Jérôme Dedecker
email: jerome.dedecker@parisdescartes.fr

Florence Merlevède
email: florence.merlevede@univ-mlv.fr

Françoise Pène
email: francoise.pene@univ-brest.fr

Rates in the strong invariance principle for ergodic automorphisms of the torus

Keywords: Invariance principles, strong approximations, ergodic automorphisms of the torus. Mathematical Subject Classification (2010): 37D30, 60F17

Let T be an ergodic automorphism of the d-dimensional torus T d . In the spirit of Le Borgne [10], we give conditions on the Fourier coefficients of a function f from T d to R under which the partial sums

satisfies a strong invariance principle. Next, reinforcing the condition on the Fourier coefficients in a natural way, we obtain explicit rates of convergence in the strong invariance principle, up to n 1/4 log n.

Introduction

We endow the d-dimensional torus T d = R d /Z d with the Lebesgue measure λ, and we denote by E(•) the expectation with respect to λ. As usual, the L p norm of a f from T d to R is denoted by f p = (E(|f | p )) 1/p . For d ≥ 2, let T be an ergodic automorphism of T d , and let f be a function from T d to R such that E(f 2 ) < ∞ and E(f ) = 0. In [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF], Le Borgne has proved that if the Fourier coefficients (c k ) k∈Z d of f are such that, for θ > 2 and every integer b > 1,

|k|≥b |c k | 2 ≤ R log -θ (b) , where |k| = max 1≤i≤d |k i | , (1.1)
then the partial sums process

[nt] i=1 f • T i , t ∈ [0, 1] (1.2)
properly normalized, satisfies both the weak and strong invariance principles. More precisely, Le Borgne has introduced in [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] an appropriate σ-field F 0 such that F 0 ⊆ T -1 (F 0 ), for which the quantities E(f

• T k |F 0 ) 2 and f • T -k -E(f • T -k |F 0 )
2 can be controlled for any positive integer k. The weak and strong invariance principles follow then, by applying Gordin's result (see [START_REF] Gordin | The central limit theorem for stationary processes[END_REF]) and Heyde's result (see [START_REF] Heyde | On the central limit theorem and iterated logarithm law for stationary processes[END_REF]) respectively.

1 Supported in part by the ANR project PERTURBATIONS 1

In Theorem 2.1 of this paper, we show that the weak and strong invariance principles still hold for functions f satisfying (1.1) with θ > 1 only, and we give a multivariate version of these results. For the weak invariance principle, this follows from an improvement of Gordin's criterion, which was already known in the univariate case (see [START_REF] Dedecker | On the weak invariance principle for non adapted sequences under projective criteria[END_REF]). For the strong invariance principle, this will follow from a new criterion for stationary sequences, presented in Theorem 4.1 of the appendix. Note that the condition (1.1) with θ > 1 is satisfied if, for a positive constant A,

|c k | 2 ≤ A d i=1 1 (1 + |k i |) log 1+α (2 + |k i |)
for some α > 1, (1.3) improving on the condition α > 2 given by Leonov in 1969 (see [START_REF] Leonov | Central limit theorem for ergodic endomorphisms of compact commutative groups[END_REF], Remark 1). Note that Leonov has also given a condition in terms of the modulus of continuity of f in L 2 . The strong invariance principle means that, enlarging T d if necessary, there exists a sequence of independent identically distributed (iid) Gaussian random variables Z i such that sup

1≤k≤n k i=1 f • T i - k i=1 Z i = o n 1/2 (log log n) 1/2 almost surely, as n → ∞.
(1.4)

It is also possible to exhibit rates of convergence in (1.4), provided that we reinforce the assumption (1.1). This has been done recently, thanks to a general result giving rates of convergence in the strong invariance principle for partial sums of stationary sequences. More precisely, let p ∈]2, 4] and q = p/(p -1). We have proved in Theorem 2.1 of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] that if there exists R > 0 such that for every integer b > 1,

|k|≥b |c k | q ≤ R log -θ (b) for some θ > p 2 -2 p(p -1) , (1.5)
and

|k|≥b |c k | 2 ≤ R log -β (b) for some β > 3p -4 p , (1.6)
then the strong approximation (1.4) holds true with an error of order o n 1/p (log n) (t+1)/2 , for t > 2/p. A condition on the ℓ q norm of (c k ) k∈Z d seems appropriate in this context, since this ℓ q -norm dominates the L p norm of f , which is required to be finite to get the rate o(n 1/p ) in the iid situation.

If we assume that the Fourier coefficients of f are such that,

|c k | q ≤ A d i=1 1 (1 + |k i |) log 1+α (2 + |k i |) , (1.7)
then the conditions (1.5) and (1.6) are both satisfied provided that α > (p 2 -2)/(p 2 -p). Now, considering (1.3), one can wonder if α > 1 in (1.7) is enough to get an approximation error of order o(n 1/p L(n)) in (1.4), where L(n) is a slowly varying function. The main result of this paper, Theorem 2.2 below, shows that the answer is positive.

Invariance principles for ergodic automorphisms of the torus

Let us first recall some probabilistic notations. A measurable function f :

T d → R m (with coordinates f 1 , ..., f m
) is said to be centered if every f i is integrable and centered. Such a function f is said to be square integrable if every f i is square integrable. Now, for every centered and square integrable functions f, g : T d → R m (with f = (f 1 , ..., f m ) and g = (g 1 , ..., g m )), we define the covariance matrix Cov(f, g) of f and g and the variance matrix Var(f ) by Cov(f, g) = (E(f i g j )) i,j=1,...,m , and Var(f ) = Cov(f, f ) .

Let us now recall some facts about ergodic automorphisms of T d . A group automorphism T of T d is the quotient map of a linear map T : R d → R d given by T (x) = S.x (. being the matrix product), where S is a d × d-matrix with integer entries and with determinant ±1. Any automorphism T of T d preserves the Lebesgue measure λ. Therefore (T d , B(T d ), T, λ) is a probability dynamical system (where B(T d ) stands for the Borel σ-algebra of T d ).

This dynamical system is ergodic if and only if no root of the unity is an eigenvalue of the matrix S associated to T . In this case, we say that T is an ergodic automorphism of T d .

An automorphism T of T d is said to be hyperbolic if the matrix S associated to T admits no eigenvalue of modulus one. With the preceding characterization of ergodic automorphisms of T d , it is clear that every hyperbolic automorphism of T d is ergodic. Ergodic automorphisms of T d are partially hyperbolic but not necessarily hyperbolic (an example of a non-hyperbolic ergodic automorphism of T d can be found in [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]).

In the next Theorem, we give weak and strong invariance principles for the partial sum process (1.2) of R m -valued functions.

