Martin Bordemann
email: martin.bordemann@uha.fr

Friedrich Wagemann
email: friedrich.wagemann@univ-nantes.fr

Global integration of Leibniz algebras

In this paper we present a global integration procedure of any real finite-dimensional Leibniz algebra into a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.

Introduction

All manifolds considered in this manuscript are assumed to be Hausdorff, finite dimensional and second countable.

A pointed rack is a pointed set (X, e) together with a binary operation ✄ : X × X → X such that for all x ∈ X, the map y → x ✄ y is bijective and such that for all x, y, z ∈ X, the self-distributivity and unit relations

x ✄ (y ✄ z) = (x ✄ y) ✄ (x ✄ z), e ⊲ x = x, and x ⊲ e = e are satisfied. All our racks will be left racks. Imitating the notion of a Lie group, the smooth version of a pointed rack is called Lie rack. One main example of a Lie rack is the conjugation Lie rack underlying a Lie group G, i.e. X = G and x ✄ y = xyx -1 .

An important class of examples of racks are the so-called augmented racks, see [START_REF] Fenn | Racks and links in codimension 2[END_REF]. An augmented rack is the data of a group G, a G-set X and a map p : X → G such that for all x ∈ X and all g ∈ G, p(g • x) = gp(x)g -1 .

The set X becomes then a rack by setting x ✄ y := p(x) • y.

Lie racks are intimately related to Leibniz algebras h. Recall that a (left) Leibniz algebra is a vector space h with a bilinear bracket [, ] : h ⊗ h → h such that for all X, Y, Z ∈ h, [X, -] acts as a derivation:

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]].
Indeed, Kinyon showed in [START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF] that the tangent space at e ∈ H of a Lie rack H carries a natural structure of a Leibniz algebra, generalizing the relation between a Lie group and its tangent Lie algebra. Conversely, every (finitedimensional real or complex) Leibniz algebra h may be integrated into a Lie rack (with underlying manifold h) using the rack product

X ✄ Y := e ad X (Y ), (1.1)
noting that the exponential of the inner derivation ad X for each X ∈ h is an automorphism. Although the assignment h, [ , ] → h, 0, ⊲ is functorial since morphisms of Leibniz algebras are easily seen to go to morphisms of pointed Lie racks, the restriction to the category of all Lie algebras would not give the usual integration as a Lie group. The purpose of the present paper is to give an integration procedure which integrates real, finite-dimensional Leibniz algebras into Lie racks in such a way that the restriction to Lie algebras gives the conjugation rack underlying the simply connected Lie group corresponding to a (real, finite-dimensional) Lie algebra.

This problem has been encountered by J.-L. Loday in 1993 [START_REF] Loday | Jean-Louis Une version non commutative des algèbres de Lie: les algèbres de Leibniz[END_REF] in the search of quantifying the lack of periodicity in algebraic K-Theory. Several attempts and constructions have been published since then. In 2010, Simon Covez [START_REF] Covez | Simon The local integration of Leibniz algebras[END_REF] solves in his thesis the local integration problem by constructing a local Lie rack integrating a given (real, finite dimensional) Leibniz algebra in such a way that in the case of Lie algebras, one obtains the conjugation Lie rack underlying the usual (simply connected) Lie group integrating it.

In this article, we construct in Section 3 a (global) Lie rack integrating a given (real, finite-dimensional) Leibniz algebra h in such a way that in the case of a Lie algebra, the construction yields the conjugation rack underlying the usual (simply connected) Lie group integrating it. More precisely, we work with augmented Leibniz algebras, i.e. Leibniz algebras h with an action of a Lie algebra g by derivations and an equivariant map p : h → g to g. We integrate the quotient Lie algebra p(h) =: g ′ into a Lie group G ′ and integrate its action on h such that the resulting (global) augmented Lie rack is an affine bundle over G ′ with typical fiber z := Ker(p). This is the content of our main theorem, Theorem 3.1.

The construction relies essentially on the existence of an open neighbourhood in the Lie group G ′ on which the exponential is a diffeomorphism and which is invariant under (the connected component of the identity) Aut 0 (G ′ ). It is well known that such a neighbourhood exists, see e.g. [START_REF] Hofmann | Commuting exponentials in a Lie group[END_REF]. On the other hand, it is the use of this neighbourhood and of the associated bump function which renders our construction non-functorial. For the moment, we do not know whether there exists a functorial construction of a Lie rack integrating a given Leibniz algebra (such that in the special case of a Lie algebra, we get back the conjugation rack underlying the usual simply connected Lie group).

2 Augmented Leibniz algebras and Lie racks 2.1 (Augmented) Leibniz algebras All modules will be considered over R and all unadorned tensor products will always be over R.

Recall that a (real) Leibniz algebra is an R-vector space h equipped with a linear map [ , ] : h ⊗ h → h, written x ⊗ y → [x, y] such that the left Leibniz identity holds for all x, y, z ∈ h

x, [y, z] = [x, y], z + y, [x, z] (2.1)
A morphism of Leibniz algebras Φ : h → h ′ is a R-linear map preserving brackets, i.e. for all x, y ∈ h we have Φ [x, y] = Φ(x), Φ(y) ′ . Recall first that each (real) Lie algebra is a (real) Leibniz algebra giving rise to a functor i from the category of all Lie algebras, RLieAlg, to the category of all Leibniz algebras, RLeib.

Furthermore, recall that each Leibniz algebra has two canonical subspaces

Q(h) := x ∈ h | ∃ N ∈ N \ {0}, ∃ λ 1 , . . . , λ N ∈ R, ∃ x 1 , . . . , x N such that x = N r=1 λ r [x r , x r ] , (2.2)
z(h) := x ∈ h | ∀ y ∈ h : [x, y] = 0 . (2.
3)

It is well-known and not hard to deduce from the Leibniz identity that both Q(h) and z(h) are two-sided abelian ideals of (h, [ , ]), that Q(h) ⊂ z(h), and that the quotient Leibniz algebras

h := h/Q(h) and h/z(h) (2.4)
are Lie algebras. Since the ideal Q(h) is clearly mapped into the ideal Q(h ′ ) by any morphism of Leibniz algebras h → h ′ (which is a priori not the case for z(h) !), there is an obvious functor h → h from the category of all Leibniz algebras to the category of all Lie algebras. It is not hard to see and not important for the sequel that the functor h → h is a left adjoint functor of the inclusion functor of the category of all Lie algebras in the category of all Leibniz algebras whence the former is a reflective subcategory of the latter, see e.g. [10, p.91] for definitions.

It is easy to observe that in both cases of the above Lie algebras, h and h/z(h), there is the following structure: Definition 2.1 A quintuple h, p, g, [ , ] g , ρ is called a g-augmented Leibniz algebra iff the following holds:

1. g, [ , ] g is a (real) Lie algebra.

2. h is an R-vector space which is a left g-module via the R-linear map ρ : g ⊗ h → h written ρξ (x) = ξ.x for all ξ ∈ g and x ∈ h.

3. p : h → g is a R-linear morphism of g-modules, i.e. for all ξ ∈ g and x ∈ h p(ξ.x) = ξ, p(x) g .

(2.5)

A morphism of augmented Leibniz algebras h, p, g, [ , ] g , ρ → h ′ , p ′ , g ′ , [ , ] ′ g , ρ′ is a pair (Φ, φ) of R-linear maps where φ : g → g ′ is a morphism of Lie al- gebras, Φ : h → h is a morphism of Lie algebra modules over φ, i.e. for all x ∈ h and ξ ∈ g Φ(ξ.x) = φ(ξ).Φ(x), (2.6
)

and p ′ • Φ = φ • p. (2.7)
The following properties are immediate from the definitions:

Proposition 2.1 Let h, p, g, [ , ]
g , ρ be an augmented Leibniz algebra. Define the following bracket on h:

[x, y] h := p(x).y.

(2.8)

h, [, ]

h is a Leibniz algebra on which g acts by derivations. If (Φ, φ) is a morphism of augmented Leibniz algebras, then Φ is a morphism of Leibniz algebras.

2. The kernel of p, Ker(p), is a g-invariant two-sided abelian ideal of h satisfying Q(h) ⊂ Ker(p) ⊂ z(h).

3. The image of p, Im(p), is an ideal of the Lie algebra g.

Proof:

We just check the Leibniz identity: Let x, y, z ∈ h, then, writing

[ , ] h = [ , ],
x, [y, z] = p(x). p(y).z

= p(x). p(y).z -p(y). p(x).z + p(y). p(x).z

= p(x), p(y) g .z + y, [x, z] (2.5) = p p(x).y .z + y, [x, z] = [x, y], z + y, [x, z] .

✷

It follows that the class of all augmented Leibniz algebras forms a category RLeibA, and there is an obvious forgetful functor from RLeibA to RLeib associating to h, p, g, [ , ] g , ρ the Leibniz algebra h, [ , ] h where the Leibniz bracket [ , ] h is defined in equation (2.8).

On the other hand, there is a functor from RLeib to RLeibA associating to each Leibniz algebra h, [ , ] the augmented Leibniz algebra h, p, h, [ , ]h, ad ′ , where p : h → h is the canonical projection and the representation ad ′ of the Lie algebra h on the Leibniz algebra h is defined by (for all x, y ∈ h)

ad ′ p(x) (y) := ad x (y) = [x, y].
(2.9)

(Augmented) Lie racks

Recall that a pointed manifold is a pair (M, e) where M is a differentiable manifold and e is a fixed element of M. Morphisms of pointed manifolds are base point preserving smooth maps.

A Lie rack is a pointed manifold (M, e) equipped with a smooth map m : M × M → M of pointed manifolds (i.e. m(e, e) = e) such that m(x, -) : M → M is a diffeomorphism for all x ∈ M and satisfying the following identities for all x, y, z ∈ M (where the standard notation is m(x, y) = x ⊲ y) e ⊲ x = x, (2.10)

x ⊲ e = e, (2.11) x ⊲ (y ⊲ z) = (x ⊲ y) ⊲ (x ⊲ z) (2.
12)

The last condition (2.12) is called the self distributivity condition. A morphism of Lie racks φ : (M, e, m) → (M ′ , e ′ , m ′ ) is a map of pointed manifolds satisfying for all x, y ∈ M the condition φ(x ⊲ y) = φ(x) ⊲ ′ φ(y).

