in France. A configuration of confined waterway with a presence of current has been reproduced in the towing tank of the laboratory. Several experimental methods have been developed for the measurement of the ship resistance, the characterization of the return current around the hull, the study of the wave reflections and the full-field characterization of the generated ship wake. To investigate the properties of the waves generated by the hulls, a fine characterization of the free surface is indispensable to lead the analysis in the spectral domain. On the basis of the theory of the waves generated by a uniformly moving source established by [START_REF] Ekman | On stationary waves in running water[END_REF] and [START_REF] Crapper | Surface waves generated by a travelling pressure point[END_REF] and the recent review of the geometrical properties of the emitted patterns in the real and spectral spaces by [START_REF] Carusotto | The Čerenkov effect revisited: from swimming ducks to zero modes in gravitational analogues[END_REF], a spectral analysis method of ship wakes has been developed. Such spectral analysis methods have been proposed by [START_REF] Mckenzie | Hydromagnetic wave interaction with the magnetopause and the bow shock[END_REF] and [START_REF] Lighthill | Waves in fluids[END_REF]. [START_REF] Griffin | Ship hull characteristics from surface wake synthetic aperture radar (SAR) imagery[END_REF] and [START_REF] Arnold-Bos | Obtaining a ship's speed and direction from its Kelvin wake spectrum using stochastic matched filtering[END_REF] have used them to estimate the velocity of a ship or the shape of its hull from synthetic aperture radar (SAR) images or high resolution optical images of the wake. However, these methods have never been validated experimentally in the field of ship wakes measured in the laboratory. In the first part of the paper, the waterway configuration set up in the towing tank of the Pprime Institute is presented and the small scale ship models representative of maritime and river ships are introduced. In the second part, the experimental measurement methods are detailed and the spectral analysis method is detailed. Then, the main results of the studies are presented and finally research perspectives will be exposed with the future project of development of the canal. along the canal of length Lc=20 m. The double-bottom gives also the possibility to generate a current by placing a circulator in the lower section, and two honeycombs structures at the extremities of the canal to laminarize the flow. The current can be generated in both directions with respect to the ship motion (co-or counter-current). The canal is equipped with bottom and lateral windows to perform optical measurements and visualizations.

THE TOWING TANK AND THE SHIP MOD-ELS

Figure 1.

Transversal section of the towing tank.

THE SMALL SCALE SHIP MODELS

In order to compare the experimental results with numerical calculations of ship wakes and ship resistance, two generic ship hulls of parabolic shape have been considered. They are based on the [START_REF] Wigley | Ship wave resistance. A comparison of mathematical theory with experimental results[END_REF] hull form with a rectangular cross-section, mathematically defined by Eq. ( 1) with n=2 for a classical Wigley hull:

= = 1 - (1)
As the block coefficient of such a Wigley hull is Cb=0.67, another Wigley-based ship hull with an exponent n=8, giving a block coefficient Cb=0.89, has been considered [START_REF] Caplier | Energy distribution in shallow water ship wakes from a spectral analysis of the wave field[END_REF]. These two ship hulls noted WH2 and WH8 because of their n-coefficients are then representative of maritime and river ships in terms of block coefficients. The ship hulls dimensions are Ls=1.20 m, B=0.18 m and H=0.15 m, and their draft is set-up to D=0.075 m for the experiments. They are shown on Figure 2. As regards the effects of scale, the ship speeds for the experiments are set high enough to avoid capillary effects (less than 5%). In this range of ship speeds, the effects of the scale on the flow characteristics are limited [START_REF] Gomit | Large scale free surface measurement for the analysis of ship waves in a towing tank[END_REF].

Figure 2.

The WH8 and WH2 ship hulls.

