André Mas

High-Dimensional Principal Projections

Keywords: Funtional principal Component Analysis, Dimension reduction, Nonparametric functional regression, Covariance operator, Perturbation theory AMS 2010 Classification: primary : 62H25, 62G08 ; secondary : 47A55

The Principal Component Analysis (PCA) is a famous technique from multivariate statistics. It is frequently carried out in dimension reduction either for functional data or in a high dimensional framework. To that aim PCA yields the eigenvectors ( ϕi) i of the covariance operator of a sample of interest. Dimension reduction is obtained by projecting on the eigenspaces spanned by the ϕi's usually endowed with nice properties in terms of optimal information. We focus on the empirical eigenprojectors in the functional PCA of a n-sample and prove several non asymptotic results. More specifically we provide an upper bound for their mean square risk. This rate does not depend on the rate of decrease of the eigenvalues which seems to be a new result. We also derive a lower bound on the risk. The latter matches the upper bound up to a log n term. The results are applied in a nonparametric functional estimation model.

Introduction

The theoretical covariance operator and its empirical analogue are objects of fundamental importance in the theory of functional data. Principal component analysis is of importance in its own right, and as a dimension reduction technique is often the first step to make finite-dimensional procedures suitable for high dimensional or functional data. In this paper we will elaborate on both aspects.

Let H denote a real separable Hilbert space endowed with inner product •, • and associated norm • and let X 1 , X 2 , ...X n be a sample of independent and identically distributed random elements with values in H. Since we are concerned with functional PCA we will for the most part be dealing with covariance operators, and a precise definition and presentation of some of their main features is in order. Whenever E X 12 < +∞, the theoretical covariance operator Σ and its empirical counterpart, Σ n , based on the sample are symmetric, positive, trace-class operators from H to H defined by :

Σ = E ((X 1 -EX 1 ) ⊗ (X 1 -EX 1 )) , (1)
Σ n = 1 n n k=1 X k -X n ⊗ X k -X n , (2)
where the tensor product between u and v in H stands for the one-rank operator from H to H defined by (u ⊗ v) (t) = u, t v, for all t in H and X n = 1 n n k=1 X k . When X is centered -which will be assumed throughout for mathematical convenience-EX = 0, Σ = E (X ⊗ X) and Σ n = 1 n n k=1 X k ⊗ X k . By (λ k , ϕ k ) we denote the k th eigenelement (eigenvalue and eigenvector) of Σ. The λ k 's are positive and simple by assumption and hence satisfy λ 1 > λ 2 > ... > 0, and (λ k ) k∈N ∈ l 1 . The eigenprojectors are of major importance in the sequel. Since the eigenvalues are of order one we may define π i = ϕ i ⊗ ϕ i the rank one projection operator onto the vector space spanned by ϕ i and P k the projector onto span{ϕ 1 , ..., ϕ k }. We add hats to denote the empirical versions of the eigenelements and we get Σ n ϕ i = λ i ϕ i for all i ∈ N, with all λ i = 0 except for a finite number not exceeding n. We also set π i = ϕ i ⊗ ϕ i and P k .

Let δ (•, •) be some distance, pseudo distance or dissimilarity measure for projectors. The initial purpose of this work is to provide sharp upper and lower non asymptotic bounds for Eδ ( π i , π i ) and for Eδ P k , P k . We intend to display the dependence in i and k in the previous expectations. We also expect these bounds to be adaptive with respect to the eigenvalues decay rate.

Secondly these results will be applied to nonparametric functional regression. [START_REF] Biau | PCA-Kernel estimation[END_REF] reduce the dimension of the regressor by exploring a fixed, finite number of principal components. This naturally causes some information loss. In this paper we will be able to let the number of components and the corresponding dimension of the panel increase with the sample size.

As has been pointed out by [START_REF] Watson | Statistics on Spheres[END_REF] for matrices and by Dauxois, Pousse and Romain (1982) for operators, the elegant techniques of perturbation theory [START_REF] Dunford | Linear Operators[END_REF] and [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]) seem to be tailored to the analysis of the spectral characteristics of random operators when they can be considered as small perturbations of an underlying target operator, as in the case of the sample covariance operators. For general theory with a special view towards applications in statistics we refer, for instance to [START_REF] Mas | Perturbation approach applied to the asymptotic study of random operators[END_REF], [START_REF] Cupidon | The Delta-method for analytic functions of random operators with application to functional data[END_REF], or [START_REF] Gilliam | The Fréchet derivative of an analytic function of a bounded operator with some applications[END_REF]. The interested reader should also note that kernel PCA was also investigated with similar tools in [START_REF] Koltchinskii | Asymptotics of spectral projections of some random matrices approximating integral operators[END_REF] and in [START_REF] Blanchard | Statistical Properties of Kernel Principal Component Analysis[END_REF] for instance. We make an intensive use of perturbation techniques in the proofs.

An essential difficulty which is typical for the infinite dimensional case is that the spacings between the eigenvalues become arbitrarily small. Relations between the spectral characteristics of the sample covariance operator such as the cumulative eigenprojector of order k and those of the population covariance operator are based on a Neumann series expansion for the sample covariance operator. For this expansion to be valid, however, the perturbation should be sufficiently small and in the present situation 'how small' depends on the spacing between the k th and the (k + 1)

st eigenvalue. Because these spacings tend to zero, for a given sample size n sufficiently small prediction errors can only be obtained for cumulative projections up to a finite order k = k n . We will see however that we can let k n → +∞ as n → +∞.

At this point let us briefly elaborate on one of the main theoretical results of the paper : the rate of convergence of the risk of the k th sample eigenprojector (i.e the eigenprojector corresponding to the k th largest eigenvalue). For each n and k an upper bound for this risk is given in Proposition 7. This upper bound is, moreover, shown to be essentially optimal apart from a logarithmic factor in Proposition 2. Apparently the upper bound for the risk depends directly on k but turns out to be independent of the particular decay of the eigenvalues of the population covariance operator as long as some mild conditions are fulfilled. This seems to be new and an improvement over similar results by Hall and Hosseini-Nasab (2005) and [START_REF] Zwald | On the convergence of eigenspaces in kernel principal component analysis[END_REF]. These authors show dependence on the inverse of the k th spacing of the eigenvalues. Since these spacings can become very small, this dependence may have a serious negative effect on the convergence rate of the risk. The reason for the discrepancy between our result and that in Hall and Hosseini-Nasab (2005) probably lies in differences of approach. Our method of proof is based on an exact expansion of the empirical eigenprojector in terms of the theoretical covariance operator and the perturbation, exploiting the associated Neumann series. As has already been observed above the norm of the perturbation operator Σ n -Σ should be sufficiently small for the expansion to be valid. Since this perturbation is random we need to single out the part of the sample space where this requirement is fulfilled. Although, indeed, the requirement for the k th eigenprojection depends on the k th spacing, very precise concentration inequalities allow us to get rid of this dependence in our final result.

The organization of this article is as follows : in the next section we provide basic material about operators and perturbations, then in Section 3 the main theorems about random projectors are given and applied to nonparametric regression in a high dimensional or functional framework. The last section collects the proofs.

2 Notations, elements of operator theory and functional calculus for operators

The space H plays a crucial role since the data will be sampled in such a space. It is clearly suited to high dimensional Euclidean data. But many functional datasets may be also embedded in a Hilbert space : either in a space of non smooth functions such as L 2 ([0, 1]) or with tunable degree of regularity like the Sobolev space of functions

H m ([0, 1]) = f ∈ L 2 ([0, 1]) : f (m) ∈ L 2 ([0, 1]) .
Another reason rather of mathematical nature highlights this choice. We focus here on the covariance structure and PCA for functional data. The trouble is that, for a Banach-valued random element X ∈ B, the covariance operator is an operator from B * , the dual of B onto B (see, for instance, [START_REF] Ledoux | Probability in Banach Spaces[END_REF] or Vakhania, Tarieladze, Chobanyan (1987)). The structure of the dual space B * may not be simple (even for rather basic examples of B such as the space of continuous functions on a compact set) except in the case when B = H is a Hilbert space where B * = H. We introduce the following two operator spaces and associated norms. The Banach space L (H, H) = L is the classical space of bounded linear operators mapping H onto itself endowed with the norm defined for each T in L by :

T ∞ = sup x∈B1 T x ,
where B 1 is the unit sphere of H. The Hilbert space L 2 is the space of Hilbert-Schmidt operators, (L 2 ⊂ L) i.e. the space of those operators T such that, for any orthonormal basis (e k ) k∈N of H,

T 2 2 = +∞ k=1 T e k 2 < +∞.
It is a well-known fact that L 2 is a separable Hilbert space whenever H is. The inner product in L 2 is T, S 2 = +∞ k=1 T e k , Se k and does not depend on the choice of the basis either. The space L 1 of finite trace (or nuclear) operators will be mentioned sometimes in the paper but is of secondary importance. We just mention that L 1 ⊂ L 2 ; The two norms mentioned above are not equivalent and

• ∞ ≤ • 2 .
Consequently the canonical injection from L 2 onto L is continuous. For further information on linear operators we refer to [START_REF] Schmeidler | Linear Operators in Hilbert Spaces[END_REF], [START_REF] Weidman | Linear Operators in Hilbert Spaces[END_REF], [START_REF] Dunford | Linear Operators[END_REF], Gohberg, Goldberg and Kaashoek (1991) amongst many others.

Finally we briefly introduce some basic facts about functional calculus for operators since all this material makes the core of the mathematical techniques involved in the proofs. The notion of functional calculus for operators is twofold. We mention the book by A.V. [START_REF] Skorohod | Random Linear Operators[END_REF] dealing with random spectral measures and leading to a specific theory. Here by functional calculus we will strictly allude to perturbation theory and the functions f (T ) we consider have to be analytic in a neighborhood of the spectrum of T .

We refer the reader to [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF], [START_REF] Weidman | Linear Operators in Hilbert Spaces[END_REF], [START_REF] Dunford | Linear Operators[END_REF], Gohberg, Goldberg and Kaashoek (1991) for a complete presentation of the perturbation-based functional calculus for operators. Roughly speaking this theory allows to define f (T ) where T is a linear operator and f is function defined on and with values in a suitable part of the complex plane. Note that the mathematical nature of f and T make them incompatible so that a special definition of f (T ) is needed. It will be in particular possible to consider the eigenprojections of T as functions of this operator. For this purpose the resolvent (ζI -T )

-1 defined for each ζ outside the spectrum of T will be needed.

Let ∂Λ k be the rectangular contour of the complex plane defined at Figure 1. And let ∂Ω k be the contour of the complex plane associated to the connected domain Ω k separating the k first eigenvalues of Σ from the others and defined in the following way : Ω k = ∪ k j=1 C j where C j is a circle centered at λ j and with radius δ j = 1 2 min (λ j -λ j+1 , λ j-1 -λ j ).

