Frédéric Bayart
email: frederic.bayart@uca.fr

Zoltan Buczolich

Yanick Heurteaux
email: yanick.heurteaux@uca.fr

F R Éd

ZOLT ÁN BUCZOLICH Éric Bayart

FAST AND SLOW POINTS OF BIRKHOFF SUMS

Keywords: 37A05, Secondary : 11K55, 28A78, 60F15 Birkhoff sum, typical/generic properties, group rotation, coboundary, law of iterated logarithm

come

Introduction

Let T = R/Z be the unit circle and let α ∈ R\Q be irrational. Denote by C 0 (T), the set of continuous functions on T with zero mean, and by S n,α f (x) the n-th Birkhoff sum, S n,α f (x) = n-1 k=0 f (x + kα). The rotation R α : x → x + α defines a uniquely ergodic transformation on T with respect to the (normalized) Lebesgue measure λ. Hence for all f ∈ C 0 (T) we know that S n,α f (x) = o(n) for all x ∈ T. The main purpose of this paper is to investigate the typical growth of S n,α f (x). There are several ways to understand this problem. We can fix α ∈ R\Q (resp. x ∈ T) and ask for the behaviour of S n,α f (x) for f in a generic subset of C 0 (T) and for a typical x ∈ T (resp. for a typical α ∈ T). We can also consider it as a problem of two variables and ask for the behaviour of S n,α f (x) for f in a generic subset of C 0 (T) and for a typical (α, x) ∈ T 2 . There are also several ways to understand the word "typical". We can look for a residual set of the parameter space or for a set of full Lebesgue measure. We shall try to put this in a general context. If we fix α ∈ R\Q, then we consider the Birkhoff sums associated to a uniquely ergodic transformation on the compact metric space T. Hence, let us fix Ω an infinite compact metric space and T : Ω → Ω an invertible continuous map such that T is uniquely ergodic. Let µ be the ergodic measure, which is regular and continuous. We will also assume that it has full support (equivalenty, that all orbits of T are dense). For x ∈ Ω and f ∈ C 0 (Ω), the Birkhoff sum S n,T f (x) is now defined by n-1 k=0 f T k x . Using ψ : N → N with ψ(n) = o(n) for f ∈ C 0 (Ω), let us define

E ψ (f ) = x ∈ Ω; lim sup n |S n,T f (x)| ψ(n) = +∞ .
The set E ψ (f ) has already been studied by several authors. In particular, it was shown by Krengel [START_REF] Krengel | On the speed of convergence in the ergodic theorem[END_REF] (when Ω = [0, 1]) and later by Liardet and Volný [START_REF] Liardet | Continuous and differentiable functions in dynamical systems[END_REF] that, for all functions f in a residual subset of C 0 (Ω), µ (E ψ (f )) = 1. We complete this result by showing that E ψ (f ) is also residual.

Theorem 1.1. Suppose that ψ : N → N satisfies ψ(n) = o(n). There exists a residual set R ⊂ C 0 (Ω) such that for any f ∈ R, E ψ (f ) is residual and of full µ-measure in Ω.

If we allow α to vary in our initial problem, then the natural framework now is that of topological groups. Hence, we fix a compact and connected metric abelian group (G, +). By Corollary 4.4 in [8, Chapter 4], G is a monothetic group, that is possesses a dense cyclic subgroup. Let µ be the Haar measure on G. It is invariant under each translation, or group rotation T u (x) = x + u. We define G 0 as the set of u ∈ G such that T u is ergodic. By well-known results of ergodic theory, u belongs to G 0 if and only if {nu; n ∈ Z} is dense in G; in this case T u is uniquely ergodic, only the Haar measure is invariant with respect to T u . Moreover, G 0 is always nonempty, it is dense and its Haar measure is equal to 1 (see Theorem 4.5 in [8, Chapter 4]).

Contrary to what happens in Theorem 1.1, the growth of S n,u f (x) for a typical (u, x) ∈ G 2 is not the same from the topological and from the probabilistic points of view. For the last one, the typical growth of S n,u f (x) has order n 1/2 .

Theorem 1.2.

(i) For all ν > 1/2 and all f ∈ L 2 0 (G),

µ ⊗ µ (u, x) ∈ G 2 ; lim sup n |S n,u f (x)| n ν ≥ 1 = 0.
(ii) There exists a residual subset R ⊂ C 0 (G) such that, for all f ∈ R,

µ ⊗ µ (u, x) ∈ G 2 ; lim sup n |S n,u f (x)| n 1/2 = +∞ = 1.
From a topological point of view, the typical growth of S n,u f (x) has order n. Indeed, for ψ :

N → N with ψ(n) = o(n), let us introduce E ψ (f ) = (u, x) ∈ G 2 ; lim sup n |S n,u f (x)| ψ(n) = +∞ . Theorem 1.3. Suppose that ψ : N → N satisfies ψ(n) = o(n). There exists a residual set R * ⊂ C 0 (G) × G 2 such that for any (f, u, x) ∈ R * we have (u, x) ∈ E ψ (f ).
We remark that, by the Kuratowski-Ulam theorem, Theorem 1.3 implies that there exists a residual set R ⊂ C 0 (G) such that, for every

f ∈ R, the set E ψ (f ) is residual in G 2 .
The last possibility is to fix x ∈ G and allow u to vary. Without loss of generality, we may assume that x = 0. Again, topologically speaking, the typical growth of S n,u f (0) is not better than o(n).

Corollary 1.4. Suppose that ψ :

N → N satisfies ψ(n) = o(n). There exists a residual set R ⊂ C 0 (G) such that for any f ∈ R, the set {u ∈ G; (u, 0) ∈ E ψ (f )} is residual in G.
We finally come back to irrational rotations where we would like to get more precise statements. Let us fix α ∈ R\Q and set

F ψ (f ) = x ∈ T; lim sup n |S n,α f (x)| ψ(n) < +∞ .
When ψ(n) = n ν , ν ∈ (0, 1), we simply denote by F ν (f ) the set F ψ (f ). We already know by the results mentioned before Theorem 1.1 that λ (F ψ (f )) = 0 for f in a residual subset of C 0 (T), where λ is the Lebesgue measure on T. It turns out that a much stronger result is true: generically, these sets have zero Hausdorff dimension! Theorem 1.5. For any ψ :

N → N with ψ(n) = o(n), there exists a residual subset R of C 0 (T) such that, for any f ∈ R, dim H (F ψ (f )) = 0.
We then do a similar study for Hölder functions f ∈ C ξ 0 (T), ξ ∈ (0, 1). Recall that a function f belongs to C ξ 0 (T) if it has zero mean and if there exists a constant C > 0 such that, for all x, y ∈ T,

|f (x) -f (y)| ≤ C|x -y| ξ .
The infimum of such constants C is denoted by Lip ξ (f ). For a function f ∈ C ξ 0 (T), we have better bounds on S n,α f (x) depending on ξ and on the arithmetical properties of α. Indeed, it is known (see [START_REF] Kuipers | Uniform distribution of sequences[END_REF]Chapter 2

, Theorem 5.4]) that |S n,α f (x)| ≤ n • Lip ξ (f ) (D * n (α)) ξ where D * n (α)
is the discrepancy of the sequence (α, 2α, . . . , nα) defined by

|D * n (α)| = sup I⊂T card{1 ≤ i ≤ n; iα ∈ I} n -|I| .
For instance, if α has type 1 (for example, if α is an irrational algebraic number), using the well-known estimates of the discrepancy, we get that |S n,α f (x)| = O(n 1-ξ+ε ) for all ε > 0. In other words, for all ν > 1 -ξ, F ν (f ) = T. We investigate the case ν ≤ 1 -ξ and we show that the Hausdorff dimension of F ν (f ) cannot always be large.

Theorem 1.6. Let ξ ∈ (0, 1). There exists f ∈ C ξ 0 (T) such that, for all ν ∈ (0, 1 -ξ),

dim H (F ν (f )) ≤ ξ 1 -ν .
This theorem is in stark contrast with Theorem 4.1 in [START_REF]On fast Birkhoff averaging[END_REF]. In this last paper, a similar study of fast Birkhoff averages of subshifts is done. In this case, the sets which correspond to F ν (f ) always have maximal dimension.

Useful lemmas

In this section, we provide lemmas which will be used several times for the proof of our main theorems. The first one allows to approximate step functions by continuous functions. In the statement of the theorem we use the standard notation 1 B (x) for the function which equals 1 if x ∈ B and equals 0 if not.

Lemma 2.1. Let Ω be a compact metric space, let µ be a continuous Borel probability measure on Ω. Let g be a step function such that Ω g(x)dµ(x) = 0 and δ > 0. Then there exists f ∈ C 0 (Ω) such that f ∞ ≤ 2 g ∞ and f = g except on a set of measure at most δ.

Proof. Let ε > 0 be very small and {a 1 , . . . , a n } be the finite set g(Ω). We can write g = n i=1 a i 1 A i where A i = {x ∈ Ω ; g(x) = a i }. Since the measure µ is regular, we can find compact sets K 1 , . . . , K n and open sets U 1 , . . . , U n such that

K i ⊂ A i ⊂ U i µ(U i ) -ε ≤ µ(A i ) ≤ µ(K i ) + ε.
By Urysohn's lemma, one may find functions

ϕ i ∈ C(Ω) such that 0 ≤ ϕ i ≤ 1, ϕ i = 1 on K i , ϕ i = 0 outside U i . We then set h = n i=1 a i ϕ i . It is clear that µ ({a i 1 A i = a i ϕ i }) ≤ µ (U i \K i ) .
Therefore,

µ ({h = g}) ≤ n i=1 µ (U i \K i ) ≤ 2nε. If k = max (-g ∞ , min (h, g ∞ )), we now have k ∞ ≤ g ∞ and µ ({k = g}) ≤ µ ({h = g}) ≤ 2nε.
The function k is continuous but is not necessarily in C 0 (Ω). Nevertheless, we observe that

Ω k(x)dµ(x) = Ω (k(x) -g(x)) dµ(x) ≤ k -g ∞ µ ({k = g}) ≤ 4nε g ∞
and we can modify k to obtain a zero mean. Let a ∈ Ω and r > 0 be such that 0 < µ (B(a, r)) ≤ µ (B(a, 2r)) < δ/2 and let ϕ 0 ∈ C(Ω) with ϕ 0 = 1 on the closed ball B(a, r), ϕ 0 = 0 outside B(a, 2r) and 0 ≤ ϕ 0 ≤ 1. We set

f = k -Ω kdµ Ω ϕ 0 dµ ϕ 0 .
Then f ∈ C 0 (Ω), f = g except on a set of measure at most 2nε + δ/2 and

f ∞ ≤ g ∞ + Ω kdµ Ω ϕ 0 dµ ≤ g ∞ + 4nε g ∞ µ (B(a, r)) .
Choosing ε > 0 sufficiently small then gives the result.

Our second lemma is a way to construct continuous functions in C 0 (Ω) with large Birkhoff sums on large subsets. We give it in our general context of a uniquely ergodic transformation T on an infinite compact metric space Ω with non-atomic ergodic measure µ. As usual, ψ : N → N satisfies ψ(n) = o(n). We denote by E c the complement of the set E.

Lemma 2.2. Let J, M ∈ N, C > 0, ε > 0. Then there exist f ∈ C 0 (Ω), m ≥ M and a compact set E ⊂ Ω such that f ∞ ≤ ε, µ(E) > 1 -ε and ∀x ∈ E, ∀j ∈ {1, . . . , J}, S m,T j f (x) ≥ Cψ(m).
Proof. Set ε = ε/3. We begin by fixing m ∈ N, any integer greater than M , and such that mε ≥ Cψ(m). Let n m to be fixed later. We then consider a Rokhlin tower associated to T , 2n and ε (see for instance [START_REF] Eisner | Operator theoretic aspects of ergodic theory[END_REF]). Namely, we consider A ⊂ Ω such that the sets T k (A), 0 ≤ k ≤ 2n -1, are pairwise disjoint and µ 2n-1 k=0 T k (A) > 1 -ε. We then consider a function g equal to ε on n-1 k=0 T k (A), equal to -ε on 2n-1 k=n T k (A) and equal to zero elsewhere. We set

F = n-1-mJ k=0 T k (A) ∪ 2n-1-mJ k=n T k (A) := F 1 ∪ F 2 .
Then, for any x ∈ F 1 , for any ≤ m -1, for any j ∈ {1, . . . , J},

T j (x) ∈ n-1 k=0 T k (A).
It follows that S m,T j g(x) = mε. In the same way, for any x ∈ F 2 , for any j ∈ {1, . . . , J}, S m,T j g(x) = -mε. Finally, for any x ∈ F , for any j ∈ {1, . . . , J}, S m,T j g(x) = mε ≥ Cψ(m).

Moreover,

µ(F ) = 2 (n -mJ) µ(A) ≥ 2 (n -mJ) • 1 -ε 2n ≥ 1 -2ε
provided n is large enough. Thanks to Lemma 2.1, we approximate g by a continuous function f ∈ C 0 (Ω) with f ∞ ≤ 2ε and f = g except on a set N of measure η > 0, with mJη < ε. Fix j ∈ {1, . . . , J}. Then

S m,T j f (x) = S m,T j g(x) except if x ∈ m-1 k=0 T -kj (N ). Let N = m-1 k=0 J j=1 T -kj (N ). Then µ(N ) ≤ mJη < ε. Moreover, S m,T j f (x) ≥ Cψ(m) for all j ∈ {1, . . . , J} and all x ∈ F ∩ N c =: E 0 . Clearly, µ(E 0 ) > 1 -3ε = 1 -ε.
We conclude by taking for E the closure of E 0 .

Fast and slow points of Birkhoff sums -I

In this section, we prove Theorems 1.1 and 1.3. Their proofs share many similarities and depend heavily on Lemma 2.2 applied in suitable situations. We will also need that if T is a uniquely ergodic transformation on Ω, then the set of C 0 (Ω)-coboundaries for T , namely the set of functions g -g • T for some g ∈ C 0 (Ω), is dense in C 0 (Ω) (see for instance [START_REF] Liardet | Continuous and differentiable functions in dynamical systems[END_REF]Lemma 1]). It is convenient to work with a coboundary since its Birkhoff sums are uniformly bounded.