THE MEASUREMENT AND THE ANALYSIS OF THE SHALLOW WATER SHIP WAKE

MEASUREMENT OF THE FREE SURFACE DE-FORMATION

The free surface deformation is measured with a stereoscopic measurement method, based on the principle of refraction of light rays through the surface of water that is developed in our laboratory [START_REF] Gomit | Free surface measurement by stereo-refraction[END_REF]. The setup is represented on Figure 3. It is composed of two Dantec SpeedSense 1040 cameras that deliver a resolution of 2320x1726 px, mounted with Nikkor AF 28 mm 1:2.8 lenses. They focus on the same zone with an opposite angle of ±15° with respect to the longitudinal axis of the canal, and ±35° with respect to the vertical axis. The common field covered by the cameras forms a rectangle of dimensions 0.75x0.90 m 2 , corresponding to the half width of the canal.

The stereoscopic system is calibrated by translating a dotted calibration plate and using a distortion model. A specklepattern is placed at the bottom of the canal and its image with the water surface at rest is recorded on the cameras for the reference. Then the ship is launched and the images of the pattern, deformed by the free surface deformations, are recorded on both cameras at a frequency of 10 frames per second with an exposure time of 10 ms. Each run is performed three times to check the repeatability of the measurement. The image pairs are then processed with a reconstruction algorithm written in C++ and based on the SLIP library [START_REF] Tremblais | Standard Library for Image Processing[END_REF] that computes the free surface deformation at each time-step. Then the whole wake is reconstructed around the ship hull with a dedicated reconstruction program, with a spatial resolution ∆X=∆Y=10 mm and a precision ∆Z=0.1 mm on the water level.

ANALYSIS IN THE REAL SPACE

An example of the wakes measured in calm water and in the presence of a counter-current with the stereo-refraction measurement method are shown on Figure 4 [START_REF] Caplier | The effects of river counter-currents on ships wakes: an experimental approach[END_REF]. These wakes are generated by the WH2 hull for a ship speed Us=0.45 m.s -1 and a water depth h=0.103 m, giving a Froude-depth number Fh,s=Us/(gh) 1/2 =0.45 and a

Froude-length number FL,s=Us/(gLs) 1/2 =0.13. The wake on the bottom of the figure is generated in a calm water configuration, whereas the top one is generated in the presence of a counter-current of velocity uc=0.20 m.s -1 , corresponding to an effective Froude-depth number in the referential of the laboratory Fh,s=(Us+uc)/(gh) 1/2 =0.65 [START_REF] Caplier | The effects of river counter-currents on ships wakes: an experimental approach[END_REF].

A qualitative analysis of these wakes in the real space shows that the transverse wavelength and amplitude are increasing in the presence of the counter-current. The waves generated by the ship in its wake are convected by the current, that results in a widening of the wash zone at the banks of the canal. For a quantitative comparison of the wakes, longitudinal or transverse cuts can be made in the wave fields. Figure 5 represents the longitudinal cuts performed for the calm water and counter-current wakes. For the same ship speed, the length of the wash zone increases from 0.9Ls to 2Ls (Ls=1.2 m is the ship length) with the counter-current, representing more than a doubling of its length. The transverse waves amplitude and wavelengths at the river banks respectively grows from A1=0.012h to A2=0.059h (h=0.103 m is the initial water depth), and from λt,1=0.1Ls to λt,2=0.3Ls, an increase of nearly 500% of the amplitude and 30% of the wavelength.

SPECTRAL ANALYSIS OF SHIP WAKES

The stereo-refraction measurement method gives a full, detailed and fine reconstruction of the ship wake. Then, it is possible to proceed the analysis in the spectral space by computing the bidimensional Fast Fourier Transform (FFT) of the wake. The method has been introduced by [START_REF] Carusotto | The Čerenkov effect revisited: from swimming ducks to zero modes in gravitational analogues[END_REF], developed by [START_REF] Gomit | Spectral analysis of ship waves in deep water from accurate measurements of the free surface elevation by optical methods[END_REF] for the analysis of ship wakes in deep water and then adapted by [START_REF] Caplier | Energy distribution in shallow water ship wakes from a spectral analysis of the wave field[END_REF] for the analysis of ship wakes generated in shallow water. An example of the spectrum of the wake generated by the WH8 hull, for a ship speed Us=0.80 m.s -1 and a water depth h=0.103 m, giving a Froude-depth number Fh,s=Us/ (gh) 1/2 =0.80 and a Froude-length number FL,s=Us/(gLs) 1/2 =0.23, is given on the right of Figure 6 (Caplier et al., 2016). The color on the spectrum represents the spectral repartition of the normalized amplitude of the FFT along the different wave lengths and directions in the wake. The high-pass or low-pass filtering of the spectrum and its reconstruction in the real space by computing an inverse bidimensional FFT allows to separate the hydraulic and undulatory components of the wake [START_REF] Gomit | Spectral analysis of ship waves in deep water from accurate measurements of the free surface elevation by optical methods[END_REF] [START_REF] Caplier | Energy distribution in shallow water ship wakes from a spectral analysis of the wave field[END_REF].