M f (x ⊲ y) = x ⊲ f (y).
A straight-forward computation shows that the pointed manifold (M, e) equipped with the gauged multiplication ⊲ f defined by 2. Let φ : (M, e, m) → (M ′ , e ′ , m ′ ) be a morphism of Lie racks. Then T e φ : h → h ′ is a morphism of Leibniz algebras.

x ⊲ f y := f (x) ⊲
Proof: 1. see the proof of Theorem 3.4 in [START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF]. 2. Since φ maps e to e ′ its tangent map T e φ maps T e M to T e ′ M ′ . We get for all x, y ∈ h = T e M where t → a(t) is a smooth curve in M with a(0) = e and (da/dt)(0) = x:

T e φ [x, y] = T e φ ∂ ∂t T e L a(t) (y) t=0 = ∂ ∂t T e φ • L a(t) (y) t=0 = ∂ ∂t T e L ′ φ(a(t)) • φ (y) t=0 = ∂ ∂t T e ′ L ′ φ(a(t)) t=0
T e φ(y)

= T e φ(x), T e φ(y) . Furthermore, recall that an augmented Lie rack (see [START_REF] Fenn | Racks and links in codimension 2[END_REF]) (M, φ, G, ℓ) consists of a pointed differentiable manifold (M, e M ), of a Lie group G, of a smooth map φ : M → G (of pointed manifolds), and of a smooth left G-

action ℓ : G × M → M (written (g, x) → ℓ(g, x) = ℓ g (x) = gx) such that for all g ∈ G, x ∈ M ge M = e M , (2.16) φ(gx) = gφ(x)g -1 .
(2.17)

It
M φ(x ⊲ y) = φ(x)φ(y)φ(x) -1 . (2.19) A morphism (Ψ, ψ) : (M, φ, G, ℓ) → (M ′ , φ ′ , G ′ , ℓ ′ ) of augmented Lie racks is a pair of maps of pointed differentiable manifolds Ψ : M → M ′ and ψ : G → G ′ such that ψ is homomorphism of Lie groups and such that for all g ∈ G φ ′ • Ψ = ψ • φ, (2.20) Ψ • ℓ g = ℓ ′ ψ(g) • Ψ. (2.21)
The class of all augmented Lie racks thus forms a category LieRackA with the obvious forgetful functor F : LieRackA → LieRack. Note that the trivial Lie rack structure of a pointed manifold (M, e) comes from an augmented Lie rack over the trivial Lie group G = {e}.

3 Global integration of Leibniz algebras

The main theorem

Instead of integrating a Leibniz algebra h, we will rather integrate an augmented Leibniz algebra. Note that this does not restrict generality, as any Leibniz algebra h gives rise to (several) augmented Leibniz algebras. Let h, p, g, [ , ] g , ρ be an augmented Leibniz algebra, and denote Im(p) =: g ′ , Ker(p) =: z and by G resp. G ′ the connected, simply connected Lie groups associated to g and g ′ .

One important ingredient in our main theorem is a smooth pointed equivariant map s : G ′ → g ′ whose tangent map at e G ′ ∈ G ′ is the identity. Such maps do exist: Lemma 3.1 There is a smooth map s : G ′ → g ′ having the following properties:

s(e G ′ ) = 0, (3.1) T e G ′ s = id g ′ , (3.2) ∀ g ∈ G, ∀ g ′ ∈ G ′ : s I ′ g (g ′ ) = A ′ g s(g ′ ) , (3.3)
i.e. the equivariance of s w.r.t. the conjugation action of G on G ′ and the adjoint action of G on g ′ (see Section 3.2).

With these preparations, the main theorem of our article reads:

Theorem 3.1 Consider p : h → g ′ as an affine bundle over g ′ with typical fibre z, and form the pull-back fibre bundle

M := s * h (3.4)
over G ′ w.r.t. any smooth pointed equivariant map s : G ′ → g ′ whose tangent map at e G ′ ∈ G ′ is the identity. Then there is a canonical G-action ℓ on M such that (M, (0, e G ′ ), ι• φ, G, ℓ is an augmented Lie rack with the following properties:

(a) The induced Leibniz algebra structure on the tangent space T (0,e G ′ ) M is isomorphic to h, [ , ] h .

(b) In the particular case g = g ′ and z = {0}, the above construction reduces to the usual conjugation Lie rack on G = G ′ .

Remark 3.2 Let us comment on the relation of our theorem to the local integration of Leibniz algebras in [START_REF] Covez | Simon The local integration of Leibniz algebras[END_REF]. The starting point in Covez' article is to view a Leibniz algebra h as an abelian extension (in the category of Leibniz algebras) of the Lie algebra h / z(h) by the antisymmetric module z(h). The Lie algebra integrates into a connected, simply connected Lie group G and therefore the action integrates also, thus the main task is to integrate the Leibniz 2-cocycle characterizing this extension. The main idea is then to see the 2-cocycle as a 1-cocycle with values in 1-cocycles, and to integrate it in two steps. This is done using a neighborhood of 1 in the Lie group G where the exponential is a diffeomorphism (by van-Est's method using paths in G).

The integration procedure for cocycles works only locally, and the outcome is thus a local Lie rack. Let us emphasize that our method is completely different from Covez' method. In contrast to Covez' integration procedure, the main point is here to integrate augmented objects into augmented objects. This permits to phrase all objects in terms of Lie objects and representations which integrate by the usual Lie Theorems, and thus gets rid of the integration of the cocycle (which is done implicitely). It should be possible to recover the formula for the integration of the cocycle from chart changes in the affine bundle.

Proof of the main theorem and Lemma 3.1

As the main assertion of the theorem concerns a G-action on M, let us prepare the proof by a careful analysis of the actions which come into play.

For the augmented Leibniz algebra h, p, g, [ , ] g , ρ we denoted by g ′ the Lie ideal p(h) of g, and let z := Ker(p) which -we recall-is a two-sided ideal of the Leibniz algebra h lying in the left centre of h. Let furthermore G (resp. G ′ ) be a connected simply connected Lie group whose Lie algebra is isomorphic to g (resp. g ′ ). Since G is connected and simply connected, its adjoint represention Ad G preserves the ideal g ′ of its Lie algebra g, whence there is a Lie group homomorphism g → A ′ g of G into Aut 0 (g ′ ), the component of the identity of the Lie group of all automorphisms of the Lie algebra g ′ . Since this latter Lie group is well known to be isomorphic to Aut 0 (G ′ ), the connected component of the identity of the topological group of all Lie group automorphisms of G ′ (which also is a Lie group), see e.g. [START_REF] Hochschild | The Automorphism Group of a Lie Group[END_REF] for details, there is a unique Lie group homomorphism

I ′ : G → Aut 0 (G ′ ) : g → g ′ → I ′ g (g ′ ) such that T e G I ′ g = A ′
g for all g ∈ G. Moreover, the injection g ′ → g induces a unique immersive Lie group homomorphism ι : G ′ → G whose image is an analytic normal subgroup of G whence G acts on ι(G ′ ) by conjugations, and we have for all g ∈ G and all

g ′ ∈ G ′ (ι • I ′ g )(g ′ ) = gι(g ′ )g -1 .
Next, let ρ : G × h → h be the unique representation of G on h such that for all ξ ∈ g and x ∈ h

d dt ρ exp(tξ) (x) t=0 = ρξ (x). (3.5)
We get for all g ∈ G and x ∈ h:

p ρ g (x) = A ′ g p(x) . (3.6)
Proof of Theorem 3.1:

The pull-back fibre bundle

M := s * h = (x, g ′ ) ∈ h × G ′ | p(x) = s(g ′ ) φ → G ′ (3.7)
over G ′ has (0, e G ′ ) as its distinguished point. Since p : h → g ′ is a surjective linear map, it is a surjective submersion whose fibre over 0 ∈ g ′ is equal to Ker(p) = z, and whose fibre over any ζ ∈ g ′ is the affine subspace p -1 ({ζ}) of h. Choosing any vector space complement b to z in h leads to differential geometric trivialization over the global chart domain g ′ of g ′ . Hence p : h → g ′ is a fibre bundle over g ′ with typical fibre z, and therefore the pull-back s * h = M is a well-defined fibre bundle over G ′ with typical fibre z. Recall that the projection φ : M → G ′ is given by the restriction of the projection on the second factor

h × G ′ → G ′ to the submanifold M ⊂ h × G ′ . Since s(e G ′ ) = 0 = p(0) it follows that the point (0, e G ′ ) is in M, and clearly φ(0, e G ′ ) = e G ′ .
There is a canonical diagonal G-action l on h × G ′ defined by lg (x, g ′ ) = ρ g (x), I ′ g (g ′ ) . For any (x, g ′ ) ∈ M, we have by definition p(x) = s(g ′ ). Therefore we get (using also the equivariance of s and p)

s I ′ g (g ′ ) = A ′ g s(g ′ ) = A ′ g p(x) = p ρ g (x) .
This proves that for any (x, g ′ ) ∈ M, lg (x, g ′ ) ∈ M. Thus l restricts to a well-defined G-action ℓ on M. Clearly ℓ g (0, e G ′ ) = (0, e G ′ ). Moreover, for any (x, g ′ ) ∈ M and g ∈ G we have

(ι • φ) ℓ g (x, g ′ ) = ι φ ρ g (x), I ′ g (g ′ ) = ι I ′ g (g ′ ) = gι(g ′ )g -1 = g (ι • φ)(x, g ′ ) g -1 ,
showing that (M, (0, e G ′ ), ι • φ, G, ℓ is an augmented Lie rack.

Property (a): According to the definition of the pull-back, the tangent space of M at (e G ′ , 0) is given by all pairs

(x, ζ) ∈ h × g ′ = T 0 h × T e G ′ G ′ such that T e G ′ s(ζ) = p(x), hence ζ = p(x) because of T e G ′ s = id g ′ . It follows that the linear map θ h : h → h × g ′ given by θ h (x) = x, p(x)
is an isomorphism of the vector space h onto the tangent space T (e G ′ ,0) M.

We get for all g ∈ G and for all y ∈ h

T (e G ′ ,0) ℓ g y, p(y) = d dt ρ g (ty), I ′ g exp tp(y) t=0 = ρ g (y), A ′ g p(y) . p(y)

t=0 = [x, y] h , p(x), p(y) g ′ = [x, y] h , p [x, y] h = θ h [x, y] h .
showing that the induced Leibniz structure from the augmented Lie rack is isomorphic with the Leibniz bracket [ , ] h on h. Property (b) is immediate.