Theorem 2.1. Let T be an ergodic automorphism of T d . For any j ∈ {1, . . . , m}, let f j : T d → R be a centered function and assume that its Fourier coefficients (c k,j ) k∈Z d satisfy the following condition: there exists a positive constant R such that for every integer b > 1,

|k|≥b |c k,j | 2 ≤ R log -θ (b) for some θ > 1 .
(2.1)

Let f = (f 1 , . . . , f m ) : T d → R m . Then the series Σ = k∈Z Cov(f, f • T k ) converges, and
lim n→∞ 1 n Var n i=1 f • T i = Σ . (2.2)
In addition,

1. The process {n -1/2 [nt] i=1 f • T i , t ∈ [0, 1]} converges in D([0, 1], R m
) equipped with the uniform topology to a Wiener process {W (t), t ∈ [0, 1]} with variance matrix Var(W (1)) = Σ.

2. Enlarging T d if necessary, there exists a sequence (Z i ) i≥1 of iid R m -valued Gaussian random variables with zero mean and variance matrix Var(Z 1 ) = Σ such that

sup 1≤k≤n k i=1 f • T i - k i=1 Z i = o n 1/2 (log log n) 1/2 almost surely, as n → ∞.
When m = 1, it is also possible to exhibit rates of convergence in (1.4) provided that we reinforce Condition (2.1). Theorem 2.2. Let T be an ergodic automorphism of T d . Let p ∈]2, 4] and q := p/(p -1). Let f : T d → R be a centered function with Fourier coefficients (c k ) k∈Z d satisfying the following conditions: there exists a positive constant R such that for every integer b > 1,

|k|≥b |c k | q ≤ R log -θ (b) for some θ > 1 , (2.3
)

and |k|≥b |c k | 2 ≤ R b -ζ for some ζ > 0 . (2.4)
Then the series

σ 2 = k∈Z E(f.f • T k ) (2.5)
converges absolutely and, enlarging T d if necessary, there exists a sequence (Z i ) i≥1 of iid Gaussian random variables with zero mean and variance σ 2 such that

sup 1≤k≤n k i=1 f • T i - k i=1 Z i = o n 1/p log n almost surely, as n → ∞. (2.6)
Observe that if (1.7) holds with α > 1 then (2.3) and (2.4) are both satisfied, so that the strong approximation (2.6) holds. However Theorem 2.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] and Theorem 2.2 above have different ranges of applicability. Indeed, let γ > 1, and define c k = ℓ -γ/q if k = (2 ℓ , 0, . . . , 0), c k = -ℓ -γ/q if k = (-2 ℓ , 0, . . . , 0) for ℓ ∈ N, and c k = 0 otherwise. Let now b and r be positive integers such that 2 r-1 < b ≤ 2 r . Since 1-γ (where λ 1 and λ 2 are two positive constants). Similarly λ 1 (log b) 1-2γ/q ≤ |k|≥b |c k | 2 ≤ λ 2 (log b) 1-2γ/q . In this situation, the conditions (1.5) and (1.6) are both satisfied provided that γ > 1 + (p 2 -2)/(p 2 -p) whereas condition (2.4) fails.

|k|≥b |c k | q = 2 ℓ≥r 1 ℓ γ , it follows that λ 1 (log b) 1-γ ≤ |k|≥b |c k | q ≤ λ 2 (log b)
To prove Theorem 2.2, we shall still use martingale approximations as done in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], but with the following modifications: Condition (2.4) allows us to consider a non stationary sequence X * ℓ = f ℓ • T ℓ , where the functions f ℓ are defined through a truncated series of the Fourier coefficients of f . For the partial sums associated to this non stationary sequence, the approximation error by a non stationary martingale can be suitably handled with the help of Condition (2.3).

Proofs of Theorems 2.1 and 2.2

As in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], we consider the filtration as defined in [START_REF] Lind | Dynamical properties of quasihyperbolic toral automorphisms[END_REF][START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] that enables to suitably approximate the partial sums n i=1 f • T i by a martingale. To be more precise, given a finite partition P of T d , we define the measurable partition P ∞ 0 by :

∀x ∈ T d , P ∞ 0 (x) := k≥0 T k P(T -k (x))
and, for every integer n, the σ-algebra F n generated by

∀x ∈ T d , P ∞ -n (x) := k≥-n T k P(T -k (x)) = T -n (P ∞ 0 (T n (x)) .
These definitions coincide with the ones of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] applied to the ergodic toral automorphism T -1 . We obviously have

F n ⊆ F n+1 = T -1 F n . Note that the sequence (f • T i ) i≥1 is non adapted to (F i ) i≥1 .
In what follows, we use the notation

E n (f ) = E(f |F n ).
3.1 Proof of Theorem 2.1

According to Theorem 4.1 and Remark 4.3 given in Appendix, it suffices to verify that condition (4.35) is satisfied. Therefore, it suffices to verify that for any j ∈ {1, . . . , m},

n≥3 log n n 1/2 (log log n) 1/2 E 0 (f j • T n ) 2 < ∞ and n≥3 log n n 1/2 (log log n) 1/2 f j -E n (f j ) 2 < ∞ . (3.1)
But, according to the proof of Propositions 4.2 and 4.3 of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] (see also [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]), for any f j satisfying (2.1),

E -n (f j ) 2 + f -E n (f j ) 2 ≪ n -θ/2 .
Since θ > 1, (3.1) is satisfied.

Proof of Theorem 2.2

Let f : T d → R be a centered function with Fourier coefficients (c k ) k∈Z d . For every nonnegative integer m, we write

f m := |k|≤m c k e 2iπ k,• . (3.2)
Notice that if f satisfies (2.4), then

f -f m 2 ≤ Rm -ζ/2 , (3.3)
and if f satisfies (2.3), then

f -f m p ≤ R(log(m)) -θ(p-1)/p . (3.4)
According to the proofs of Propositions 4.2, 4.3 and 4.4 of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], there exist c ≥ 1 and γ, λ

∈ (0, 1) such that, setting b(n) := [γ -n ], we have sup m≤b(n) ( E -n (f m ) p + f m -E n (f m ) p ) ≪ λ n (3.
E -N (f m f m • T ℓ ) -E(f m f m • T ℓ ) p/2 ≪ λ n (3.6)
(according to (4.61) and (4.62) of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF]). Moreover, according to the proof of Propositions 4.2 and 4.3 of [6], we have, for any f satisfying (2.3),

sup m≥1 ( E -n (f m ) p + f m -E n (f m ) p ) ≪ n -θ(p-1)/p , (3.7)
and

E -n (f ) p + f -E n (f ) p ≪ n -θ(p-1)/p . (3.8)
In addition, according to the proof of Proposition 4.4 of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], there exists a positive integer c, such that for any f satisfying (2.3) and (2.4),

max 1≤k≤n E -nc (S 2 k (f )) -E(S 2 k (f )) p/2 ≪ n 2-2θ(p-1)/p . (3.9)
For any f satisfying (2.4), using the arguments developed in the proofs of Propositions 4.2 and 4.3 of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], we infer that there exists β ∈ (0, 1) such that sup

m≥1 ( E -n (f m ) 2 + f m -E n (f m ) 2 ) ≪ β n , (3.10)
and

E -n (f ) 2 + f -E n (f ) 2 ≪ β n . (3.11) Let us write P ℓ (•) = E ℓ (•) -E ℓ-1 (•). Now, let α be a positive real such that αζ ≥ 3 -2/p. We then define d * 1 := k∈Z P 1 f 1 • T k , X * 1 := f 1 • T ,
and, for every j ≥ 0 and every ℓ ∈ {2 j + 1, ..., 2 j+1 },

d * ℓ := k∈Z P ℓ f [2 αj ] • T k , X * ℓ := f [2 αj ] • T ℓ .
For every positive integer n, we define

M * n (f ) := n ℓ=1 d * ℓ and S * n (f ) := n ℓ=1 X * ℓ .
The conclusion of Theorem 2.2 comes from the three following lemmas.