Proof of Theorem 1.1. Let (h l ) be a dense sequence of coboundaries in C 0 (Ω) and let C l > 0 be such that sup n S n,T h l ∞ ≤ C l . Let f l , E l and m l be given by Lemma 2.2 for

C = l + C l + 1, M = l, J = 1, ε = 1/l. We set g l = h l + f l and we observe that, for x ∈ E l , |S m l ,T g l (x)| ≥ (l + C l + 1)ψ(m l ) -C l ≥ (l + 1)ψ(m l ).
Since E l is compact and g l is continuous, we can choose δ l > 0 and an open set F l ⊂ Ω containing E l such that, for any f ∈ B(g l , δ l ), for any x ∈ F l , (1)

|S m l ,T f (x)| ≥ lψ(m l ).
Let R = L≥1 l≥L B(g l , δ l ) which is a residual set in C 0 (Ω) and pick f ∈ R. There exists an increasing sequence (l k ) going to +∞ such that f ∈ B(g l k , δ l k ) for all k. We set

F = lim sup F l k = K≥1 k≥K F l k . Since µ(F l k ) ≥ µ(E l k ) ≥ 1 -1
l k the set F has full measure. Moreover, since µ has full support and µ k≥K F l k = 1 for all K, F is also residual in Ω. Finally if x belongs to F , then ( 1) is true for infinitely many l, which shows Theorem 1.1.

In the next proof Ω is replaced by the compact connected metric abelian group G and we consider uniquely ergodic translations T v . We recall that for these translations, all non-constant characters γ are C 0 -coboundaries: they can be written as

γ = γ 0 • T v -γ 0 , where γ 0 = 1 γ(v)-1 γ. Proof of Theorem 1.3. Since G is compact we can choose a sequence (h l ) of trigonometric polynomials which is dense in C 0 (G) (see [10, Section 1.5.2]). Let v ∈ G 0 , that is T v is ergodic. Since h l is a C 0 -coboundary for all T jv , j = 1, . . . , l, there exists C l > 0 such that sup n sup j∈{1,...,l} S n,jv h l ∞ ≤ C l .
Let f l , E l and m l be given by Lemma 2.2 for

T = T v , C = C l + l + 1, M = l, J = l, ε = 1/l. Set g l = h l + f l and observe that, for x ∈ E l , j ∈ {1, . . . , l}, |S m l ,jv g l (x)| ≥ (l + C l + 1)ψ(m l ) -C l ≥ (l + 1)ψ(m l ).
Since {jv; j = 1, . . . , l} × E l is compact in G × G and g l is continuous, we can choose δ l > 0 and an open set

H l ⊂ G × G such that {jv; j = 1, . . . , l} × E l ⊂ H l and, for any (f, u, x) ∈ B(g l , δ l ) × H l , (2)
|S m l ,u f (x)| > lψ(m l ).
We now observe that 2) is true for infinitely many integers l.

l≥L {g l } × T j v; j = 1, . . . , l × E l is dense in C 0 (G) × G × G for any L ≥ 1. Hence, R * = L≥1 l≥L B(g l , δ l ) × H l is a residual subset of C 0 (G) × G 2 and any (f, u, x) ∈ R * satisfies that (u, x) belongs to E ψ (f ) since (
Proof of Corollary 1.4. This corollary follows easily from Theorem 1.3 and from the Kuratowski-Ulam theorem. Indeed, we know that there exist a residual set R ⊂ C 0 (G) and

x ∈ G such that, for all f ∈ R, {u ∈ G; (u, x) ∈ E ψ (f )} is residual. Now, setting R = {f (• -x); f ∈ R}, for any f ∈ R , {u ∈ G; (u, 0) ∈ E ψ (f )} is residual.

Fast and slow points of Birkhoff sums -II

We turn to the proof of Theorem 1.2. Its first part heavily depends on the following Menshov-Rademacher inequality (see for instance [START_REF] Doob | Stochastic processes[END_REF]Chapter 4]).

Lemma 4.1. Let X 1 , . . . , X N be a sequence of orthonormal random variables and c 1 , . . . , c N be a sequence of real numbers. Then

E   max 1≤n≤N   n j=1 c j X j   2   ≤ log 2 2 (4N ) N n=1 c 2 n .
Proof of Theorem 1.2 part (i). Recall that G f (x)dµ(x) = 0. Without loss of generality, we suppose f 2 = 1 and we consider X k (u, x) = f (x + ku) as a random variable on the probability space (G 2 , µ ⊗ µ). Next we show that (X k ) k≥1 is an orthonormal sequence. Indeed, let γ∈ Ĝ f (γ)γ be the Fourier expansion of f . Then, for k, j ≥ 1,

G 2 X k X j dµ ⊗ dµ = γ,γ ∈ Ĝ f (γ) f (γ ) G γ(x)γ (x)dµ(x) G γ(ku)γ (ju)dµ(u). Now, G γ(x)γ (x)dµ(x) is zero provided γ = γ and is equal to 1 otherwise. Moreover, let us fix γ ∈ Ĝ and set γ k (u) = γ(ku), γ j (u) = γ(ju). Then G γ k γ j dµ = 0 except if γ k = γ j , namely except if γ k-j = 1.
If k = j, using that Ĝ is torsion-free since G is compact and connected, this can only happen if γ = 1. Therefore, we have shown that

G 2 X k X j dµ ⊗ dµ = γ | f (γ)| 2 = 1 if k = j | f (1)| 2 = 0 otherwise.
Applying Lemma 4.1 with c j = 1 yields

(3)

G 2 max 1≤n≤N |S n,u f (x)| 2 dµ(u) ⊗ dµ(x) ≤ log 2 2 (4N )N.
Let ν > 1/2 and for k ≥ 1,

E k = (u, x) ∈ G 2 ; ∃n ∈ {2 k , . . . , 2 k+1 -1}, |S n,u f (x)| ≥ n ν .
Using Markov's inequality and (3), we get

µ⊗µ(E k ) ≤ µ⊗µ max 1≤n≤2 k+1 |S n,u f (x)| ≥ 2 νk ≤ 1 2 2kν log 2 2 4 • 2 k+1 • 2 k+1 ≤ Ck 2 2 k(1-2ν) . Since k µ ⊗ µ(E k ) < ∞, the Borel-Cantelli lemma implies that µ ⊗ µ(lim sup k E k ) = 0
and the conclusion follows.

Remark 4.2. In fact, the same proof shows that, for any ε > 0,

µ ⊗ µ (u, x) ∈ G 2 ; lim sup n |S n,u f (x)| n 1 2 log 3 2 +ε (n) ≥ 1 = 0.
To prove the second part of Theorem 1.2, we shall use both a Baire category and a probabilistic argument. The probabilistic part is based on the the following lemma, which is a consequence of the proof of the law of the iterated logarithm done in [START_REF] Billingsley | Probability and measure[END_REF] (the important point here is that we need a choice of N which does not depend on the particular choice of the sequence).

We recall that a random variable X : (Ω, A, P ) → R has a Rademacher distribution if P (X = 1) = P (X = -1) = 1/2.

Lemma 4.3. Let ε > 0 and M ∈ N. There exists N ≥ M such that, for any sequence (Y k ) of independent Rademacher variables defined on the same probability space (Ω, A, P ),

P sup M ≤n≤N | n k=1 Y k (ω)| √ n log log n > 1 2 > 1 -ε.
The following lemma is the key point of our proof.

Lemma 4.4. Let ε ∈ (0, 1), C > 0 and M ∈ N. There exist f ∈ C 0 (G), N > M and F ⊂ G 2 with f ∞ ≤ ε, µ ⊗ µ(F ) > 1 -ε and (u, x) ∈ F =⇒ sup M ≤n≤N |S n,u f (x)| n 1/2 ≥ C.
Proof. Without loss of generality, we may assume that √ log log M > 2C/ε. Lemma 4.3 gives us a value of N associated to ε and M . We then consider a sequence (X k ) of independent Rademacher variables defined on the same probability space (Ω, A, P ). We select a neighbourhood O of 0 ∈ G so that, setting

E O = u ∈ G; (j -j)u /
∈ 2O for all 0 ≤ j, j ≤ N, j = j , we have µ(E O ) > 1 -ε. This is possible since, denoting by (O l ) a basis of neighbourhoods of 0 in G, we have

G 0 ⊂ {u ∈ G; ku = 0 for all k ∈ Z\{0}} ⊂ l E O l .
By compactness of G, G is contained in a finite union (

x 1 + O) ∪ • • • ∪ (x K + O). We set A 1 = x 1 + O and, for 2 ≤ k ≤ K, A k = (x k + O)\(A 1 ∪ • • • ∪ A k-1
). The sets A 1 , . . . , A k provide a Borelian partition of G.

We then split each A k into a disjoint sum

A k = B k ∪ B k with µ(B k ) = µ(B k ) = µ(A k )/2. For 1 ≤ k ≤ K define ϕ k by ϕ k = 1 B k -1 B k . We finally put g(x, ω) = K k=1 εX k (ω)ϕ k (x) so that S n,u g(x, ω) = ε n-1 j=0 K k=1 X k (ω)ϕ k (x + ju).
Let us fix u ∈ E O . For all x ∈ G and all j ∈ {0, . . . , N -1}, there exists exactly one integer k ∈ {1, . . . , K}, that we will denote by k(j, u, x), such that ϕ k (x + ju) = 0. Hence, for (u, x)

∈ E O × G and n ≤ N , S n,u g(x, ω) = ε n-1 j=0 X k(j,u,x) (ω)ϕ k(j,u,x) (x + ju).
Moreover, for j = j , the integers k(j, u, x) and k(j , u, x) are different: otherwise, (j -j )u would belong to 2O.

Applying Lemma 4.3 to the sequence X k(j,u,x) ϕ k(j,u,x) (x + ju) 0≤j≤N -1 which is a sequence of independent Rademacher variables, we get the existence of Ω u,x ⊂ Ω such that

P (Ω u,x ) > 1 -ε and (u, x, ω) ∈ E O × G × Ω u,x =⇒ sup M ≤n≤N |S n,u g(x, ω)| √ n log log n ≥ ε 2 . Hence (u, x, ω) ∈ E O × G × Ω u,x =⇒ sup M ≤n≤N |S n,u g(x, ω)| √ n ≥ ε 2 log log M > C.
Keeping in mind that µ(E O ) > 1 -ε holds as well, by Fubini's theorem we can select and fix ω ∈ Ω such that

(4) µ ⊗ µ (u, x) ∈ G 2 ; sup M ≤n≤N |S n,u g(x, ω)| √ n > C > (1 -ε) 2 > 1 -2ε.
Given δ > 0, according to Lemma 2.1, the function g = g(•, ω) can be approximated by a continuous function f ∈ C 0 (G) such that f ∞ ≤ 2ε and which coincides with g except in a set of measure less than δ/N . It follows that for every u ∈ G and for any n ∈ {M, . . . , N }, S n,u f (x) = S n,u g(x) except in a set of measure less than δ. Finally, if δ is sufficiently small, inequality ( 4) is still satisfied if we replace g by f .

Proof of Theorem 1.2, part (ii). Let (h l ) be a sequence of trigonometric polynomials dense in C 0 (G). For all l ≥ 1 and all u ∈ G 0 , since h l is a C 0 -coboundary for T u , we know that sup n S n,u h l ∞ < +∞. We then find G l ⊂ G 0 with µ(G l ) > 1 -1/l and C l > 0 such that, for all u ∈ G l , sup n S n,u h l ∞ ≤ C l . We apply Lemma 4.4 with ε = 1/l, C = l + C l + 1 and M l = l. We get a function f l ∈ C 0 (G), an integer N l ≥ M l and a set F l ⊂ G 2 . We define g l = h l + f l and E l = F l ∩ (G l × G) so that µ ⊗ µ(E l ) ≥ 1 -2/l. The way we constructed all these objects ensures that, for any (u, x) ∈ E l , sup

M l ≤n≤N l |S n,u g l (x)| n 1/2 ≥ l + 1.
This yields the existence of a δ l > 0 such that, for any f ∈ B(g l , δ l ) and any (u, x) ∈ E l , sup

M l ≤n≤N l |S n,u f (x)| n 1/2 ≥ l.
We finally consider the residual set R = L≥1 l≥L B(g l , δ l ) and we pick f ∈ R. There exists an increasing sequence (l k ) such that f ∈ B(g l k , δ l k ). Let E = lim sup k E l k which has full measure and pick (u, x) ∈ E. There exists a subsequence (l k ) of (l k ) such that (u, x) ∈ E l k for all k. We then have sup

M l k ≤n≤N l k |S n,u f (x)| n 1/2 ≥ l k
which allows us to conclude.

Remark 4.5. The proof gives slightly more than announced: there exists a residual set R ⊂ C 0 (G) such that, for all ε ∈ (0, 1/2) and all f ∈ R,

µ ⊗ µ (u, x) ∈ G 2 ; lim sup n |S n,u f (x)| n 1/2 (log log n) 1 2 -ε = +∞ = 1.

Fast and slow points for irrational rotations on the circle

Throughout this section, we fix α ∈ R\Q.

A partition of T.

To get an estimate of the Hausdorff dimension of F ψ (f ), which is more precise than the result already obtained on its measure, we will need a refinement of Rokhlin towers specific to irrational rotations. We shall use the following system of partitions of T associated to the irrational rotation R α , as it is described for instance in [11, Lecture 9, Theorem 1]. Let (p n /q n ) be the n-th convergent of α in its continued fraction expansion. Define

∆ (n) 0 = [0, {q n α}) if n is even [{q n α}, 1) if n is odd. Denote also ∆ (n) j = R j α ∆ (n) 0
. For any n ≥ 1, the intervals ∆ (n) j , 0 ≤ j < q n+1 and ∆ (n+1) j , 0 ≤ j < q n , are pairwise disjoint and their union is the whole T. We shall denote by d n the length of ∆

(n) 0 . It is well known that 1 2q n+1 ≤ d n ≤ 1 q n+1 .

Continuous functions.

The main step towards the proof of Theorem 1.5 is the following lemma which improves partly Lemma 2.2.