Proof of Lemma 3.1:

For any strictly positive real number τ , let S τ i be the open strip

S τ i := {z = α + iβ ∈ C | |β| < τ }. (3.8)
Furthermore, we set

U τ i := {ξ ∈ g ′ | all the eigenvalues of ad ξ lie in S τ i }, (3.9)
and recall for any ξ ∈ g ′ the definition of its adjoint representation ad ξ : η → [ξ, η] for all η ∈ g ′ . Furthermore, set

V τ i := exp U τ i ⊂ G ′ . (3.10)
We now claim the following assertions:

1. The subset U τ i is an open Aut 0 (g ′ )-invariant neighbourhood of 0 ∈ g ′ such that for all ξ ∈ U τ i and for all η in the nilradical of g ′ , the element ξ + η still lies in U τ i .

The restriction of the exponential map to

U τ i is a diffeomorphism onto V τ i which is an open Aut 0 (G ′ )-invariant neighbourhood of the unit element e G ′ ∈ G ′ .
3. Let τ ′ be any real number such that 0 < τ ′ < τ ≤ π. Then there is a smooth Aut 0 (g ′ )-invariant real-valued function γ :

g ′ → R such that (a) γ(g ′ ) ⊂ [0, 1], (b) For all ξ ∈ U τ ′ i , we have γ(ξ) = 1, (c) The support of γ is contained in U τ i .
The first and second parts are well-known and can be recovered from similar assertions like e.g. Lemma 2.5 in [START_REF] Hofmann | Commuting exponentials in a Lie group[END_REF] (see also the remarks on the bottom of p. 325 loc. cit.). The main idea is the use of the map χ :

g ′ → C[λ] 1 n ∼ = C n , where C[λ] 1
n denotes the space of unitary degree n polynomials in λ with complex coefficients, which is defined for all ξ ∈ g ′ by χ(ξ)(λ) := χ ad ξ (λ)

where we have written χ for the characteristic polynomial. Thus the set

U τ i = χ-1 T (S ×n τ i ) is an open Aut 0 (g ′ )-invariant subset of g ′ . Here T : C n → C[λ]
) → (λ -z 1 ) . . . (λ -z n ) which projects onto a (well-known, but not unsurprising) homeomorphism T : C n / S n → C[λ] 1
n . The invariance of χ under translation by elements ξ in the nilradical stems from the fact that sufficient powers of ad ξ are zero and thus their traces do not contribute to χ. In the second part of the statement, one uses the formula for the derivative of the exponential function and the Inverse Function Theorem. Injectivity can be shown using a beautiful argument of Lazard-Tits [8, Section 2.3]. The reasoning for the last part is also standard.

By the above Assertions 1. and 2., there are two open neighbourhoods U ′ (π/2)i ⊂ U ′ πi of 0 ∈ g ′ which are both Aut 0 (g ′ )-invariant and on which the restriction of the exponential map exp G ′ is a diffeomorphism onto the Aut 0 (G ′ )-invariant open neighbourhoods V ′ (π/2)i ⊂ V ′ πi of the unit element e G ′ of G ′ . Moreover, by Assertion 3., there is an Aut 0 (g ′ )-invariant bump function γ ′ : g ′ → [0, 1] whose support is in U ′ πi and which is equal to 1 on U ′ (π/2)i . Let us define the following map s :

G ′ → g ′ by s(g ′ ) := γ ′ exp -1 G ′ (g ′ ) exp -1 G ′ (g ′ ) ∀ g ′ ∈ V ′ πi 0 ∀ g ′ ∈ V ′ πi (3.11)
It is clear that s is a well-defined smooth map G ′ → g ′ . Moreover s(e G ′ ) = 0 by the properties of the exponential map, and for all

ζ ∈ g ′ = T e G ′ G ′ T e G ′ s(ζ) = d ds γ ′ exp -1 G ′ (exp G ′ (sζ)) exp -1 G ′ (exp G ′ (sζ)) | s=0 = d ds (sζ) | s=0 = ζ
because the bump function γ ′ is constant equal to 1 near 0. Hence T e G ′ s = id g ′ . Finally, since for each g ∈ G and ζ ∈ g ′

I ′ g exp G ′ (ζ) = exp G ′ A ′ g (ζ)
and since γ ′ is invariant under the action g → A ′ g of G on g ′ we get for all g ∈ G and g ′ ∈ G ′ s I ′ g (g ′ ) = A ′ g s(g ′ ) , proving the lemma.

A counterexample

Let us conclude with an example showing that the neighborhoods U τ i do not depend functorially on the Lie algebra g. Namely, given a strictly positive real number τ , we give an example of a homomorphism of Lie algebras f : g → g ′ such that F : G → G ′ does not send the neighborhood V τ i ⊂ G to the neighborhood V ′ τ i ⊂ G ′ where G and G ′ are the simply connected Lie groups corresponding to g and g ′ respectively, F is the Lie group homomorphism existing by Lie's Second Theorem and

V τ i := exp U τ i ⊂ G.
In order to construct such an example, observe that for the abelian Lie algebra g = R n , we have that G = R n is the additive group and U τ i = g for any strictly positive real τ . Furthermore, the exponential map is the identity of R n and V τ i = G is equally true for all strictly positive real τ . Thus for abelian Lie group/Lie algebra, the neighborhoods are in some sense maximal.

On the other hand, consider g ′ := su(2, C), the Lie algebra of skewhermitian matrices of trace zero. It is well-known (see e.g. [START_REF] Duistermaat | Universitext[END_REF] The eigenvalues of the infinitesimal generator x of this rotation matrix are 0 and ±iα. Thus x is an element of the neighborhood U ′ τ i if and only if α ≤ τ . Now let us construct the counter-example: Let τ > 0 be fixed in advance. Let n = 1 and f : R → su(2, C) be given by sending 1 ∈ R to some x ∈ su(2, C) such that the above angle α is strictly bigger than τ . The homomorphism f integrates to a 1-parameter subgroup in SU(2, C), i.e. a Lie group homomorphism F : R → SU(2, C). By construction, F does not send V τ i = R into V ′ τ i .

✷

  Let RLeib f d denote the category of all finite-dimensional real Leibniz algebras. The preceding proposition shows that there is a functor T * R : LieRack → RLeib f d which associates to any Lie rack (M, e, ⊲) its tangent space T * R(M) := T e M at the distinguished point e ∈ M equipped with the Leibniz bracket equation (2.15).

  θ h (x), θ h (y) = (x, p(x) , y, p(y)

1 ) 1 -

 11 Section 1.2B) that the adjoint representation of SU(2, C) on su(2, C) is given by rotations, namely for a, b ∈ C, a matrix a b b ā acts by rotation with angle α := arccos(2(Re a) 2 -(Re a) 2 (Im a) 2 + (Re b) 2 + (Im b) 2 .

  y is a Lie rack (M, e, ⊲ f ). We refer for more exotic examples to[START_REF] Fenn | Racks and links in codimension 2[END_REF]. The following relation to Leibniz algebras is due to M. Kinyon[START_REF] Kinyon | Leibniz algebras, Lie racks, and digroups[END_REF]:

	Proposition

2.2 Let (M, e, m) be a Lie rack and h = T e M. Define the following bracket [ , ] on h by [x, y] = ∂ ∂t T e L a(t) (y) t=0 (2.15) where t → a(t) is any smooth curve defined on an open real interval containing 0 satisfying a(0) = e and (da/dt)(0) = x ∈ h. Then we have the following 1. h, [ , ] is a real Leibniz algebra.

  is a routine check that the multiplication ⊲ on M defined for all x, y ∈ M

	by
	x ⊲ y := ℓ φ(x) (y)	(2.18)
	satisfies all the axioms (2.10), (2.11), and (2.12) of a Lie rack, thus making
	(M, e

M , ⊲) into a Lie rack such that the map φ is a morphism of Lie racks, i.e. for all x, y ∈

  1 n is the continuous, closed and open map between C n and C[λ] 1 n given by (z 1 , . . . , z n

Acknowledgements

FW thanks Université de Haute Alsace (Mulhouse, France) for several research visits where the subject of this article has been discussed. The authors would like to thank the referee for bringing to their attention reference [6] and the many useful remarks of which we benefited greatly.


Davide Giraudo

Alfredas Račkauskas

Weak invariance principle in Besov spaces for stationary martingale differences

Keywords: invariance principle, martingale differences, stationary sequences, Besov spaces AMS MSC 2010: 60F17, 60G10, 60G42

The classical Donsker weak invariance principle is extended to a Besov spaces framework. Polygonal line processes build from partial sums of stationary martingale differences as well independent and identically distributed random variables are considered. The results obtained are shown to be optimal.

Introduction and main results

By weak invariance principle in a topological function space, say, E we understand the weak convergence of a sequence of probability measures induced on E by normalized polygonal line processes build from partial sums of random variables. The choice of the space E is important due to possible statistical applications via continuous mappings. Since stronger topology generates more continuous functionals, it is beneficial to have the weak invariance principle proved in as strong as possible topological framework. Classical Donsker's weak invariance principle considers the space E = C[0, 1] and polygonal line processes build from partial sums of i.i.d. centred random variables with finite second moment. An intensive research has been done in order to extend Donsker's result to a stronger topological framework as well to a larger class of random variables (see, e.g., [START_REF] Merlevède | Recent advances in invariance principles for stationary sequences[END_REF], [START_REF] Račkauskas | Necessary and sufficient condition for the Hölderian functional central limit theorem[END_REF], [START_REF] Giraudo | Holderian weak invariance principle for stationary mixing sequences[END_REF] and references therein).

In this paper we consider weak invariance principle in Besov spaces for a class of strictly stationary sequence of martingale differences. To be more precise, let us first introduce some notation and definitions used throughout the paper.

Let (Ω, F , P) be a probability space and T : Ω → Ω be a bijective bi-measurable transformation preserving the probability P. The quadruple (Ω, F , P, T) is referred to as dynamical system (see, e.g., [START_REF] Petersen | Ergodic theory[END_REF] for some background material). We assume that there is a sub-σ-algebra F 0 ⊂ F such that TF 0 ⊂ F 0 and by I we denote the σ-algebra of the sets A ∈ F such that T -1 A = A.

Next we consider a strictly stationary sequence (X j , j 0) constructed as X j := f • T j , where f : Ω → R is F 0 -measurable. We define also a non-decreasing filtration F k = T -k F 0 , k 1. Note that (X j , F j , j 0) is then a martingale differences sequence provided E (f | TF 0 ) = 0. Set S f,0 := 0, S f,n :=

n-1 j=0 f • T j , n 1.
a Normandie Univ, UNIROUEN, CNRS, LMRS, 76000 Rouen, France, davide.giraudo1@univ-rouen.fr. b Corresponding author c Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania, alfredas.rackauskas@mif.vu.lt.