Lemma 3.1. We have |S n (f ) -S * n (f )| = o(n 1/p (log n)) almost surely. Lemma 3.2. We have |S * n (f ) -M * n (f )| = o(n 1/p (log n)) almost surely. Lemma 3.3. The conclusion of Theorem 2.2 holds with M * n (f ) replacing S n (f ).
Proof of Lemma 3.1. For any nonnegative integer j, let

D j := sup 1≤k≤2 j k+2 j ℓ=2 j +1 (X ℓ -X * ℓ ) . Let N ∈ N * and let k ∈]1, 2 N ]. We first notice that D j ≥ | 2 j+1 ℓ=2 j +1 (X ℓ -X * ℓ )|, so if K is the integer such that 2 K-1 < k ≤ 2 K , then S k -S * k ≤ |X 1 -X * 1 | + K-1 j=0 D j . Consequently, since K ≤ N , max 1≤k≤2 N |S k -S * k | ≤ |X 1 -X * 1 | + N -1 j=0 D j . (3.12)
Therefore, by standard arguments, Lemma 3.1 will follow if we can prove that D j = o j 2 j/p almost surely. This will hold true as soon as

j≥1 D j q q 2 jq/p j q < ∞ for some q ∈ [1, p] . (3.13)
We shall verify (3.13) for q = 2. Notice that

D j 2 = sup 1≤k≤2 j k+2 j ℓ=2 j +1 (f -f [2 αj ] ) • T ℓ 2 ≤ 2 j f -f [2 αj ] 2 .
Hence, by using (2.4), D j 2 2 ≪ 2 2j 2 -ζαj , which together with the fact that αζ ≥ 2 -2/p implies (3.13) with q = 2, and then Lemma 3.1.

Proof of Lemma 3.2. Without loss of generality, we assume that θ < (p 2 -2)/(p(p -1)). Following the beginning of the proof of Lemma 3.1, Lemma 3.2 will be proven if (3.13) holds with D j defined by

D j = sup 1≤k≤2 j k+2 j ℓ=2 j +1 (X * ℓ -d * ℓ ) . (3.14)
With this aim, setting, for every k ∈ {1, ..., 2 j },

R j,k = k ℓ=1 f [2 αj ] • T ℓ - m∈Z P ℓ f [2 αj ] • T m = k+2 j ℓ=2 j +1 (X * ℓ -d * ℓ ) • T -2 j ,
we first observe that

D j p = sup 1≤k≤2 j |R j,k | p ≪ 2 j/p j k=0 2 -k/p R j,2 k p , (3.15)
(where for the inequality we have used inequality [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] in [START_REF] Wu | Strong invariance principles for dependent random variables[END_REF]). Now, according to the proof of Proposition 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] with

X ℓ = f [2 αj ]
• T ℓ and using again stationarity, we get that for any integer k ≥ 0 and any integer

N ≥ 2 k , max 1≤m≤2 k R j,m p ≪ N ℓ=1 E -ℓ (f [2 αj ] ) p + N -1 ℓ=0 f [2 αj ] -E ℓ (f [2 αj ] ) p + 2 k m=1 ℓ≥m+N P -ℓ (f [2 αj ] ) • T ℓ 2 p 1/2 + 2 k m=1 ℓ≥m+N P ℓ (f [2 αj ] ) • T -ℓ 2 p 1/2 . (3.16)
Let us first consider the case where [2 αj ] ≤ b(2 k ). Starting from (3.16) with N = 2 k and using the fact that

P -ℓ (f [2 αj ] ) p ≤ 2 E -ℓ (f [2 αj ] ) p and that P ℓ (f [2 αj ] ) p ≤ f [2 αj ] -E ℓ-1 (f [2 αj ] ) p , we get that max 1≤m≤2 k R j,m p ≪ 2 k ℓ=1 E -ℓ (f [2 αj ] ) p + 2 k -1 ℓ=0 f [2 αj ] -E ℓ (f [2 αj ] ) p + 2 k/2 ℓ≥2 k +1 E -ℓ (f [2 αj ] ) p + 2 k/2 ℓ≥2 k f [2 αj ] -E ℓ (f [2 αj ] ) p .
Therefore, taking into account the upper bound (3.7) for the two first terms in the right hand side, and the upper bound (3.5) to handle the two last terms (since

[2 αj ] ≤ b(2 k )), we derive that max 1≤m≤2 k R j,m p ≪ 2 k(1-θ(p-1)/p) (3.17)
(recall that θ(p -1)/p < 1). On the other hand, starting from (3.16) with N = 2[2 kp/2 ] and using Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF], we get that max

1≤m≤2 k R j,m p ≪ 2[2 kp/2 ] ℓ=1 E -ℓ (f [2 αj ] ) p + 2[2 kp/2 ] ℓ=0 f [2 αj ] -E ℓ (f [2 αj ] ) p + 2 k/2 ℓ≥[2 kp/2 ] E -ℓ (f [2 αj ] ) p ℓ 1/p + 2 k/2 ℓ≥[2 kp/2 ] f [2 αj ] -E ℓ (f [2 αj ] ) p ℓ 1/p .
Therefore, it follows from (3.7) that max

1≤m≤2 k R j,m p ≪ 2 (kp(1-θ(p-1)/p))/2 . (3.18) Let C = [α -1 (log(γ -1
))/(log 2)] and j 0 = (log 2) -1 (log j -log C) .

(3.19) Clearly, if j 0 ≤ k then [2 αj ] ≤ b(2 k
). Therefore using the upper bound (3.18) when k < j 0 and the upper bound (3.17) when k ≥ j 0 , we get that for any positive integer j

j k=0 2 -k/p R j,2 k p ≪ j p/2 j 1/p j θ(p-1)/2 , since θ < (p 2 -2)/(p(p -1)). Now, since θ > 1, it follows that j≥1 j -p j k=0 2 -k/p R j,2 k p p < ∞ .
From (3.15), this implies that (3.13) holds with D j defined by (3.14) and q = p. This ends the proof of lemma 3.2.