Lemma 5.1. Let M ∈ N, C > 0, s ∈ (0, 1), δ > 0 and ε > 0. Then there exist f ∈ C 0 (T) with f ∞ ≤ ε, a compact set E ⊂ T, and an integer m ≥ M such that

(5) ∀x ∈ E, |S m,α f (x)| ≥ Cψ(m); (6)
H s δ (E c ) < ε.
Proof. Let m ≥ M be such that mε > Cψ(m). Let also n be a large integer and consider the partition of T described in Section 5.1:

T = 0≤j<q n+1 ∆ (n) j ∪ 0≤j<qn ∆ (n+1) j
, where the convergents of α are p n /q n . Since it will be easier to deal with even numbers we put q n = 2 q n /2 , n ∈ N which is the greatest even integer less than q n . Hence q n and q n+1 are even. We define a continuous function f with zero mean such that

• on ∆ (n) j , 0 ≤ j < q n+1
2 and on ∆ (n+1) j

, 0 ≤ j < qn 2 , f = ε except on two very small intervals of size η > 0 where f is affine to ensure that f vanishes at the boundary of ∆ (n) j and ∆ (n+1) j .

• on ∆ (n) j , q n+1 2 ≤ j < q n+1 and on ∆ (n+1) j , qn 2 ≤ j < q n , f = -ε except on two very small intervals of size η > 0 where f is affine to ensure that f vanishes at the boundary of ∆

(n) j and ∆ (n+1) j . • if x ∈ 0≤j< q n+1 ∆ (n) j ∪ 0≤j< qn ∆ (n+1) j we set f (x) = 0. We set Γ (n) j (resp. Γ (n+1) j ) the (largest) subinterval of ∆ (n) j (resp. ∆ (n+1) j
) such that |f | = ε and we let

E = 0≤j< q n+1 2 -m Γ (n) j ∪ q n+1 2 ≤j< q n+1 -m Γ (n) j ∪ 0≤j< qn 2 -m Γ (n+1) j ∪ qn 2 ≤j< qn-m Γ (n+1) j . If x belongs to E, then f (x + jα) = f (x) for all j = 0, . . . , m -1 and |f (x)| = ε. Therefore, we have |S m,α f (x)| = mε > Cψ(m).
On the other hand, E c is the union of at most

• (2m + 2) intervals of size d n ; • (2m + 2) intervals of size d n+1 ; • 2( q n+1 + q n ) intervals of size η.
Hence we have

H s δ (E c ) ≤ (2m + 2)d s n + (2m + 2)d s n+1 + 2( q n+1 + q n )η s < ε
if we choose n sufficiently large and then η sufficiently small.

Proof of Theorem 1.5. We mimic the proof of Theorem 1.1. Recall that

F ψ (f ) = x ∈ T; lim sup n |S n,α f (x)| ψ(n) < +∞ .
Let (h l ) be a sequence of coboundaries which is dense in C 0 (T). Then for any l ≥ 1, there exists C l > 0 such that sup n S n,α h l ∞ ≤ C l . Let f l , E l and m l be given by Lemma 5.1 for C = l + C l + 1, M = l and ε = s = δ = 1/l. We set g l = h l + f l and observe that, for

x ∈ E l , |S m l ,α g l (x)| ≥ (l + C l + 1)ψ(m l ) -C l ≥ (l + 1)ψ(m l ).
There exists δ l > 0 such that, for any f ∈ B(g l , δ l ) and any x ∈ E l ,

|S m l ,α f (x)| ≥ lψ(m l ). Since the sequence (g l ) is dense in C 0 (T), R = L≥1 l≥L B(g l , δ l ) is a residual subset of C 0 (T). Pick f ∈ R.
There exists an increasing sequence (l k ) such that f ∈ B(g l k , δ l k ). We set E = lim sup E l k and observe that, for any x ∈ E,

lim sup n |S n,α f (x)| ψ(n) = +∞.
Moreover, E c = K≥1 k≥K E c l k . For any s ∈ (0, 1), the properties of the sets E l ensure that H s k≥K E c l k = 0. Since F ψ ⊂ E c , we conclude that dim H (F ψ ) ≤ s and therefore dim H (F ψ ) = 0. 5.3. Hölder functions. We now modify the previous construction to adapt it to Hölder continuous functions.

Lemma 5.2. Let M ∈ N, ν ∈ (0, 1), ξ ∈ (0, 1) with ν + ξ < 1, A > 0, ξ 1-ν < s ≤ 1, δ > 0, ε > 0. There exist a continuous function f ∈ C 0 (T) with f ∞ ≤ 1, Lip ξ (f ) ≤ 1, an integer N ≥ M , and a compact set E ⊂ T such that (7) ∀x ∈ E, ∃m ∈ {M, . . . , N }, |S m,α f (x)| ≥ Am ν , (8)
H s δ (E c ) < ε.
Proof. The construction of f will be more or less difficult depending on the arithmetical properties of α. Let (p n /q n ) be the nth convergent of α in its continued fraction expansion.

For each n ≥ 0, there exists τ n ≥ 1 such that q n+2 = q τn n . We define

τ := lim inf n τ n ∈ [1, +∞].
We then fix ν ∈ (0, 1) such that ν > ν, ξ + ν < 1 and

(9) ξ 1 -ν < s.
If moreover τ < 1-ν ξ , we also require that τ < 1-ν ξ . Let n be a large integer and consider the partition of T described in Section 5.1:

T = 0≤j<q n+1 ∆ (n) j ∪ 0≤j<qn ∆ (n+1) j .
Again for ease of notation we suppose that q n and q n+1 are even; if not, a modification similar to the one used in the proof of Lemma 5.1 can be used.

First case: τ ≥ 1-ν ξ . Then, for n large enough, τ n s > 1 + η for some fixed η > 0. We fix such an n and we then define f as follows:

• on ∆ (n) j = (a j , b j ), 0 ≤ j < q n+1
2 , f is equal to (x -a j ) ξ on a j , a j +b j 2

, equal to

(b j -x) ξ on a j +b j 2 , b j . • On ∆ (n) j = (a j , b j ), q n+1 2 ≤ j < q n+1 , f is equal to -(x -a j ) ξ on a j , a j +b j 2 , equal to -(b j -x) ξ on a j +b j 2 , b j . • f is equal to 0 otherwise. It is then clear that f ∞ ≤ 1, Lip ξ (f ) ≤ 1 and
T f dλ = 0. Recalling that d n = b j -a j for 0 ≤ j < q n+1 we then set

δ 0 = ξ 1 -ν ∈ (0, 1), γ 0 = 1 δ 0 = 1 -ν ξ > 1, Γ j = (a j + d γ 0 n , b j -d γ 0 n ) , 0 ≤ j < q n+1 , E 0 = q n+1 2 -1-q δ 0 n+1 j=0 Γ j ∪ q n+1 -1-q δ 0 n+1 j= q n+1 2 Γ j .
Observe that if y ∈ Γ j , then |f (y)| ≥ d γ 0 ξ n and that R α (Γ j ) ⊂ Γ j+1 , 0 ≤ j < q n+1 -1. It follows that, for x ∈ E 0 with constants C which do not depend on n and may change from line to line

S q δ 0 n+1 ,α f (x) ≥ C q δ 0 n+1 d γ 0 ξ n (10)
≥ Cq δ 0 n+1 q -γ 0 ξ n+1 ≥ Cq δ 0 1- γ 0 δ 0 ξ n+1 ≥ Cq δ 0 ν n+1 ≥ A q δ 0 n+1 ν
provided n is large enough. Thus ( 7) is satisfied with m = q δ 0 n+1 and E = E 0 for large values of n. Moreover,

E c 0 is contained in the union of • 2 q δ 0 n+1 + 2 intervals of size d n (the intervals ∆ (n) j
which are not considered);

• 2q n+1 intervals of size d γ 0 n (the extreme parts of the intervals ∆

(n) j ); • q n intervals of size d n+1 (the intervals of the following generation ∆ (n+1) j

).

Hence, for n large enough, H s δ (E c ) ≤ C q δ 0 n+1 q -s n+1 + q n+1 q -γ 0 s n+1 + q n q -τns n .

Since δ 0 -s < 0, 1 -γ 0 s < 0 and 1 -τ n s < -η, (8) is also satisfied provided n is large enough.

Second case: τ < 1-ν ξ . This time, the intervals coming from j ∆ (n+1) j are too long to be neglected with respect to the H s -measure. By the choice of ν , we know that there exist integers n as large as we want such that

(11) 1 ≤ √ τ n ≤ τ n < 1 -ν ξ ;
we will fix such an n later. We keep the same values for δ 0 , γ 0 , Γ j and E 0 and the same definition for f on 0≤j<q n+1 ∆

(n) j
as in the first case. On the other hand, we define f on ∆

(n+1) j = (u j , v j ) by imposing f (x) = (x -u j ) ξ on u j , u j +v j 2 , f (x) = (v j -x) ξ on u j +v j 2 , v j if 0 ≤ j < q n /2 and f (x) = -(x -u j ) ξ on u j , u j +v j 2 , f (x) = -(v j -x) ξ on u j +v j 2 , v j if q n /2 ≤ j < q n . We then set δ 1,n = δ 1 = √ τ n ξ 1 -ν ∈ (0, 1), γ 1,n = γ 1 = 1 δ 1 = 1 √ τ n 1 -ν ξ > 1, Θ j = u j + d γ 1 n+1 , v j -d γ 1 n+1 , E 1 = qn 2 -1-q δ 1 n j=0 Θ j ∪ qn-1-q δ 1 n j= qn 2 Θ j ,
and E = E 0 ∪ E 1 . Remember that d n+1 ≈ q -τn n . We can still use [START_REF] Rudin | Fourier analysis on groups[END_REF] and can deduce analogously for any x ∈ E 1 ,

S q δ 1 n ,α f (x) ≥ Cq δ 1 n d γ 1 ξ n+1 (12)
≥ Cq δ 1 -τnγ 1 ξ n ≥ C q δ 1 n ν ≥ A q δ 1 n ν
provided n is large enough. From now on we can fix a sufficiently large n. The set E c consists of at most

• 2 q δ 0 n+1 + 2 intervals of size d n ; • 2q n+1 intervals of size d γ 0 n ; • 2 q δ 1 n + 2 intervals of size d n+1 ; • 2q n intervals of size d γ 1 n+1 . Thus, H s δ (E c ) ≤ C q δ 0 -s n+1 + q 1-γ 0 s n+1 + q δ 1 -τns n + q 1-τnγ 1 s n .
By using ( 9) and ( 11) we conclude exactly as before since

δ 1 -τ n s ≤ τ n ξ 1 -ν -s ≤ 1 -ν ξ ξ 1 -ν -s < 0 and 1 -τ n γ 1 s ≤ 1 - 1 -ν ξ s < 0.
Proof of Theorem 1.6. We will prove slightly more than announced. Let E ξ be the closed subspace of C ξ 0 (T) defined by

E ξ = f ∈ C 0 (T); ∀x, y ∈ T, |f (x) -f (y)| ≤ |x -y| ξ = f ∈ C 0 (T); Lip ξ (f ) ≤ 1 .
The space E ξ , equipped with the norm of the uniform convergence is now again a separable complete metric space. We will prove that, for all functions f in a residual subset of E ξ , for all ν ∈ (0,

1 -ξ), dim H (F ν (f )) ≤ ξ 1-ν . Since F ν (f ) ⊂ F ν (f ) provided ν ≤ ν,
it is sufficient to prove this inequality for ν belonging to a sequence (ν k ) which is dense in (0, 1 -ξ). Now, the countable intersection of residual sets remaining residual, we just have to prove that, for a fixed ν ∈ (0, 1 -ξ), all functions f in a residual subset of E ξ satisfy dim H (F ν ) ≤ ξ 1-ν . Let (h l ) be a sequence of C 0 -coboundaries which is dense in E ξ and with Lip ξ (h l ) ≤ 1 -1 l . For any l ≥ 1, there exists C l > 0 such that sup n S n,α h l ∞ ≤ C l . Let f l , N l and E l be given by Lemma 5.2 with s = ξ 1-ν + 1 l , δ = ε = 1 l , M = l and A = l(C l + l + 1). We set g l = h l + 1 l f l so that g l ∈ E ξ , (g l ) is dense in E ξ and, for any x in the compact set E l , there exists m ∈ {l, . . . , N l } with

|S m,α g l (x)| ≥ (l + C l + 1)m ν -C l ≥ (l + 1)m ν .
We can then find δ l > 0 such that, for all f ∈ B(g l , δ l ) and all x ∈ E l , there exists m ∈ {l, . . . , N l } with

|S m,α f (x)| ≥ lm ν and H s 1/l (E c l ) < 1 l .
We set R = L≥1 l≥L B(g l , δ l ) ∩ E ξ which is a residual subset of E ξ . Pick f ∈ R. There exists an increasing sequence (l k ) such that f ∈ B(g l k , δ l k ). We set E = lim sup k E l k and observe that, for any x ∈ E,

lim sup n |S n,α f (x)| n ν = +∞ so that F ν ⊂ E c . Now the construction of the sets E l ensures that dim H (E c ) ≤ ξ 1 -ν .
Question 5.3. Is the value 1-ξ ν optimal? In particular, it does not depend on the type of α, which may look surprizing. (ii) for all γ < ν, there exists a residual subset R of C 0 (G) such that, for all f ∈ R,

µ u ∈ G; lim sup n S n,u f (0) n γ = +∞ = 1?
It can be shown that ν = 1/2 works for (ii). Indeed, Lemma 4.4 and Fubini's theorem imply that, for all ε ∈ (0, 1), all C > 0 and all M ∈ N, there exist

x ∈ G, f ∈ C 0 (G), N > M and E ⊂ G with f ∞ < ε, µ(E) > 1 -ε and sup M ≤n≤N Sn,uf (x) n 1/2
≥ C for u ∈ E. Translating f if necessary, we may assume that x = 0. We then conclude exactly as in the proof of Theorem 1.2. Second, Theorem 1.5 improves Theorem 1.1 for rotations of the circle by replacing nowhere dense sets with the more precise notion of sets with zero Hausdorff dimension. There are also enhancements of meager sets, for instance σ-porous sets (see [START_REF] Zajiček | Porosity and σ-porosity[END_REF]) Question 6.2. Does there exist a residual subset R of C 0 (T) such that, for any f ∈ R, E ψ (f ) is σ-porous?