Our main object of investigation is the sequence of polygonal line processes ζ f,n := (ζ f,n (t), t ∈ [0, 1]), n 1, defined by ζ f,n (t) := S f,⌊nt⌋ + (nt -⌊nt⌋)f • T ⌊nt⌋ , where for a real number a 0, ⌊a⌋ := max{k : k ∈ {0, 1, . . .}, k a}. To define the paths space under consideration let L p ([0, 1]) be the space of Lebesgue integrable functions with exponent p (1 p < ∞) and the norm

x p = 1 0 |x(t)| p d t 1/p , x ∈ L p ([0, 1]).
If x ∈ L p ([0, 1]) its L p -modulus of smoothness is defined as

ω p (x, δ) = sup |h| δ I h |x(t + h) -x(t)| p d t 1/p , δ ∈ [0, 1],
where

I h = [0, 1] ∩ [-h, 1 -h]. Let α ∈ [0, 1). The Besov space B o p,α = B o p,α ([0, 1]
) is defined by

B o p,α := x ∈ L p ([0, 1]) : lim δ→0 δ -α ω p (x, δ) = 0 .
Endowed with the norm

x p,α = x p + sup δ∈(0,1)

δ -α ω p (x, δ), x ∈ B o p,α , the space B o p,α is a separable Banach space and the following embeddings are continuous:

B o p,α ֒→ B o p,β , for 0 β < α; B o p,α ֒→ B o q,α , for 1 q < p < ∞.

Each B o p,α (0, 1) where p 1, 0 α < 1/2, supports a standard Wiener process W = (W (t), 0 t 1) (see, e.g., [START_REF] Roynette | Mouvement brownien et espaces de Besov[END_REF]). We note also, that any polygonal line process belongs to each of B o p,α , p 1, α ∈ [0, 1).

As usually D -→ denotes convergence in distribution.

Theorem 1. Let p > 2, 1/p < α < 1/2 and q(p, α) := 1/(1/2 -α + 1/p).

(1)

Let (f • T i , T -i F 0 , i 0
) be a martingale differences sequence. Assume that the following two conditions hold :

(i) lim t→∞ t q(p,α) P {|f | t} = 0; (ii) E E f 2 | TF 0 q(p,α)/2 < ∞. Then the convergence n -1/2 ζ f,n D ----→ n→∞ E (f 2 | I) W holds in the space B o p,α , where W is indepen- dent of E f 2 | I .
Let us note that condition (i) is stronger for the function f than its square integrability since (i) yields E (|f | r ) < ∞ for any r < q(p, α) and q(p, α) > 2 when α > 1/p. However as shows our next result, condition (i) is necessary and sufficient for independent identically distributed (i.i.d.) random variables. To formulate the result let Y, Y 1 , Y 2 , . . . be mean zero i.i.d. random variables with finite variance

σ 2 = E Y 2 > 0. Let ξ n = (ξ n (t), t ∈ [0, 1]), be defined by ξ n (t) = ⌊nt⌋ k=1 X k + (nt -⌊nt⌋)X ⌊nt⌋+1 .
Theorem 2. Let p > 2, 1/p < α < 1/2 and let q(p, α) be defined by [START_REF] Billingsley | Convergence of probability measures[END_REF]. Then

n -1/2 σ -1 ξ n D ----→ n→∞ W in the space B o p,α (2)
if and only if lim t→∞ t q(p,α) P {|Y | t} = 0.

(3) Since B o ∞,α matches Hölder spaces we see, that Theorems 2 and 1 complement the weak invariance principle obtained in a Hölderian framework by [START_REF] Račkauskas | Necessary and sufficient condition for the Hölderian functional central limit theorem[END_REF] and [START_REF] Giraudo | Holderian weak invariance principle for stationary mixing sequences[END_REF]. Concerning condition (ii) of Theorem 1 we prove a need of certain extra assumption by a counterexample which for any dynamical system with positive entropy constructs a function f that satisfies the condition (i) but the convergence of polygonal line processes fails. Precise result reads as follows.

Theorem 3. Let p > 2, 1/p < α < 1/2 and q(p, α) be given by [START_REF] Billingsley | Convergence of probability measures[END_REF]. For each dynamical system (Ω, F , P, T) with positive entropy, there exists a function m : Ω → R and a σ-algebra F 0 for which TF 0 ⊂ F 0 such that:

(i) the sequence m • T i , T -i F 0 , i 0 is a martingale difference sequence;
(ii) the convergence lim t→+∞ t q(p,α) P |m| t = 0 takes place;

(iii) the sequence n -1/2 ζ m,n is not tight in B o p,α .
As it is seen from our next results the case where 0 α 1/p is indeed quite different from the previously considered case where 1/p < α < 1/2.

Theorem 4. Let p 1 and α ∈ [0, 1/2) ∩ [0, 1/p]. Let f • T i , T -i F 0 , i 0 be a martingale differences sequence. If E f 2 < ∞ then n -1/2 ζ f,n D ----→ n→∞ E (f 2 |I)W in the space B o p,α , where W is independent of E f 2 | I .
Theorem 5. Let α and p be as in Theorem 4. Then

n -1/2 σ -1 ξ n D ----→ n→∞ W in the space B o p,α . (4)
Let us note that the finiteness of the second moment E Y 2 is necessary for the convergence (4). The rest of the paper is organized as follows. In Section 2 we shortly present needed information on structure of Besov spaces and tightness of measures on these spaces. Section 3 contains proofs of Theorems 1, 2 and 3 whereas Section 4 is devoted to the proofs of Theorems 4 and 5. Finally, in Section 6 we discuss possible applications of invariance principle in the Besov framework.

Some functional analysis and probabilistic tools

We denote by D j the set of dyadic numbers in [0, 1] of level j, i.e.

D 0 = {0, 1}, D j = (2l -1)2 -j ; 1 l 2 j-1 , j 1. Set D = j 0 D j
and write for r ∈ D j , r -:= r -2 -j , r + := r + 2 -j .

The triangular Faber-Schauder functions Λ r for r ∈ D j , j > 0, are

Λ r (t) =    2 j (t -r -) if t ∈ (r -, r]; 2 j (r + -t) if t ∈ (r, r + ]; 0 else.
When j = 0, we just take the restriction to [0, 1] in the above formula, so

Λ 0 (t) = 1 -t, Λ 1 (t) = t, t ∈ [0, 1].
Theorem 6 ([3]). Let p > 1 and 1/p < α < 1. The Faber-Schauder system {Λ r , r ∈ D} is the Schauder basis for B o p,α : each x ∈ B 0 p,α has the unique representation,

x = r∈D λ r (x)Λ r ,
where

λ r (x) := x(r) - x(r + ) + x(r -) 2 , r ∈ D j , j 1
and in the special case j = 0, λ 0 (x) := x(0), λ 1 (x) := x(1).

Moreover the norm is equivalent to the sequential norm:

x p,α ∼ x seq p,α := sup j 0 2 jα-i/p r∈Dj |λ r (x)| p 1/p .
The Schmidt orthogonalization procedure (with respect to inner product in L 2 (0, 1)) applied to Faber-Schauder system leads to the Franklin system {f k , k 0}:

f k (t) = k i=0 c ik Λ i (t), t ∈ [0, 1],
with c kk > 0 for k 0, where the matrix (c ik ) is uniquely determined.

Theorem 7 ([3]

). The Franklin system {f n , n 0} is the basis for B o p,α , p 1, 0 α < 1: each x ∈ B o p,α has the unique representation,

x = ∞ k=0 x k f k ,
where

x k = x, f k := 1 0 x(t)f k (t)d t, k 0.
The following proposition is proved in [START_REF] Boufoussi | A Kolmogorov and tightness criterion in modular Besov spaces and an application to a class of Gaussian processes[END_REF] for α > 1/p but similar arguments works as well for any 0 α < 1. Proposition 8. Let p 1 and 0 α < 1. The set K ⊂ B o p,α is relatively compact if and only if

(i) sup x∈K ||x|| p < ∞, (ii) lim δ→0 sup x∈K δ -α ω p,α (x, δ) = 0.
Proof. One easily checks that (i) and (ii) yields relative compactness of K in L p ([0, 1]). Therefore for any sequence (x n ) n 1 of K there exists a subsequence, which we denote also (x n ), converging in L p ([0, 1]) to some x ∈ L p ([0, 1]). To finish the proof it suffices to prove that

(a) x ∈ B o p,α ; (b) (x n ) n 1 is a Cauchy sequence in B o p,α .
Taking a.s. convergence subsequence (x n ′ ) and applying Fatou lemma we easily obtain for any 0 < δ 1,

ω p (x, δ) lim inf n ′ ω p (x n ′ , δ) sup n ω p (x n , δ). (5)
This yields (a). To prove (b) observe that for each ε > 0 there exists δ ε > 0 such that δ -α sup n ω p (x n , δ) < ε when δ < δ ε , hence, for n, m 1

||x n -x m || p,α = ||x n -x m || p + max [ sup 0<δ δε ; sup δε<δ 1 ]δ -α ω p (x n -x m , δ) ||x n -x m || p + ε + 2δ -α ε ||x n -x
(i) lim b→∞ sup n 1 P ||Z n || p > b = 0; (ii) for each ε > 0 lim δ→0 sup n 1 P ω p,α (Z n , δ) ε = 0.
Proof. See, e.g., the proof of Theorem 8.2. in [START_REF] Billingsley | Convergence of probability measures[END_REF].

Theorem 10. Let 1 > α > 1/p. The sequence (Z n ) of random elements in the Besov space B o p,α (0, 1) is tight if and only if the following two conditions are satisfied:

(i) lim b→∞ sup n P ||Z n || p > b = 0; (ii) for each ε > 0, lim J→∞ lim sup n→∞ P sup j J 2 jα-j/p r∈Dj |λ r (ζ n )| p 1/p > ε = 0.
Proof. It is just a corollary of tightness criterion established in [START_REF] Ch | Tightness in Schauder decomposable Banach spaces[END_REF] for Schauder decomposable Banach spaces as Besov spaces B o p,α with α > 1/p, are such.

3 Proofs: the case α > 1/p

We start this section with some auxiliary results which could be helpful when dealing with weak invariance principle for stationary sequences. Throughout we denote

W f,n = n -1/2 ζ n .