Proof of Lemma 3.3. Let M n = n ℓ=1 d ℓ where d ℓ = d 0 • T ℓ with d 0 = i∈Z P 0 (f • T i ).
Notice that the upper bound (3.8) and the fact that θ > 1 imply in particular that

n>1 n -1/p E -n (f ) p < ∞ and n>1 n -1/p f -E n (f ) p < ∞ ,
and then that k∈Z P 0 (X k ) p < ∞ (use for instance Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] to see this). Therefore

k∈Z P 0 (X k ) 2 < ∞. Using (4.2) of Lemma 4.1, we get that S n (f ) -M n 2 = o( √ n) . (3.20)
From (2.2) of Theorem 2.1, we know that n -1 S n (f ) 2 2 converges to σ 2 . It follows from (3.20) that

σ 2 = n -1 E(M 2 n ) = E(d 2 0
). We shall prove now that

M * n -M n 2 = O n 1/p . (3.21)
Let N be the positive integer such that 2 N -1 < n ≤ 2 N . Since M * n -M n , is a martingale, we have that

M * n -M n 2 2 = n ℓ=1 E((d * ℓ -d ℓ ) 2 ) ≤ E((d * 1 -d 1 ) 2 ) + N -1 j=0 2 j+1 ℓ=2 j +1 E((d * ℓ -d ℓ ) 2 ) . (3.22)
By stationarity, for any ℓ ∈ [2 j + 1, 2 j+1 ] ∩ N we get that

d * ℓ -d ℓ 2 = i∈Z P 0 ((f -f [2 αj ] ) • T i ) 2 ≤ 2 j+3 f -f [2 αj ] 2 + i≥2 j+1 P -i (f -f [2 αj ] ) 2 + i≥2 j+1 P i (f -f [2 αj ] ) 2 .
According to

(3.3) f -f [2 αj ] 2 ≤ R2 -ζαj/2 . (3.23)
On the other hand, by Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF],

i≥2 j+1 P -i (f -f [2 αj ] ) 2 ≪ k≥2 j k -1/2 E -k (f -f [2 αj ] ) 2 and i≥2 j+1 P i (f -f [2 αj ] ) 2 ≪ k≥2 j k -1/2 f -f [2 αj ] + E k (f -f [2 αj ] ) 2 .
Using the estimate (3.10) and (3.11), it follows that

i≥2 j+1 P -i (f -f [2 αj ] ) 2 + P -i (f -f [2 αj ] ) 2 = O(β 2 j ) . (3.24)
Combining the upper bounds (3.23) and (3.24) with the fact that αζ ≥ 3 -2/p, it follows that

d * ℓ -d ℓ 2 ≪ 2 -j(p-2)/(2p) .
Using this estimate in (3.22), we obtain that M * n -M n 2 2 ≪ n 2/p , proving (3.21). Now, let us recall Theorem 2.1 in [START_REF] Shao | Almost sure invariance principles for mixing sequences of random variables, Stochastic Process[END_REF] (used with a n = n 2/p (log n)): if there exists a finite constant K such that sup

k≥1 d * k p ≤ K , (3.25)
and if

n i=1 E((d * i ) 2 |F i-1 ) -E((d * i ) 2 ) = o n 2/p (log n) a.s. , (3.26)
then, since E((M * n ) 2 ) ∼ nσ 2 , enlarging T d if necessary, there exists a sequence (Z * ℓ ) ℓ≥1 of independent Gaussian random variables with zero mean and variance

E(Z * ℓ ) 2 = E(d * ℓ ) 2 = (σ * ℓ ) 2 such that sup 1≤k≤n M * k - k ℓ=1 Z * ℓ = o n 1/p (log n) almost surely, as n → ∞. (3.27)
Let (δ k ) k≥1 be a sequence of iid Gaussian random variables with mean zero and variance σ 2 , independent of the sequence (Z * ℓ ) ℓ≥1 . We now construct a sequence (Z ℓ ) ℓ≥1 as follows. If σ * ℓ = 0, then

Z ℓ = δ ℓ , else Z ℓ = (σ/σ * ℓ )Z * ℓ .
By construction, the Z ℓ 's are iid Gaussian random variables with mean zero and variance σ 2 . Let G ℓ = Z ℓ -Z * ℓ and note that (G ℓ ) ℓ≥1 is a sequence of independent Gaussian random variables with mean zero and variances Var(G ℓ ) = (σ -σ * ℓ ) 2 . Notice now that

v 2 n = Var n i=1 G i = n i=1 d i 2 -d * i 2 2 ≤ M n -M * n 2 2 .
From the basic inequality

P max 1≤k≤n k i=1 G i > x ≤ 2 exp - x 2 2v 2 n ,
and the fact that by (3.21), v 2 n ≪ n 2/p , it follows that for any ε > 0,

n>1 n -1 P max 1≤k≤n k i=1 G i > εn 1/p (log n) < ∞ , showing that max 1≤k≤n | k i=1 G i | = o(n
M * k - k ℓ=1 Z ℓ = o(n 1/p (log n)) almost surely, as n → ∞.
It remains to show that (3.25) and (3.26) are satisfied. We start with (3.25). Notice that d * 1 p ≤ k∈Z P -k (f 1 ) p and that, for every j ≥ 0 and every ℓ ∈ {2 j + 1, ..., 2 j+1 },

d * ℓ p ≤ k∈Z P -k (f [2 αj ] ) p .
By Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF],

k∈Z

P -k (g) p ≪ k≥1 k -1/p E -k (g) p + k≥1 k -1/p g -E k (g) p ,
with constants non depending on g. Hence, using the estimate (3.7) and the fact that θ > 1, we get that, for every j ≥ 0 and every ℓ ∈ {2 j + 1, ..., 2 j+1 }, there exists a constant K non depending on j such that d * ℓ p ≤ K. This ends the proof of (3.25). To prove (3.26), we proceed as follows. Following the beginning of the proof of Lemma 3.1, we infer that (3.26) will be proven if we can show that

D j := sup 1≤ℓ≤2 j 2 j +ℓ i=2 j +1 (E((d * i ) 2 |F i-1 ) -E((d * i ) 2 )) = o j 2 2j/p a.s. . (3.28)
This will hold true as soon as

j≥1 D j p/2 p/2 2 j j p/2 < ∞ . (3.29)
For any j fixed and any i ∈ Z, let d j,i = k∈Z P i f [2 αj ] • T k . By stationarity

D j p/2 := sup 1≤ℓ≤2 j ℓ i=1 (E(d 2 j,i |F i-1 ) -E(d 2 j,i )) p/2
.