3 0

λ m+1 λ m • • • λ 1 2λ 1 -1 1 ∂Λ k δ m Figure 1: Rectangular contour 0 λ k • • • λ 2 λ 1 2δ k 2δ 2 2δ 1 ∂Ω k Figure 2: Contour made of disjoint circles
Results from perturbation theory yield :

P k = 1 2πι ∂Ω k (ζI -Σ) -1 dζ = 1 2πι k j=1 ∂Cj (ζI -Σ) -1 dζ = k j=1 π j , (3)
P k = 1 2πι ∂Λ k (ζI -Σ) -1 dζ,
where ι 2 = -1 and π j is the orthogonal projector associated with the single eigenvector ϕ j . This may be rewritten π j = ϕ j ⊗ ϕ j when λ j is of order one. The empirical counterpart of (3) involves a random contour, say, ∂ Ω k (resp. ∂ Λ k ) similar in shape to ∂Ω k (resp. ∂Λ k ) which contains the k first eigenvalues of Σ n . In fact only the left edge of ∂ Λ k may be considered as random :

P k = 1 2πι ∂ Ω k (ζI -Σ n ) -1 dζ, (4)
and P k as defined in (4) equals the sum of all eigenprojections of Σ n with eigenvalues inside ∂ Ω k each denoted π j .

Main results

We state first the assumptions needed to derive our results. Throughout the sequel the symbols c and C will denote universal constants.

Assumptions on X

In this Hilbert setting, the distribution of a centered random element X may be characterized in a very simple way. Indeed if = d denotes equality in distribution, we have :

X = d +∞ j=1 λ j η j ϕ j (•) (5)
where the η j 's are centered non-correlated real random variables with unit variance. The above decomposition is often referred to as the Karhunen-Loève development or development of X with respect to its reproducing kernel Hilbert space (RKHS). Besides the η j 's are independent when X is a Gaussian random function. For definition and studies of RKHS we refer to Berlinet, Thomas-Agnan (2004). Consider X a centered random function with Karhunen-Loève development [START_REF] Bosq | Linear processes in function spaces[END_REF]. We assume that the η i 's are independent, that the eigenvalues λ i are all of order one and that the sequence (λ i ) i∈N is decreasing. We need higher moment assumptions because we need to apply Bernstein's exponential inequality to functionals of Σ n -Σ. More specifically we assume that for all j, ∈ N there exists a constant b such that

max j E |η j | 2 ≤ !b -1 . (6)
These assumptions hold for instance for Gaussian X (with b = 4 for instance) and when the η j 's are compactly supported.

Remark 1 Assuming the independence of the η i 's is a rather strong assumption. It stems here from the use of well-known exponential inequalities. The computations may be possible however by replacing independance by cross moments conditions or by invoking exponential bounds for mixing sequences. These issues go beyond the scope of the article.

Assumptions on the spectrum of Σ Let the function λ : R + → R + be defined by λ (j) = λ j for any j ∈ N and by continuous interpolation between j and j + 1. From the assumption above we already know that j λ j < +∞. Indeed the summability of the eigenvalues of Σ is ensured whenever E X 2 < +∞. Moreover, we assume that one of the two following assumptions hold for the eigenvalues :

λ j = cj -1-α , (7)
λ j = c exp (-αj) , (8)
where c and α are some strictly positive constants. The spacings δ j can be deduced easily. These last conditions are rather mild and match a large class of eigenvalues. The generalization to other classes of eigenvalues sequences like Laurent series λ j = cj -1-α (log j) -β or even λ j = cj -1 (log j) -1-β with β > 0 is straightforward and will be omitted.

Eigenprojectors

As announced in the Introduction we aim at estimating the risk for principal projectors and recall that in our framework π k is associated to the k th principal space of the functional PCA and defined by π k = ϕ k ⊗ ϕ k where ϕ k is the k-th empirical eigenvector associated with the k-th empirical eigenvalue.

Introducing risk or loss functions for projectors should be carried out with caution. Indeed proximity between projection operators is intimately connected with proximity between subspaces which in turn may be slightly more intricate to measure than for vectors or even functions. To briefly illustrate these facts consider a fixed basis in H say (ϕ j ) j∈N and consider P k that projects on span{ϕ j : 1 ≤ j ≤ k}. Obviously for all x in H we have P k x → x when k increases to infinity but P k -I ∞ = 1 for all k. The same arguments holds when comparing P k and P k+1 for large k. Consequently the choice of a proximity measure between π k and π k for instance may have a major influence on the final rates. We provide below a lower bound and several upper bounds for the differences π k -π k and P k -P k and investigate the usual sup norm and evaluation semi-norms at fixed or random points. In the following a series plays a crucial role. Let :

a k =   i =k λ i |λ i -λ k | + λ k δ k   .
Our main results may be generally stated by means of a k and with no specific assumption on the eigenvalues. We will prove in Lemma 15 that under [START_REF] Chen | Single and multiple index functional regression models with nonparametric link[END_REF] or ( 8), a k ≤ ck for some positive constant c. This will provide a simpler statement of the theorems. We start with a lower bound for π k .

Proposition 2

The following lower bound holds :

E π k -π k 2 ∞ ≥ 1 2n j =k λ k λ j / (λ j -λ k ) 2 -4 a 4 k n 2 .
When ( 7) or ( 8) holds it becomes :

E π k -π k 2 ∞ ≥ c 3 k 2 n .
We notice that the lower bound seems to be non-sensitive to the eigenvalue decay rate.

Remark 3 The proof of Proposition 2 is strongly connected to functional calculus techniques for operators. We write

π k = f k (Σ n ) and π k = f k (Σ)
. Then considering Σ n as a (random) perturbation of Σ the lower bound is derived through some development up to the second order of

π k -π k , namely f k (Σ n ) -f k (Σ) = g 1 k (Σ n -Σ) + g 2 k (Σ n -Σ, Σ n -Σ) . We show that the second order term g 2 k (•, •) is negligible with respect to the first g 1 k (Σ n -Σ) and we compute the exact value of E g 1 k (Σ n -Σ) 2 ∞ .
Remark 4 Dauxois, Pousse and Romain (1982) in their pioneering work derived the asymptotic distribution of projectors such as π k . The covariance operator of the Gaussian limiting distribution appears at page 145 and is detailed on page 146 of their article. It is intimately connected with the first order term of our lower bound given above.

Now we are ready to turn to the issue of the upper bound. We start with the main results about the cumulative projectors P k and P k . But for the sake of clarity we need to distinguish three situations.

Case 1 : (7) with α > 1 or (8) hold, Case 2 : (7) holds with 0 < α ≤ 1, Case 3 : (7) holds.

Theorem 5 Let (6) hold. Take u =X i or u =X n+1 or u nonrandom with the additional assumption that sup i | u, ϕ i | 2 /λ i < 1 then for all n ≥ 2 and k ≥ 2:

E P k -P k u 2 ≤ κc 0 exp -c 1 n a 2 k + c 3 • a 2 k n log 2 n •   k j=1 jδ j   2 where κ = (2b -1) i λ 2 i + (trΓ) 2 .
Then we can derive :

In Case 1 :

E P k -P k u 2 ≤ c 4 k 2 log 2 n n In Case 2 : E P k -P k u 2 ≤ c 4 k 2 log 2 k log 2 n n
In Case 3 :

E P k -P k 2 ∞ ≤ c 4 k 2 log 2 k log 2 n n Remark 6
It is important to notice that we could not assess Theorem 5 in sup norm in Cases 1 and 2, when the eigenvalues decay very rapidly i.e. with exponential rate. Actually this result may be attainable but we could not get it with our method of proof. Roughly speaking, the assumption sup i | u, ϕ i | /λ i < 1 is crucial to get the bound above with a fast decay of the λ p 's. This assumption prevents us to obtain the uniform bound. This pathology is closely connected with the inverse problems aspects of the estimation of ϕ k hence of π k . However the forthcoming Proposition 7 overcomes this problem when P k is replaced by π k .

The two next results may be viewed as consequences of Theorem 5. Their proofs are consequently omitted.

Proposition 7 Let (6) hold. Then :

E π k -π k 2 ∞ ≤ c 0 exp -c 1 n a 2 k + c 3 • a 2 k n log 2 n.
When [START_REF] Chen | Single and multiple index functional regression models with nonparametric link[END_REF] or [START_REF] Cupidon | The Delta-method for analytic functions of random operators with application to functional data[END_REF] hold this bound becomes :

E π k -π k 2 ∞ ≤ exp -c 1 log 2 n + c 2 k 2 log 2 n n
This latter bound is rather sharp when compared with the lower bound derived in Proposition 2 at least in our framework and with our set of assumptions. It differs from a log 2 n term.

We turn to the eigenvectors in the next Corollary and recall that π k = ϕ k ⊗ ϕ k where ϕ k is determined up to a sign. In order to fix this point ϕ k must be chosen so that the sign of ϕ k , ϕ k is positive (see [START_REF] Hall | On properties of functional principal components analysis[END_REF]).

Corollary 8 Select ϕ k so that the sign of ϕ k , ϕ k is positive the under the assumptions of Proposition 7 :

E ϕ k -ϕ k 2 ≤ 4 exp -c 1 log 2 n + c 2 k 2 log 2 n n .
This result could be compared with Theorem 1 p.114 in Hall and Hosseini-Nasab (2006). These authors obtained in a similar framework but under milder moment assumptions

ϕ k -ϕ k ≤ c/ (δ k √ n)
with high probability. The corollary above may be viewed as some kind of improvement because the right hand side does not depend on the eigenvalue decay rate and is always sharper since k 2 δ k → 0 when k → +∞. In the framework of Kernel PCA, Bousquet, Blanchard and Zwald (2007) obtain similar results involving δ -1 k in their Theorem 3.7. It would be an interesting issue to know if our approach extends to Kernal PCA but it is out of the scope of the current paper. Their bounds however were somewhat surprising in the following sense : if the eigenvalues decay quickly to zero so do the δ k 's and this means that the data are close to a finite dimensional sample. Then we would expect a rate always closer to √ n for the convergence rate of these eigenprojectors. Yet these bound based on δ -1 k get always worse. In the case of situation (8), resp. ( 7) these bounds are O e k / √ n resp. O k 2+α / √ n . This problem does not remain here, at least under the assumptions we consider for the λ k 's.

Remark 9

The method of proof adapts to projections on any set of eigenvectors chosen amongst the k first. This will not significantly improve the rate of convergence however. This fact, underlined by a referee, paves the way to introducing some kind of high-pass filters (with an obvious Fourier analogy) by projecting on (ϕ m , ...ϕ k ) with m = αk , α ∈ (1/2, 1) for instance. This may be of interest when studying spectrometric curves for which the high-frequency components are higly informative.