Auxiliary results

Lemma 11. Let p 1 and α > 1/p. Assume that Z is a random element in boths spaces C[0, 1] and B o p,α . Then for any stationary sequence f • T j if

(i) W f,n D ----→ n→∞ Z in C[0, 1], and
(ii) (W f,n ) is tight in B o p,α , then W f,n D ----→ n→∞ Z in B o p,α .
Proof. From (ii) we have that each subsequence of (W f,n ) has further subsequence that converges

in distribution. If W f,n ′ D ----→ n→∞ Y ′ and W f,n ′′ D ----→ n→∞ Y ′′ then we have that λ r (W f,n ′ ) D ----→ n→∞ λ r (Y ′ ) and λ r (W f,n ′′ ) D ----→ n→∞ λ r (Y ′′
) for any dyadic number r. But (i) gives that both λ r (Y ′ ) and λ r (Y ′′ ) have the same distribution as λ r (Z). Since Schauder coefficients (λ r (Z)) determines the distribution of Z we can conclude that Y ′ and Y ′′ are equally distributed with Z. This ends the proof.

For polygonal line processes build from any stationary sequence the tightness conditions given in Theorem 10 can be simplified.

Theorem 12. Let p 1 and α > 1/p. The sequence (W f,n ) is tight in B o p,α provided that lim J→+∞ lim sup N →+∞ N j=J 2 j 1 0 x p-1 P 2 -(N -j)/2 max 1 k 2 N -j |S f,k | > x2 j/q(p,α) dx = 0. ( 6
)
Proof. Assume that f satisfies [START_REF] Giraudo | Holderian weak invariance principle under a Hannan type condition[END_REF]. We have to show that (W f,n ) satisfies the conditions of Theorem 10. First we check its condition (i). Since

||W f,n || p sup 0 t 1 |W f,n (t)| = n -1/2 max 0 t 1 |S f,k |, the proof of (i) reduces to lim b→+∞ sup N 1 P 2 -N/2 max 1 k 2 N |S f,k | > b = 0. (7)
Notice that ( 6) implies (by considering the term of index J in the sum) that lim J→+∞ lim sup

N →+∞ 2 J P 2 -(N -J)/2 max 1 k 2 N -J |S f,k | > 1 2 2 J/q(p,α) = 0,
and consequently lim J→+∞ lim sup

N →+∞ 2 J P 2 -N/2 max 1 k 2 N |S f,k | > 2 J/q(p,α) = 0.
For a fixed ε, we choose J 0 such that lim sup

N →+∞ 2 J0 P 2 -N/2 max 1 k 2 N |S f,k | > 2 J0/q(p,α) < 2ε.
There exists an integer N 0 such that for N N 0 ,

P 2 -N/2 max 1 k 2 N |S f,k | > 2 J0/q(p,α) < ε. Since sup N N0 P 2 -N/2 max 1 k 2 N |S f,k | > b → 0 as b goes to infinity, we can choose b ′ 0 such that max N N0 P 2 -N/2 max 1 k 2 N |S f,k | > b ′ 0 < ε. Taking b 0 := max 2 J0/q(p,α) /2, b ′ 0 , we have for b b 0 , sup N 1 P 2 -N/2 max 1 k 2 N |S f,k | > b < ε,
which proves [START_REF] Gut | An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game[END_REF] and the same time (i) of Theorem 10. Now, let us prove condition (ii) of Theorem 10. Since

N j=J 2 j 1 0 x p-1 P 2 -(N -j)/2 max 1 k 2 N -j |S f,k | > x2 j/q(p,α) dx 2 N 3/4 1/2 x p-1 P |f | > x2 N/q(p,α) dx 2 N (1/2) p-1 P |f | > 3 4 2 N/q(p,α) ,
we infer that condition (6) implies

lim t→+∞ t q(p,α) P |f | > t = 0. ( 8
)
We first prove that for each positive ε,

lim sup n→∞ P sup j ⌊log n⌋+1 2 jα-j/p r∈Dj |λ r (W f,n )| p 1/p > ε = 0.
We shall actually prove that lim sup n→∞ P sup

j ⌊log n⌋+1 2 jα-j/p r∈Dj W f,n (r + ) -W f,n (r) p 1/p > ε = 0, (9)
since the differences W f,n (r) -W f,n (r -) can be treated similarly. To this aim, define for a fixed j ⌊log n⌋ + 1 the sets

I k := r ∈ D j , k n r < r + < k + 1 n , 0 k n -1;
and

J k := r ∈ D j , k n r < k + 1 n r + < k + 2 n , 0 k n -2.
Assume that r belongs to I k . Then ⌊nr⌋ = ⌊nr + ⌋ = k. We thus have

W f,n (r + ) -W f,n (r) = nr + -nr f • T k / √ n = n 1/2 2 -j f • T k . ( 10
)
Now, assume that r belongs to J k . Then

W f,n (r + ) -W f,n (r) W f,n (r + ) -W f,n ((k + 1)/n) + |W f,n ((k + 1)/n) -W f,n (r)| = n -1/2 nr + -(k + 1) f • T k+1 + f • T k -(nr -k) f • T k ,
and using the fact that 0 nr + -(k + 1) nr + -nr = n2 j and 0 1 -

(nr -k) (nr + -k) - (nr -k) = n2 -j , we get W f,n (r + ) -W f,n (r) √ n2 -j f • T k + f • T k+1 . ( 11
) Since D j = {1 -2 -j } ∪ n-1 k=0 I k ∪ n-2 k=0 J k and for r = 1 -2 -j , W f,n (r + ) -W f,n (r) n -1/2 2 -j |f • T n | ,
we have in view of ( 10) and [START_REF] Račkauskas | Hölder norm test statistics for epidemic change[END_REF],

r∈Dj W f,n (r + ) -W f,n (r) p 1/p n -1/2 2 -j |f • T n | + n 1/2 2 -j n-1 k=0 Card (I k ) f • T k p 1/p + n 1/2 2 -j n-2 k=0 Card (J k ) f • T k + f • T k+1 p 1/p .
We now have to bound Card (I k ) and Card (J k ). Let 1 l 2 j . If (2l -1)2 -j belongs to I k , then we should have 2 j k/n 2l -1 < 2l < 2 j (k + 1)/n hence 2 j-1 k/n l < 2 j-1 (k + 1)/n and it follows that I k cannot have more than 2 j /n elements. If (2l -1)2 -j belongs to J k , then we should have 2 j (k + 1)/n 2l < 2 j (k + 2)/n and we deduce that the cardinal of J k does not exceed 2 j /n. Therefore, we have

r∈Dj W f,n (r + ) -W f,n (r) p 1/p 3n 1/2 2 -j 2 j /n 1/p n k=0 f • T k p 1/p . and sup j ⌊log n⌋+1 2 jα-j/p r∈Dj |λ r (W f,n )| p 1/p 3 sup j ⌊log n⌋+1 n 1/2 2 -j 2 jα-j/p n k=0 f • T k p 1/p n -1/2+α-1/p n k=0 f • T k p 1/p = n -1/q(p,α) n k=0 f • T k p 1/p .
We thus have to prove that the latter term goes to zero in probability as n goes to infinity.

Lemma 13. Let f be a function such that (8) holds. Then

n -1/q(p,α) n k=0 f • T k p 1/p P ----→ n→∞ 0.
Proof. For fixed δ and n, define f ′ := f 1 |f | δn 1/q(p,α) and f ′′ = f -f ′ . By Markov's inequality, we have with q = q(p, α)

P n -1/q n k=0 f ′ • T k p 1/p > ε ε -p n -p/q n k=0 E f ′ • T k p 2ε -p n 1-p/q E |f ′ | p . (12)
Now, note that

E |f ′ | p = p δn 1/q 0 t p-1 P(|f ′ | > t)dt p δn 1/q 0 t p-q-1 dt • sup s>0 s q P |f | > s = p p -q δ (p-q)/q n (p-q)/q • sup s>0 s q P |f | > s ,
hence by [START_REF] Račkauskas | Necessary and sufficient condition for the Hölderian functional central limit theorem[END_REF], we have

P n -1/q n k=0 f ′ • T k p 1/p > ε ε -p 2p p -q δ (p-q)/q . ( 13
)
Notice also that

P n -1/q n k=0 f ′′ • T k p 1/p > ε (n + 1)P(|f | > δn 1/q ). ( 14
)
The combination of ( 13) and ( 14) gives lim sup

n→+∞ P n -1/q n k=0 f • T k p 1/p > ε ε -p 2p p -q δ (p-q)/q
and since δ is arbitrary and p > q, this concludes the proof of Lemma 13.

An application of the Lemma 13 gives [START_REF] Merlevède | Recent advances in invariance principles for stationary sequences[END_REF]. Now, we have to prove that lim J→∞ lim sup n→∞ P sup

J j ⌊log n⌋ 2 jα-j/p r∈Dj W f,n (r + ) -W f,n (r) p 1/p > ε = 0.
It suffices to prove that lim J→∞ lim sup

n→∞ P n -1/2 sup J j ⌊log n⌋ 2 jα-j/p r∈Dj S ⌊nr + ⌋ -S ⌊nr⌋ p 1/p > ε = 0. ( 15
)
Indeed, we have

W f,n (r + ) -W f,n (r) n -1/2 S f,⌊nr + ⌋ -S f,⌊nr⌋ + f • T ⌊nr + ⌋ + f • T ⌊nr⌋ ,
and for j ⌊log n⌋, 2 j n, so that the set {⌊nr + ⌋, ⌊nr⌋, r ∈ D j } consists of distinct elements. Therefore, sup

1 j ⌊log n⌋ 2 jα-j/p r∈Dj n -1/2 f • T ⌊nr + ⌋ + f • T ⌊nr⌋ p 1/p > ε 2n -1/q n k=0 f • T k 1/p ,
and this quantity goes to zero in probability by Lemma 13. The proof of (15) reduces to establish that for each positive ε,

lim J→+∞ lim n→+∞ ⌊log n⌋ j=J P (A n,j ) = 0, ( 16
)
where

A n,j :=    2 j-1 l=1 S ⌊n2l2 -j ⌋ (f ) -S ⌊n(2l-1)2 -j ⌋ (f ) p > εn p/2 2 j(1-pα)    .
We now bound P (A n,j ) by splitting the probability over the set B n,j :=

2 j-1 l=1 S f,⌊n2l2 -j ⌋ -S f,⌊n(2l-1)2 -j ⌋ > n 1/2 2 j(1/p-α) .
One bounds P (A n,j ∩ B n,j ) by P(B n,j ), which can in turn be bounded by

2 j-1 l=1 P S f,⌊n2l2 -j ⌋ -S f,⌊n(2l-1)2 -j ⌋ > n 1/2 2 j(1/p-α)
and thanks to stationarity and the fact that