In the spirit of Theorem 1.2, the next step would be to perform a multifractal analysis of the exceptional sets. Precisely, let f ∈ C 0 (T) and ν ∈ (1/2, 1). Let us set

E -(ν, f ) = (α, x) ∈ T 2 ; lim sup n log |S n,α f (x)| log n ≥ ν .
These sets have Lebesgue measure zero.

Question 6.3. Can we majorize the Hausdorff dimension of E -(ν, f )?

We could also replace everywhere the lim sup by lim inf.

Question 6.4. Let ψ : N → N with ψ(n) = o(n). Does there exist f ∈ C 0 (Ω) such that {x ∈ Ω; lim inf n |S n,T f (x)|/|ψ(n)| = +∞} is residual? has full measure? 6.2.
Other sums. The study of S n,α f (x) is a particular case of the series n≥1 a n f (x + nα). In the particular case a n = 1/n this series, also called the one-sided ergodic Hilbert transform, was thoroughly investigated in [START_REF] Fan | Everywhere divergence of one-sided ergodic Hilbert transform[END_REF].

In [START_REF] Fan | Everywhere divergence of one-sided ergodic Hilbert transform[END_REF], the authors show that for any non-polynomial function f ∈ C 2 0 (T) with values in R, there exists a residual set R f of irrational numbers depending on f such that, for every

α ∈ R f , lim sup N N n=1 f (x + nα) n = +∞
for almost every x ∈ T and they ask if this holds for every x ∈ T (they show that this is the case if f (n) = 0 when n ≤ 0). We provide a counterexample.

Example 6.5. Let a ∈ (0, 1) and f ∈ C 2 0 (T) be defined by its Fourier coefficients f (0) = 0, f (n) = ia n for n > 0, f (n) = -ia -n for n < 0. A small computation shows that

f (x) = iae 2πix 1 -ae 2πix - iae -2πix 1 -ae -2πix = -2a sin(2πx) 1 -2a cos(2πx) + a 2 .
We shall prove that the one-sided ergodic Hilbert transform of f is bounded at x = 0. Indeed, setting

G N (t) = N n=1 e 2πint n ,
it is easy to show that

N n=1 f (nα) n = k>0 ia k G N (kα) - k>0 ia k G N (-kα) = i k>0 a k G N (kα) -G N (kα) = -2 k>0 a k m G N (kα) .
Now, it is well-known that the imaginary part of G N (t), namely N n=1 sin(2πnt) n is uniformly bounded in N and t (see e.g. [6, p.4]). Question 6.6. Can we investigate, in the spirit of this paper and of [START_REF] Fan | Everywhere divergence of one-sided ergodic Hilbert transform[END_REF], the case a n = n -a , with 0 < a < 1? 6.3. Coboundaries in C ξ 0 (T). The natural norm in C ξ 0 (T) is given by ( 13)

f ξ = sup x∈T |f (x)| + sup x,y∈T x =y |f (x) -f (y)| |x -y| ξ .
One may wonder whether, in Theorem 1.6, we have residuality in (C ξ 0 (T), ξ ) instead of in (E ξ , ∞ ). A natural way to do that would be to prove that the coboundaries are dense in C ξ 0 (T). This is not the case, which shows again that C ξ 0 (T) is a weird space. In C ξ 0 (T) we denote the ball of radius r centered at f ∈ C ξ 0 (T) by B ξ 0 (f, r), that is g ∈ B ξ 0 (f, r) if and only if g -f ξ < r. We shall prove the following precise statement.

Theorem 6.7. For any α ∈ R \ Q for any ξ ∈ (0, 1) there exists f ∈ C ξ 0 (T) such that for any g ∈ B ξ 0 (f, 0.1) the function g is not a C 0 (and hence not a

C ξ 0 )-coboundary, that is there is no u ∈ C 0 (T) such that g = u • R α -u. Hence C 0 -coboundaries are not dense in C ξ 0 (T). Proof. By induction we select n 1 = 1, n k ∈ N, J k ⊂ [n k , n k+1 ) ∩ Z with the following properties. If we let h k = k n k+1 1/ξ then the intervals (14) {[jα -h k , jα + 3h k ]; j ∈ J k , k ∈ N} are pairwise disjoint
(all these intervals are considered mod 1 on T),

 j∈J k [jα -h k , jα + 3h k ]   < 1 100 k+2 , (16) m k def = #J k > 0.99 • n k+1 . (15) λ 
For this property we can use that k <k j∈J k [jα -h k , jα + 3h k ] is a union of intervals, which by (15) are of total measure less than 1/200 and the sequence (jα) is uniformly distributed on T, especially if we suppose that the n k s are denominators of suitable convergents of α and recall Subsection 5.1. We also suppose that J k is maximal possible, by this we mean that if j ∈ [n k , n k+1 ) ∩ Z and j ∈ J k then It is obvious that f ∈ C ξ 0 (T) with Lip ξ (f ) ≤ 1. Suppose that g ∈ B ξ 0 (f, 0.1) and proceeding towards a contradiction suppose that g = u • R α -u with a u ∈ C 0 (T). Then there exists K u such that |u| ≤ K u . Clearly, for any x ∈ T and any n ∈ N, we have We will prove in (28) and (29) that for any function g ∈ B ξ 0 (f, 0.1), its Birkhoff sums are not bounded and this will provide a contradiction. Suppose k is fixed. Since g ∈ B ξ 0 (f, 0.1) we have for any j ∈ J k This and (19) imply that for j ∈ J k (22) g(jα) -g(jα + 2h k ) ≥ 0.9 • 2 ξ h ξ k > 0.9h ξ k = 0.9 k n k+1 .

Next we consider the cases when j ∈ J k , j ∈ [n k , n k+1 ). Then (17) applies. Suppose first that there exists k < k, j ∈ J k , such that jα, jα + 2h k ∈ [j α -h k , j α + 3h k ]. The construction of f on [j α -h k , j α + 3h k ] ensures that In case one of jα, jα + 2h k belongs to a [j α -h k , j α + 3h k ], k < k, j ∈ J k and the other is not an element of any such interval then f (x) = 0 at some x in [jα, jα + 2h k ] and a combination of ( 23

6. Miscellaneous remarks 6 . 1 .

 61 Open questions. Our study suggests further questions. The first one is related to Corollary 1.4. Question 6.1. Does there exist ν ∈ [1/2, 1] such that (i) for all γ > ν, for all f ∈ C 0 (G), µ u ∈ G; lim sup n S n,u f (0)

  h k , jα + 3h k ] ∩ k <k j ∈J k [j α -h k , j α + 3h k ] = ∅.By the definition of h k and (16) we have(18) m k • h ξ k > 0.99 • k. Next we define f . On an interval [jα -h k , jα + 3h k ], j ∈ J k , k ∈ N we define f in the following way: f (jα -h k ) = f (jα + h k ) = f (jα + 3h k ) = 0 and (19) f (jα) = h ξ k , f (jα + 2h k ) = -h ξ k , otherwise f is linear on each [jα + nh k , jα + (n + 1)h k ] with n ∈ [-1, 0, 1, 2]. If x ∈ ∪ k∈N ∪ j∈J k [jα -h k , jα + 3h k ] then we set f (x) = 0.

  + jα) = |u(x + (n + 1)α) -u(x)| ≤ 2K u .

  (21) |f (jα) -g(jα) -(f (jα + 2h k ) -g(jα + 2h k ))| |2h k | ξ < 0.1.

  (23) |f (jα) -f (jα + 2h k )| ≤ 2h k (h k ) ξ-1 < 0.001 • h ξ k provided n k+1 was choosen sufficiently large. If jα ∈ k <k j ∈J k [j α-h k , j α+3h k ] then either f (jα) = 0, or jα ∈ k >k j ∈J k [j αh k , j α + 3h k ].In this latter case |f (jα)| ≤ h ξ k with k > k and we can suppose by the inductive definition of the h k that h k < 0.0005 1/ξ • h k . Thus (24) |f (jα)| ≤ 0.0005 • h ξ k . Similarly if jα + 2h k ∈ k <k j ∈J k [j α -h k , j α + 3h k ] we can suppose that (25) |f (jα + 2h k )| ≤ 0.0005 • h ξ k .

  ) and (24), or (25) is applicable. Summarizing, we have finally shown that for all j ∈[n k , n k+1 )\J k , (26) |f (jα) -f (jα + 2h k )| < 0.002 • h ξ k .It is clear that for large k this will contradict (20).Next suppose that the negation of (28) and the negation of (29) hold. -g(jα+ 2h k )) ≥ #J k • 0.9 k n k+1 > 0.99 • n k+1 • 0.9h ξ k . |g(jα) -g(jα + 2h k )| < 0.01 • n k+1 • 0.102 • h ξ k • 2 ξ .

	(see (18) as well), or
	(29)	n k+1 -1 j=n k	g(jα + 2h k ) ≤ -	1 4	n k+1 h ξ k < -	k 4	.
	This implies
	(30)	n k+1 -1 j=n k	(g(jα) -g(jα + 2h k )) < 2 •	k 4	=	1 2	n k+1 h ξ k .
	By (22) and (16)
	(31) (g(jα) On the other hand, by (16) and (27) j∈J k
	n k+1 -1
	(32)
		j=n k
		j ∈J k
	Since g ∈ B ξ 0 (f, 0.1) by (21) and (26) we obtain
	(27)	|g(jα) -g(jα + 2h k )| < 0.102 • h ξ k • 2 ξ .
	We claim that either
	(28)		n k+1 -1 j=n k	g(jα) ≥	1 4	n k+1 h ξ k >	k 4

Now (31) and (32) contradict (30).

. The first and the third author were partially supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). The second author was supported by the Hungarian National Research, Development and Innovation Office-NKFIH, Grant 124003. He also thanks the Rényi Institute where he was a visiting researcher during the academic year 2017-18.


D Andrieu

C Kaiser

A Ourednik
email: andre.ourednik@epfl.ch

J Lévy

Advanced cartogram construction using a constraint based framework

Introduction

The visualization of social phenomena through classical thematic mapping often leads to unsatisfying representations. Classical political referendum maps, for instance, lead to an underestimation of the opinions expressed by populations of large urban agglomerations. Classical maps of BIP, religious belief, or social segregation give a distorted image of the world by linking the relative importance of quantitative and qualitative values of such variables to the Euclidian topographical extent of the statistical areas where these values are measured. Cartograms are a well-known technique used to compensate for this inconvenience by breaking the link between statistical regions and their topographical areas. Consequently, this liberates one visual variable (that of polygon size) for a more relevant use, such as the representation of the relative social importance of these regions (usually measured by the size of their populations), while leaving intact their topological relations. Although well-known, the cartogram technique is still underused. We relate this fact to three restricting factors: (a) a difficult lecture for a large public used to topographical space geometries (b) the inability of cartograms to represent complex spatial layers and (c) the unavailability of user-friendly stand-alone cartogram software. The goal of our work, as discussed in the present paper, is to compensate for these flaws.

State of the art of cartogram production and diffusion.

The production of a cartogram is actually a map deformation problem. The goal of a cartogram algorithm is to equalize density across space with given geometric boundaries and density values associated to spatial partitions delimited by these boundaries. However, there is no unique solution to this problem. The aim is to deform the initial geometric boundaries to minimize the size error, while simultaneously changing the polygon shape and respecting the topology. Polygon shapes should accordingly keep their original forms as much as possible to remain recognizable. The total area of the region under study should stay constant. The deformation source is generally a polygon with an associated statistical value that represents the importance of the polygon area. A variant of this cartogram deformation type is displayed when the deformation source consists of one or several points located in a polygon, with several statistical values conjointly defining the polygon deformation. As illustrated in the graphic below (representing the agglomeration of Lausanne, Switzerland), this deformation can also be constrained by other spatial information layers, such as transportation network lines.

Several cartogram algorithms have already been proposed (e.g. [START_REF] Dougenik | An algorithm to construct continuous area cartograms[END_REF][START_REF] Vs | A new technique for constructing continuous cartograms[END_REF][START_REF] Dorling | Area cartograms: Their use and creation[END_REF][START_REF] Kocmoud | Constructing Continuous Cartograms: A Constraint-Based Approach[END_REF][START_REF] Gastner | Diffusion-based method for producing density equalizing maps[END_REF][START_REF] Keim | Medial-axis-based cartograms[END_REF]. However, according to our present knowledge, these algorithms are restricted to the polygon deformation case, or to a variant thereof, adapted for raster images. Additionally, there are certain stand-alone programs that implement several of these algorithms (e.g. Mapresso [http://www.mapresso.com]). These programs are nevertheless limited to only one polygon layer at a time and no real constraints help the user with the deformation control for a given object. For this reason, we decided to create a flexible constraint-based cartogram creation framework in order to have a user-friendly application that should run with an acceptable speed, even for complex layers.

Our cartogram framework

Our cartogram framework is a cross-platform, open-source application written in Java, primarily designed as an independent application using the ESRI Shapefile format for input and output. An integration of the algorithm into other GIS is still possible in the future. The most important feature of the cartogram framework is the internal data structure for the representation of the geometric objects. There is a point database common to all layers and every geometric object exclusively contains a reference to the corresponding point in the point database. Therefore, if we displace a given point, all the geometric objects containing it are automatically modified. The computational amount is reduced, which differs from more classical approaches that store each polygon separately in memory. In order to control the deformation and keep the shape of the polygons and lines as recognizable as possible, we have introduced geometric deformation constraints. The topology of the polygons and lines is respected, even if more than one layer is deformed at the same time. Moving a point can be considered as applying a force vector to that point. Instead of simply applying this force vector without control, we verify whether the topology is still intact after its application. There are several topological constraints, like self-intersection for polygons or closing (two opposites lines are grouped together). If the topology is not guaranteed, we try to apply a smaller force vector. In the case of failure of the application of a smaller force vector, the deformation is completely rejected. This mechanism guarantees the integrity of the topology. Another pertinent mechanism is the influence of a force vector application on the neighborhood. Instead of simply applying the force vector to one point, the force also interacts on the adjacent points. The force of the vector applied to the neighborhood diminishes with the distance. The framework can thus work with complex polygons through the weighting of several polygon parts. We call this feature «multi-polygon respect». In order to guarantee the convergence to a possible solution of the cartogram problem, the application of a force vector that increases the global size error is rejected. Therefore, with our cartogram framework, even the use of random force vectors should theoretically lead to a solution of the cartogram problem. Several cartogram algorithms can then be feasibly attempted. The implementation of several algorithms is possible and different results can be compared. At the present stage, we have implemented a small set of different algorithms that have been specially adapted for the use within our cartogram framework. In order to optimize user-friendliness, we try to make a reasonable estimation for each adjustable parameter. An experimented user may, however, change these parameters in order to optimize the cartogram creation process.