Application to high-dimensional kernel estimation

We propose to apply the preceding results in the specific context of a two-step estimation algorithm adapted to the regression when X lies in a high-dimensional or functional space. Let (Y, X) be a couple of random elements and let r (x) = E (Y |X = x) be the regression function evaluated at a fixed x. From the sample (Y i , X i ) 1≤i≤n ∈ R×H where H stands like above for a Hilbert space of functions or the Euclidean space R d where d is large (typically with respect to the sample size) we investigate the classical issue of the estimation of r (x). General regression estimation has been intensively studied in the past years [START_REF] Masry | Nonparametric regression estimation for dependent functional data: Asymptotic normality[END_REF], [START_REF] Ferraty | Recent Advances on Functional Data analysis and Related Fields[END_REF]Vieu. (2004, 2006), Ferraty, Mas, Vieu (2007), Ferraty, Laksaci, Tadj, Vieu (2010). Several methods may be carried to solve numerical or implementation issues like the computation of small ball probabilites, a crucial step in asymptotic theory. However the complexity of the general model leads to degenerate minimax rates of convergence, typically of order [log n]

-α for some positive α -which may be seen as the ultimate side effect of the curse of dimensionality-as shown in [START_REF] Mas | Lower bound in regression for functional data by small ball probability representation in Hilbert space[END_REF]. This paves the way to different strategies for nonlinear models involving functional data. Such alternatives are additive models, multiple indices models or mixed approach such as projection pursuit. Several authors studied these variants of the general regression : Amato, Antoniadis, De Feis (2006) and [START_REF] Ferraty | Recent Advances on Functional Data analysis and Related Fields[END_REF] for the single index model, Chen, Hall and Müller (2011) for the multiple index model and Ferraty, Goïa, Salinelli, Vieu (2012) for the Functional Projection Pursuit Regression amongst others.

However here the method is the following. First compute the (potentially functional) PCA of the sample (X i ) 1≤i≤n and retrieve the projectors P k with a prescribed k. Here k will be chosen so that k d. Then compute an estimate of r (x) inspired from the classical Nadarya-Watson approach :

r N W (x) = n i=1 Y i K ( X i -x /h) n i=1 K ( X i -x /h) .
This is not really fair to call r N W (x) above an estimate of r (x) since the norm • cannot be practically calculated and should be numerically approximated for instance when H =L 2 ([0, 1]). By replacing the theoretical norm on the space H by the 'projection' semi-norm • k defined by

u k = P k u R k
the estimate becomes :

r (x) = n i=1 Y i K P k (X i -x) /h n i=1 K P k (X i -x) /h .
This proposal is intuitive and rather simple to implement through standard statistical packages from most softwares. The problems arise from the dependence between X i and P k : the numerator and denominator of r (x) are not sums of independent random variables and the dependence structure is, in our opinion rather confusing.

We introduce :

r * (x) = n i=1 Y i K ( P k (X i -x) /h) n i=1 K ( P k (X i -x) /h) .
Contrary to r (x) the random variable r * (x) is not an estimate since P k is the population projector of the PCA and is consequently unknown. But its numerator and denominator ar both sums of independent random variables. It may be viewed as an oracle in the sense that the risk for r * (x) is expected to remain below the risk for r (x). In order to investigate r (x) Biau and Mas (2012) introduce τ n the minimax rate of convergence in the non-parametric regression model over a wellchosen class of functions and show that :

r (x) -r * (x) τ n → n→+∞ 0, (9)
as the sample size increases and for a fixed dimension of projection D (denoted here by k and that may increase with the sample size in our framework). We have to adapt their method here because the dimension k is not fixed. Besides in (9) constants depending on D are used for bounding ( r (x) -r * (x)) /τ n . These constants are made explicit here and we must take them into account because they usually tend to infinity when k does.

Remark 10 A more classical route would be to investigate the risk for the estimate r (x) and by the way to prove the convergence of r (x) . Considering (9) circumvents this difficult issue which is beyond the scope of the paper.

For later use let :

S * = n i=1 K ( P k (X i -x) ) , S n = n i=1 K P k (X i -x) , Z * = n i=1 Y i K ( P k (X i -x) /h) , Z n = n i=1 Y i K P k (X i -x) /h .
We introduce now the set of assumptions we need to derive our main Proposition.

Assumptions on the small ball probabilities [A1] :

F k (s) = P ( P k (X i -x) < s) with F k (s) > 0 in a right neighborhood of 0 and sup h∈[0,1] F k (hu) F k (h) ≤ c F u p with p ≤ k.
Assumptions on the kernel K [A2] : The kernel K is bounded above and below on its support [0, 1] with 0

< inf u∈[0,1] K (u) =K≤ |K| ∞ , u p K (u) ∈ L 1 ([0, 1)) where p appears in [A1].
Assumptions on the regression function [A3] : r is bounded. At this point some comments are needed about the assumptions : [A2] and [A3] are classical. The reader familiar with nonparametric methods will remark that we do not need regularity assumptions about r. This is due to the fact that below we are only interested in bounding the difference r (x) -r * (x) and not r (x) -r (x). As a consequence the approximation of r -in its functional analysis sense-is not our aim here and ensuing smoothness conditions are no more required. Assumption [A1] though not quite standard is rather mild. Indeed we consider there the cumulative density function (in a neighborhood of zero) of the positive real random variable P k (X i -x) which in turn is the norm of the R k valued random vector P k (X i -x). It is natural to think that the local behaviour of F k (s) is polynomial with degree less than p in a neighborhood of 0. Smoothness of the density of P k X i evaluated at P k x would ensure this fact. And if F k (s) ∼ s p L (s) with L a slowly-varying function at 0 then [A1] holds.

Proposition 11

The following non asymptotic bound holds under the assumptions ( 6), ( 8) or [START_REF] Chen | Single and multiple index functional regression models with nonparametric link[END_REF] with α > 1 and [A1-3] above :

E [ r (x) -r * (x)] 2 ≤ c 6 k 4 nh 2 log 2 n log h √ n k 2 . ( 10
)
A first difference appears with Biau and Mas (2012). The bound of Proposition 11 displays explicitly the dimension k whereas this dimension was fixed and did not appear in the work of these authors. A comparison with their Corollary 4.1 shows that, up to a log n term, the numerator k 4 log k is new. It accounts for the price to pay to let the dimension increase with the sample size.

We remind that in a non asymptotic framework the minimax rate for the risk in nonparametric regression (see for instance the monograph by [START_REF] Tsybakov | Introduction a l'estimation non-paramétrique[END_REF] and references therein) is up to constants τ * n (α, k) = n -2α/(2α+k) where α is a smoothness parameter unimportant here. When a Nadarya-Watson estimate is selected the optimal bandwidth is an h * = O n -1/(2α+k) , and we

can write τ * n (p, k) = n • (h * ) k -1
.

The next Proposition develops the comparison between E [ r (x) -r * (x)] 2 and τ * n (p, k).

Proposition 12 Under the assumptions of the previous proposition and assuming that k > 2, 0 ≤ h ≤ h max < 1 and that h τ log 3 n is uniformly bounded with respect to n for any τ > 0 there exists a constant c 7 (h max ) such that :

sup h,k nh k E [ r (x) -r * (x)] 2 ≤ c 7 .
In an asymptotic perspective nh k E [ r (x) -r * (x)] 2 → n→+∞ 0 for any bandwidth h depending either on k n or n such that h → 0.

The proof of Proposition 12 is omitted. For the first part it suffices to see that

h k-2 k 4 log 3 n ≤ h k-2-τ k 4 h τ log 3 n h k-2-τ max k 4 ≤ c 7
where here and elsewhere x y means that x ≤ cy for some constant c. The second part follows from straigthforward calculations.

Mathematical derivations

We introduce the following events for later use. Let 0 < c 0 < 1/2

A k = λ k ∈ Ω k ∩ λ k+1 / ∈ Ω k , B k (c 0 ) = sup ζ∈∂Ω k (ζI -Σ) -1/2 (Σ n -Σ) (ζI -Σ) -1/2 ∞ < c 0 .
The event B k (c 0 ) will be introduced within the proof of our main result. Lemma 18 will detail the connection between B k (c 0 ) and A k . When A k holds the k first empirical eigenvalues (and only the k first) are inside Ω k . Then subject to these events we can derive the linearization :

P k -P k = 1 2πι ∂Ω k (ζI -Σ n ) -1 -(ζI -Σ) -1 dζ. (11)
The formula above comes down to removing the random contour ∂ Ω k in the definition of P k at (4). But we have to condition with repect to A k first. On A k the strategy will be to see that

E P k -P k ∞ 1 A k ≤ 2P
A k and to bound accurately the latter probability. Thanks to lemma 18 we will restrict ourselves to bounding P B k (c 0 ) . We also assuume in all the sequel that a k log n/ √ n < 1/2.

Proof of Theorem 5

The proof takes two steps. First we apply the linearization formula ( 11) above but we have to control P A k non asymptotically which is achieved at Lemma 16 below. Second we will use perturbation theory to bound accurately the linearized difference between P k and P k .

We state or recall now two technical Lemmas.

Lemma 15 When ( 7) or ( 8) hold there exists a constant c α such that for all k ≥ 1

i =k λ i |λ i -λ k | + λ k δ k ≤ c α k. (12)
The proof stems from classical Riemann approximations. The two next Lemmas aim at proving that the events A k and B k (c 0 ) hold with a high probability. We derive in fact two exponential inequalities of the same kind showing finally that P B k (t)

P A k exp -k 2 /n . Lemma 16 Denote a k = i =k λi |λi-λ k | + λ k δ k
We have :

P B k (t) ≤ 2 exp - nt 2 2a 2 k 1 (2b -1) + 256b 3 (2b-1)a k t .
Remark 17 Earlier work focus on a probabilistic control of Σ n -Σ to obtain bounds on the projectors. Here the idea consists in studying a normalized -through perturbation and resolventvariant of the latter to derive a more precise result.