⌊n2l2 -j ⌋ -⌊n(2l -1)2 -j ⌋ n2 -j + 1 2n2 -j , (17)
we obtain

P (A n,j ∩ B n,j ) 2 j-1 P max 1 k ⌊2n2 -j ⌋ |S f,k | > n 1/2 2 j(1/p-α) 2 j-1 2 p-1 1 1/2 t p-1 P max 1 k ⌊2n2 -j ⌋ |S f,k | > tn 1/2 2 j(1/p-α) dt. ( 18
)
Now, in order to bound P A n,j ∩ B c n,j , we start by the pointwise inequalities

εn p/2 2 j(1-pα) 1(A n,j ∩ B c n,j ) 2 j-1 l=1 S f,⌊n2l2 -j ⌋ -S f,⌊n(2l-1)2 -j ⌋ p 1(A n,j ∩ B c n,j ) 2 j-1 l=1 S f,⌊n2l2 -j ⌋ -S f,⌊n(2l-1)2 -j ⌋ p 1 S f,⌊n2l2 -j ⌋ -S f,⌊n(2l-1)2 -j ⌋ n 1/2 2 j(1/p-α) .
Integrating and using the fact that for a non-negative random variable Y and a positive R,

E (Y p 1 {Y R}) = pR p 1 0 t p-1 P (Y > Rt) dt,
we derive by stationarity and (17) that

P A n,j ∩ B c n,j p ε 2 j-1 1 0 t p-1 P max 1 k ⌊2n2 -j ⌋ |S f,k | > tn 1/2 2 j(1/p-α) dt. ( 19
)
Let us denote by K a constant depending only on p and ε which may change from line to line. By (18) and ( 19), we derive that

⌊log n⌋ j=J P (A n,j ) K ⌊log n⌋ j=J 2 j 1 0 t p-1 P max 1 k ⌊n2 1-j ⌋ |S f,k | > tn 1/2 2 j(1/p-α) dt. If 2 N n < 2 N +1
, then we have

⌊log n⌋ j=J P (A n,j ) K N j=J 2 j 1 0 t p-1 P max 1 k 2 N +2-j |S f,k | > t2 N/2 2 j(1/p-α) dt, hence ⌊log n⌋ j=J P (A n,j ) K N +2 j=J 2 j 1 0 s p-1 P max 1 k 2 N +2-j |S f,k | > s2 N +2 2 2 j(1/p-α) ds.
Splitting the integral into two parts, we infer that lim sup

n→+∞ ⌊log n⌋ j=J P (A n,j ) K lim sup N →+∞ N +2 j=J 2 j 1 0 s p-1 P max 1 k 2 N +2-j |S f,k | > s2 (N +2)/2 2 j(1/p-α) ds (20)
and the limit of the latter quantity as J goes to infinity is zero by [START_REF] Giraudo | Holderian weak invariance principle under a Hannan type condition[END_REF]. This concludes the proof of Theorem 12.

Remark 14. Using deviation inequalities, similar results as those found for the Hölderian weak invariance principle for stationary mixing and τ -dependent sequences in [START_REF] Giraudo | Holderian weak invariance principle for stationary mixing sequences[END_REF] can be found for Besov spaces.

Lemma 15 (Proposition 3.5 in [START_REF] Giraudo | Hölderian invariance principle for martingale difference random fields[END_REF] ). For any q > 2, there exists a constant c(q) such that if f • T i i 0 is a martingale differences sequence with respect to the filtration T -i F 0 i 0 then for each integer n 1,

P 1 √ n max 1 i n |S f,i | t c(q)n 1 0 P |f | √ nut u q-1 du+ + c(q) ∞ 0 P E f 2 | TF 0 1/2 > vt min v, v q-1 dv. (21)

Proof of Theorem 1

Acording to Lemma 11 we need only to prove that the sequence n -1/2 ζ f,n n 1 is tight in B o p,α . To this aim, we have to check the condition (6) of Theorem 12. For fixed N and J such that N J, j ∈ J, . . . , N and x ∈ [0, 1], we have by (21) of Lemma 15,

P 2 -N -j 2 max 1 k 2 N -j |S f,k | > x2 j q(p,α) c(q)2 N -j 1 0 P |f | 2 N -j 2 x2 j q(p,α) u u q-1 du + c(q) ∞ 0 P E f 2 | TF 0 1/2 > vx2 j q(p,α)
min v, v q-1 dv, ( 22) from which we infer that

N j=J 2 j 1 0 x p-1 P 2 -N -j 2 max 1 k 2 N -j |S f,k | > x2 j q(p,α) dx c(q)2 N N j=J 1 0 x p-1 1 0 P |f | 2 N 2 x2 j(1/p-α) u dxu q-1 du + c(q) ∞ 0 1 0 x p-1 N j=J 2 j P E f 2 | TF 0 1/2 > vx2 j q(p,α) min v, v q-1 dvdx =: A(N, J) + B(N, J). (23)
Using the fact that P |f | t t -q(p,α) sup s t s q(p,α) P |f | s , we derive the bound

A(N, J) c(q)2 N (1-q(p,α)/2) N j=0 2 j(1/p-α) 1 0 1 0
x p-q(p,α)-1 u q-q(p,α)-1 sup s q(p,α) P |f | s , s 2 N 2 xu2 j(1/p-α) dxdu.

Since j N , we have 2 N 2 xu2 j(1/p-α) xu2 N/q(p,α) and accounting the inequality N j=0 2 j(1/p-α) 2 N (1/p-α) /(1 -2 1/p-α ), we obtain A(N, J) c(q) 1 0 1 0 x p-q(p,α)-1 u q-q(p,α)-1 sup s q(p,α) P |f | s , s 2 N q(p,α) xu dxdu.

Since p > q(p, α) and q > q(p, α), the integral 1 0 1 0 u q-q(p,α)-1 x p-q(p,α)-1 dxdu is convergent and we infer by the monotone convergence theorem that ∀J 1, lim

N →+∞ A(N, J) = 0. (24)
Now, in order to control B(N, J), we use the following elementary inequality: if Y is a non-negative random variable, then for each J 1, α) .

j J 2 j P Y 2 j/q(p,α) 2E Y q(p,α) 1 Y 2 J/q(p,
Applying this to Y := E f 2 | TF 0 1/2 /(vx), we obtain that

B(N, J) c(q) ∞ 0 1 0 x p-q(α)-1 E E f 2 | TF 0 q/2 1 E f 2 | TF 0 1/2 vx2 J/q(p,α)
• min v, v q-1 v -q(p,α) dvdx.

Here again, we conclude by monotone convergence that lim

J→+∞ sup N 1 B(N, J) = 0, (25)
since the integrals 1 0 x p-q(α)-1 dx and +∞ 0 min v, v q-1 v -q(p,α) dx are finite (as q > q(α)).

Tightness of the sequence (W f,n ) n 1 now follows from Theorem 12 and the combination of ( 23), ( 24) and (25). Acounting Lemma 11 this concludes the proof of Theorem 1.

Proof of Theorem 2

Sufficiency of the condition is contained in Theorem 1. Indeed, we represent the sequence (Y j ) j 0 by f • T j j∈Z , that is, f • T j j∈Z is an i.i.d. sequence and Y j has the same distribution as f • T j . To this aim we define Ω = R Z , F = B Z and P = P Z Y , where P Y is the distribution of Y 0 . Let f ((ω j )) = ω 0 for (ω j ) ∈ R Z and let T : Ω → Ω be the shift operator: T((ω j )) = (ω j+1 ). Next let F 0 := σ f • T j , j 0 . Then TF 0 ⊂ F 0 and E (f | TF 0 ) = 0, since f is independent of TF 0 and centered. Moreover, E f 2 | TF 0 = E f 2 , again by independence. Therefore, condition (ii) of ( 1) is satisfied. Since I is trivial, E f 2 | I = E f 2 , which gives the convergence (2).

Let us prove the necessity of (3) for the invariance principle in B o p,α . Since the space B o p,α is a separable Banach space, the sequence (W 2 n ) n 1 is tight in B o p,α . Using Theorem 1 in [START_REF] Rosenbaum | First order p-variations and Besov spaces[END_REF], we can find for any positive η a number J 0 such that lim sup

n→∞ P sup j J0 2 jα-j/p r∈Dj W 2 n (r + ) -W 2 n (r) p 1/p > ε η.
Therefore, if n is large enough, we have

P 2 nα-n/p r∈Dn W 2 n r + -W 2 n (r) p 1/p > ε 2η. Since r∈Dn W 2 n (r + ) -W 2 n (r) p = 2 -np/2 2 n-1 l=1 |S 2l -S 2l-1 | p = 2 -np/2 2 n-1 l=1 |X 2l-1 | p ,
we have the convergence in probability of the sequence 2 nα-j/p 2 -np/2 2 n-1 l=1 |X 2l-1 | p to 0. By [START_REF] Gut | An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game[END_REF], this implies that 2 nα-j/p 2 -np/2 P |X 1 | > 2 n → 0, hence (3) holds. This ends the proof of Theorem 2.

Proof of Theorem 3

We first start by a lemma which guarantees the lake of tightness of the partial sum process.

Lemma 16. Let 1/p < α < 1/2 and let f be a function such that there exist increasing sequences of real numbers (n l ) l 1 and (k l ) l 1 satisfying the following properties: k l /n l → 0 as l goes to infinity and

inf l 1 P 1 n q(p,α) l max 1 k k l 1 k α n l -k i=0 |S f,i+k -S f,i | p 1/p > 1 > 0,
where q(p, α) is given by (1). Then the sequence

(W n (f )) n 1 is not tight in B o p,α .
Proof. If the sequence (W n (f )) n 1 was tight in B o p,α , then we would be able to extract a weakly convergence subsequence of (W n l (f )) l 1 . Therefore, we can assume without loss of generality that (W n l (f )) l 1 converges in distribution in B p,α . Consequently, the sequence sup |t| k l /n l t -α ω p (W n l , t) l 1 should convergence to 0 in probability as l goes to infinity. But

sup |t| k l /n l t -α ω p (W n l , t) c(p) n q(p,α) l max 1 k k l 1 k α n l -k i=0 |S f,i+k -S f,i | p 1/p
for some constant depending only on p (this can be seen by restricting the supremum over the t of the form k/n l where 1 k k l ).

Let us recall the statement of Lemma 3.8 in [START_REF] Lesigne | Large deviations for martingales[END_REF].

Lemma 17. Let (Ω, F , P, T) be an ergodic probability measure preserving system of positive entropy.

There exists two T-invariant sub-σ-algebras B and C of F and a function g : Ω → R such that:

• the σ-algebras B and C are independent;

• the function g is B-measurable, takes the values -1, 0 and 1, has zero mean and the process (g • T n ) n∈Z is independent;

• the dynamical system (Ω, C, P, T) is aperiodic.