Conclusion

Cartograms do represent geographic space with a non-metric dimension. Despite the "deformation" with respect to the topographical and Euclidian map, cartogram space conserves a map's topology, while reserving other visual variables for the representation of complex social phenomena. Our framework allows for the production of polygon and line-based cartograms, as well as a combination of both through the simultaneous deformation of several information layers. Moreover, cartogram creation becomes accessible to scientists non-specialized in geomatics that prefer to focus on the perception of space, rather than the technical aspects of cartogram creation. Technically, the innovation of our framework consists in the integration of several layers, in the treatment of complex forms, as well as in the possibility of considering other forms than polygons (especially lines) as a deformation source or constraint.

Acknowledgements

This work has been supported by the Swiss National Foundation through the National Research Project 54 "Our Inhabited Space", whose "module 2" is dedicated to the cartographical representation of the contemporary inhabited space.


Corinne Berzin
email: corinne.berzin@upmf-grenoble.fr

José R León

Joaquín Ortega
email: jortega@ivic.ve.work

Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

Keywords: Non-linear functionals, Empirical process, kernel density estimation, crossings, Kullback-Leibler deviation, regularization by convolution

, and a strong approximation theorem, we prove several results for the p-deviation in estimation of the derivatives of the density. We also study the asymptotic behaviour of the number of crossings of the Smoothed Empirical Process defined by Yukich [17] and of a modified version of the Kullback deviation.

Introduction

The problem of estimating the density s(x) of simple random sample X 1 , X 2 , . . . , X n is a classical problem in non-parametric statistics. The most popular method is kernel estimation, introduced by Parzen and Rosenblatt in the 50's. The study of consistency properties and asymptotic normality of these estimators was one of the important subjects of study during the following two decades.

Among these works was Bickel & Rosenblatt [START_REF] Bickel | On some global measures of the deviation of density function estimates[END_REF] where the authors use strong convergence results to establish the asymptotic normality of the L 2 distance between the estimator and the real density. This work, source of inspiration for many others, allows the substitution of the Empirical Process by the Brownian Bridge in the functional under study, that is, the L 2 distance between the estimator and the real density. This replacement can be controlled using the strong approximation theorem of Komlós, Major and Tusnády [START_REF] Cs | Strong Approximations in Probability and Statistics[END_REF]. One of the important consequences of their result is that it gives L 2 confidence bands and provides the tools for hypothesis tests for contiguity alternatives for functions close to the density in the null hypothesis. Silverman [START_REF] Silverman | Weak and strong uniform consistency of the kernel estimate of a density and its derivatives[END_REF] extends these methods to the estimation of the density's derivatives.

In this work we generalize Bickel & Rosenblatt's result to kernel estimators of the density and of the derivative of the density, not only for the L 2 norm but also for L p norms. The first work where this problem was studied was Csörgő y Horváth [START_REF] Horv Áth | Central limit theorems for L p -norms of density estimators[END_REF], also using the strong approximation method. From then on there has been much work done trying to simplify the hypothesis for the density and the kernel, as, for example, Horváth [START_REF] Áth | On L p -norms of multivariate density estimators[END_REF], Beirlant & Mason [START_REF] Beirlant | On the asymptotic normality of L p -norms of empirical functionals[END_REF] and finally the work of Giné, Mason & Zaitsev [START_REF] Gin | The L 1 -norm Density Estimator Process[END_REF], where a limit theorem uniform with respect to the regularization kernel is obtained. In the last three papers mentioned the technique of Poissonization is used, which proves to be stronger than strong approximation.

Our method is interesting because, apart from allowing us to study the derivative of the density, handles the asymptotic convergence of functionals of the type

1 0 G( √ nh( ln (u) -l(u)))du
where ln is an estimator of the density or its derivatives. The p norm case corresponds to G(x) = |x| p . Using this viewpoint we can look at the asymptotics for a variant of the Kullback deviation. The results are obtained passing to functionals of the Brownian Bridge via the strong approximation theorem and then studying the asymptotic behaviour of these functionals using methods developed in a previous article [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF].

With these same methods we can also study the number of crossings of a level by the Smoothed Empirical Process, defined by Yukich [START_REF] Yukich | A note on limit theorems for perturbed empirical processes[END_REF], as we explain in the next section, where we describe more precisely our results.

Description of the Results

Let X 1 , X 2 , . . . , X n be a random sample from the distribution F with density s(u), F n be the empirical distribution of the sample and β F n (u) = √ n(F n (u) -F (u)) be the empirical process. The kernel density estimator is

ŝn (u) = ŝn,h (u) = 1 h ∞ -∞ ϕ( u -v h ) dF n (v)
where ϕ is the estimation kernel and h depends on n: h = h(n).

1.-Define √ n[ŝ n (u) -E(ŝn(u))] = 1 h ∞ -∞ ϕ( u -v h ) dβ F n (v)
Integrating by parts

√ nh[ŝ n (u) -E(ŝn(u))] = √ h βn,h (u)
where βn,h is the derivative of the ϕ-regularization of the empirical process β n,h . If s has two continuous and bounded derivatives it is known that the optimal window is h(n) = O(n -1/5 ). The window is sub-optimal if h = o(n -1/5 ), otherwise it is super-optimal. It is also known that

lim n→∞ √ nh[E(ŝ n (u)) -s(u)] = α(u)
where α(u) is a known function of the unknown density for an optimal window and α(u) ≡ 0 for suboptimal windows. Consider the function g p (x) = |x| p -E |N | p , p ≥ 1 where N is a standard Gaussian r.v. The results that we will show imply for sub-optimal windows the following theorem of Csörgő & Horváth [START_REF] Horv Áth | Central limit theorems for L p -norms of density estimators[END_REF]:

1 √ h 1 0 | √ nh(ŝ n (u) -s(u))| p -E[| √ nh(ŝ n (u) -s(u))| p ] du → N (0, v 2 p )
Indeed the integral above is equivalent to

1 √ h 1 0 g p ( √ h βn,h (u) [Var( √ h βn,h (u))] 1/2 )| Var( √ h βn,h (u))| p/2 du.
Using now the KMT theorem we can replace βn,h (u) by the derivative of the regularization of the F-B.B. and get that the last integral converges to N (0, v 2 p ). In section 6 we obtain results along this line for the derivatives of the density and for a general function in place of |x| p for the sub-optimal and optimal cases.

Let

β F n (u) = √ n(F n (u) -F (u)
) be the empirical process and define

ν n,h (u) = ŝn,h (u) -s(u). If ϕ = 1 [-1,1] /2 then √ nhν n,h (u) = β F n (u + h) -β F n (u -h) 2 √ h + o(1)
as long as the density is sufficiently regular and h is chosen adequately. We want to study the asymptotic behaviour of the sums

1 n [nt] i=1 g n ( i n , √ nhν n,h ( i n )) and 1 n [nt] i=1 f (β F n ( i n ))g n ( i n , √ nhν n,h ( i n ))
where g n is a sequence of functions that satisfy certain properties that will be made explicit later on. These sums are generalizations of the so-called divisible statistics studied by Khmaladze [START_REF] Khmaladze | Martingale limit theorems for divisible statistics[END_REF] and they correspond to the following integrals

t 0 g n (u, √ nhν n,h (u))du and t 0 f (β F n (u))g n (u, √ nhν n,h (u))du. ( 1
)
Using the Komlós, Major & Tusnády (KMT) strong approximation theorem ( [START_REF] Cs | Strong Approximations in Probability and Statistics[END_REF], see section 9 for the statement) we show that if g n → g then

lim n→∞ t 0 g n (u, √ nhν n,h (u))du = t 0 E[g(u, s(u) 2 N )]du lim n→∞ t 0 f (β F n (u))g n (u, √ nhν n,h (u))du = t 0 f (b F (u))E[g(u, s(u) 2 N )]du
where b F (u) is the F-Brownian bridge (F-B.B.), N is a standard Gaussian variable and the convergence is in L 2 (Ω) for the first limit and in law for the second. We also study the speed of convergence in both cases. Using a similar result for the F-B.B. shown in a previous work [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF], we get that there exists a Brownian motion (B.M.) W independent of b F and a positive function σ g (.), which depends only on g such that

lim n→∞ 1 √ h t 0 {f (β F n (u))g n (u, √ nhν n,h (u)) -f (b F (u))E[g(u, s(u) 2 N )]}du = t 0 f (b F (u))d W (σ g (u)).
with convergence in law. Making f ≡ 1 one gets the same result for the first kind of integrals in [START_REF] Aldous | On mixing and stability of limit theorems[END_REF]. We consider these problems for general kernel estimators of the density and also several applications of these results which we now describe briefly.

2.-We consider in section 7 the modified Kullback deviation between the kernel estimator of the density ŝn and s. Let

K(ŝ n , s) = 1-h h ŝn (u) ln( ŝn (u) s(u) ) du.
We show that

n √ h[K(ŝ n , s) - 1 2 1-h h 1 s(u) E[ŝn(u) -s(u)] 2 du - 1-h h (ŝ n (u) -s(u)) du]
is equivalent, in the sub-optimal and optimal cases, to

1 2 √ h 1 0 s -1 (u) [ √ nh(ŝ n (u) -s(u))] 2 -E[ √ nh(ŝ n (u) -s(u))] 2 du
which is similar to the previous result for p = 2 and converges to a normal distribution. We show a similar result for the super-optimal case but with a different normalization.

3.-Define as in Yukich [START_REF] Yukich | A note on limit theorems for perturbed empirical processes[END_REF] the Smoothed Empirical Process:

βn,h (u) = √ n( Fn (u) -F (u))
where Fn (u) = u -∞ ŝn (v) dv.

(u) = ∞ -∞ ϕ(v)F (u -hv) dv, he shows that, if √ nh 2 → a, then βn,h (u) D → b F (u) + a 2 K 2 ṡ(u), where K 2 = ∞ -∞ ϕ(v)v 2 dv.
Let Ñ h n (x) be the number of times that the process βn,h (•) crosses level x before time 1, and define ˜ h n (•) as a modification of the occupation measure for βn,h (see Corollary 2 in section 9 for an exact definition). By a formula of Banach-Kac [START_REF] Banach | Sur les lignes rectifiables et les surfaces dont l'aire est finie[END_REF][START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF]

∞ -∞ f (x) Ñ h n (x) dx = 1 0 f ( βn,h (u))| ˙ β n,h (u)|du. Let Λ h = ( πh 2||ϕ|| 2
2 ) 1 2 , using results included in a previous work [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF] we obtain in section 9 the following:

a) If √ nh 2 → 0 then ξh n (f ) = 1 √ h ∞ -∞ f (x)(Λ h Ñn (x) -˜ h n (x))dx D → V
where V is a r.v. which, conditional on F ∞ , the σ-field generated by {b F (u), 0 ≤ u ≤ 1}, has the same law as σ 0

1 0 f (b F (u)) s(u)d W (u), W (•) is a B.M. independent of b F (•) (see Theorem 3 in [4]
) and σ 0 is defined in the next section. b) On the other hand if

√ nh 2 → a then Λ h ∞ -∞ f (x) Ñ h n (x) dx D → 1 0 f (b F (u) + a 2 K 2 ṡ(u)) s(u)du.
Moreover, we show that ξh n (f ) converges weakly to a r.v. Ṽ such that

L( Ṽ /F ∞ ) = σ 0 1 0 f (b F (u) + a 2 K 2 ṡ(u)) s(u) d W (u).
c) In section 9 we also treat the case √ nh 2 → +∞. In this situation we need a different normalization sequence for the process βn,h .

Hypothesis and notation

Let F be a distribution function with bounded support and density s. To simplify the notation we shall suppose that its support is [0, 1] i.e. F (0) = 0 and F (1) = 1. The F-Brownian Motion is defined as

W F (t) = W (F (t)), where W is a standard B. M. With this definition one has E(W F (u)W F (v)) = F (u ∧ v). The F-B.B. is defined as b F (t) = W F (t) -F (t)W (1) and then E(b F (u)b F (v)) = F (u ∧ v)[1 -F (u ∨ v)].
For each t and h > 0 we define the regularized processes b

F h (t) = ϕ h * b F (t) and W F h (t) = ϕ h * W F (t) where ϕ h (t) = 1 h ϕ( t h
) and * denotes the convolution. We use h instead of ε (as in [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF]) for the regularization parameter to be consistent with the usual notation in density estimation.

We shall use the Hermite polynomials, which can be defined by exp(tx

-t 2 2 ) = ∞ n=0 H n (x) t n n! .
They form an orthogonal system for the standard Gaussian measure φ(x)dx and, if h

∈ L 2 (φ(x) dx), h(x) = ∞ n=0 ĥn H n (x) and ||h|| 2 2,φ = ∞ n=0 n! ĥ2 n .
Mehler's formula [START_REF] Breuer | Central Limit Theorems for non-linear functionals of Gaussian fields[END_REF] gives a simple form to compute the covariance between two L 2 functions of Gaussian r.v.'s: If (X, Y ) is a Gaussian random vector having correlation ρ then

E[h(X)k(Y )] = ∞ n=0 ĥn kn n!ρ n .
(

) 2
We have the following hypothesis (H1) For the kernel ϕ:

1 -1 ϕ(t) dt = 1, ϕ ≥ 0, ϕ is absolutely continuous with support in [-1, 1]. Define ψ(u) = ϕ * φ(u) where φ(u) = ϕ(-u) and θ(u) = ψ(u)||ϕ|| -2 2 , u ∈ R. Let K 2 = ∞ -∞ ϕ(v) v 2 dv
. Further conditions will be required for some applications.