Proof of Lemma 16 :

In this subsection, we use Bernstein's exponential inequality for Hilbertvalued random variables (see for instance [START_REF] Bosq | Linear processes in function spaces[END_REF] p.49) and references therein). First denote

µ k = λ k -δ k Π = Σ n -Σ and Π (µ k ) = (µ k I -Σ) -1/2 (Σ n -Σ) (µ k I -Σ) -1/2
and rewrite :

(ζI -Σ) -1/2 (Σ n -Σ) (ζI -Σ) -1/2 = G k (ζ) Π (µ k ) G k (ζ) , with G k (ζ) = (ζI -Σ) -1/2 (µ k I -Σ) 1/2 we get : sup ζ∈∂Ω k (ζI -Σ) -1/2 (Σ n -Σ) (ζI -Σ) -1/2 ∞ ≤ sup ζ∈∂Ω k G k (ζ) 2 ∞ • Π (µ k ) ∞ ≤ 9 Π (µ k ) ∞ since sup k sup ζ∈∂Ω k G k (ζ) ∞ ≤ 3 whatever the contour chosen. Then : Π (µ k ) 2 ∞ ≤ Π (µ k ) 2 2 = +∞ p,q =k (Σ n -Σ) (ϕ p ) , ϕ q 2 |µ k -λ p | |µ k -λ q | + (Σ n -Σ) (ϕ k ) , ϕ k 2 δ 2 k + +∞ p =k (Σ n -Σ) (ϕ p ) , ϕ k 2 |µ k -λ p | δ k ≤ 4 +∞ p,q =k (Σ n -Σ) (ϕ p ) , ϕ q 2 |λ k -λ p | |λ k -λ q | + (Σ n -Σ) (ϕ k ) , ϕ k 2 δ 2 k (13) + 2 +∞ p =k (Σ n -Σ) (ϕ p ) , ϕ k 2 |λ k -λ p | δ k , since for p = k |µ k -λ p | ≥ |λ k -λ p | -δ k ≥ |λ k -λ p | /2.
Finally, ignoring the constants 4 and 2 in (13) and 9 that appeared above, our only task is to give an exponential bound for the probability :

P   +∞ p,q =k (Σ n -Σ) (ϕ p ) , ϕ q 2 |λ k -λ p | |λ k -λ q | + (Σ n -Σ) (ϕ k ) , ϕ k 2 δ 2 k + +∞ p =k (Σ n -Σ) (ϕ p ) , ϕ k 2 |λ k -λ p | δ k > t   .
Consider the linear bounded symmetric operator s k defined in the basis

(ϕ i ) 1≤i by s k (ϕ i ) = ϕ i / |λ k -λ i | for i = k and s k (ϕ k ) = ϕ k / √ δ k . It is plain that : +∞ p,q =k (Σ n -Σ) (ϕ p ) , ϕ q 2 |λ k -λ p | |λ k -λ q | + (Σ n -Σ) (ϕ k ) , ϕ k 2 δ 2 k • +∞ p =k (Σ n -Σ) (ϕ p ) , ϕ k 2 |λ k -λ p | δ k = s k (Σ n -Σ) s k 2 2 , with s k (Σ n -Σ) s k = (1/n) n i=1 Z i,k with Z i,k = s k X i ⊗ s k X i -s k Σs k hence EZ i,k = 0.
In order to apply Theorem 2.5 in [START_REF] Bosq | Linear processes in function spaces[END_REF] mentioned above we have to identify l k and b k such that for all integer m, n

-m+1 E Z 1,k m 2 ≤ m!l 2 k b m-2 k /2.
We claim that the previous inequality holds with b k = a k 256b 3 / [(2b -1) n] and l k = (2b -1) a 2 k /n. We prove it now. Let m = 2 we get E Z 1,k

2 2 = E s k X 1 4 -s k Σs k 2 2 where s k Σs k 2 2 = +∞ i=1 λ 2 i / |λ k -λ i | 2 . Set abusively |λ i -λ k | = δ k when i = k along the next five lines then : E s k X 1 4 = E +∞ i=1 λ i η 2 i / |λ k -λ i | 2 = +∞ i=1 λ 2 i Eη 4 i |λ k -λ i | 2 + +∞ i =j λ i λ j |λ k -λ i | |λ k -λ j | ≤ 2b +∞ i=1 λ 2 i |λ k -λ i | 2 + +∞ i λ i |λ k -λ i | 2 - +∞ i=1 λ 2 i |λ k -λ i | 2 .
Then

E s k X 1 4 -s k Σs k 2 2 = 2 (b -1) +∞ i=1 λ 2 i |λ k -λ i | 2 + +∞ i λ i |λ k -λ i | 2 ≤ (2b -1) +∞ i λ i |λ k -λ i | 2 .
From this we see that we can choose l 2 k = (2b -1) a 2 k /n. By classical properties of the norm in Hilbert spaces and Minkowski's inequality we get :

E Z 1,k m 2 ≤ 2 m/2 E s k X 1 4 + s k Σs k 2 2 m/2 ≤ 2 m E s k X 1 2m + s k Σs k m 2 .
We keep this inequality in mind. Now we bound more generally E s k X 1 2m . We have to compute : i1,...,im

λ i1 ...λ im |λ k -λ i1 | ... |λ k -λ im | E η 2 i1 ...η 2 im
where we set abusively |λ j -λ p | = δ j when p = j for the sake of simplicity. We focus on E η 2 i1 ...η 2 im and recall that we assumed just above (6) that the η j 's are independent. In order to bound accurately this expectation we consider the rearrangement of the η i 's involved and introduce D as the numbers of distincts indices in the product η 2 i1 ...η 2 im . Clearly 1 ≤ D ≤ m. We write

η 2 i1 ...η 2 im = Π D l=1 η 2α l p l
where p l are all distinct indices (hence the η p l are independent) and the sequence of exponents (α 1 , ..., α D ) adds up to m.

Hence E η 2 i1 ...η 2 im = Π D l=1 Eη 2α l p l ≤ Π D l=1 (α l !) • b α l -1 = b m-D Π D l=1 α l !. Since Π D l=1 α l !/m! ≤ 1 we get : E η 2 i1 ...η 2 im ≤ m!b m 1 b D .
From the moment bound [START_REF] Cardot | CLT in functional linear models[END_REF] and by Jensen's inequality we get

1 = E |η 1 | 2 2 ≤ E |η 1 | 4 ≤ max j E |η j | 4 ≤ 2b hence 1/b D ≤ 2 D ≤ 2 m .
Collecting all these facts we get first :

E s k X 1 2m ≤ m! (2b) m a m k . From s k Σs k m 2 = +∞ i=1,i =k λ 2 i / |λ k -λ i | 2 + λ 2 k /δ 2 k m/2
we can derive the moment inequality involving Z 1,k :

E Z 1,k m 2 ≤ 2 m   m! (2b) m a m k +   +∞ i=1,i =k λ 2 i / |λ k -λ i | 2 + λ 2 k /δ 2 k   m/2    ≤ 2 m m! (2b) m a m k   1 + +∞ i=1,i =k λ 2 i / |λ k -λ i | 2 + λ 2 k /δ 2 k m/2 m! (2b) m a m k    ≤ 2 m+1 m! (2b) m a m k
It is now a simple computation to identify this last term with b k and we get b

k = a k 256b 3 / [(2b -1) n] .
The exponential inequality in the Lemma is derived from Theorem 2.21 p.49, equation (2.21) in [START_REF] Bosq | Linear processes in function spaces[END_REF].

Lemma 18 For some c 0 < 1/2 we have :

P A k ≤ 3P B k (c 0 ) .

Proof of Lemma 18 :

The proof of the Lemma consists essentially in proving A k ⊂ B * k (c 0 ) for some c 0 where B * k (c 0 ) is a set containing at least B k (c 0 ). The proof is similar to Lemmas E1 and E2 pages 7 and 8 of the supplementary material of Hilgert, Mas, Verzelen (2013). The contour used by these authors differs slightly from Ω k (another circle was added to make sure that any λ j j ≤ k falls inside the contour). We recall that the choice of the contour does not change the projector whenever k eigenvalues are located inside (see for instance Lemma XVIII.2.31 p.2255 of the part III in [START_REF] Dunford | Linear Operators[END_REF]).

Proof of Theorem 5:

The outline of the proof is the following. We split the situation in two sets. We first choose an accurate l k,n = a k √ n log n < 1/2 and consider B k (l k,n ). When B k (l k,n ) does not hold we can bound roughly P k -P k ∞ by 2P B k (l k,n ) and use Lemma 16 above to get the exponential inequality

P B k (l k,n ) ≤ 2 exp - log 2 n 2 1 (2b -1) + 256b 3 (2b-1) log n √ n ≤ 2 exp -c log 2 n
where c is a positive constant. Now when B k (l k,n ) holds we take advantage of the linearization formula [START_REF] Ferraty | Recent Advances on Functional Data analysis and Related Fields[END_REF].

Recall that Π = Σ n -Σ and Π (µ k ) = (µ k I -Σ) -1/2 (Σ n -Σ) (µ k I -Σ) -1/2 Denote R (ζ) = (ζI -Σ) -1 and R (ζ) = (ζI -Σ n ) -1 . It is easy to see that R (ζ) = R 1/2 (ζ) T (ζ) R 1/2 (ζ) with T (ζ) = (ζI -Σ) 1/2 R (ζ) (ζI -Σ) 1/2 , hence T (ζ) -I = T (ζ) R 1/2 (ζ) ΠR 1/2 (ζ) and T (ζ) = I -R 1/2 (ζ) ΠR 1/2 (ζ) -1 whenever sup ζ∈∂Ω k Π (ζ) ∞ ≤ l k,n < 1. Then : R (ζ) -R (ζ) = R 1/2 (ζ) I -Π (ζ) -1 Π (ζ) R 1/2 (ζ) . (14)
We underline here that at this point the contour considered in the lines below is given in Figure 2. As announced earlier :

P k -P k u ≤ 2 u 1 B k (l k,n ) + 1 2πι ∂Ω k R 1/2 (ζ) I -Π (ζ) -1 Π (ζ) R 1/2 (ζ) udζ 1 B k (l k,n ) .
(15) We focus on the second term in the left hand side of the above equation :

1 2πι ∂Ω k R 1/2 (ζ) I -Π (ζ) -1 Π (ζ) R 1/2 (ζ) udζ 1 B k (l k,n ) ≤ l k,n 2π (1 -l k,n ) ∂Ω k R 1/2 (ζ) ∞ R 1/2 (ζ) u dζ.
At this point we must consider different situations depending on the nature of u.

