Jérôme Dedecker

Florence Merlevède

Françoise Pene

Françoise Pène

Rates of convergence in the

Keywords: Mathematics Subject Classification (2010). 60F17; 37D30. Keywords. almost sure invariance principle, strong approximations, non adapted sequences, ergodic automorphisms of the torus

Introduction and notations

Let (Ω, A, P) be a probability space, and T : Ω → Ω be a bijective bimeasurable transformation preserving the probability P. 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 ). The L p norm of a random variable X is denoted by X p = (E(|X| p )) 1/p . Let X 0 be a real-valued and square integrable random variable such that E(X 0 ) = 0, and define the stationary sequence (X i ) i∈Z by X i = X 0 • T i . Define then the partial sum by S n = X 1 + X 2 + • • • + X n . According to the Birkhoff-Khinchine theorem, S n satisfies a strong law of large numbers. One can go further in the study of the statistical properties of S n . We study here the rate of convergence in the almost sure invariance principle (ASIP). More precisely, we give conditions under which there exists a sequence of F. Pène is partially supported by the French ANR projects MEMEMO2 and PERTURBATIONS.

independent identically distributed (iid) Gaussian random variables (Z i ) i≥1 such that sup 1≤k≤n k i=1 (X i -Z i ) = o(n 1/p L(n)) almost surely, (1.1) for p ∈]2, 4] and L an explicit slowly varying function. Let us recall that, in the iid case, Komlós, Major and Tusnády [START_REF] Komlós | An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrscheinlichkeitstheorie und Verw[END_REF] and Major [START_REF] Major | The approximation of partial sums of independent RV's[END_REF] obtained an ASIP with the optimal rate o(n 1/p ) in (1.1) as soon as the random variables admit a moment of order p.

Since the seminal paper by Philipp and Stout [START_REF] Philipp | Almost sure invariance principle for partial sums of weakly dependent random variables[END_REF], many authors have considered this problem in a dependent context, but most of the papers deal with the adapted case, when X 0 is F 0 measurable (for instance, F 0 is the past σ-algebra σ(X i , i ≤ 0)). Unfortunately, it is quite common to encounter dynamical systems for which the natural filtration does not permit the control of any quantity involving terms of the type E(X n |F 0 ) p .

In this paper, we shall not assume that X 0 is F 0 -measurable, and we shall give conditions on the quantities E(X n |F 0 ) p , X -n -E(X -n |F 0 ) p and E(S 2 n |F -n ) -E(S 2 n ) p/2 for (1.1) to hold (see Theorems 3.1 and 3.2 of Section 3). These conditions are in the same spirit as those given by Gordin [START_REF] Gordin | The central limit theorem for stationary processes[END_REF] for p = 2 to get the usual central limit theorem. Our proof is based on the approximation

d i = k∈Z (E(X k |F i ) -E(X k |F i-1 ))
introduced by Gordin [START_REF] Gordin | The central limit theorem for stationary processes[END_REF] and Heyde [START_REF] Heyde | On the central limit theorem for stationary processes[END_REF]. In the adapted case, similar conditions are given in the recent paper [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF], together with a long list of applications.

In the non adapted case, it is easy to see that our results apply to a large class of two-sided functions of iid sequences, or two-sided functions of absolutely regular sequences. But they also apply to very complicated dynamical systems, for which such a representation by functions of absolutely regular sequences is not available. In the next section, we consider the case where T is an ergodic automorphism of the d-dimensional torus T d , and P is the Lebesgue measure on T d . In this context, we use the σ-algebras F i considered by Le Borgne [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]. As a consequence of Theorem 2.1, we obtain that (1.1) holds for p = 4 and X i = f • T i , where f : T d → R, as soon as the Fourier coefficients (c k ) k∈Z d of f are such that

|c k | ≤ A d i=1 1 (1 + |k i |) 3/4 log α (2 + |k i |)
for some α > 13/8.

We also get that there exists a positive ε such that sup

1≤k≤n k i=1 (X i -Z i ) = o(n 1/2-ε ) almost surely,
as soon as

|c k | ≤ A d i=1 1 (1 + |k i |) δ for some δ > 1/2.
These rates of convergence in the almost sure invariance principle complement the results by Leonov [START_REF] Leonov | Central limit theorem for ergofic endomorphisms of compact commutative groups[END_REF] and Le Borgne [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] for the central limit theorem and the almost sure invariance principle respectively. Let us mention that Dolgopyat [START_REF] Dolgopyat | Limit theorems for partially hyperbolic systems[END_REF] established an ASIP with the rate o(n 1/2-ε ) (for some ε > 0) valid for ergodic automorphisms of the torus and f a Hölder continuous function. Thanks to the decorrelation estimates obtained in [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF], the rate for Hölder observables can be improved by applying the general result of Gouëzel in [START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF] to get the rate o(n 1/4+ε ) for every ε > 0, and by applying the results of the present paper to get the rate o(n 1/4 L(n)). To our knowledge, the present work gives the first strong approximation results for such partially hyperbolic transformations T for unbounded (and then non continuous) functions f .