MEASUREMENT OF THE RETURN CURRENT

The stereoscopic system can also be used to measure the return current generated around the hull, by computing the displacements of floating particles placed at the surface of the water. The images of these markers can then be processed by a stereo-PIV algorithm to calculate the flow velocity [START_REF] Chatellier | A parametric PIV/DIC method for the measurement of free surface flows[END_REF]. The result is given on Figure 7 on which the return current is measured around the WH2 hull for a ship speed Us=0.70 m.s -1 and a water depth h=0.103 m, giving a Froude-depth number Fh,s=Us/ (gh) 1/2 =0.70 and a Froude-length number FL,s=Us/(gLs) 1/2 =0.20. The return current is calculated in the whole field around the hull, it reaches its maximum of amplitude (30% of the ship speed) at the middle of the hull, and extends along the whole width of the waterway. Then its direction alternates between the crests and troughs of the transverse waves, and its amplitude decreases slowly.

THE SHIP RESISTANCE AND ITS VISUAL FOOTPRINT

Resistance trials have been carried out in the towing tank of the Pprime Institute. The measurement of the ship drag force is performed with a Kistler 9272 multi-component dynamometer installed between the ship hull and the towing mast (Figure 8). The six-component piezoelectric sensor measures the axial forces in each direction as well as the torques. The ship resistance is taken as the axial force FX opposed to the motion of the ship. The maximum measured amplitude is FX=20 N and the uncertainty is I=0.80 N [START_REF] Caplier | Etude expérimentale des effets de hauteur d'eau finie, de confinement latéral et de courant sur les sillages et la résistance à l'avancement des navires[END_REF]. Then the non-dimensional drag coefficient Cd can be derived with Equation 2, where Sw is the wetted surface area of the ship:

= (2)
Figure 9 represents the drag coefficients of the WH2 and WH8 hulls measured in a shallow water configuration (h=0.103 m) at different ship speeds (the Froude-depth number of the ship Fh,s is between 0.60 and 1.31, and the Froude-length number of the ship FL,s is between 0.18 and 1.38). The evolution of the ship resistance is bounded by the critical Froude-depth numbers Fh1 and Fh2 calculated by Schijf's theory [START_REF] Schijf | Protection of embankments and bed in inland and maritime waters, and in overflow or weirs[END_REF]. These critical Froude-depth numbers are calculated by Equations 3 and 4:

= 2!"# $%&'( )* + , (3)
= 2!"# -)$%&'( )* + , (4)
They depend on the blockage ratio m (cross section of the ship to cross section of the canal). In the present experiments, m=0.0874. A sudden increase of the drag coefficient is observed for Fh,s=Fh1=0.64 for both hulls. This increase referred as "the resistance crisis" appears at the boundary between the sub-critical and the transcritical regime. Finally, there is a decrease of the ship resistance at the entrance in the supercritical regime for Fh,s=Fh2=1.37. Parallel visualizations have been made during the resistance trials, with a high-speed camera Photron Fastcam SA1 of resolution 1024x1024 px equipped with a Sigma 28 mm F1.8 DG ASP lens (Figure 8). The camera installed on the side of the canal, and captures images of the wave amplitudes at the window during the passage of the ship at a frequency of 125 fps. Then a dedicated algorithm assembles the images to give a picture-like representation of the wave amplitudes. Figure 10 represents the visualizations performed during the transition between the subcritical and transcritical regime (Froude-depth number Fh,s from 0.60 to 0.85 and Froudelength number FL,s from 0.19 to 0.22). These visualizations highlight an increase of the transverse wavelength and amplitude at the boundary between the subcritical and transcritical regimes. A wave breaking is also visible on some waves. So when a ship navigates in a confined waterway, it experiences a sudden increase of its fuel consumption, and the waves that it generates grow, break and reflect on the river banks. That process is destructive for the river banks and needs to be investigated in a future work, to prevent its appearance and to design appropriate bank protections.