Observe now that, for any j fixed, (d j,i ) i∈Z is a stationary sequence of martingale differences in L p . Let

M j,n := n i=1 d j,i .
Applying Theorem 3 in [START_REF] Wu | Moderate deviations for stationary processes[END_REF] (since 1 < p/2 ≤ 2) and using the martingale property of the sequence (M j,n ) n≥1 , we get that

E sup 1≤ℓ≤2 j ℓ i=1 (E(d 2 j,i |F i-1 )-E(d 2 j,i )) p/2 ≪ 2 j d 2 j,1 p/2 p/2 +2 j j-1 k=0 E(M 2 j,2 k |F 0 ) -E(M 2 j,2 k ) p/2 2 2k/p p/2 . Using the fact that d 2 j,1 p/2 = d j,1 2
p ≤ K where K does not depend on j, the convergence (3.29) will be then proven if we can show that j≥1

1 j p/2 j-1 k=0 2 -2k/p E 0 (M 2 j,2 k ) -E(M 2 j,2 k ) p/2 p/2 < ∞ .
According to the arguments developed in the proof of Theorems 3.1 and 3.2 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF] (see (3.19) and (3.20) of [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF]), since (M j,k ) k≥1 is a sequence of martingales, we infer that this last convergence will be satisfied as soon as there exists a positive integer c such that

j≥1 1 j p/2 j k=0 2 -2k/p E -c2 k (M 2 j,2 k ) -E(M 2 j,2 k ) p/2 p/2 < ∞ . (3.30)
We shall prove in what follows that this convergence holds as soon as c is chosen in such a way that (3.9) holds true.

For any positive integer n, let

S j,n = n ℓ=1 f [2 αj ] • T ℓ and R j,n = S j,n -M j,n .
We first write that

E -c2 k (M 2 j,2 k ) -E(M 2 j,2 k ) p/2 ≤ E -c2 k (S 2 j,2 k ) -E(S 2 j,2 k ) p/2 + 4 E -c2 k (S j,2 k R j,2 k ) p/2 + 2 R j,2 k 2
p . (3.31) Let j 0 be defined as in (3.19). Using the upper bound (3.18) when k < j 0 and the upper bound (3.17) when k ≥ j 0 , we get that for any positive integer j,

j k=0 2 -2k/p max 1≤m≤2 k R j,m 2 p ≪ j p j 2/p j θ(p-1) ,
since we can assume without loss of generality that θ < (p 2 -1)/(p(p -1)). Now, since θ > 1, it follows that

j≥1 1 j p/2 j k=0 2 -2k/p max 1≤m≤2 k R j,m 2 p p/2 < ∞ . (3.32)
On an other hand, c being chosen such that (3.9) holds true, the upper bound in (3.9) together with the fact that θ > 1, implies that

j k=0 2 -2k/p max 1≤m≤2 k E -c2 k (S 2 j,m ) -E(S 2 j,m ) p/2 ≪ j k=0
2 -2k/p 2 2k(1-θ(p-1)/p) = O(1) .

Therefore,

j≥1 1 j p/2 j k=0 2 -2k/p max 1≤m≤2 k E -c2 k (S 2 j,m ) -E(S
1 j p/2 j k=0 2 -2k/p E -c2 k (S j,2 k R j,2 k ) p/2 p/2 < ∞ . (3.34)
With this aim, we use Inequality (3.24) in [START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF]

(taking n = 2 k , u n = [2 k/2 ], r = c2 k ). Therefore, E -c2 k (S j,2 k R j,2 k ) p/2 ≪ 2 k/4 E 0 (S j,2 k ) 2 + S j,2 k -E 2 k (S j,2 k ) 2 + max m={2 k ,2 k -[2 k/2 ]} R j,m 2 p + 2 k/2 E -[2 k/2 ] (S j,2 k ) 2 + S j,2 k -E 2 k +[2 k/2 ] (S j,2 k ) 2 + max m={2 k ,[2 k/2 ]} E -c2 k (S 2 j,m ) -E(S 2 j,m ) p/2 + 2 k |ℓ|≥2 k P 0 (f [2 jα ] • T ℓ ) 2 . (3.35)
By stationarity,

E 0 (S j,2 k ) 2 + S j,2 k -E 2 k (S j,2 k ) 2 ≤ 2 k ℓ=1 E -ℓ (f [2 jα ] ) 2 + 2 k -1 ℓ=0 f [2 jα ] -E ℓ (f [2 jα ] ) 2 .
Hence by using (3.10),

2 k/4 E 0 (S j,2 k ) 2 + S j,2 k -E 2 k (S j,2 k ) 2 ≪ 2 k/4 . (3.36)
Using again (3.10) and the stationarity, we get that

E -[2 k/2 ] (S j,2 k ) 2 + S j,2 k -E 2 k +[2 k/2 ] (S j,2 k ) 2 ≤ 2 k ℓ=1 E -([2 k/2 ]+ℓ) (f [2 jα ] ) 2 + 2 k -1 ℓ=0 f [2 jα ] -E [2 k/2 ]+ℓ (f [2 jα ] ) 2 ≪ β 2 k/2 . Therefore 2 k/2 E -[2 k/2 ] (S j,2 k ) 2 + S j,2 k -E 2 k +[2 k/2 ] (S j,2 k ) 2 = O(1) . (3.37)
On an other hand, using again (3.10) and the stationarity,

|ℓ|≥2 k P 0 (f [2 jα ] • T ℓ ) 2 ≤ ℓ≥2 k E -ℓ (f [2 jα ] ) 2 + f [2 jα ] -E ℓ-1 (f [2 jα ] ) 2 ≪ β 2 k , which implies that 2 k |ℓ|≥2 k P 0 (f [2 jα ] • T ℓ ) 2 = O(1) . ( 3

Appendix

Let (Ω, A, P) be a probability space, and T : Ω → Ω be a bijective bimeasurable transformation preserving the probability P. Let us denote by | • | m the euclidean norm on R m and by < •, • > m the associated scalar product. For a σ-algebra F 0 satisfying F 0 ⊆ T -1 (F 0 ), we define the nondecreasing filtration (F i ) i∈Z by

F i = T -i (F 0 ). Let F -∞ = k∈Z F k and F ∞ = k∈Z F k . For a random variable X with values in R m , we denote by X p,m = (E(|X| p m )) 1/p its norm in L p (R m ).
In what follows X 0 is a random variable with values in R m , and we define the stationary sequence

(X i ) i∈Z by X i = X 0 • T i . We shall use the notations E k (X) = E(X|F k ), E ∞ (X) = E(X|F ∞ ), E -∞ (X) = E(X|F -∞ ), and P k (X) = E k (X) -E k-1 (X).
The aim of this section is to collect some results about invariance principles for stationary sequences that are non necessarily adapted to the underlying filtration. We start with a martingale approximation result. The estimate (4.2) of Proposition 4.1 below is a generalization of Item 2 of Theorem 1 in [START_REF] Wu | Strong invariance principles for dependent random variables[END_REF] to the multidimensional case and to the case where the variables are non necessarily adapted to the filtration under consideration. The convergence (4.3) is new. Notice that the proof of the next lemma is based on algebraic computations and on Burkholder's inequality. Burkholder's inequality being also valid in Hilbert spaces (see [START_REF] Burkholder | Sharp inequalities for martingales and stochastic integrals[END_REF]), the approximation lemma below is then also valid for variables taking values in a separable Hilbert (2, p). Assume that E -∞ (X 0 ) = 0 almost surely, that E ∞ (X 0 ) = X 0 almost surely, and that i∈Z P 0 (X i ) p,m < ∞ .