First notice that R 1/2 (ζ) ∞ = sup p |ζ -λ p | -1/2 ≤ c |ζ -λ k | -1/2
due to the design of the contour ∂Ω k . Taking expectation and applying Cauchy-Schwartz inequality to E u 2 1 B k (l k,n ) we get :

E P k -P k u 2 ≤ 4E u 4 1/2 P 1/2 B k (l k,n ) + 2 l k,n 2π (1 -l k,n ) 2 E ∂Ω k |ζ -λ k | -1/2 R 1/2 (ζ) u dζ 2 ≤ c exp -c log 2 n + 2 π 2 l 2 k,n E ∂Ω k |ζ -λ k | -1/2 R 1/2 (ζ) u dζ 2 , (16)
for some c and c . Now we split the contour ∂Ω k and compute the integral in the brackets for each circle ∂C j . Consider two cases. First if j = k :

E ∂C k |ζ -λ k | -1/2 R 1/2 (ζ) u dζ = |δ k | -1/2 ∂C k E +∞ p=1 u, ϕ p 2 |ζ -λ p | dζ ≤ |δ k | 1/2 sup ζ∈∂C k +∞ p=1 λ p |ζ -λ p | ≤ c |δ k | 1/2 +∞ p=1,p =k λ p |λ k -λ p | + λ k δ k ≤ c |kδ k | 1/2 since sup i | u, ϕ i | 2 /λ i < 1. Now if j = k we have quite similarly : E ∂Cj |ζ -λ k | -1/2 R 1/2 (ζ) u dζ ≤ c ∂Cj |λ j -λ k | -1/2 E R 1/2 (ζ) u dζ ≤ |λ j -λ k | -1/2 ∂Cj E +∞ p=1 u, ϕ p 2 |ζ -λ p | dζ ≤ |λ j -λ k | -1/2 δ j E +∞ p=1,p =j u, ϕ p 2 |λ j -λ p | + u, ϕ j 2 δ j ≤ |λ j -λ k | -1/2 δ j +∞ p=1,p =j λ p |λ j -λ p | + λ j δ j ≤ |λ j -λ k | -1/2 δ j j ≤ jδ j .
Plugging these bounds into ( 16) we obtain :

E P k -P k u 2 ≤ c exp -c log 2 n + C • l 2 k,n   k j=1 jδ j   2 . ( 17
)
Obviously this bound is of interest when k j=1

jδ j is reasonably low (typically bounded uniformly with respect to k or O (log k)) which in turns occurs when λ j = c exp (-αj) or when λ j = cj -1-α with α ≥ 1. However when λ j = cj -1-α with 0 < α < 1 or when λ j = cj -1 log 1+α j k j=1 jδ j ≥ k (1-α)/2 and gets large when k does the bound above loses some interest. It is possible to circumvent this problem just by changing the contour used to define and compute the projectors. It turns out that this new contours yields a control of the norm

E P k -P k 2 ∞
, which is a deeper result than the pointwise control above in the special situation : λ j = cj -1-α with 0 < α.

Indeed consider the new contour ∂Λ k obtained as the boundary of the rectangle of the complex plane with left edge at x = λ k -δ k , right edge at λ 1 + 1 and horizontal edge at y = ±1. This contour is drawn at Figure 1. The initial formula [START_REF] Ferraty | Recent Advances on Functional Data analysis and Related Fields[END_REF]

remains unchanged P k -P k = 1 2πι ∂Λ k (ζI -Σ n ) -1 -(ζI -Σ) -1 dζ.
It is also obvious that all the preliminary computations carried out to control the probability of the events

A k = λ k ∈ Ω k ∩ λ k+1 / ∈ Ω k hence B k (l kn )
remain true when changing the contour. Consequently we can turn to the bound [START_REF] Ferraty | Nonparametric models for functional data with applications in regression, time series prediction and curve discrimination[END_REF]. Denote ∂Λ k = ∂Λ h± k ∪ ∂Λ r k ∪ ∂Λ l k where ∂Λ r k denotes the right edge, ∂Λ l k the left edge and ∂Λ h± k the union of the two disjoint horizontal edges. It is plain that

∂Λ h± k ∪∂Λ r k |ζ -λ k | -1/2 R 1/2 (ζ) u dζ ≤ C
where C is some constant since |ζ -

λ k | -1/2 ≤ |1 -λ k | -1/2 for ζ ∈ ∂Λ h± k ∪ ∂Λ r k .
The remaining term is :

∂Λ l k |ζ -λ k | -1/2 R 1/2 (ζ) u dζ = 1 -1 +∞ p=1 u, ϕ p 2 |ιy -δ k | (ιy + |λ k -λ p | -δ k ) dy ≤ c 1 0 +∞ p=1 u, ϕ p 2 (y + δ k ) (y + |λ k -λ p |) dy
where c is a universal constant and we used bounds like |ιy -δ k | ≥ (y + δ k ) /2. We have :

1 0 +∞ p=1 u, ϕ p 2 (y + δ k ) (y + |λ k -λ p |) dy = 1/δ k 0 +∞ p=1 δ k u, ϕ p 2 (s + 1) (δ k s + |λ k -λ p |) ds ≤ 1 0 +∞ p=1 δ k u, ϕ p 2 (δ k s + |λ k -λ p |) ds + 1/δ k 1 +∞ p=1 δ k u, ϕ p 2 δ k s 2 ds ≤ 1 0 +∞ p=1 δ k u, ϕ p 2 |λ k -λ p | ds + ln (1/δ k ) +∞ p=1 u, ϕ p 2 ≤ 2 ln (1/δ k ) +∞ p=1 u, ϕ p 2 ,
and ( 16) becomes :

E P k -P k u 2 ≤ 4 exp -c log 2 n + cl 2 k,n ln 2 (1/δ k ) E u 2 .
This time the bound obtained is of interest when the assumption in ( 7) holds. Then ln (1/δ k ) ≤ c ln k.

As a consequence of the line above and of [START_REF] Ferraty | Advances on nonparametric regression from functional data[END_REF] we see that for all nonrandom u with norm 1,

P k -P k u 2 ≤ 21 B k (l k,n ) + cl 2 k,n ln 2 (1/δ k ) hence that P k -P k 2 ∞ ≤ 21 B k (l k,n ) + cl 2 k,n ln 2 (1/δ k ) .
This finishes the proof of Theorem 5.

Proof of Corollary 8 :

First notice that (

π k -π k ) (ϕ k ) 2 ≤ π k -π k 2 ∞ where ( π k -π k ) (ϕ k ) = ϕ k , ϕ k ϕ k -ϕ k but we also have ( π k -π k ) ϕ k , ϕ k = ϕ k , ϕ k 2 -1. Now ϕ k -ϕ k ≤ ϕ k -ϕ k , ϕ k ϕ k + ϕ k , ϕ k ϕ k -ϕ k = (1 -ϕ k , ϕ k ) ϕ k + π k (ϕ k ) -π k (ϕ k ) ≤ 1 -ϕ k , ϕ k + π k -π k ∞ since ϕ k , ϕ k ≤ 1. At last ϕ k -ϕ k 2 ≤ 2 (1 -ϕ k , ϕ k ) 2 + 2 π k -π k 2
∞ and we obtain the desired bound.

Proof of Proposition 2.

From all that was done above we restrict ourselves to proving the lower bound for (

π k -π k ) when sup ζ∈C k (ζI -Σ) -1/2 (Σ n -Σ) (ζI -Σ) -1/2 ∞ < a k / √
n that is when we can readily linearize π k -π k up to order 2 :

π k -π k = 1 2πι C k (ζI -Σ n ) -1 -(ζI -Σ) -1 dζ = 1 2πι C k (ζI -Σ) -1 (Σ n -Σ) (ζI -Σ) -1 dζ + T k,n with T k,n = 1 2πι C k R 1/2 (ζ) I -Π (ζ) -1 Π 2 (ζ) R 1/2 (ζ) dζ.
So that we get :

π k -π k = S k (Σ n -Σ) π k + π k (Σ n -Σ) S k + T k,n (18)
with S k = j =k ϕ j ⊗ ϕ j / (λ j -λ k ) . The computation of S k appears in several papers dealing with projectors or resolvent-based computations (e.g. p.347 in Cardot, Mas, Sarda (2007)). The second order term is T k,n with :

T k,n ∞ ≤ δ k Π (ζ) 2 ∞ 1 -Π (ζ) ∞ sup ζ∈C k R (ζ) ∞ = Π (ζ) 2 ∞ 1 -Π (ζ) ∞ ≤ a k √ n a k √ n 1 -a k √ n = d k,n , since we assume that sup ζ∈C k Π (ζ) ∞ < a k / √ n < 1/2.
We turn to the first order term namely

S k (Σ n -Σ) π k + π k (Σ n -Σ) S k .It is plain that : S k (Σ n -Σ) π k + π k (Σ n -Σ) S k ∞ ≥ S k (Σ n -Σ) π k ϕ k + π k (Σ n -Σ) S k ϕ k = S k (Σ n -Σ) π k ϕ k = S k Σ n ϕ k since π k = ϕ k ⊗ ϕ k and S k Σ n ϕ k 2 = 1 n 2 n i=1 X i , ϕ k S k X i 2 .
Then from [START_REF] Gilliam | The Fréchet derivative of an analytic function of a bounded operator with some applications[END_REF] we get :

S k Σ n ϕ k ≤ π k -π k ∞ + d k,n , S k Σ n ϕ k 2 ≤ 2 π k -π k 2 ∞ + 2d 2 k,n .
Taking expectation we finally get

E π k -π k 2 ∞ ≥ (1/2) E S k Σ n ϕ k 2 -d 2 k,n .
Now from calculations similar to those carried out within the proof of Lemma 16 we get :

E S k Σ n ϕ k 2 = 1 n 2 E n i=1 S k [ X i , ϕ k X i ] 2 = 1 n 2 E n i,j=1 X i , ϕ k X j , ϕ k S k (X i ) , S k (X j ) = = 1 n E X 1 , ϕ k 2 S k (X 1 ) 2 + n -1 2n E [ X 1 , ϕ k X 2 , ϕ k S k (X 1 ) , S k (X 2 ) ] .
Conditioning by X 2 and taking expectation with respect to X 1 it is simple to see that the second term in the right hand side above is null. The first one is :

1 n E X 1 , ϕ k 2 S k (X 1 ) 2 = λ k n E S k (X 1 ) 2 = 1 n λ k j =k λ j / (λ j -λ k ) 2 because S k (X 1 ) = ϕ k j =k ϕ j , X 1 / (λ j -λ k ). Hence : E π k -π k 2 ∞ ≥ 1 2n λ k j =k λ j / (λ j -λ k ) 2 -d 2 k,n .
Besides when (7) holds a

2 k λ k j =k λ j / (λ j -λ k ) 2 k 2
. This together with the fact that d 2 k,n is negligible with respect to a 2 k /n yields the desired result :

E π k -π k 2 ∞ ≥ c a 2 k n ,
for some absolute constant c.

Derivation of results of Section 3.2

The next two lemmas aim at providing the reader with basic inequalities which will be extensively used in the sequel.

Lemma 19

We have for all h > 0

KF k (h) ≤ EK ( P k (X i -x) /h) ≤ c F u k |K (u)| du F k (h) ≤ cF k (h) , KnF k (h) ≤ ES * ≤ cnF k (h) .
The lower bound in the equation above is obtained from :

EK ( P k (X i -x) /h) ≥ EK ( P k (X i -x) /h) 1I { P k (Xi-x) <h} .
The upper bound is derived from Fubini's theorem :

EK ( P k (X i -x) /h) = -K (u) F k (hu) du ≤ F k (h) |K (u)| F k (hu) F k (h) du ≤ c F u p |K (u)| du F k (h)
We refer to Ferraty, Mas, Vieu (2007) and [START_REF] Biau | PCA-Kernel estimation[END_REF] where this method was already used in an asymptotic framework.