CONCLUSIONS AND PERSPECTIVES

During the past few years, several experimental studies have been made in the towing tank of the Pprime Institute, to investigate the effects of confinement on ships wakes and drag and to understand interaction between the ship and the waterway. Measurement methods have been developed and adapted for this topic, to obtain a full and detailed measurement of the ship wake and to determine the drag forces applying on the hull. These methods give high quality results that allow to perform a fine analysis of the wave properties in both real and spectral spaces. The hydrodynamic effects appearing in the waterway can also be quantified and analyzed. The comparison of the resistance diagrams with the visualizations of the wakes allows to identify the visual footprint of the crisis of ship resistance. The towing tank of the Pprime Institute is currently under work, and will be lengthened of one third of its actual length, to reach 30 m. The towing carriage and the current generation will also be improved. These great improvements on the facility will open the way to future experimental studies, in order to continue the investigations on this path of research.

Figure 1 .Figure 2 .

 12 Figure 1. Study Flowchart

  αb Banks inclination angle (°) AFFT Normalized amplitude of the Fast Fourier Transform of the ship wake (-) A1 Normalized amplitude of the transverse waves in calm water (-) A2 Normalized amplitude of the transverse waves in counter-current (-) B Beam (width) of the ship hull (m) Cb Block coefficient of the ship hull (-) Cd Drag coefficient of the ship hull (-) D Draft of the ship hull (m) Fh1 Critical Froude number (sub.→trans.) (-) Fh2 Critical Froude number (trans.→sup.) (-) Fh,s Froude-depth number of the ship (-) FL,s Froude-length number of the ship (-) FX Axial component of ship resistance (N) g Gravitational acceleration (m.s -2 ) H Height of the ship hull (m) h Water depth (m) hb Height of the inclined bank (m) hbottom Height of the double bottom (m) I Expanded uncertainty (k=2) of the ship resistance measurement (N) kx,y Wavenumber along the x (longitudinal) or y (transversal) axis (m -1 ) Ls Length of the ship (m) Lc Length of the canal (m) L1 Normalized length of the wash zone in calm water (-) L2 Normalized length of the wash zone in counter-current (-) λt,1 Normalized wavelength of the transverse waves in calm water (-) λt,2 Normalized wavelength of the transverse waves in counter-current (-) m Blockage ratio of the waterway (-) ρ Density of water (kg/m 3 ) Sw Wetted surface area of the ship (m 2 ) Us Ship speed (m.s -1 ) uc Velocity of the river current (m.s -1 ) ur Velocity of the return current (m.s -1 ) W Large width of the waterway (m) w Small width of the waterway (m) wb Width of the inclined bank (m) X Longitudinal dimension of the wake (m) ∆X Spatial resolution along the X axis (m) Y Transversal dimension of the wake (m) ∆Y Spatial resolution along the Y axis (m) Z Vertical dimension of the wake (m) ∆Z Spatial resolution along the Z axis (m)

Figure 1

 1 Figure 1 represents the cross-section of the towing tank of the Pprime Institute. It is composed of a bottom trapezoidal section of small width w=1.10 m and large width W=1.50 m, and a top rectangular section of same width. The inclination of the banks is αb=45° and their width and height are wb=hb=0.20 m. A double bottom of height hbottom=0.38 m can be installed in the towing tank to reduce the water depth to reproduce a shallow waterway of rectangular cross-section of depth h. The water depth has been set up to h=0.103 m for the experiments. The rail mounted towing carriage displaces the ship hull at a speed Us up to 2.35 m.s -1along the canal of length Lc=20 m. The double-bottom gives also the possibility to generate a current by placing a circulator in the lower section, and two honeycombs structures at the extremities of the canal to laminarize the flow. The current can be generated in both directions with respect to the ship motion (co-or counter-current). The canal is equipped with bottom and lateral windows to perform optical measurements and visualizations.