(4.1)

Let d 0 = i∈Z P 0 (X i ), M n := n i=1 d 0 • T i and R n := S n -M n . For any positive integer n, R n p ′ p,m ≪ n k=1 |ℓ|≥k P ℓ (X 0 ) p,m p ′ . (4.2)
In addition,

max 1≤k≤n |R k | m p = o(n 1/p ′ ) as n → ∞. (4.3)
Remark 4.1. The constant appearing in (4.2) depends only on p and not on (Ω, A, P, T, X 0 , F 0 ).

Proof of Proposition 4.1. It will be useful to note that P i (X j ) = P i-j (X 0 ) • T j = P 0 (X j-i ) • T i almost surely. The following decomposition is valid:

R n = n k=1 X k - n j=1 P j (X k ) - n k=1 j≥n+1 P k (X j ) - n k=1 ∞ j=0 P k (X -j ) = E 0 (S n ) - n k=1 j≥n+1 P k (X j ) + S n -E n (S n ) - n k=1 ∞ j=0 P k (X -j ) . (4.4)
Applying Burkholder's inequality for multivariate martingales, and using the stationarity, we obtain that there exists a positive constant c p such that, for any positive integer n,

n k=1 j≥n+1 P k (X j ) p ′ p,m ≤ c p n k=1 j≥n+1 P k (X j ) p ′ p,m ≤ c p n k=1 j≥k P 0 (X j ) p,m p ′ , (4.5)
and n k=1 j≥0

P k (X -j ) p ′ p,m ≤ c p n k=1 j≥0 P k (X -j ) p ′ p,m = c p n k=1 j≥k P 0 (X -j ) p,m p ′ . ( 4.6)
On an other hand, since E -∞ (X 0 ) = 0 almost surely, we have E 0 (S n ) = k≥0 P -k (S n ) almost surely. Hence by Burkholder's inequality for multivariate martingales together with stationarity, there exists a positive constant c p depending only on p such that

E 0 (S n ) p ′ p,m ≤ c p k≥0 P -k (S n ) p ′ p,m ≤ c p k≥0 n ℓ=1 P -k (X ℓ ) p,m p ′ ≤ c p n-1 k=0 ℓ≥k+1 P -ℓ (X 0 ) p,m p ′ + c p k≥n k+n ℓ=k+1 P -ℓ (X 0 ) p,m p ′ ≤ c p n-1 k=0 ℓ≥k+1 P -ℓ (X 0 ) p,m p ′ + c p i≥n+1 P -i (X 0 ) p,m p ′ -1 k≥n k+n ℓ=k+1 P -ℓ (X 0 ) p,m ≤ c p n-1 k=0 ℓ≥k+1 P -ℓ (X 0 ) p,m p ′ + c p n ℓ≥n+1 P -ℓ (X 0 ) p,m p ′ . Therefore E 0 (S n ) p ′ p,m ≤ 2c p n k=1 ℓ≥k P -ℓ (X 0 ) p,m p ′ . ( 4.7)
We handle now the quantity S n -E n (S n ) p ′ p,m . Since E ∞ (X 0 ) = X 0 almost surely, we first write that S n -E n (S n ) = k≥n+1 P k (S n ). Hence, applying Burkholder's inequality for multivariate martingales and using the stationarity, we infer that there exists a positive constant c p depending only on p such that

S n -E n (S n ) p ′ p,m ≤ c p k≥n+1 P k (S n ) p ′ p,m ≤ c p k≥n+1 n ℓ=1 P k (X ℓ ) p,m p ′ ≤ c p 2n k=n+1 k-1 ℓ=k-n P ℓ (X 0 ) p,m p ′ + c p k≥2n+1 k-1 ℓ=k-n P ℓ (X 0 ) p,m p ′ ≤ c p n k=1 ℓ≥k P ℓ (X 0 ) p,m p ′ + c p i≥n+1 P i (X 0 ) p,m p ′ -1 k≥2n+1 k-1 ℓ=k-n P ℓ (X 0 ) p,m ≤ c p n k=1 ℓ≥k P ℓ (X 0 ) p,m p ′ + c p n ℓ≥n+1 P ℓ (X 0 ) p,m p ′ . Therefore S n -E n (S n ) p ′ p,m ≤ 2c p n k=1 ℓ≥k P ℓ (X 0 ) p,m p ′ . ( 4.8)
Starting from (4.4) and taking into account the upper bounds (

We turn now to the proof of (4.3). Let r be some fixed positive integer. Since

M k = k i=1 d 0 • T i and d 0 • T i = j∈Z P i (X j ), the following decomposition holds: R k = k i=1 X i - k i=1 r+i ℓ=-r-i P i (X ℓ ) - k i=1 ℓ≥r+i+1 P i (X ℓ ) - k i=1 ℓ≥r+i+1 P i (X -ℓ ) . (4.9)
Applying Burkholder's inequality for multivariate martingales and using the stationarity, we infer that there exists a positive constant c p depending only on p such that, for any positive integer n,

max r≤k≤n k i=1 ℓ≥r+i+1 P i (X ℓ ) m p ′ p ≤ c p n i=1 ℓ≥r+i+1 P i (X ℓ ) p ′ p,m ≤ c p n j≥r+1 P 0 (X j ) p,m p ′ , (4.10)
since

P i (X ℓ ) = P 0 (X ℓ-i ) • T i . Similarly max r≤k≤n k i=1 ℓ≥r+i+1 P i (X -ℓ ) m p ′ p ≤ c p n j≥r+1 P 0 (X -j ) p,m p ′ . ( 4.11)
We write now that

k i=1 X i - k i=1 r+i ℓ=-r-i P i (X ℓ ) = k i=1 X i - k i=1 r+i ℓ=1 P i (X ℓ ) - k i=1 r+i ℓ=0 P i (X -ℓ ) . (4.12)
The following decomposition holds

k i=1 r+i ℓ=1 P i (X ℓ ) = k-r i=1 r+i ℓ=1 P i (X ℓ ) + k i=k-r+1 r+i ℓ=1 P i (X ℓ ) = k ℓ=1 k-r i=1 1 i≥ℓ-r P i (X ℓ ) + k+r ℓ=1 k i=k-r+1 1 i≥ℓ-r P i (X ℓ ) . (4.13) Now, k ℓ=1 k-r i=1 1 i≥ℓ-r P i (X ℓ ) = r ℓ=1 k-r i=1 P i (X ℓ ) + k ℓ=r+1 k-r i=ℓ-r P i (X ℓ ) = E k-r (S r ) -E 0 (S r ) + E k-r (S k -S r ) - k ℓ=r+1 E ℓ-r-1 (X ℓ ) = E k-r (S k ) -E 0 (S r ) - k ℓ=r+1 E ℓ-r-1 (X ℓ ) , (4.14)
and k+r ℓ=1 k i=k-r+1