(H2) For the function s: s ∈ C 2 [0, 1], 0 < s(x) for all x ∈ [0, 1].

Let g(x, y) be a function in L 2 (φ(y)dy) a.s. continuous in the first variable and with polynomial growth in the second variable (|g(x, y)| ≤ K P (|y|) for x in a set compact) that satisfies the following two conditions:

i) E g(x, N ) = 0, 0 ≤ x. ii) E [N g(x, N )] = 0, 0 ≤ x. Define g(x, y) = ∞ k=2 c k (x)H k (y) the Hermite expansion of g, where we suppose that ∀x > 0, ∃k ≥ 2 such that c k (x) > 0, ( σb h (u)) 2 = Var( ḃF h (u))
and

g h (u, y) = g( √ h σb h (u), y), (3)
and for 0 ≤ t ≤ 1

S b F h (t) = 1 √ h t 0 g h (u, ξ b h (u))du with ξ b h (u) = ḃF h (u) σb h (u) (4)
Define also

σ 2 (t) = t 0 κ(u) du (5)
where

κ(u) = ∞ k=2 k!c 2 k ( s(u)||ϕ|| 2 ) 2 -2 θ k (w) dw = 2 -2 E[g( s(u)||ϕ|| 2 , X)g( s(u)||ϕ|| 2 , θ(w)X + 1 -θ 2 (w)Y )] dw
and X,Y are independent, standard Gaussian variables. We shall write

l q (u, x) = π 2 x + qK 2 s(u) 2 s(u)||ϕ|| 2 = +∞ k=0 d k,q (u)H k (x)
and

σ 2 q (u) = 2 -2 ∞ n=2 d 2 n,q (u)n!θ n (w)dw, for u ∈ [0, 1],
x ∈ R and q ≥ 0 (note that σ 2 0 (u) doesn't depend on u so we will note

σ 2 0 (u) = σ 2 0 ). Also define c n = 2 K 2 √ nh 2 . We have ḃF h (t) = 1 h t/h -∞ b F (t -hy) dϕ(y). We introduce Z b F h (f ) = h -1 2 ∞ -∞ f (x)[Λ h N b F h (x) -b F (x)] dx with Λ h = πh 2 ||ϕ|| -1 2 ,
where

N b F h (x) is the number of times that the process b F h (•) crosses level x before time 1 and b F (•) is a modification of the local time for the F-B.B. on [0, 1] that satisfies, for any continuous function f , ∞ -∞ f (x) b F (x) dx = 1 0 f (b F (u)) s(u) du
In what follows we shall drop out the indices F when no confusion is possible. Throughout the paper, Const shall stand for a generic constant, whose value may change during a proof, N denotes a standard Gaussian r.v.

Stable Convergence

We shall use the notion of stable convergence, which we describe now following [START_REF] Aldous | On mixing and stability of limit theorems[END_REF][START_REF] Hall | Martingale Limit Theory and its Applications[END_REF] and [START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF].

Let X n be a sequence of r. v.'s defined over (Ω, F, P ) and taking values in C[0, 1], and let G ⊂ F be a sub-σ-field. Let X be another r. v. defined over an extension ( Ω, F, P ) of the original probability space, with values in

C[0, 1]. The sequence X n converges G-stably to X if lim n E(Y h(X n )) = Ē(Y h(X)),
for all bounded and continuous functions h : C[0, 1] → R and all G-measurable and bounded r.v. Y .

Stable convergence is invariant under absolutely continuous changes of the measure.

Preliminary Results

We shall say that a function f is locally Lipschitz if

|f (x) -f (y)| ≤ |P (x, y)||x -y| (6)
where P is a polynomial. . We need two results from [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF] which we state for completeness.

Theorem 1 Under H1, H2, i) and ii)

S b h (t)→ W (σ 2 (t)) stably in C[0, 1]
where W is a B.M. and σ 2 (t) is given by ( 5). Furthermore the vector process

(b F h (•), S b h (•)) converges weakly in C[0, 1] × C[0, 1] towards (b F (•), W (σ 2 (•))) and the processes b F (•) and W (σ 2 (•)) are independent. Corollary 1 Under H1, H2 and if f ∈ C 2 with f bounded, Z b h (f ) converges stably as h → 0 towards a r.v. V ∈ L 2 (Ω) and the conditional distribution L(V |F ∞ ) is Gaussian with zero mean and random variance equal to σ 2 0 1 0 f 2 (b F (u))s(u)du. (σ 2
0 has been defined in previous section 3). Remark 1. Conditionally on F ∞ , V has the same distribution as σ 0

1 0 f (b F (u)) s(u) d W (u) (remem- ber that F ∞ is the σ-field generated by {b F (u), 0 ≤ u ≤ 1}). Remark 2.
Although the theorem has been stated assuming that in the definition of S b h (t), g h (u, y) is of the form g( √ h σb h (u), y), (see ( 3) and ( 4)), a slight modification of the proof shows that the result is true for other forms of the function g h (u, y) that will appear in sections 6 and 9.

5 Convergence in L 2 for the Empirical Process

Let β F n (u) = √ n(F n (u) -F (u)
) be the empirical process associated to a n-sample X 1 , . . . , X n of the continuous distribution F , where F n is the corresponding empirical d. f. and F satisfies the conditions of section 3. The transformation

U i = F (X i ), i = 1, . . . , n gives a n-sample of the uniform distribution U[0, 1]. If E n (v) is the associated empirical distribution function, the corresponding empirical process is α n (v) = √ n (E n (v) -v), 0 ≤ v ≤ 1. Note that α n (F (u)) = β F n (u) and if b(t) denotes the standard B.B. sup t∈[0,1] √ n |α n (t) -b(t)| = √ n ||β F n -b F || ∞
The following approximation theorem is due to Komlós, Major and Tusnády (1975) [START_REF] Cs | Strong Approximations in Probability and Statistics[END_REF]. Theorem [KMT]For any integer n there exists some B.B. b n such that for some absolute positive constants C, Λ, γ the following inequality holds:

P ( sup t∈[0,1] √ n|α n (t) -b n (t)| ≥ x + C ln(n)) ≤ Λ exp(-γx)
for any positive x. Consequently

sup t∈[0,1] |α n (t) -b n (t)| = O( ln(n) √ n ) a.s. Let β n,h (u) = 1 h ∞ -∞ ϕ( u -v h )β F n (v)
||β n,h -b n,h || ∞ ≤ Const ln(n) √ n and || βn,h -ḃn,h || ∞ ≤ Const ln(n) h √ n ,
where the constant is random.

Lemma 1 For p ≥ 1, E( √ n ||β F n -b F n || ∞ ) p ≤ Const (ln(n)) p (7) E( √ n ||β n,h -b n,h || ∞ ) p ≤ Const (ln(n)) p (8) E( √ n || βn,h -ḃn,h || ∞ ) p ≤ Const h -p (ln(n)) p (9)
Proof. We only consider the case p = 2, the proof is similar for general p.

Define Y n (u) = √ n|β F n (u) - b F n (u)| and let A λ = {||Y n || ∞ ≥ λ + C ln(n)}, where λ ≥ 0 and C is the constant in the KMT theorem. We have E(||Yn|| 2 ∞ ) = E(||Yn|| 2 ∞ 1 A c λ ) + E(||Yn|| 2 ∞ 1 A λ ) ≤ (λ + C ln(n)) 2 + E(||Yn|| 2 ∞ 1 A λ ) (10)
Call G n the distribution function of ||Y n || ∞ , then, integrating by parts and using KMT E(||Yn||

2 ∞ 1 A λ ) = ∞ λ+C ln(n) u 2 dG n (u) = (λ + C ln(n)) 2 (1 -G n (λ + C ln(n))) + ∞ λ+C ln(n) 2u(1 -G n (u)) du (11)
but by the inequality in the KMT theorem

1 -G n (λ + C ln(n)) ≤ Λe -γλ ; 1 -G n (u) ≤ Λe -γ(u-C ln(n)) thus (11) ≤ Λ(λ + C ln(n)) 2 e -γλ + 2Λ ∞ λ+C ln(n) ue -γ(u-C ln(n)) du ≤ Const(ln(n)) 2 + Const n γC (λ + C ln(n))e -γ(λ-C ln(n)) + e -γ(λ-C ln(n)) ≤ Const(ln(n)) 2 (12)
Using ( 12) in [START_REF] Hall | Central limit theorem for integrated square error of multivariate nonparametric density estimators[END_REF] we get [START_REF] Horv Áth | Central limit theorems for L p -norms of density estimators[END_REF]. We shall now prove (9), the proof of ( 8) is similar but simpler.

E(n || βn,h -ḃn,h || 2 ∞ ) = E(n sup x | ∞ -∞ 1 h 2 φ( x -y h )(β F n (y) -b F n (y)) dy| 2 ) ≤ E(n (sup x ∞ -∞ 1 h | φ(u)| |β F n (x -uh) -b F n (x -uh)| du) 2 ) ≤ Const 1 h 2 E ||Y n || 2 ∞ ≤ Const (ln(n)) 2 h 2 .
Theorem 2 Let g be an even function verifying [START_REF] Breuer | Central Limit Theorems for non-linear functionals of Gaussian fields[END_REF], under H1 and H2 and if

ln(n) √ hn → 0 as h → 0, lim n→∞ 1 0 g( √ h βn,h (u))du = 1 0 E{g( s(u)N ||ϕ|| 2 )}du in L 2
Proof. We have

E[ 1 0 g( √ h βn,h (u))du - 1 0 E{g( s(u)||ϕ|| 2 N )}du] 2 ≤Const E[ 1 0 g( √ h βn,h (u))du - 1 0 g( √ h ḃn,h (u))du] 2 + Const E[ 1 0 g( √ h ḃn,h (u))du - 1 0 E{g( s(u)||ϕ|| 2 N )}du] 2 (13)
Using the Lipschitz property for g, Hölder's inequality and lemma 1 the first term is bounded by

Const (ln(n)) 2 nh .
Up to a constant the second term is equal to

E[ 1 0 g( √ h ḃF h (u)) -E(g( s(u)||ϕ|| 2 N )] 2 du ≤ Const E[ 1 0 (g(ξ b h (u) √ h σb h (u)) -g(ξ b h (u)||ϕ|| 2 s(u))) du] 2 + Const E[ 1 0 (g(ξ b h (u)||ϕ|| 2 s(u)) -E[g(||ϕ||2 s(u)N )]) du] 2 .
For the first of these two terms, use the fact that

| √ h σb h (u) -||ϕ|| 2 s(u)| ≤ Const h
for h ≤ u ≤ 1-h, the Lipschitz property and Hölder's inequality to obtain that it is bounded by Const h 2 .

For the second term we consider the function

g(u, y) = g(y||ϕ|| 2 s(u)) -E[g( s(u)||ϕ|| 2 N )].
It verifies i) and ii), hence by Theorem 1 the second term is O(h). We thus obtain that

(13) ≤ Const (ln(n)) 2 hn + O(h)
and the theorem holds.

Estimation of a Density's Derivatives

In [START_REF] Horv Áth | Central limit theorems for L p -norms of density estimators[END_REF] Csörgő and Horváth consider the asymptotic behaviour of the p-th order risk for the kernel estimator of a density. In previous works the case p = 2 was established for sub-optimal windows by Bickel & Rosenblatt [START_REF] Bickel | On some global measures of the deviation of density function estimates[END_REF] and in all generality by Hall [START_REF] Hall | Central limit theorem for integrated square error of multivariate nonparametric density estimators[END_REF].

With our method we can consider the same problem as Csörgő and Horváth but with a general G instead of G(u) = |u| p and for the estimation of the derivatives of the density.

Assume that s is in C m+2 [0, 1] and satisfies the conditions of section 3. Define the estimator for the mth-derivative as in [START_REF] Silverman | Weak and strong uniform consistency of the kernel estimate of a density and its derivatives[END_REF] by

ŝ(m) n (u) = 1 nh m+1 n k=1 ϕ (m) ( u -X k h )
with ϕ even and in C m+1 . We shall use the notation s

(m) n (u) = E[ŝ (m) n (u)].
It is easy to show, using a Taylor expansion of order m + 2 for s, that for 0 < u < 1 the bias is

E[ŝ (m) n (u) -s (m) (u)] = (-1) m+2 (m + 2)! h 2 ∞ -∞ ϕ (m) (v)u m+2 s (m+2) (θ 1 ) dv,
where θ 1 depends on u, v and n. Thus

h -2 (E[ŝ (m) n (u) -s (m) (u)]) → K 2 s (m+2) (u)/2 as n goes to infinity (remember that K 2 = ∞ -∞ v 2 ϕ(v) dv).
The variance term can also be calculated, giving

nh 2m+1 E(ŝ (m) n (u) -s (m) n (u)) 2 = ∞ -∞ [ϕ (m) (v)] 2 s(u -hv) dv -h[ ∞ -∞ ϕ (m) (v) s(u -hv) dv] 2 → ∞ -∞ (ϕ (m) (v)) 2 dv s(u) = ||ϕ (m) || 2 2 s(u) i.e. nh 2m+1 σ 2 n,m (u) → ||ϕ (m) || 2 2 s(u)
as n → ∞. Let now G be a locally Lipschitz function (see [START_REF] Breuer | Central Limit Theorems for non-linear functionals of Gaussian fields[END_REF] for the definition), we want to study

I (m) n = 1 √ h 1-h h G ŝ(m) n (u) -s (m) (u) σ n,m (u) -E G ŝ(m) n (u) -s (m) (u) σ n,m (u) du.
It is well known, and can also be shown using the previous expressions for the bias and variance, that the optimal estimation window for the mean square error is

h = O(n -1/2m+5 ) i.e. √ nh m+ 5 2 → C m and in this case √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nh m+ 1 2 σ n,m (u) → f m (u) = K 2 C m s (m+2) (u) 2||ϕ (m) || 2 s(u) (14)
as n → ∞. The window is sub-optimal if h = o(n -1/2m+5 ).

Let us introduce some notation before stating the next result.