Figure 3 .

 3 Figure 3. Schematic representation of the experimental setup for the free surface measurement by stereo-refraction.

Figure 4 .

 4 Figure 4. The wakes generated in calm water (bottom) and in the presence of a counter-current (top) by the WH2 hull, measured with the stereorefraction measurement method.

Figure 5 .Figure 6 .

 56 Figure 5. Longitudinal cuts in the wakes generated in calm water and in the presence of a counter-current at a transversal position Y/B=3.5. (L1 ; A1 ; λt,1) and (L2 ; A2 ; λt,2) represent respectively the width of the reflection zone, and the amplitude and the wavelength of the transverse waves, in calm water and in counter-current.

Figure 7 .

 7 Figure 7. The return current ur measured around the WH2 hull for a Froude-depth number Fh,s=0.70 with the stereocorrelation measurement method.

Figure 8 .

 8 Figure 8. Measurement of the ship resistance with a multi-component dynamometer and lateral visualizations with a high-speed camera.

Figure 9 .Figure 10 .

 910 Figure 9. Drag coefficients of the ship hulls measured in a shallow water configuration for different ship speeds. The black dotted lines correspond to the critical Froude numbers Fh1=0.64 and Fh2=1.37 linked to the blockage of the waterway.

Funding source:

This work was supported by the French Government under the "investissement d'avenir" ("investments for the future") program managed by the Agence Nationale de la Recherche (ANR, fr: National Agency for research), (reference: Méditerranée infection 10-IAHU-03). This work was also supported by Région Provence-Alpes-Côte d'Azur and European funding FEDER PRIMMI (Fonds Européen de Développement Régional -Plateformes de Recherche et d'Innovation Mutualisées Méditerranée Infection).

Transparency declaration

No funding source

No relationship with industry

No conflict of interest AUTHORS BIOGRAPHY Clément Caplier holds the current position of Postdoctoral Researcher at the University of Poitiers -Institut Pprime, France. He is specialized in optical measurement techniques for hydrodynamics. His previous experience includes an experimental thesis on the effects of finite water depth, lateral confinement and current on ships wakes and drag.

Guillaume Gomit holds the current position of Assistant

Professor at the University of Poitiers -Institut Pprime, France. He obtained his PhD from the University of Poitiers in 2013 for his thesis on the optical measurement of ship waves in towing tanks. His research interests range from experimental hydrodynamic to rheology and transport of cohesive sediments.

Germain Rousseaux holds the current position of Senior Scientist at CNRS. He is a physicist doing interdisciplinary studies. His previous experience includes a PhD thesis on sand ripples formation with specializations on wave-current interactions (wave blocking, tidal bore...) and wake physics. He is interested with wave resistance theory and its applications to wash waves reduction by shapes optimization.

Damien Calluaud holds the current position of Assistant

Professor at the University of Poitiers -Institut Pprime, France. He develops some stereoscopic and free surface measurements. His main interests are the ship waves, the bank erosion, the sediment transport and the ecohydraulics.

Ludovic Chatellier holds the current position of Assistant

Professor the University of Poitiers -Institut Pprime, France. He is involved in renewable energy, hydrodynamics, ecohydraulics and optical metrology. His previous experience includes free-surface measurements, flow and wake analysis, development of energy harvesters and fish protection devices.

Laurent David is Professor of fluid mechanics in the university of Poitiers. He is at the head of the team HYDEE (Hydrodynamics and Environmental flows). Since twenty years, he develops 2D and 3D optical measurements (TR-PIV, surface and volumic PIV). His research interests are free surface flows, turbulent and unsteady flows, fluid structure interactions and ecohydraulics.


Didier Raoult
email: didier.raoult@gmail.com

Philippe Parola

Prof. Didier Raoult

Vaccination against the big three killers: an illusion?