Lemma 20 For all t > 0,

P S * -ES * ES * > t ≤ 2 exp -nF k (h) t 2 K 2 /2 |K| ∞ (c + tK) .
The proof is a consequence of Bernstein's theorem, of the boundedness of K and of Lemma 19 hence omitted.

Preliminary facts for the proof of Proposition 11 :

We deal with r (x) -r * (x) = Z S -Z * S * . It is simple to get the four following lines given here without proof for further purpose :

r (x) -r * (x) = r (x) S * -S S * - Z * -Z S * , max {| r (x)| , |r * (x)|} ≤ |r| ∞ + n i=1 |ε i | , E n i=1 ε i K ( P k (X i -x) /h) |X 1 , ..., X n 2 = σ 2 ε n i=1 K 2 ( P k (X i -x) /h) ≤ σ 2 ε |K| ∞ S * .
We have :

r (x) -r * (x) = [ r (x) -r * (x)] S * -S S * + r * (x) S * -S S * - Z * -Z S * . (19)
It is straightforward to see that

In view of ( 21) and ( 20) the conclusion will follow from an appropriate control of both terms :

E r * (x) S * -S 2 1 G , E Z * -Z 2 1
G . These are addressed in the forthcoming Lemmas. We also notice that :

E | r (x) -r * (x)| 2 1 G ≤ 50 9K 2 n 2 F 2 k (h) E r * (x) S * -S 2 1 G + E Z * -Z 2 1 G . (22)
Lemma 21

E r * (x) S * -S Taking into account that G holds we get the announced result.

Lemma 22

We have

E S * -S 2 ≤ A • n k 4 h 2 F 2 k (h) log 2 n log h √ n k 2 , E Z * -Z 2 ≤ |r| ∞ B • n k 4 h 2 F 2 k (h) log 2 n log h √ n k 2 ,
where A and B are constants depending only on K.

Proof :

We deal first with S * -S and limit ourselves to carrying out computations when K is the naive kernel defined by K (u) = 1 0≤u≤1 . The interested reader will easily check that the derivation may be adapted to more general kernels under assumption A2. The method by Biau and Mas (2012) is because the second term is negligible with respect to the first. The optimal η is chosen now so that :

n 2 η 2 k 2 F 2 k (h) h 2
n 2 F 2 (h) exp -nη 2 /M k 2 log 2 n , which yields :

η * k √ n log n log 1/2 h √ n k 2 .
Plugging this optimal η * in [START_REF] Koltchinskii | Asymptotics of spectral projections of some random matrices approximating integral operators[END_REF] gives the first result of the Lemma.

Recall that we left initially Z * -Z to treat S * -S. We have :

E Z * -Z 2 = E n i=1 r (X i ) K ( P k (X i -x) /h) -K P k (X i -x) /h 2 + σ 2 ε n i=1 E K ( P k (X i -x) /h) -K P k (X i -x) /h 2 .
Focusing here again on the specific important case of a naive kernel we check that n i=1 E K ( P k (X i -x) /h) -K P k (X i -x) /h 2 = nE 1 P k (X1-x) ≤h, P k (X1-x) >h -1 P k (X1-x) >h, P k (X1-x) ≤h 2 may be treated with exactly the same tools as above so that we can upper bound it up to constants with nηkF k (h) /h + nF (h) exp -nη 2 /M k 2 log 2 n . These terms are unimportant since they are negligible with respect to the contribution of the other part of the decomposition of Z * -Z namely :

E n i=1 r (X i ) K ( P k (X i -x) /h) -K P k (X i -x) /h 2 = E n i=1
r (X i ) 1 P k (Xi-x) ≤h, P k (Xi-x) >h -1 P k (Xi-x) >h, P k (Xi-x) ≤h and we are faced with exactly the same computations as above for S * -S.

Proof of Proposition 11:

Combining [START_REF] Gohberg | Classes of Linear Operators Vol I & II, Operator Theory : Advances and Applications[END_REF] with Lemma 22 we get :

E | r (x) -r * (x)| 2 1 G ≤ 50 9K 2 n 2 F 2 k (h) E r * (x) S * -S 2 1 G + E Z * -Z 2 1 G ≤ C • k 4 nh 2 log 2 n log h √ n k 2 .
This finishes the proof of the proposition.

Remark 13 2 H

 132 The statement of Proposition 12 allows independent choices of h (tending to 0) and k (going to infinity) by comparingE [ r (x) -r * (x)]2 only with the variance part of the estimate in the new projected model Y = r • P k (x) + ε.Remark 14As pointed out by a referee our results remain true when Y is functional-valued, say Y ∈ H where H is some Hilbert space. Obviously the mean squares such as E [ r (x) -r * (x)] 2 should then be replaced by E r (x) -r * (x) where • H is the norm on H.

> 1 / 5 .2 1 G ≤ 2 |r| 2 ∞ 2 ∞ 2 ( 20 )> 1 /5 |r| 2 ∞> 1 / 5 ≤ 3 Z

 1512222012153 that G ⊂ S * -ES * ES * We split r (x) -r * (x) in two parts. We start with :E | r (x) -r * (x)| + nEε 2 + n (n -1) (E |ε|) ≤ 2P S * -ES * ES * + nEε 2 + n (n -1) (E |ε|) 2 ,where the last line is controlled by the exponential inequality in Lemma 20. When nF k (h) / log n is larger than a prescribed and universal constant we get n 2 P S * -ES * ES * 1/n and the line above becomes negligible.On the other hand it is simple to get :| r (x) -r * (x)| 1 G ≤ * -Z ES * 1 G .

2 1 2 |X 1 ,ε 2 |X 1 ,

 12121 {| S * -ES * ES * |≤1/3} |X 1 , ..., X n ≤ S * -Keep in mind that G holds : E r * (x) S * -S i K ( P k (X i -x) /h)

2 + 2 n i=1 r 2 ≤ 2 |r| ∞ n i=1 1 P 2 + 2 |r| ∞ n i=1 1 P

 22i=12121 i ) 1 P k (Xi-x) ≤h, P k (Xi-x) >h (X i ) 1 P k (Xi-x) >h, P k (Xi-x) ≤h k (Xi-x) ≤h, P k (Xi-x) >h k (Xi-x) >h, P k (Xi-x) ≤h 2 ,

Corresponding author, mas@math.univ-montp2.fr

h.ruymgaart@ttu.edu

Acknowledgements : We are very grateful to two referees whose remarks and careful reading considerably enhanced the paper.

not sufficient to get the bound annouced and we have to follow another route with sharper bound :

Then :

Let 0 < η < h to be fixed later. We treat only the first series since our method applies to the other :

The distribution of the new first series above is Binomial hence

We turn to the other term. From :

and the fact that E exp (λ X ) < +∞ for some λ we get that

bounded by a n -p for some large p (depending on a good choice of M ). Then


André Mas

Testing for the mean of random curves : a penalization approach

Keywords: Random curves, functional data, weak convergence, hypothesis testing, local alternatives AMS Mathematics Subject Classi…cation : 62E20, 62G10, 62G20

Testing for the mean of random curves : a penalization approach

Introduction

1.1 The functional statistics setting.

Inference on random curves is undoubtedly a soaring area in nonparametric statistics. Modern computational techniques make it possible to deal with "high dimensional random vectors". Data that are obtained from an underlying continuous-time process, for instance, are extremely common in …nance, climatology, medicine, etc. Random curves collected from (independent or not) experiments are also likely to be studied by non parametric techniques. Some references are [START_REF] Franck | A statistical view of some chemometrics regression tools[END_REF], Cavallini, Montanari, Loggini, Lessi and Cacciari (1994), [START_REF] Besse | Autoregressive forecasting of some functional climatic variations[END_REF] for applications in respectively chemometrics, industry and meteorology.

This trend revealed that, conversely to probabilists, statisticians sometimes lack theoretical results for studying random functions. Many authors anyway paid attention to these topics, developping methods to link this rather formal framework with the statistician's "everyday's life". Amongst these are [START_REF] Kneip | Statistical tools to analyze data representing a sample of curves[END_REF], [START_REF] Ramsay | Functional Data Analysis[END_REF]. [START_REF] Dauxois | Asymptotic theory for the principal component analysis of a random vector function: some applications to statistical inference[END_REF], in an earlier paper, investigated the principal component analysis for Hilbert-valued random variables. Recently several authors generalized to the functional framework standard models in …nite dimension : the linear regression model in [START_REF] Cardot | Functional linear model[END_REF], the autoregressive model for time series in [START_REF] Bosq | Linear processes in function spaces[END_REF], the in…nite moving average model in [START_REF] Mas | Weak convergence for the covariance operators of a Hilbertian linear process[END_REF].

We propose an asymptotic procedure to test for the mean of a random function. Let X 1 ; :::; X n be an i.i.d. sample of random curves with mean m (note that m is also a curve) de…ned on some abstract probability space ( ; A; P ). We should write X i (!; t) where for …xed ! 2 ; X i (!; :) is the path of the curve and where for …xed t X i (:; t) is a real random variable. But for the sake of simplicity both indices will be dropped.

In this work we will suppose that the data are truly in…nite-dimensional or functional. In practice, the random curves are observed only at discrete times. A version of the underlying process is then rebuilt by interpolation techniques such as splines, wavelets, etc. These issues (as well as the problem of estimating the curve m) will not be considered in the sequel since they give rise to a considerable literature. We refer for instance to [START_REF] Rice | Estimating the mean and covariance structure nonparametrically when the data are curves[END_REF].

The test may be written :

H 0 : m = m 0 H a : m 6 = m 0
where m 0 = m 0 ( ) is a …xed known function. The principle of such a test is not new. Several authors developped procedures in the framework of di¤usion processes : X 1 is supposed to satisfy the stochastic di¤erential equation

dX (t) = m (t) dt + "dW (t)
where W is the standard Wiener process and " is the noise level. We refer to [START_REF] Spokoiny | Adaptive hypothesis testing using wavelets[END_REF], [START_REF] Lepski | Asymptotically exact nonparametric hypothesis tesing in sup-norm and at a …xed point[END_REF] for instance. The function m is supposed to belong to a class of functions (Sobolev, Hölder, or more generally Besov) indexed by some smoothness parameter. This gaussian and typical framework allows these authors to derive very interesting results : adaptive (w.r.t. the function class) tests but also optimal rates of testing -optimal in a sense that is de…ned by introducing the notion of minimax rate of testing-when the noise level decays to zero. [START_REF] Fan | Test of signi…cance based on wavelet thresholding and Neyman's truncation[END_REF] and Fan and Lin (1998) also proposed, in connection with high-dimensional ANOVA, an adaptive Neyman test.