σ 2 so,m = ∞ k=2 k! 1 k! +∞ -∞ H k (y)φ(y)G(y) dy 2 [ 2 -2 θ k m (ω) dω] with θ m (ω) = ∞ -∞ ϕ (m) (y) ϕ (m) (y + ω) dy ||ϕ (m) || -2 2 , σ 2 o,m = 1 0 ∞ k=2 k! 1 k! +∞ -∞ H k (y)φ(y)G(y + f m (u)) dy 2 [ 2 -2 θ k m (ω) dω] du, ζ so (u) = [ +∞ -∞ -z G(z) φ(z) dz] ṡ(u) 2 s(u)||ϕ|| 2 s(u) , ζ o (u) = +∞ -∞ G(y) φ(y -f 0 (u)) 1 η 2 (u) (y -f 0 (u)) 2 ḟ0 (u)η(u) -ḟ0 (u)η(u) -(y -f 0 (u)) η(u) dy where η 2 (u) = ||ϕ|| 2 2 s(u) (15)
(see [START_REF] Kac | On the average number of real roots of a random algebraic equation[END_REF] for the definition of f m ). We also define the following integral operator

Ka, a = 1 0 1 0 F (u ∧ v)(1 -F (u ∨ v))a(u)a(v) du dv (16)
Theorem 3 Let G be a function verifying [START_REF] Breuer | Central Limit Theorems for non-linear functionals of Gaussian fields[END_REF], under H1, H2, if s ∈ C (m+3) [0, 1], ϕ is an even function and ln(n) √ nh → 0 as h → 0, then

I (m) n D → N (0, σ 2 G,m ) where if m = 0 σ 2 G,m = σ 2 so,m
in the sub-optimal case σ 2 o,m in the optimal case and if m = 0

σ 2 G,m = σ 2 so,m + Kζ so , ζ so in the sub-optimal case σ 2 o,m + Kζ o , ζ o in the optimal case
Remark 1: For m = 0 and G(u) = |u| p the result was proved by Csörgő & Horváth in [START_REF] Horv Áth | Central limit theorems for L p -norms of density estimators[END_REF]. Note that, if G is even, ζ so (u) = 0 and then Kζ so , ζ so = 0.

Remark 2: In the sub-optimal case, if m = 0 and G is even we can drop the hypothesis that s ∈ C m+3 ∈ [0, 1] and use the less restrictive condition s ∈ C m+2 ∈ [0, 1]. The same conclusion holds if m = 0. Proof.

I (m) n = 1 √ h 1-h h G h m+ 1 2 β (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -E G h m+ 1 2 β (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du where β (m+1) n,h (u) = 1 h m+2 ∞ -∞ ϕ (m+1) ( u -v h )β F n (v) dv thus h m+ 1 2 β (m+1) n,h (u) = 1 h √ h ∞ -∞ ϕ (m+1) ( u -v h )β F n (v) dv = √ h 1 h ∞ -∞ ϕ (m+1) (v)β F n (u -hv) dv and then h m+ 1 2 β (m+1) n,h
(u) has the same form as √ h βF n,h (u) replacing φ by ϕ (m+1) . We have:

I (m) n = 1 √ h 1-h h G h m+ 1 2 b (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -E G h m+ 1 2 b (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du + 1 √ h 1-h h G h m+ 1 2 β (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -G h m+ 1 2 b (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du + 1 √ h 1-h h E G h m+ 1 2 b (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -E G h m+ 1 2 β (m+1) n,h (u) √ nσ n,m (u)h m+ 1 2 + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du = (I) + (II) + (III)
We need the following lemma, which is similar to Lemma 1.

Lemma 2 For p ≥ 1, E( √ n ||β (m+1) n,h -b (m+1) n,h || ∞ ) p ≤ Const ( ln(n) h m+1 ) p
Proof. The proof is similar to that of Lemma 1, putting ϕ (m+1) instead of φ .

Using the Lipschitz property (6) for G and Lemma 2, we can show that

|(III)| ≤ Const ln(n) √ nh . In the same manner we obtain E[(II) 2 ] ≤ Const ( ln(n) √ nh ) 2 .
We have then proved that asymptotically I (m) n ∼ (I) and also (I)

D = 1 √ h 1-h h G ξ b,(m) h (u) + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -E G ξ b,(m) h (u) + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du = 1 √ h 1-h h G h (u, ξ b,(m) h (u)) du + E n,m
where ξ b,(m) h

(u) = h m+ 1 2 b F,(m+1) h (u) √ nσ n,m (u)h m+ 1 2 b F,(m+1) h (u) = 1 h m+2 ∞ -∞ ϕ (m+1) ( u -v h ) b F (v) dv.
Then

G h (u, y) = G(y + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2
)

-E G N + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 -y E N G N + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 and E n,m = 1 √ h 1-h h ξ b,(m) h (u) E N G N + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1 2 du.
Note that

h m+ 1 2 b F,(m+1) h (u) = √ h 1 h ∞ -∞ ϕ (m+1) (v)b F (u -hv) dv = √ h ḃF h (u)
and then h m+ 12 b F,(m+1) h

(u) is of the same form as √ h ḃF h (u), using ϕ (m) instead of ϕ, and

ξ b,(m) h (u) = ḃF h (u) σb h (u) D = N (0, 1)
Observe that G h (u, y) is not of the form g( √ h σb h (u), y) but as we remarked at the end of section 4, Theorem 1 is still true and

1 √ h 1-h h G h (u, ξ b,(m) h (u)) du → N (0, σ 2 G,m )
where

σ 2 G,m = σ 2 so,m
in the sub-optimal case σ 2 o,m in the optimal case On the other hand

E n,m = 1-h h q n,m (u) h m b F,(m+1) h (u) √ nσ n,m (u)h m+ 1 2 du and q n,m (u) = ∞ -∞ y φ(y) G(y + √ nh m+ 1 2 (s (m) n (u) -s (m) (u)) √ nσ n,m (u)h m+ 1
Integrating by parts we can see that when m = 0, this expression converges a.s. to zero because

h m b F,(m) h (u) a.s. → b F (u) ∞ -∞ ϕ (m) (v) dv = 0 and q n.m (u) √ nσ n,m (u)h m+ 1 2
≤ Const so we only have to consider the case when m = 0 and then

E n,0 = 1-h h q n,0 (u) ḃF h (u) √ nhσ n,0 (u) du with q n,0 (u) = ∞ -∞ y φ(y) G(y + √ nh(s n (u) -s(u)) √ nhσ n,0 (u) ) dy.
Integrating by parts the last expression one obtains that E n,0 converges in law to -

1 0 b F (u)ζ so (u)du in the sub-optimal case (resp. to - 1 0 b F (u)ζ o (u)
du in the optimal one) and the theorem plus the remarks follow.

Kullback Deviation

Consider the "modified" Kullback deviation between the kernel estimator of the density ŝn and the true density s verifying the conditions of section 3 and H2. The deviation is 'modified' since we consider the integration over the interval [h, 1 -h] for h > 0 and not over [0, 1]. Let

K(ŝ n , s) = 1-h h ŝn (u) ln( ŝn (u) s(u) ) du.
We suppose in this section that ϕ is even.

Let us show that this definition makes sense under the condition nh 1+a → +∞ as h → 0, for some a > 0.

For u ∈ [h, 1 -h] we can decompose ŝn (u) s(u) as ŝn (u)

s(u) = 1 + 1 √ n βn,h (u) -ḃn,h (u) s(u) + 1 √ n ḃn,h (u) s(u) + s n (u) -s(u) s(u) ,
by Lemma 1 and since s is bounded below we know that

sup{ 1 √ n βn,h (u) -ḃn,h (u) s(u) , u ∈ [h, 1 -h]} ≤ C 1 (ω) ln(n) hn .
On the other hand we have the following inequality

sup{| s n (u) -s(u) s(u) |, u ∈ [h, 1 -h]} ≤ Const h 2 and since ∞ -∞ φ(v) dv = 0 we get ḃn,h (u) = 1 h ∞ -∞ [b F n (u -hv) -b F n (u)] φ(v) dv.
Using the modulus of continuity for the B.B. we finally obtain that

sup{| 1 √ n ḃn,h (u) s(u) |, u ∈ [h, 1 -h]} ≤ C 2 (ω) 1 √ nh 1 2 +δ .
We have then shown that for u

∈ [h, 1 -h], ŝn (u) s(u) = 1 + H n (u) and sup{|H n (u)|, u ∈ [h, 1 -h]} ≤ C(ω) [ ln(n) hn + h 2 + 1 √ nh 1
2 +δ ] which gives the required result if δ is small enough.

Let us now introduce some notation before proving the next theorem. Let

w 2 k = ||ϕ|| 4 2 2 [ 2 -2 θ 2 (ω) dω], c(u) = s"(u) s(u) and ã(u) = 1 2 C 0 K 2 c(u)
(' denotes the derivative)

Theorem 4 Under H1, H2 and the assumptions, nh 2+a → +∞ as h → 0, for some a > 0, and ϕ is an even function,

n √ h K(ŝ n , s) - 1 2 1-h h 1 s(u) E[ŝn(u) -s(u)] 2 du - 1-h h (ŝ n (u) -s(u)) du D → N (0, σ 2 2 )
where

σ 2 2 = w 2 k
in the sub-optimal case w 2 k + Kã, ã in the optimal case (see [START_REF] Silverman | Weak and strong uniform consistency of the kernel estimate of a density and its derivatives[END_REF] for the definition of Kã, ã )

Remark: The term 1-h h (ŝ n (u) -s(u)
) du tends to zero more slowly than the required normalization n √ h so it cannot be dropped. Proof.

K(ŝ n , s) = 1-h h ŝn (u) ln(1 + ŝn (u) -s(u) s(u) ) du
It is easy to show, using a Taylor expansion of order three for the logarithm, that

K(ŝ n , s) = 1 2 1-h h (ŝ n (u) -s(u)) 2 s(u) du + 1-h h (ŝ n (u) -s(u)) du + - 1 2 1-h h (ŝ n (u) -s(u)) 3 s 2 (u) du + 1 3 1-h h ŝn (u) ŝn (u) -s(u) s(u) 3 1 (1 + θ(u, n, ω)) 3 du
where θ(u, n, ω) is a point between 0 and ŝn (u) -s(u) s(u) , and then

n √ h K(ŝ n , s) - 1 2 1-h h 1 s(u) E[ŝn(u) -s(u)] 2 du - 1-h h (ŝ n (u) -s(u)) du (17) = 1 2 √ h 1-h h 1 s(u) √ nh(ŝ n (u) -s(u)) 2 -E( √ nh(ŝ n (u) -s(u)) 2 ) du - n √ h 2 1-h h (ŝ n (u) -s(u)) 3 s 2 (u) du + n √ h 3 1-h h ŝn (u) ŝn (u) -s(u) s(u) 3 1 (1 + θ(u, n, ω)) 3 du = (I) + (II) + (III).
By the results of the previous section we know that (I) → N (0, σ 2

2 ) in law when n → ∞ (In fact, this is not a straightforward application of Theorem 5 with G(x) = x 2 , because we don't consider here the normalization √ nhσ n,0 (u) and furthermore, we have the term 1 s(u) which does not appear in Theorem 5. However, the idea of the proof is the same and using Mehler's formula (2) the variance is easily computable because of the very simple form of G). Thus to finish it is sufficient to prove that (II) and (III) tend to zero in probability when n → ∞. For this it is enough to prove that E[(II) 2 ] → 0 and that (III) tends to zero a.s., but

E[(II) 2 ] ≤ Const 1 nh 2 1-h h E[ √ nh(ŝ n (u) -s(u))] 6 du and E[ √ nh(ŝ n (u) -s(u))] 6 ≤ Const then E[(II) 2 ] ≤ Const 1
nh 2 and the last term tends to zero because of the hypothesis. Let us look more closely at the last term of [START_REF] Yukich | A note on limit theorems for perturbed empirical processes[END_REF]. We have already seen that

|ŝ n (u) -s(u)| ≤ C 3 (ω) [ ln(n) hn + h 2 + 1 √ nh 1 2 +δ ] ≤ C 3 (ω) a(h) with a(h) → 0 when h → 0, then |θ(u, n, ω)| ≤ C 3 (ω) a(h) and |ŝ n (u)| ≤ d(ω). Finally n √ h |ŝ n (u)| ŝn (u) -s(u) s(u) 3 ≤ C (ω) [ ln 3 (n) n 2 h 5 2 + nh 13 2 + 1 √ nh 1+3δ ]
and then under the hypothesis and for δ small enough, (III) → 0 a.s. Hence the result follows.

We consider also the super-optimal case; following the same arguments as before we get the following Theorem 5 Under H1, H2, if s ∈ C 3 [0, 1], ϕ is an even function, nh 5 → +∞ as h → 0 and nh 8 → 0,then

2 √ n K 2 h 2 K(ŝ n , s) - 1 2 1-h h 1 s(u) E[ŝn(u) -s(u)] 2 du - 1-h h (ŝ n (u) -s(u)) du D → = - 1 0 b F (u)c(u) du ≡ N (0, Kc, c ) (Remember that c(u) = s"(u) s(u)
).