1 i≥ℓ-r P i (X ℓ ) = k ℓ=1 k i=k-r+1 P i (X ℓ ) + k+r ℓ=k+1 k i=ℓ-r P i (X ℓ ) = E k (S k ) -E k-r (S k ) + E k (S k+r -S k ) - k+r ℓ=k+1 E ℓ-r-1 (X ℓ ) = E k (S k+r ) -E k-r (S k ) - k+r ℓ=k+1 E ℓ-r-1 (X ℓ ) . (4.15)
Therefore starting from (4.12), and considering the decompositions To handle the fourth term in the right-hand side of (4.17) we proceed as follows. Since E -∞ (X ℓ ) = 0 almost surely, we first write that

E ℓ-r-1 (X ℓ ) = ∞ j=r+1 P ℓ-j (X ℓ ) . Then max r≤k≤n k+r ℓ=r+1 E ℓ-r-1 (X ℓ ) m ≤ ∞ j=r+1 max r≤k≤n k+r ℓ=r+1 P ℓ-j (X ℓ ) m .
Let now

u i = P 0 (X i ) p,m , C r = i≥r+1 u i and α i = C -1 r u i .
By using the facts that for any p ≥ 1, x → x p is convex and that α i ≥ 0 with i≥r+1 α i = 1 and writing j a j = j α j (a j /α j ), we obtain that max

r≤k≤n k+r ℓ=r+1 E ℓ-r-1 (X ℓ ) m p p ≤ ∞ j=r+1 α 1-p j E max r≤k≤n k+r ℓ=r+1 P ℓ-j (X ℓ ) p m .
Applying Burkholder's inequality for multivariate martingales and using the stationarity, we infer that there exists a positive constant c p depending only on p such that, for any positive integer n,

max r≤k≤n k+r ℓ=r+1 P ℓ-j (X ℓ ) m p ′ p ≤ c p n+r ℓ=r+1 P ℓ-j (X ℓ ) p ′ p,m ≤ c p n P 0 (X j ) p ′ p,m . So, overall max r≤k≤n k+r ℓ=r+1 E ℓ-r-1 (X ℓ ) m p p ≤ (c p n) p/p ′ ∞ j=r+1 α 1-p j u p j = (c p n) p/p ′ ∞ j=r+1 P 0 (X j ) p p . (4.19)
We handle now the second term in the right-hand side of (4.17). We first write that

S k -E k (S k+r ) = S k -S k-r -E k (S k+r -S k-r ) + S k-r -E k (S k-r ) . (4.20) Let Y r = 0 i=-(r-1) X i - r i=-(r-1) E 0 (X i ) .
With this notation,

S k -S k-r -E k (S k+r -S k-r ) = Y r • T k .
Hence, for any positive real A,

max r≤k≤n S k -S k-r -E k (S k+r -S k-r ) m p p ≤ 2 p A p + 2 p max r≤k≤n |Y r 1 |Yr|m>A • T k | m p p ≤ 2 p A p + 2 p n Y r 1 |Yr|m>A p p,m .
Since Y r p,m ≤ K r where K r is a constant depending on r, we get that lim

A→∞ lim sup n→∞ 1 n max r≤k≤n S k -S k-r -E k (S k+r -S k-r ) m p ′ p = 0 . ( 4

.21)

We deal now with the term max r≤k≤n |S k-r -E k (S k-r )| m p . Since E ∞ (X 0 ) = X 0 almost surely, we have that, almost surely

S k-r -E k (S k-r ) = k-r ℓ=1 ℓ-k-1 j=-∞ P ℓ-j (X ℓ ) = -k j=-∞ k-r ℓ=1 P ℓ-j (X ℓ ) + -r-1 j=-k+1 k-r ℓ=k+j+1 P ℓ-j (X ℓ ) . Therefore max r≤k≤n |S k-r -E k (S k-r )| m ≤ -r j=-∞ max r≤k≤n k-r ℓ=1 P ℓ-j (X ℓ ) m + -r-1 j=-n+1 max r≤k≤n k-r ℓ=k+j+1 P ℓ-j (X ℓ ) m ≤ -r j=-∞ max r≤k≤n k-r ℓ=1 P ℓ-j (X ℓ ) m + -r-1 j=-n+1 max r≤k≤n k-r ℓ=r-n P ℓ-j (X ℓ ) m + -r-1 j=-n+1 max r≤k≤n k+j ℓ=r-n P ℓ-j (X ℓ ) m . (4.22) Let u i = P 0 (X i ) p,m , C r = -r
i=-∞ u i and α i = C -1 r u i . As before, using the facts that for any p ≥ 1, x → x p is convex and that α

i ≥ 0 with -r i=-∞ α i = 1, we obtain that -r j=-∞ max r≤k≤n k-r ℓ=1 P ℓ-j (X ℓ ) m p p ≤ -r j=-∞ α 1-p j E max r≤k≤n k-r ℓ=1 P ℓ-j (X ℓ ) p m .
Applying Burkholder's inequality for multivariate martingales and using the stationarity, we infer that there exists a positive constant c p depending only on p such that, for any positive integer n,

-r j=-∞ max r≤k≤n k-r ℓ=1 P ℓ-j (X ℓ ) m p p ≤ (c p n) p/p ′ i≥r P 0 (X -i ) p p . (4.23)
With similar arguments, we derive that

|S k -E k (S k+r )| m p ≤ n 1/p ′ i≥r P 0 (X -i ) p + o(n 1/p ′ ) . ( 4
|R k | m p ≤ max 1≤k≤r |R k | m p + max r≤k≤n |R k | m p ≪ r 2 + n 1/p ′ |j|≥r P 0 (X j ) p,m + o(n 1/p ′ ) .
Letting first n tend to infinity and next r tend to infinity, (4.3) follows.

Starting from Proposition 4.1 one can prove the following theorem concerning the weak and strong invariance principles for non-adapted sequences.

Theorem 4.1. Let X 0 be a zero mean random variable in L 2 (R m ) and F 0 a σ-algebra satisfying Remark 4.2. The weak invariance principle (Item 1 of Theorem 4.1) still holds if T is not ergodic, but in that case the limiting distribution is a mixture of Brownian motion (this has been proved in [START_REF] Dedecker | On the weak invariance principle for non adapted sequences under projective criteria[END_REF] when m = 1). This weak invariance principle can be also extended to separable Hilbert spaces, with the appropriate covariance operator. In the adapted case (i.e. X 0 is F 0 -measurable), the non ergodic Hilbert-valued version of Item 1 has been proved in [START_REF] Dedecker | The conditional central limit theorem in Hilbert spaces[END_REF].