Here adaptivity means that the procedure will automatically reduce the dimension of the model. All these authors apply wavelet thresholding techniques.

A new feature of the paper relies on the removal of the assumption of Gaussian observations, which was almost systematic in the above-mentioned articles. This framework makes the test truly new because we need to carry out alternative techniques based on lower moment assumptions, enlarging the class of process which may be investigated. The drawback is : we lose optimality properties. The procedure remains asymptotic and does not rely on a …nite dimensional approximation of m: The limiting distribution is however extremely simple. Some may view it as goodness of …t test. It also turns out that the main technical problems arising may be expressed in terms of an ill-posed inverse problem.

The setup of the paper is closer to a recent work by Cardot, Ferraty, [START_REF] Cardot | Testing hypotheses in the functional linear model[END_REF]. The authors propose a test for the regression operator in a linear model with functional inputs. The second serious point in this work is the mild condition on the rate of decrease of the eigenvalues of the covariance operator.

It should also be stressed that our problem does not depend on parameters intrinsec to the model or to spaces of functions. All the calculations just rely on the Hilbertian structure of the space, no matter which space is chosen. Consequently minimaxity or adaptivity make no sense.

From now on, the studied random curves will always be seen as random variables de…ned on the abstract probability space ( ; A; P ) and with values in an in…nite dimensional, real and separable Hilbert space H endowed with inner product h:; :i and norm k k : The Hilbert space setting enables to consider di¤erent sorts of basis and also to derive similar results for the sequence space l 2 . It also makes computations easier (especially as far as the central limit theorem is concerned). In the special case when H = L 2 ([0; 1]) and for all random functions u and v in H (u and v are consequently de…ned on [0; 1]) hu; vi is a random number :

hu; vi (!) = Z 1 0 u(!; t)v(!; t)dt: (1)
Spaces of smooth functions, such as Sobolev spaces, may (and are often) prefered to L 2 spaces for stability reasons. We refer to [START_REF] Silverman | Smoothed principal component Analysis by choice of norm[END_REF] for developments of this approach. Formally, this does not change much the inner product de…ned in (1) : Lebesgue's measure dt is replaced with another non-…nite measure d (t) (about this well-known fact see Rudin (1973) Ch 8.8 : Sobolev Spaces).

Preliminary facts about operators

Let X 1 ; :::; X n be an i.i.d. sample of Hilbert-valued random variables with mean m: We are …rst going to assume that the covariance operator of these random variables exists. Assumptions about the second order structure of X 1 will appear in the next paragraph. We denote (resp. n ) the covariance operator of X 1 (resp. the empirical covariance operator of the sample). These operators are bounded linear selfadjoint and positive mappings from H to H: They are de…ned this way : for all x in H;

(x) = E [hX 1 m; xi (X 1 m)] ; n (x) = 1 n n X k=1 [hX k m; xi (X k m)] :
In the following, for all u; v in H; u v just stands for the rank-one operator de…ned for all x in H by u v(x) = hu; xi v: With these notations, we get :

= E [(X 1 m) (X 1 m)] ; n = 1 n n X k=1 [(X k m) (X k m)] :
We will now assume that the random variable X 1 has a …nite strong moment of order two i.e. E kX 1 k 2 < +1: This means that is a trace class operator on H: If ::: 0 denotes the ordered sequence of its real and positive eigenvalues (associated to the eigenvectors e 1 ; e 2 ; ::::) it also implies that the vector ( 1 ; 2 ; :::) 2 l 1 : For further information about these facts we refer to Vakhania, Tarieladze and Chobanyan (1987).

We will always suppose that is one to one : the sequence of its strictly positive eigenvalues is in…nite and that the set ker = fh 2 H : h = 0g is reduced to f0g. Under the moment assumptions on X mentioned just above, it is well-known that is a positive selfadjoint compact operator. For any linear operator T de…ned on and with values in H; kT k 1 stands for the usual norm of continuous linears operators i.e. kT k 1 = sup kT xk for x in the unit ball of H. If we go further in the moments assumption, we even know that p n k n k is bounded in probability whenever E kX 1 k 4 is …nite.

The operator 1=2 plays a key-role in the sequel. It is de…ned by

1=2 = +1 X k=1 p k (e k e k )
and is a compact operator. When is one to one, its inverse 1=2 is de…ned on a domain D in H (D is a dense vector subspace of H). The linear mapping 1=2 is unbounded (i.e.

1=2 is continuous at no point of D) which also means

1=2 1 = +1 . As a consequence
1=2 X 1 cannot be considered as a (bounded) random variable. More information about these topics may be found in Dunford and Schwartz (1988).

2 Formulating the problem.

An original linear inverse problem

Now, in order to understand the particularities of the situation, let us consider …rst what happens in the …nite dimensional setting. Let X 1 ; :::; X n be a sample of i.i.d. vectors of R p with mean m 2 R p : The test is the following

H 0 : m = m 0 H a : m 6 = m 0 :
The operator n is replaced with a square matrix of size p; say M n (p); the empirical covariance matrix of the sample, which is often supposed to be invertible for n large enough. Then the test statistics (usually chi-square) is derived from :

M n (p) 1=2 1 p n n X k=1 (X k m 0 )
! which converges under the null hypothesis to a gaussian random variable whose covariance matrix is the identity.

In our functional framework we still have : S n = n 1=2 P n k=1 (X k m 0 ) converges weakly to a gaussian r.v. G by the central limit theorem for Hilbert space valued random elements but the distribution of G depends on the unknown i 's (see Fact 2 of Appendix 1). These eigenvalues may be viewed as nuisance parameters. Besides n ; conversely to M n is never invertible in H even for large n since its range (the linear span of X 1 ; ::

:; X n ) is of …nite dimension. Considering " 1=2 n
S n " makes no sense, as well as " 1=2 G" as was mentioned above. Consequently our goal is double. First we should approximate the operator 1=2 by a pseudo inverse, say L n ; built from n and hence random. Then we will have to study weak convergence for L n S n : This approach will be made more precise in the next section. Anyway we should expect the norm of L n to be a non decreasing sequence tending to +1: We could also say that the sequence of operators L n is unstable or ill-conditioned. Finally, copying the …nite dimensional approach to our test procedure leads us to a non standard ill-posed inverse problem : G will be approximated by S n and 1=2 by L n but S n and L n are connected via the sample X 1 ; :::; X n . Fortunately we will see that it is possible to propose a convergent test procedure.

Approximating 1=2 by a sequence of bounded operators is the purpose of the forthcoming paragraph.

Pseudo-inverse estimators.

The problem of approximating inverses of selfadjoint compact operators is absolutely not new. It is adressed in [START_REF] Nashed | Generalized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations[END_REF], [START_REF] Arsenin | Solutions of ill-posed problems[END_REF], [START_REF] Groetsch | Inverse Problems in the Mathematical Sciences[END_REF] amongst many others. The main point is always to regularize a matrix M (resp. an operator S) which is invertible but "not by much"(resp. unbounded). This property implies that for any vector x; M x (resp. Sx) may have large variations even when x does not vary much. Numerous procedures were proposed. We will keep one wich is suited to our problem and may be easily implemented.

The penalized regularization procedure is based on ( + I) 1=2 . It is a ridge-type method and depends on the index which is a strictly postive real number The penalization term is I where I is the identity operator on H: The operator ( + I) 1=2 is continuous for all strictly postive with norm 1= p : Another way to regularize 1=2 consists in deleting all the terms beyond a threshold p: In the basis e i we get which is often used has several drawbacks :

1=2 (p) = diag
The S.T. estimator is based on the functional Principal Component Analysis of n . It is necessary to estimate the eigenvectors and eigenvalues of this random operator before projecting the sample X 1 m; :::; X n m on these eigenvectors (to obtain principal components as a by-product). The estimation procedure is consequently not that simple and usually entails serious stability problems. Estimating the penalized estimate just requires a nonrandom basis, (e.g. spline, Fourier, wavelet). The data are then projected on a known vector space.

As it may be seen in Cardot, Ferraty and Sarda (1999), and [START_REF] Bosq | Modelization, non-parametric estimation and prediction for continuous time processes[END_REF], convergence rates of estimates steming from the S.T. procedure always depend on the speed of decay of the (unknown) eigenvalues and eigenvectors of : Several assumption are usually made, restricting the generality of the results. Speed of convergence of estimates involving the S.T. estimator are usually not good, due to the very slow rate of uniform convergence of the empirical eigenvectors of n to the eigenvectors of for large values of p. Also note that, conversely to the S.T. estimator, the norm of the penalized one, 1= p n , is nonrandom and does not depend on the rate of decay of the eigenvalues of or on the observed data. All the previous reasons explain the …nal choice of a penalization procedure for regularizing the inverse of the square root of the covarariance operator.

However even if the regularized operator does not depend explicitely on the eigenvalues, it should be stressed that the test statistics depends on them. A good knowledge of the rate of decay of the eigenvalues is even crucial to compute smoothing parameters intrinsic to our estimates (the above-mentioned n as well as k n , mentioned below-see next section.)

Main results

Weak convergence (resp. convergence in probability) will be denoted w ! (resp. P !) in the sequel and will be considered with respect to the norm of the space H. As mentioned above, we denote D the domain of the operator 1=2 i.e. the set of points x in H for which 1=2 x has a …nite norm. Clearly if x is expressed in the basis e i , x = P x k e k . We have :

D = ( x 2 H : X k x 2 k k < +1
)

:
Before giving the main results, we recapitulate the two assumptions needed :

H1 : E kX 1 k 4 < +1: H2 : is one to one.
We set once and for all :

S n = 1 p n n X k=1 (X k m 0 ) :
and we know that S n w ! G where G is the H valued gaussian random element with mean 0 and covariance operator : For further purpose we …nally recall that the following well-known decomposition for G holds almost surely on an appropriate probability space

G = +1 X p=1 p p p e p
where the p 's are gaussian centered random variables with unit variance. N (0; 3) : Remark 3.2 : Note that Theorem 3.1 which just asserts the existence of the test procedure is valid without any assumptions on the distribution of the data or on the spectrum of : The crucial step in the proofs relies on a result by Mas (2002a) on the estimate of closeness of distributions for images of measures on Banach spaces by general families of mappings. The distance between distributions is expressed in terms of the Prokhorov metric. We refer to the Appendix for the main properties of this metric as well as references.