Crossings of the Empirical Process

Denote by N h n (x) the number of crossings before time 1 of the process β n,h at level x. Having in mind the result for the F-B.B. (see Corollary 1) we have the following Theorem 6 Let f be a function in C 2 with f bounded. Under H1, H2 and if

ln(n) √ nh → 0 as h → 0 then V h (f ) = Λ h ∞ -∞ f (x)N h n (x) dx - ∞ -∞ f (x) h n (x) dx → 0 in L 2
where we define,

∞ -∞ f (x) h n (x) dx as the 'modified occupation measure' ∞ -∞ f (x) h n (x)dx = 1 0 f (β n,h (u)) s(u) du (Remember that Λ h = πh 2 ||ϕ|| -1 2 ) Proof. The Banach-Kac [2] [14] formula gives ∞ -∞ f (x) N h n (x) dx = 1 0 f (β n,h (u))| βn,h (u)|du.
To study the behaviour of this r.v. we apply the previous results to the function g

(x) = |x|, to get that if h → 0 √ h 1 0 | βn,h (u)|du → ||ϕ|| 2 2 π 1 0 s(u)du in L 2 and then ∞ -∞ √ h N h n (x) dx → ||ϕ|| 2 2 π ∞ -∞ b (x) dx in L 2 .
We consider, since f is continuous, the following normalization for the integrated number of crossings

Λ h ∞ -∞ f (x)N h n (x)dx = Λ h 1 0 f (β n,h (u))| βn,h (u)|du. Thus E[Vh(f )] 2 = E 1 0 f (β n,h (u))Λ h | βn,h (u)|du - 1 0 f (β n,h (u)) s(u)du 2 ≤ Const E 1 0 [f (β n,h (u)) -f (b n,h (u))]Λ h | βn,h (u)|du 2 + E 1 0 f (b n,h (u))Λ h {| βn,h (u)| -| ḃn,h (u)|}du 2 + E 1 0 f (b n,h (u))(Λ h | ḃn,h (u)| -s(u)) du 2 + E 1 0 (f (b n,h (u)) -f (β n,h (u)) s(u) du 2
Using the Lipschitz property ( 6), Hölder's inequality for f and Lemma 1, the first and fourth term are

O(( ln(n) √ n ) 2 ) and the second term is O(( ln(n) √ nh ) 2 ). The third term is E[ 1 0 f (b F h (u))(Λ h | ḃF h (u)| -s(u)) du] 2 ≤ hE[Z b h (f )] 2 + E[ 1 0 (f (b F (u)) -f (b F h (u))) s(u)) du] 2
where Z b h (f ) was defined in section 3; by Corollary 1, we know that hE[Z b h (f )] 2 = O(h). Moreover, working in the same manner as in that corollary for the second term we get that

E[ 1 0 (f (b F (u)) -f (b F h (u))) s(u)) du] 2 = o(h), thus E[Vh(f )] 2 ≤ Const [( ln(n) √ n ) 2 + ( ln(n) √ nh ) 2 + h + o(h)].
Hence the theorem holds. Remark: Using this result one has the following convergence

Λ h ∞ -∞ f (x)N h n (x) dx D → ∞ -∞ f (x) b (x) dx.
We only sketch the proof: Let be a

h (f ) = Λ h ∞ -∞ f (x)N h n (x) dx, ζ h (f ) = ∞ -∞ f (x) h n (x)dx, Y (f ) = ∞ -∞ f (x) b (x) dx, Y n (f ) = ∞ -∞ f (x) bn (x)
dx and d a distance that metrizes the convergence in law. We have the following inequality

d(a h (f ), Y (f )) ≤ d(a h (f ), ζ h (f )) + d(ζ h (f ), Y (f ))
By the previous result we know that d(a h (f ),

ζ h (f )) → 0. It is easy to prove that ||ζ h (f ) -Y n (f )|| 2 → 0 and since d(ζ h (f ), Y (f )) = d(ζ h (f ), Y n (f )) ≤ ||ζ h (f ) -Y n (f )|| 2 , the remark follows.
Theorem 7 Under the hypothesis of Theorem 6,

ξ n (f ) = 1 √ h ∞ -∞ f (x) Λ h N h n (x) -h n (x)
dx→V stably where V is as in Corollary 1.

Proof. We decompose the expression for ξ n (f ) as:

ξ n (f ) = 1 √ h 1 0 f (β n,h (u))Λ h | βn,h (u)|du - 1 0 f (β n,h (u)) s(u)du = 1 √ h 1 0 [f (β n,h (u)) -f (b n,h (u))]Λ h | βn,h (u)|du + 1 0 f (b n,h (u))Λ h {| βn,h (u)| -| ḃn,h (u)|}du + 1 0 f (b n,h (u))(Λ h | ḃn,h (u)| -s(u)) du + 1 0 (f (b n,h (u) -f (β n,h (u)) s(u) du = (I) + (II) + (III) + (IV )
We have already seen that

E(I) 2 + E(II) 2 + E(IV ) 2 ≤ Const ( ln(n) √ nh ) 2 + ( ln(n) √ nh ) 2
Hence it only remains to consider (III), but

(III) = 1 √ h 1 0 f (b n,h (u))(Λ h | ḃn,h (u)| -s(u)) du D = 1 √ h 1 0 f (b F h (u))(Λ h | ḃF h (u)| -s(u)) du Thus (III) D = 1 √ h 1 0 f (b F h (u))Λ h | ḃF h (u)| du - 1 0 f (b F (u)) s(u) du + 1 √ h 1 0 f (b F (u)) -f (b F h (u)) s(u) du = (III) 1 + (III) 2
In the proof of the last theorem we showed that E[(III2)] 2 = o(1). On the other hand, the term (III) 1 is exactly Z b h (f ) and converges stably to V when h tends to zero. Thus the theorem follows.

Smoothed Empirical Process

We can obtain an extension of the previous result by considering the Smoothed Empirical Process:

βn,h (u) = β n,h (u) + √ n[ Fn (u) -F (u)]
where Fn (u

) = E[ Fn (u)], Fn (u) = u -∞
ŝn (v)dv with F satisfying the conditions of section 3, s verifying H2 and ϕ even. Note that βn,h has a bias. It is obvious that

βn,h (u) = √ n[ Fn (u) -F (u)]
and hence this process is the same as that defined by Yukich in [START_REF] Yukich | A note on limit theorems for perturbed empirical processes[END_REF]. He proved that if √ nh 2 → 0 then βn,h (u) → b F (u) in law. Moreover using Taylor's development for

√ n( Fn (u) -F (u)), he gets that, if √ nh 2 → a, then βn,h (u) D → b F (u) + a 2 K 2 ṡ(u), where K 2 = ∞ -∞ ϕ(v)v 2 dv.
Working as in the previous section we can obtain the convergence results for Ñ h n (x), the number of crossings on [h, 1 -h] for the process βn,h at level x, i.e. when

√ nh 2 → a ≥ 0, Λ h ∞ -∞ f (x) Ñ h n (x) dx D → 1 0 f (b F (u) + a 2 K 2 ṡ(u)) s(u)du.
When √ nh 2 → +∞, using a Taylor development for 2 K2h 2 ( Fn (u) -F (u)) 0 < u < 1, we get that this term tends to ṡ(u). So it is natural to consider c n βn,h (remember that c n = 2 K2 √ nh 2 ) and to study the convergence results for N n,h (x), the number of times in [h, 1 -h] that the process c n βn,h crosses level x. We obtain the following.

Corollary 2 Suppose that f satisfies (6), s ∈ C 2 [0, 1] (instead of H2), H1 holds, ϕ is an even function and

√ nh 2 → +∞ as h → 0 then a)If nh 5 → +∞, ∞ -∞ f (x)N n,h (x)dx P → 1 0 f ( ṡ(u))|s(u)| du = ∞ -∞ f (x)N ( ṡ, x) dx
where N ( ṡ, x) stands for the number of crossings of level x by ṡ in [0, 1]. b) If √ nh 5 → b (b can take the value 0 and if it is the case we will suppose furthermore that H2 holds and s

∈ C 3 [0, 1]), √ nh 5 ∞ -∞ f (x)N n,h (x)dx D → 2 K 2 ||ϕ|| 2 1 0 f ( ṡ(u)) 2 π d 0,b (u) s(u) du.
(See section 3 for the definition of d 0,b (u) and note that d 0,0 (u) = 1).

Proof. The proof consists in applying the KMT theorem and Theorem 1 of [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF].

Let us define the following modification of the occupation measure

∞ -∞ f (x) ˜ h n (x) dx = 1-h h f ( βn,h (u)) s(u) du and ξn (f ) = 1 √ h ∞ -∞ f (x)[Λ h Ñ h n (x) -˜ h n (x)] dx (Remember that Ñ h n (x)
L( Ṽ |F ∞ ) = σ 0 1 0 f (b F (u) + aK 2 2 ṡ(u)) s(u) d W (u)
Proof. We recall that F ∞ is the σ-algebra generated by {b

F (u), 0 ≤ u ≤ 1}. ξn (f ) = 1 √ h [ 1-h h f ( βn,h (u))Λ h | ˙ β n,h (u)| du - 1-h h f ( βn,h (u)) s(u) du] = 1 √ h [ 1-h h f (β n,h (u) + a n (u))Λ h | βn,h (u) + ȧn (u)| du - 1-h h f (β n,h (u) + a n (u)) s(u) du] where a n (u) = √ n[ Fn (u) -F (u)] = √ n ∞ -∞ ϕ(v) [F (u -hv) -F (u)] dv. We decompose ξn (f ) as ξn (f ) = 1 √ h [ 1-h h f (b n,h (u) + a n (u))Λ h | ḃn,h (u) + ȧn (u)| du - 1-h h f (b n,h (u) + a n (u)) s(u) du] + 1 √ h 1-h h [f (β n,h (u) + a n (u)) -f (b n,h (u) + a n (u))]Λ h | βn,h (u) + ȧn (u)| du + 1 √ h 1-h h f (b n,h (u) + a n (u))Λ h [| βn,h (u) + ȧn (u)| -| ḃn,h (u) + ȧn (u)|] du + 1 √ h 1-h h [f (b n,h (u) + a n (u)) -f (β n,h (u) + a n (u))] s(u) du = (I) + (II) + (III) + (IV ).
Consider the last three terms. Using the Lipschitz property (6) for f , Hölder's inequality, Lemma 1 and the fact that, by using a Taylor development for F and s respectively, a n (u) and ȧn (u) are bounded by By an elementary calculation we can see that -a|) in the optimal one. The convergence to zero, in probability, for the term (I) 3 has been proved for the sub-optimal case. Hence we must only consider the optimal case, and then (I) 3 is asymptotically equivalent to

√ nh 2 on [h, 1 -h], it is easy to see that E(II) 2 and E(IV ) 2 are O(( ln(n) √ nh ) 2 ) and E(III) 2 is O(( ln(n) √ nh ) 2 ). ξn (f ) is then equivalent to (I) and (I) D = 1 √ h 1-h h f (b F h (u) + a n (u)){Λ h | ḃF h (u) + ȧn (u)| -s(u)} du = 1 √ h 1-h h f (b F h (u) + a n (u))G h (u,
|a n (u) - a 2 K 2 ṡ(u)| ≤ Const{| √ nh 2 -a| + √ nh 3 },
we thus obtain that (I) 3 is asymptotically equivalent to

1-h h f (b F (u) + a 2 K 2 ṡ(u)) s(u) s(u) √ h ḃF h (u) du.
By Theorem 1 in [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF] this quantity tends to zero in probability when h tends to zero. Thus (I) 3 tends to zero in probability and ξn (f ) ∼ (I) 1 = 1 √ h 1-h h f (b F h (u) + a n (u))G h (u, ξ b h (u)) du. following the same arguments as in the proof of Corollary 1 (see remark 2 at the end of section 4), we finally get that the last term tends stably to the required limit.

We consider now the case where √ nh 2 → +∞ and we get, in the next corollary, the speed at which the convergence takes place. √ h → 0 (q ≥ 0), under H2, s ∈ C 3 [0, 1] and assuming that f is in C 2 with bounded second derivative,

1 √ h ||ϕ|| -1 2 √ nh 5 ∞ -∞ f (x)N n,h (x) dx - 2 K 2 1-h h f (c n βn,h (u)) 2 π d 0,q (u) s(u) du D → 2 K 2 ||ϕ|| -1 2 1 0 f ( ṡ(u)) 2 π d 1,q (u) db F (u) + 2 K 2 2 π 1 0 σ q (u)f ( ṡ(u)) s(u)d W (u)
where W is a B. M. independent of b F .

  dv, with ϕ verifying H1, be the regularized empirical process, and denote by βn,h (u) its derivative. Defining b F n (•) = b n (F (•)), b n,h = b n ϕ h and using the KMT theorem, it is easy to see that

2 N 2 N 2 √ 2 N 2 N 2 N 2 N| π 2 N 2 2 π 2 N

 22222222222 h (u) + a n (u))[k h (u) E | π + p n (u)| -s(u)] du h (u) + a n (u))k h (u)ξ b h (u) E[N | π + p n (u)|] du = (I) 1 + (I) 2 + (I) 3 where k h (u) = √ h σb h (u)||ϕ|| -1 2 , p n (u) = π h ȧn (u) √ h σb h (u)andG h (u, y) = k h (u)[| π 2 y + p n (u)| -E | π + p n (u)| -y E[N | π + p n (u)|].Consider the second term (I) 2 .|k h (u) E | π + p n (u)| -s(u)| ≤ |k h (u)[E | π + p n (u)| -1]| + |k h (u) -s(u)|We have already seen in section 5 that on[h, 1 -h], |k h (u) -s(u)| ≤ Const h. Furthermore, a simple calculation shows that [E + p n (u)| -1] = π 2 [E |N + d n (u)| -{d n (u)[Φ(0) -Φ(-d n (u))] + [φ(-d n (u)) -φ(0)]} where d n (u) = p n (u),φ and Φ are the standard Gaussian density and distribution function respectively. So we get that | E | π 2 N + p n (u)| -1| ≤ Const h and therefore |I 2 | ≤ Const √ h 1-h h |f (b F h (u) + a n (u))| du. We thus obtain that E[(I)2] 2 ≤ Const h. Consider now the third term. F h (u) + a n (u)) ḃF h (u) E[N | π + p n (u)|] du.

Corollary 4 f 1 0f

 41 Assume that f verifies (6), s ∈ C 3 [0, 1], H1 holds, ϕ is an even function and nh 4 → +∞ as h → 0, then a) If nh 6 → +∞ and |s| is bounded below on [0, 1], then (x)N n,h (x) dx -1-h h f (c n βn,h (u))|c n √ n( Ḟn (u) -Ḟ (u))| du D → ( ṡ(u))sign(s(u))db F (u) b) If nh 5 -q 2

  Using the Lipschitz property (6) for f and the fact that sup0≤u≤1 ||b F h (u) -b F (u)|| 2 = O(h) and

	1-h h	1 ||ϕ|| 2	f (b F h (u) + a n (u))	aK 2 2||ϕ|| 2	s(u) s(u)	√	h ḃF
	sup
	h≤u≤1-h

h (u) du.

) dy.

Remark: In the special case where q = 0, d 1,0 (u) = 0 and the first term of the limit vanishes. Proof. The proof consists, as before, in an application of the KMT theorem, Theorem 1 of [START_REF] Le | Non-linear functionals of the brownian bridge and their applications[END_REF], Theorem 1 and arguments similar to those used to prove Corollary 1.