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

For 30 years, the largest investment in life sciences has been in seeking an immunization against the "Big Three": AIDS, tuberculosis and malaria. This choice appears logical as these diseases are among the largest killers among infectious diseases, accounting probably together for >1.750.000 deaths per year [START_REF] Parola | The return of the big three killers[END_REF][START_REF] Who | Disease burden and mortality estimates Cause specific mortality[END_REF]. Attempting to develop a vaccine prevention strategy that yielded extraordinary results on smallpox and which was very significant for poliomyelitis, diphtheria, mumps and measles, could have yielded spectacular results. However, this attempt, despite the increasing levels of financial and scientific investment with hundreds of publications every year, has not produced any tangible and applicable results. Despite the annual declarations of new vaccine strategies for these three diseases, real applicable strategies have never followed. This does not seem surprising to us, and we think that we need to change our strategy in the current state of our knowledge.

Malaria is a non-immunizing disease [START_REF] Ashley | Malaria[END_REF]. People who are exposed to malaria can receive several hundred infective bites per year, suffered several malaria episodes with different strains, and the low immunity that has been eventually acquired, disappears very quickly after a several months in a malaria-free country. We are currently experiencing this in France with African students and Comorian migrants. Vaccine development continues but none of the malaria vaccine candidates developed to date have been shown to provide long-lasting benefits at a population level. Even what appeared to some researchers in the recent years as promising results, the RTS,S/AS01 malaria vaccine administered at months 0, 1, and 2 to 5-17 months children confered, when boosted, about 32% protection against severe malaria for a few months [START_REF] Rts | Efficacy and safety of RTS,S/AS01 malaria vaccine with or without a booster dose in infants and children in Africa: final results of a phase 3, individually randomised, controlled trial[END_REF].We think that when several episodes of a natural disease fail to determine immunity for more than a few months, the chance of finding a vaccine is extremely low.

Likewise for HIV infection, none of the efforts have had any effect despite the spectacular announcements. Twelve late-stage trials worldwide are testing experimental vaccines [START_REF] Robinson | HIV/AIDS Vaccines: 2018[END_REF]. After 30 years still there is no preventive HIV vaccine [START_REF] Maxmen | Promising HIV vaccines could stall without coordinated research[END_REF]. Current research efforts are mainly addressed to "curative" vaccine in HIV infected people, with aim to reach "functional" cure. Anyway, HIV infection is not an immunizing disease. The only forms of real resistance known to retroviruses are either natural resistance, such as in gorillas for HIV [START_REF] Ayouba | Origin of the HIV-1 group O epidemic in western lowland gorillas[END_REF], or, for a number of animals, the internalization and neutralization of the retrovirus that protects against subsequent viral infections. This is called endogenization, which is a phenomenon perfectly observed in Koalas [START_REF] Greenwood | Transmission, Evolution, and Endogenization: Lessons Learned from Recent Retroviral Invasions[END_REF].

Finally, with regard to tuberculosis, BCG is a theoretically ideal vaccine, a live vaccine, a variant of tuberculosis agent. A mild disease is inoculated, that gives a perfectly effective immune response that can be tested and provides robust immunity. It is effective in protecting infants and children against severe miliary and meningeal TB, although its protection is variably lost in adluts [START_REF] Zhu | Tuberculosis vaccines: Opportunities and challenges[END_REF]. Nevertheless, there are currently several million new tuberculosis cases per year, including 1.2 million deaths, which indicates that more than a century of vaccination policy by the BCG has not really cleaned the situation. More than 20 novel TB vaccine candidates are now in clinical trials [START_REF] Zhu | Tuberculosis vaccines: Opportunities and challenges[END_REF][START_REF]Tubersulosis vaccine initiative[END_REF]. We'll see what will be the future of this renewed research effort.

What has changed the prevalence of these diseases is, for malaria, long-lasting insecticide-treated bednets, and for the three diseases, the treatment of carriers in order to significantly reduce the reservoir. The fight against the big three killers is still ongoing, but strategies based on mechanical and therapeutic prevention coupled with shortened treatments are more likely in the next 20 years to help further reduce the prevalence of the three major killers, rather than the hypothetical discovery of a miracle vaccine that would contradict the natural observation of the cycle of these three diseases.

Authors report no conflicts of interest. No external funding was received.