Weak convergence

The next theorem determines explicitely both sequences k n and n such that b T n converges. The "price to pay" is quite low and relies on the forthcoming uniform bound on the eigenvalues : H3 : for some M; " > 0; p M= p (log p) 1+" Theorem 3.2 Let 0 < < 1=2: Under H1, H2 and H3 when k n = n 1=2 and n = (ln n) "=(1+ ") ; and when H 0 holds :

b T n w ! n!+1 N (0; 3) : Remark 3.
3 Assumption H3 may be viewed as prior information on the eigenvalues. The estimates b 0 p s obviously provide a good knowledge of the unknown p 's. If however more information is available and if the bound provided in H3 can be made more precise, one may get deeply di¤erent rates of increase (resp. of decrease) for k n (resp. n ). An inspection of the proof would lead to the following table :   Rate of decay for the eigenvalues exponential geometric assumption H3 k n log geometric geometric n geometric geometric log

The point is : the respective "speed" of k n and n are inverted. When the problem is "higly ill-conditionned" (i.e an exponential rate of decrease for the eigenvalues) k n should increase slowly. Conversely when the eigenvalues are assumed to converge "slowly" to zero (assumption H3) k n may be large and the penalization parameter may also decrease slowly. The reduction of dimension induced by k n is implicitely governed by n which depends on the "illness" of the inverse problem.

Study of the power for local alternatives

For the sake of completeness, the test procedure is given below :

Fix m 0 and compute b T n where S n = n 1=2 P n k=1 (X k m 0 ) and

n = 1 n n X k=1 (X k m 0 ) (X k m 0 ) :
Fix a level of signi…cance and compute u such that P (jN (0; 1)j u ) = 1

If b

T n p 3u H 0 is accepted otherwise, it is rejected.

A very classical way to investigate the power of the test consists in studying the power function against sequences of alternatives tending to the null. Let us take

H a ( n ) : m = m 0 + n t:
where t 2 H and n > 0 with n # 0: We are interested in computing the "optimal" rate of decay for n i.e. the fastest rate of decay for which the test distinguishes H a ( n ) from H 0 at a prescribed level.

At this point we need to recall a notation. Remind that the set

D = ( x 2 H : X k x 2 k k < +1
) ; considerations made above prove that the power of the test is directly connected to the "ill-posedness" situation. The eventual bad rate is not only due to the in…nite dimensional setting but gets worse because of the inverse problem.

is

Proofs

Along the proofs C and M will stand for …xed constants whose values do not matter. Remind that the notation Y n = O P (t n ) means that the sequence of random variables Y n t 1 n is bounded in probability. Let us denote

e T n = 1 d n ( n + n ) 1=2 S n 2 c n where c n = kn X p=1 p p + n ; d n = v u u t kn X p=1 p p + n 2 Proposition 4.1 If k n = ( n p n) 1; e T n b T n P ! 0: When H1 holds k n k 1 = O P (1= p n) : We turn to ( + n I) 1=2 S n P ( + n I) 1=2 S n > M 1 M 2 E ( + n I) 1=2 S n 2 = 1 M 2 E ( + n I) 1=2 X 1 2 = 1 M 2 +1 X p=1 p p + n c n + m n M 2
and ( + n I) 1=2 S n = O P ( p m n + c n ) and we can conclude as announced

1 d n ( n + n I) 1=2 S n 2 ( + n I) 1=2 S n 2 = O P p m n + c n 3=2 n d n p n ! (4)
For the next Proposition we need to introduce kn the projector on the k n …rst eigenvectors of Proposition 4.3 : It is possible to …nd coupled sequences n and k n such that the limiting distribution of

e T n = 1 d n ( + n I) 1=2 S n 2 c n
is the same (if it exists) as the limiting distribution of

T G;n = 1 d n kn ( + n I) 1=2 G 2 c n :
Proof of the Proposition : The proof of this lemma is based on a recent result by Mas (2002a). It should be …rst stressed that calculations show that e T n cannot be written as a sum of n random variables. Indeed

e T n = 1 d n ( + n I) 1=2 S n 2 c n :
If the square norm is developped, n 2 terms will appear. Consequently, it seems that it is not possible to prove the weak convergence of e T n by standard methods. The famous theorems of preservation of weak convergence by mappings (e.g. Billingsley, 1968 p.30 and 31) cannnot be applied either. We introduce E the Prokhorov metric for measures on a Banach space E: Reformulating the Proposition and identi…ying a random variable on H with the measure it induces we will get the desired result if R e

T n ; T G;n ! 0:

Or equivalently

1 d n R ( + n I) 1=2 S n 2 ; kn ( + n I) 1=2 G 2 ! 0:
First we bound

H ( + n I) 1=2 S n ; kn ( + n I) 1=2 G H kn ( + n I) 1=2 S n ; kn ( + n I) 1=2 G + H ( + n I) 1=2 S n ; kn ( + n I) 1=2 S n :
The last term may be bounded by m 1=3 n . We refer to the Appendix for further explanations about this bound. The …rst term may be bounded, following Theorem 1 p.236 in Vakhania (1977)

We are going to cope with each term separatedly. On a one hand it is always possible to …x two conjugate sequences k n and n (depending on each other) such that m n ! 0:

On the other hand, once either k n or n is determined, it is possible to choose the other such that k

  is a bounded operator with norm 1=2 p and is known as the spectral truncation (S.T.) of order p of 1=2 . But 1=2 (p)

2 whereRemark 3 . 1 2 b c n Theorem 3 . 1 :

 231231 Let the sequences k n (of integers) and n (of nonnegative numbers) respectively increase to in…nity and decrease to zero We set b c n = the b p are the estimated eigenvalues of n : The random term b c n (resp. b d n ) is a bias term (resp. a normalizing term). The b p 's are directly estimated from the (smooth) PCA of the X i 's hence from the spectrum of n : Asymptotic results (almost sure convergence, weak convergence) are given in Dauxois, Pousse and Romain (1982). The test statistic is bT n = 1 b d n ( n + n ) 1=2S n Under H1 and H2 there exists two conjugated sequences k n and n such that, when H 0 holds (m = m 0 ) :

  Proof of the Proposition :

  the domain of 1=2 : Consequently 1=2 t is …nite if and only if t 2 D. The optimal rate of convergence of the sequence of local alternatives to the null is n = p d n =n: Let us pick a t in H and distinguish two cases.

	Proposition 3.1 First case : If t = 2 D the test statistic b T n diverges.for the sequence of alternatives H a p d n =n
	Second case : If t 2 D, b T n converges weakly to a N alternatives H a p d n =n :	1=2 t	2 ; 3 for the sequence of
	Remark 3.4 Since d n	C 1 n

  by : For the sake of clarity we denote H = H ( + n I) 1=2 S n ; kn ( + n I) 1=2 G :We will get the …nal result whenever we can express the Prokhorov distance between kU k 2 and kV k 2 only by means of the Prokhorov distance between U and V: This was precisely done in Case 1 of paragraph 4.1 in Mas (2002a). We have : Since obviously the function s ! s p ln s is non decreasing in a positive neighborhood of zero, the …nal bound is

		C	k	1=4 n jln nj 1=2 n 1=8 3=8
	H	m 1=3 n + C	k	1=4 n jln nj 1=2 n n 1=8 3=8	= n :
	R e T n ; T G;n ln R e C d n H p T n ; T G;n C n p ln n d n	:

n : H for a su¢ ciently large n:

  Dkn ( + n I) 1=2 G; kn ( + n I) 1=2 t EThe …rst term obviously converges weakly to a N (0; 3) ; the second is deterministic and converges to 1=2 t 2 : The last term has the same distribution as a centered gaussian real random variable whose variance is (apply Lebesgue's dominted convergence theorem) which ensures that s n tends to zero as d n " +1 and concludes the proof.

	expression. But
	1 d n		kn ( + n I) 1=2 G	p	n n t	2	c n
	=	1 d n	kn ( + n I) 1=2 G	2	c n
	+		kn ( + n I) 1=2 t	2
		2 p d n
					s 2 n =	4 d 2 n	kn X p=1	p ( p + n ) 2 t 2 p
	The series converges to	P +1 p=1	t 2 p p

Mailing address : André Mas, Laboratoire de Probabilités-Statistiques, CC051, Université Montpellier 2, Place Eugène Bataillon, 34095 Montpellier Cedex 5.

We have :

Proof of the Proposition :

This last term may be bounded by

We have successively :

The central limit theorem for triangular arrays of independent random variables yields the …nal result (see Billinglsey (1968) p.42).

Proof of Theorem 3.1 : Collecting Propositions 4.1, 4.2, 4.3 we have to check that the three rates of convergence are not contradictory, namely those obtained in (2), ( 4), [START_REF] Bosq | Modelization, non-parametric estimation and prediction for continuous time processes[END_REF]. The conclusion will then follow from Proposition (4.4).

We …rst go back to [START_REF] Bosq | Linear processes in function spaces[END_REF]. This conditions still implies an underlying link between k n and n : It does not seem possible to compare condition [START_REF] Cardot | Functional linear model[END_REF] and condition (4) but both will be full…lled whenever

and ( 2) is obvioulsy respected in this case: Note also that p c n p k n : Consequently we are again in the situation of Proposition 4.3 but ( 7) is replaced with (8) and Theorem 3.1 follows.

Proof of Theorem 3.2 : Assumption H3 implies that

Setting ln k n =

1=" n where > 0 implies that the term above tends to zero. The expression in ( 8) is now

Taking at last, for instance k n = n 1=2 gives the …nal result.

Proof of Proposition 3.1 :

The method of the proof for Theorem 3.1 and especially the use of Prokhorov metric directly yields the following : under H a ( n ) the distribution of b T n is asymptotically the same as the distribution of

We prove just below that taking n = d n = p n is su¢ cient for the sequences of alternatives to be detected by the test procedure.

In fact we have to consider two cases. The …rst one is quite simple : if t = 2 D, kn ( + n I) 1=2 t does not converge at all and the test statistic diverge if n = p d n =n. The second case is not that much complicated. If t 2 D, kn ( + n I) 1=2 t converges to 1=2 t and we have to determine the limiting distribution of the above displayed Appendix De…nition and some properties of the Prokhorov metric Along the proofs we used a metric de…ned on spaces of measures. This (Prokhorov) distance metrizes the topology of weak convergence for probability measures on a separable Banach space E: It will be denoted E . We refer to [START_REF] Billingsley | Convergence of probability measures[END_REF], [START_REF] Dudley | Distances of Probability Measures and Random Variables[END_REF] and [START_REF] Araujo | The central limit theorem for real and Banach valued random variables[END_REF] for an introduction to the main features of E : The de…nition is the following (it is given for two measures 1 and 2 on E but is immediatly adaptable to random elements) :

where F is the family of closed sets in E and A " stands for the "-neighborhood of A (with respect to the norm on E). In fact E may be de…ned for other family of sets than F:

Within the proofs we need the follwing property of E : First, Theorem 1 in Dudley (1968) implies that H (X; Y ) inf f" > 0 : P (kX Y k > ") "g

Hence if k denotes the projector on the k …rst eigenvectors of the covariance operator of the H valued random variable X P X k X > "

by Tchebytchev inequality and

which was the result needed to get m n in the proof of Proposition 4.3.