In the sequel, we shall assume for simplicity that P g = 1 = P g = -1 = 1/2. The construction follows the lines of that of Theorem 2.1 in [START_REF] Giraudo | Holderian weak invariance principle under a Hannan type condition[END_REF]. We define three increasing sequences of positive integers (I l ) l 1 , (J l ) l 1 , (N l ) l 1 and a sequence of real numbers (L l ) l 1 such that

∞ l=1 1 L l < ∞ and
L l is a continuity point of the cumulative distribution function of the random variable 2 -1 Y 1/2,1 , which is defined in (45). Now, we define a sequence of real numbers (J l ) l 1 in such a way that for each l 1,

J l P Y 1/2,1 > 2L l -7/8 1/16. ( 26
)
Now, by Proposition 22, we can choose for each l 1 an integer I l such that

∀n I l , P Y n,1/2,1 (g) > 2L l -P Y 1/2,1 > 2L l 1 lJ l . ( 27
)
Let K l := 2 I l +J l . We define the sequence (N l ) l 1 in such a way that for each l 1,

1 N 1/q(p,α) l K 1-α l l-1 u=1 N u 2 Iu 1/q(p,α)
1 and (28)

N l +∞ u=l+1 K u /N u < 1 16 . ( 29
)
Using Rokhlin's lemma, we can find for any integer l 1 a measurable set C l ∈ C such that the sets T i C l , i = 0, . . . , N l -1 are pairwise disjoint and

P N l -1 i=0 T i C l > 1/2. We define for l 1 f l := 1 L l J l j=1 N l 2 I l +j 1/q(p,α) 1 2 J l +j+1 i=2 J l +j +1 T N l -i C l and
(30)

f := +∞ l=1 f l , m := g • f. ( 31
)
Note that P (f l = 0) K l /N l , hence by ( 29) and the Borel-Cantelli lemma, the function f is well defined almost everywhere. Define

F 0 := σ g • T i , i 0 ∨ C.
Proposition 18. The σ-algebra F 0 satisfies TF 0 ⊂ F 0 . The function m defined by (30) and (31) is F 0 -measurable and satisfies E [m | TF 0 ] = 0 and lim t→+∞ t q(p,α) P {|m| > t} = 0.

A proof can be found in [START_REF] Giraudo | Holderian weak invariance principle under a Hannan type condition[END_REF] It remains to prove that the sequence (W n (m)) n 1 is not tight in B o p,α . To this aim, we shall check the conditions of Lemma 16. We first show the following intermediate step.

Lemma 19. For each integer l 1,

P 1 N 1/q(p,α) l max 1 k K l k -α N l -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p > 2 > 1 8 .
Let l 1 be fixed. Assume that ω belongs to T N l -i0 for some i 0 ∈ {K l , . . . , N l -1}. Let i be such that i 0 -2 I l +j+1 i i 0 -2 I l +j + 1 for some j ∈ {1, . . . , J l }. We have

f l • T i (ω) = 1 L l N l 2 I l +j 1/q(p,α) .
Consequently, for any k such that 2 I l +j-1 < k 2 I l +j and each i such that i 0 -2

I l +j+1 i i 0 -k -2 I l +j + 1, we have |S gf l ,i+k -S gf l ,i | = 1 L l N l 2 I l +j 1/q(p,α) |S g,i+k -S g,i | . It thus follows that 1 N 1/q(p,α) l max 1 k K l k -α N -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p 1 T N l -i0 C l max 1 j J l max 2 I l +j-1 <k 2 I l +j k -α 1 L l 1 2 I l +j 1/q(p,α) i0-k-2 I l +j +1 i=i0-2 I l +j+1 |S g,i+k -S g,i | p 1/p 1 T N l -i0 C l ,
and using disjointness of the sets T N l -i0 C l , K l i 0 N l -1, we infer that

P 1 N 1/q(p,α) l max 1 k K l k -α N -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p > 2 P 1 N 1/q(p,α) l max 1 k K l k -α N -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p > 2 ∩ N l -1 i0=K l T N l -i0 C l = N l -1 i0=K l P 1 N 1/q(p,α) l max 1 k K l k -α N -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p > 2 ∩ T N l -i0 C l N l -1 i0=K l P(A i0 ∩ T N l -i0 C l ) (32)
where

A i0 = max 1 j J l max 2 I l +j-1 <k 2 I l +j k -α 1 L l 1 2 I l +j 1/q(p,α) i0-k-2 I l +j +1 i=i0-2 I l +j+1 |S g,i+k -S g,i | p 1/p > 2 .
Since T is measure-preserving, the events A i0 ∩ T N l -i0 C l , K l i 0 N l -1 have the same probability, which is equal to P(A K l ∩T N l -K l C l ). The events A K l and T N l -K l C l belong respectively to B and C, hence they are independent. In view of (32), we obtain

P 1 N 1/q(p,α) l max 1 k K l k -α N l -k i=0 |S gf l ,i+k -S gf l ,i | p 1/p > 2 (N l -K l ) P(A K l )P(C l ) P(A K l )/2. (33)
Now, in order to control the latter term, we shall use the following lemma:

Lemma 20. Let (H l ) l 1 be an increasing sequence of integers. Assume that for each l 1, the family of events (A l,j ) 1 j H l is independent and that H l j=1 P (A l,j ) ∈ [3/4, 1]. Then for each l 1,

P H l j=1 A l,j 1/4.
Proof of Lemma 20. By Bonferroni's inequality, we have for any l 1,

P H l j=1 A l,j H l j=1 P (A l,j ) - 1 i<j H l P (A l,i ∩ A l,j ) .
Using independence of (A l,j ) 1 j H l , we derive that

P   H l j=1 A l,j   H l j=1 P (A l,j ) - 1 2   2
1 i<j H l P (A l,i ) P (A l,j )   = H l j=1 P (A l,j ) - 1 2      H l j=1 P (A l,j )   2 - H l j=1 (P (A l,j )) 2    H l j=1 P (A l,j ) - 1 2   H l j=1 P (A l,j )   2 3/4 -1/2 = 1/4.
We now use Lemma 20 with the choices H l = J l and

A l,j :=      max 2 I l +j-1 <k 2 I l +j k -α 1 L l 1 2 I l +j 1/q(p,α)   K l -k-2 I l +j +1 i=K l -2 I l +j+1 |S g,i+k -S g,i | p   1/p > 2      .
We indeed have, with the notations of (45) and by (27),

J l j=1 P (A l,j ) -J l P Y 1/2,1 > 2L l J l j=1 1 lJ l = 1/

Proofs: the case α 1/p

We start with the following lemma which reduces the proof of convergence to that of tightness.

Lemma 21. Let p 1 and 0 α min{1/2, 1/p}. Assume that Z is a random element in B o p,α . Then for any stationary sequence f • T j if

(i) W f,n D ----→ n→∞ Z in L p [0, 1],
and

(ii) (W f,n ) is tight in B o p,α , then W f,n D ----→ n→∞ Z in B o p,α .
Proof. From (ii) we have that each subsequence of (W f,n ) has further subsequence that converges

in distribution. If W f,n ′ D ----→ n→∞ Y ′ and W f,n ′′ D ----→ n→∞ Y ′′ then we have that for Franklin basis (f k ) it holds that W f,n ′ , f k D ----→ n→∞ Y ′ , f k and W f,n ′′ , f k D ----→ n→∞ Y ′′ ,
I n (δ, ε) = 0, (37)
where

I n (δ, ε) = P δ -α sup |h| δ 1 0 |W f,n (t + h) -W f,n (t)| p d t 1/p > ε .
Since the function W f,n (t), 0 t 1 is affine in each interval ((k -1)/n, k/n], it holds for s, t ∈

[(k -1)/n, k/n], |W f,n (s) -W f,n (t)| n 1/2 |s -t| • f • T k . ( 38
)
This observation leads to

ω p (W f,n , δ) c p [U f,n (δ) + V f,n (δ)] ,
where c p > 0 is a constant depending on p only,

U f,n (δ) := min{δ, n -1 }n 1/2 1 n n k=1 |f • T k | p 1/p , V f,n (δ) := n -1/2 max 1 ℓ ⌊nδ⌋   1 n n-ℓ k=0 k+ℓ j=k+1 f • T j p   1/p .
As a consequence in order to establish (37) we have to prove

lim δ→0 sup n 1 P δ -α U f,n (δ) ε = 0, (39) lim δ→0 sup n 1/δ P δ -α V f,n (δ) > ε = 0. ( 40
)
Consider first (39) and start with p = 2 and 0 α < 1/2. By Chebyshev inequality

P δ -α U f,n (δ) ε ε -2 δ -2α min{δ 2 , n -2 }E n k=1 |f • T k | 2 ε -2 δ -2α min{δ 2 , n -2 }nE f 2 ε -2 δ 1-2α
and (39) follows in this case. Now let p > 2 and 0 α 1/p. For this case we shall use truncation.

Set for τ > 0,

f ′ = f 1(|f | τ max{n, δ -1 }), f ′′ = f -f ′ . Then P(δ -α U f,n (δ) > ε) nP(|f | τ max{n, δ -1 }) + P(δ -α U f ′ ,n (δ) > ε) and, since nP(|f | τ max{n, δ -1 }) τ -2 E (f ) 2 1(|f | √ δ -1 )
we reduce the proof of (39) to

lim δ→0 sup n 1 P δ -α U f ′ ,n (δ) ε = 0. ( 41
)
We have by Chebyshev inequality,

P δ -α U f ′ ,n (δ) ε ε -p δ -pα min{δ p , n -p }n p/2 E n -1 n k=1 |f ′ • T k | p ε -p δ -pα min{δ p , n -p }n p/2 E ((f ′ ) p ) ε -p δ -pα min{δ p , n -p }n p/2 τ p-2 (max{n, δ -1 }) (p-2)/2 E f 2 ε -p τ p-2 (min{δ, n -1 }) 1-pα E f 2 .
Hence lim

δ→0 sup n 1 P δ -α U f ′ ,n (δ) ε ε -p τ p-2 .
Since τ > 0 is arbitrary, the limit is indeed zero, and the proof of (41) is completed.

To prove (40) we start again with the case p = 2 and 0 α < 1/2. In this case Chebyshev inequality along with stationarity and Doob-Kolmogorov inequality yields

P(δ -α V f,n (δ) > ε) ε -2 δ -2α E (V f,n (δ)) 2 ε -2 δ -2α n -1 E max 1 ℓ ⌊nδ⌋ ℓ j=1 f • T j 2 ε -2 δ 1-2α E f 2
and (40) follows. This ends the proof of (40) in the case p = 2.