F 0 ⊆ T -1 (F 0 ). For any i ∈ Z, let X i = X 0 • T i and F i = T -i (F 0 ). Let S n = X 1 + • • • + X n . Assume that T is ergodic, that E -∞ (X 0 ) = 0 almost surely, and that E ∞ (X 0 ) = X 0 almost surely. 1. Assume that n∈Z P 0 (X n ) 2,m < ∞ . (4.27) Then n -1 Var(S n ) converges to C = k∈Z Cov X 0 , X k . ( 4
Proof of Theorem 4.1. Let S m,1 = {x ∈ R m : |x| m = 1}. For a matrix A from R m to R m , let |A| m = sup x∈Sm,1 |Ax| m . By stationarity n -1 Var(S n ) = n -1 |k|<n (n -|k|)Cov(X 0 , X k ). Hence n -1 Cov(S n ) converges to C provided that k∈Z |Cov(X 0 , X k )| m < ∞. Since E -∞ (X k ) = 0 almost surely and E ∞ (X k ) = X k almost surely, it follows that X k = i∈Z P i (X k ) almost surely. Moreover Cov(P i (X 0 ), P j (X k )) = 0 for i = j. Hence, Cov(X 0 , X k ) = i∈Z Cov(P i (X 0 ), P i (X k )) , and consequently |Cov(X 0 , X k )| m ≤ i∈Z P i (X 0 ) 2,m P i (X k ) 2,m . By (4.27) it follows that k∈Z |Cov(X 0 , X k )| m ≤ k∈Z i∈Z P i (X 0 ) 2,m P i (X k ) 2,m = i∈Z P 0 (X i ) 2,m 2 < ∞ , which proves the convergence of n -1 Var(S n ) to C.

Let now d 0 := j∈Z P 0 (X j ). Since (4.27) is assumed, d 0 belongs to L 2 (R m ). In addition E(d 0 |F -1 ) = 0 almost surely. Let d i := d 0 • T i for all i ∈ Z. Then (d i ) i∈Z is a stationary ergodic sequence of martingale differences in L 2 (R m ). Let (4.31)

Since n -1 Var(S n ) converges to C, it follows that Var(d 0 ) = C. Therefore, Item 1 of Theorem 4.1 follows from the weak invariance principle for partial sums of stationary multivariate martingale differences in L 2 (R m ) (see [START_REF] Dedecker | The conditional central limit theorem in Hilbert spaces[END_REF] for the non ergodic Hilbert-valued version) together with the maximal martingale approximation given in (4.31).

We turn now to the proof of Item 2. According to Theorem 3.1 in [START_REF] Berger | An almost sure invariance principle for stationary ergodic sequences of Banach space valued random variables[END_REF] (that is the generalization of the Strassen's invariance principle [START_REF] Strassen | Almost sure behavior of sums of independent random variables and martingales[END_REF] for real martingales with ergodic increments to the multivariate case), enlarging the probability space if necessary, there exists a sequence (Z i ) i≥1 of iid Gaussian random variables in R m with zero mean and covariance Var(Z 1 ) = C such that sup

5 )(

 5 according to (4.50), (4.51) and (4.53) of[START_REF] Dedecker | Rates of convergence in the strong invariance principle for non adapted sequences[END_REF]

  .38) Starting from (3.35) and taking into account the convergence (3.32) and (3.33), and the upper bounds (3.36), (3.37) and (3.38), we then derive that (3.34) holds. This ends the proof of (3.26) and therefore of Lemma 3.3.

  space, H, by replacing the norm | • | m by the norm on H, let say | • | H . Proposition 4.1. Let p ∈ [1, ∞[ and p ′ = min

PP

  i (X ℓ ) = S k -E k (S k+r ) + E 0 (S r ) + k+r ℓ=r+1 E ℓ-r-1 (X ℓ )i (X -ℓ ) . (4.16)The decomposition (4.9) together with the upper bounds (4.10), (4.11) and (4.16) imply thatmax r≤k≤n |R k | m p ′ p ≪ n |j|≥r+1 P 0 (X j ) p,m p ′ + max r≤k≤n S k -E k (S k+r ) m p ′p + E 0 (S r ) p ′ s inequality for multivariate martingales and using the stationarity, there exists a positive constant c p depending only on p such that, for any positive integer n,

  from (4.22) and considering the upper bounds (4.23) and (4.24), we get thatmax r≤k≤n |S k-r -E k (S k-r )| m p ≤ n 1/p ′ i≥r P 0 (X -i ) p . (4.25) From the decomposition (4.20) together with (4.21) and (4.25), it follows that max r≤k≤n

=

  ) = C given by (4.28), o (n log log n) 1/2 almost surely, as n → ∞. (4.30)

M n := n i=1 d|R k | m 2 2 =

 i=12 i and R n := S n -M n . Using (4.27), it follows from (4.3) of Lemma 4.1 that max 1≤k≤n o(n) .

= 1 / 2 n 3 / 2 (

 1232 o (n log log n) 1/2 almost surely, as n → ∞.Therefore the strong approximation result (4.30) will follow if we can show that|R n | m = o (n log log n) 1/2 almost surely, as n → ∞. (4.32) Since R n = n i=1 (f -d 0 ) • T i , (4.32) will follow by Theorem 4.7 in [3] if we can prove that n>3 R n 2 n 3/2 (log log n) 1/2 < ∞ . Using (4.2) of Lemma 4.1, this last convergence will hold provided that n>3 n k=1 |ℓ|≥k P ℓ (X 0 ) 2,m 2 log log n) 1/2 < ∞ . (4.33)

  1/p (log n)) almost surely. Therefore starting from (3.27), we conclude that if (3.25) and (3.26) hold then Lemma 3.3 does; namely, enlarging T d if necessary, there exists a sequence (Z ℓ ) ℓ≥1 of iid Gaussian random variables with zero mean and variance σ 2 such that sup 1≤k≤n

  Then, enlarging the probability space if necessary, there exists a sequence (Z i ) i≥1 of iid Gaussian random variables in R m with zero mean and variance matrix Var(Z i

.28)

In addition the process {n -1/2 S [nt] , t ∈ [0, 1]} converges in D([0, 1], R m

) equipped with the uniform topology to a Wiener process {W (t), t ∈ [0, 1]} with variance matrix Var(W (1)) = C.

2. Assume that

n≥3 log n( P 0 (X n ) 2,m + P 0 (X -n ) 2,m ) (log log n) 1/2 < ∞ .

(4.29)

Notice that n>3 n k=1

. Now, using the subadditivity of x → x 1/2 , it follows that (4.33) will be satisfied as soon as

which holds as soon as (4.29) does (changing the order of summation in ℓ j k ). This ends the proof of Item 2 of Theorem 4.1.

For the sake of applications, we now give sufficient conditions for (4.27) and (4.29) to hold.

Remark 4.3. The condition (4.27) is satisfied if we assume that

and the condition (4.29) holds if we assume that

The proof of the remark above is omitted since it uses exactly the arguments developed to prove Remarks 3.3 and 3.6 in [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF] (see Section 5.5 of [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF]). Notice that the conditions (4.34) or (4.35) imply clearly that E -∞ (X 0 ) = 0 almost surely and that E ∞ (X 0 ) = X 0 almost surely.