Assume that p > 2 and α 1/p. Let us fix ε > 0. Define for any δ ∈ (0, 1) and any integer n 1/δ the events

A n,δ :=   δ -α n -1/2 max 1 ℓ ⌊nδ⌋   1 n n-ℓ k=0 k+ℓ-1 j=k f • T j p   1/p > ε    B n,δ,τ := max 1 ℓ ⌊nδ⌋ n -1/2 max 1 k n-l |S f,k+l -S f,k | ε p p-2 τ p p-2
, where τ is an arbitrary but fixed positive number. We have the bound

P (B n,δ,τ ) P sup 0 s<t 1,|t-s|<δ n -1/2 |ζ f,n (t) -ζ f,n (s)| ε p p-2 τ p p-2 .
Since the sequence (ζ f,n ) is tight in the space C[0, 1] (see [START_REF] Billingsley | Convergence of probability measures[END_REF]), we have

lim δ→0 sup n 1/δ P (B n,δ,τ ) = 0. (42)
Now, note that on A n,j ∩ B c n,j,τ , we have for any 0 ℓ ⌊nδ⌋,

n-ℓ k=0 k+ℓ-1 j=k f • T j p n-ℓ k=0 k+ℓ-1 j=k f • T j 2 max 1 ℓ ⌊nδ⌋ n -1/2 max 1 k n-l |S f,k+ℓ -S f,k | p-2 n (p-2)/2 (43) n-ℓ k=0 k+ℓ-1 j=k f • T j 2 n (p-2)/2 ε p τ p . (44)
Therefore, we have

ε < δ -α n -1/2 max 1 ℓ ⌊nδ⌋   1 n n-ℓ k=0 k+ℓ-1 j=k f • T j p   1/p δ -α n -1/2 n -1/p n (p-2)/2 ε p τ p 1/p max 1 ℓ ⌊nδ⌋    n-ℓ k=0 k+ℓ-1 j=k f • T j 2    1/p = δ -α n -2/p ετ max 1 ℓ ⌊nδ⌋    n-ℓ k=0 k+ℓ-1 j=k f • T j 2    1/p
, and we infer that

P A n,δ ∩ B c n,δ,τ P   δ -αp n -2 τ p max 1 ℓ ⌊nδ⌋ n-ℓ k=0 k+ℓ-1 j=k f • T j 2 > 1    .
By Markov's inequality, stationarity and Doob's inequality, we have

P A n,δ ∩ B c n,δ,τ δ -αp n -2 τ p E    max 1 ℓ ⌊nδ⌋ n-ℓ k=0 k+ℓ-1 j=k f • T j 2    δ -αp n -2 τ p E    n k=0 max 1 ℓ ⌊nδ⌋ k+ℓ-1 j=k f • T j 2    = δ -αp n -1 τ p E max 1 ℓ ⌊nδ⌋ |S f,ℓ | 2 2δ 1-αp τ p E f 2 .
Since pα 1, we get

P A n,δ ∩ B c n,δ 2τ p E f 2
and since τ is arbitrary, we get lim δ→0 sup n 1/δ P (A n,δ ) = 0 in view of (42). This concludes the proof of (40) and that of Theorem 4.

Proof of Theorem 5

It follows from Theorem 4 and the same arguments as used in the proof of Theorem 2.

Some applications

As already was mentioned in the introduction, a choice of functional spaces for polygonal line processes is usually inspired by possible applications in statistics via continuous mappings: if

W f,n D ----→ n→∞ W in the space B o p,α , then T (W f,n ) → T (W )
for any continuous function T : B o p,α → R. This general observation can be used, for example, to analyse so called k-scan processes

S (i) f,k = i+k-1 j=i f • T j , i = 1, . . . , n -k + 1. Proposition 22. Let f be a function such that the sequence (W n (f )) n 1 converges to a standard Brownian motion W in B o p,α , where 1/p < α < 1/2. For each a, b ∈ [0, 1] such that a < b, we define Y n,a,b (f ) := n -1/q(p,α) max ⌊an⌋<k ⌊bn⌋+1 1 k α n-k i=0 S (i) f,k p 1/p .
Then the following convergence holds: Now, take q > p. Using Hölder's inequality with the exponents q/p and q/(q -p), we have N -(p-q)/q W q,α N -(p-q)/q .

Y n,a,b (f )
Since W q,α is almost surely finite, we get from (48) that (47) holds.

Statistics based on k-scan processes can be used to detect epidemic change in the mean of a sample of size n (see, e.g., [START_REF] Račkauskas | Hölder norm test statistics for epidemic change[END_REF], and reference therein). More precisely, given a sample X 1 , X 2 , . . . , X n , consider the model X i = µ1 (k * ,m * ] (i) + ε i , i = 1, . . . , n, where (ε i ) is a stationary sequence, µ = 0 and k * , m * are unknown parameters of the model. We want to test the null hypothesis H 0 : µ = 0 against the alternative µ = 0. To this aim one can consider the statistics

T n = n -1/q(p,α) max 0<k n 1 k α n-k i=0 |X i + • • • + X i+k | p 1/p .
Under null its limit is defined by Proposition 22 provided (ε i ) satisfies the weak invariance principle in the Besov space B o p,α . Under alternative then we see that

T n = 1 - h * n 1/p h * n 1-α n 1/2 + O P (1)
as n → ∞, where h * = m * -k * is the duration of the epidemic state.

  Let use define a functional F : B o p,α → R by F (x) := sup a<t b t -α It |x(s + t) -x(s)| p ds 1/p . Then F is continuous with respect to the topology of B o p,α and F (W ) = Y a,b . We thus have F (W n (f )) D ----→ n→∞ Y a,b . To conclude that (45) holds, it suffices to prove thatZ n := F (W n (f )) -Y n,a,b (f ) → 0 in probability as n → +∞.First note that Z n is non-negative. Second, we haveF (W n (f )) max ⌊an⌋ k ⌊bn⌋+1 max k/n t<(k+1)/n

δ[ 1 - 1 /

 11 δ -α ω p (W n (f ), δ) k+1)/n |W n (f, s + k/n) -W n (s)| p ds -α ω p (W n (f ), δ) . (46)By Theorem 9, the second term in the right hand side of (46) goes to zero in probability. Therefore, it suffices to prove that max probability as n → +∞.(47) To see this, we start from the inequality max|W n (f, s + t) -W n (f, s)| p ds |W n (f, s) -W n (f, s -t)| p ds 1/p .Let N be a fixed integer. The functionalG N : B o p,α → R, G N (x) = sup 0<t<1 t -α [1-1/N,1]∩I-t |x(s) -x(s -t)| p ds 1/p is continuous. Therefore, if ε > 0 is a continuity point of the cumulative distribution function of G N (W ) for each N , k+1)/n |W n (f, s + k/n) -W n (f, s)| p ds N,1]∩I-t |W (s) -W (s -t)| p ds 1/p > ε . (48)

  |W (s) -W (s -t)| q ds 1/q

  m || p and we complete the proof since lim n,m→∞ ||x n -x m || p = 0. Zn ) converges weakly to P Z : lim n→∞ f d P Zn = f d P Z for each bounded continuous f : B o p,α → R.

	Consider stochastic processes Z, (Z n ) n 1 with paths space B o p,α which is endowed with Borel σ-p,α ). Let P Z , P Zn , n 1, be the corresponding distributions. As generally accepted the algebra B(B o sequence (Z n ) converges in distribution to Z in B o p,α (denoted Z n D ----→ n→∞ Z in B o p,α ) provided (P The sequence (Z n ) is tight in B o p,α p,α if for each ε > 0 there is a relatively compact set K ε ⊂ B o such that inf n 1 P(Z n ∈ K ε ) > 1 -ε. Due to the well known Prohorov's theorem convergence in distribution in a separable metric space is coherent with tightness. Indeed, to prove convergence in
	distribution one has to establish tightness and to ensure uniqueness of the limiting distributions.
	The following tightness criterion is obtained from Proposition 8.
	Theorem 9. The sequence (Z n ) of random processes with paths in B o p,α (0, 1) is tight if and only if
	the following two conditions are satisfied:

  f k for any k. But (i) gives that both Y ′ , f k and Y ′′ , f k have the same distribution as Z, f k . Since coefficients Z, f k determines the distribution of Z we can conclude that Y ′ and Y ′′ are equally distributed with Z. This ends the proof.

	4.1 Proof of Theorem 4
	Due to continuity of the embedding B o 2,α ֒→ B o p,α if 1	p	2 and 0	α < 1/2. it is enough to
	prove the case where either p = 2 and 0 α < 1/2 or p > 2 and 0 α 1/p.
	Recall W f,n = n -1/2 ζ f,n . We shall prove for each ε > 0
	lim δ→0	sup n 1

  It|W n (f, s + t) -W n (f, s)| p ds Let k be an integer such that ⌊an⌋ k ⌊bn⌋ + 1 and let t be a real number such that k/n t < (k + 1)/n. ThenIt |W n (s + t) -W n (s)| p ds |W n (s + k/n) -W n (s)| p ds |W n (f, s + k/n) -W n (s)| p ds |W n (f, s + t) -W n (s + k/n)| p ds |W n (f, s + k/n) -W n (s)| p ds 1/p + ω p (W n (f ), t -k/n) ,which implies thatF (W n (f )) -Y n,a,b (f ) max |W n (f, s + k/n) -W n (s)| p ds

				1/p						1/p
				-	I k/n
	1-k/n				1/p		1-t		1/p
						+
	1-t							0
	1-k/n 1-(k+1)/n	|W n (f, s + k/n) -W n (s)|	p ds	1/p	+ ω p (W n (f ), t -k/n)
			1-k/n
	=
			1-(k+1)/n
					k	-α	1-k/n		1/p
	⌊an⌋ k ⌊bn⌋+1	n		1-(k+1)/n
	+	max ⌊an⌋ k ⌊bn⌋+1
										1/p
						t -α
	max ⌊an⌋ k ⌊bn⌋+1	max k/n t<(k+1)/n	k n	-α	It	|W n (f, s + t) -W n (f, s)|	p ds	1/p	.

We get, in view of (33) the lower bound

This concludes the proof of Lemma 19. Now, we prove that for any l 1,

We first prove that 1

where

Now, by definition of f u , for each ω ∈ Ω, the following inequality holds:

1/q(p,α) .

Consequently,

, and this term does not exceed 1 by (28). This proves (35) Now, defining f ′′ l := +∞ u=l+1 gf u , we have

By constructing of f u , we have P f u = 0 K u /N u , hence