To conclude, let us mention some previous works in the context of dynamical systems: several results have been established with the rate o(n 1/2-ε ) for some ε > 0 (see [START_REF] Hofbauer | Ergodic properties of invariant measures for piecewise monotonic transformations[END_REF][START_REF] Denker | Approximation by Brownian motion for Gibbs measures and flows under a function[END_REF][START_REF] Dolgopyat | Limit theorems for partially hyperbolic systems[END_REF][START_REF] Nagayama | Almost sure invariance principle for dynamical systems with stretched exponential mixing rates[END_REF][START_REF] Melbourne | Almost sure invariance principle for nonuniformly hyperbolic systems[END_REF]). Results giving a rate in o(n 1/4+ε ) for every ε > 0 can be found in [START_REF] Melbourne | Central limit theorems and invariance principles for timeone maps of hyperbolic flows[END_REF][START_REF] Field | Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions[END_REF][START_REF] Melbourne | A vector-valued almost sure invariance principle for hyperbolic dynamical systems[END_REF][START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF]. Most of these results hold for bounded functions f .

Let us reiterate that we can reach the rate o(n 1/4 L(n)) instead of o(n 1/4+ε ) for every ε > 0. Moreover, our conditions giving the rate o(n 1/p L(n)) are related to moments of order p of f . Such results are not very common in the context of dynamical systems (let us mention [START_REF] Gouëzel | Almost sure invariance principle for dynamical systems by spectral methods[END_REF] in the particular case of Gibbs-Markov maps, and [START_REF] Dedecker | The almost sure invariance principle for unbounded functions of expanding maps[END_REF][START_REF] Merlevède | Strong approximation of partial sums under dependence conditions with application to dynamical systems[END_REF] for generalized Pommeau-Manneville maps).

ASIP with rates for ergodic automorphisms of the torus

A probability dynamical system (Ω, F, P, T ) is given by a probability space (Ω, F, P) and a measurable P-preserving transformation T : Ω → Ω. Such a dynamical system is said to be ergodic if the only A ∈ F such that T -1 A = A a.s. are the sets of probability 0 or 1.

If (Ω, F, P, T ) is ergodic, the study of the stochastic properties of the stationary sequence (f • T k ) k≥1 starts with the Birkhoff-Khintchine theorem [START_REF] Birkhoff | Proof of the ergodic theorem[END_REF][START_REF] Khintchine | Zu Birkhoffs Lösung des Ergodenproblems[END_REF]. This theorem ensures that, for every integrable function f : Ω → R, the sequence n -1 n k=1 f • T k converges almost surely to E(f ). This means that (f • T k ) k≥1 satisfies a strong law of large number. A natural question is then to investigate further stochastic properties of (f

• T k ) k≥1 .
We illustrate our general Theorems 3.1 and 3.2 by a concrete example of invertible non hyperbolic dynamical system, which is actually partially hyperbolic. We shall prove a strong invariance principle for a large class of unbounded functions f , with a rate depending on the rate of convergence to zero of the Fourier coefficients of f . In this context, we use the σ-algebras F i considered by Le Borgne [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]. The stationary sequence (T i ) i∈Z is non adapted to this filtration (F i ) i∈Z , in the sense that T i is not F i measurable. This is an important difference with the classical probabilistic situation, where the study of stationary sequences can often be done with the help of a natural "past" filtration (think of stationary Markov chains, or of causal linear processes).

Let d ≥ 2. We consider a group automorphism T of the torus T d = R d /Z d . For every x ∈ R d , we write x its class in T d . We recall that T is the quotient map of a linear map T : R d → R d given by T (x) = S • x, where S is a d × d-matrix with integer entries and with determinant 1 or -1. The map x → S • x preserves the infinite Lebesgue measure λ on R d and T preserves the probability Lebesgue measure λ. We suppose that T is ergodic, which is equivalent to the fact that no eigenvalue of S is a root of the unity. In this case, it is known that the spectral radius of S is larger than one (and so S admits at least an eigenvalue of modulus larger than one and at least an eigenvalue of modulus smaller than one). This hypothesis holds true in the case of hyperbolic automorphisms of the torus (i.e. in the case when no eigenvalue of S has modulus one) but is much weaker. Indeed, as mentioned in [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF], the following matrix gives an example of an ergodic non hyperbolic automorphism of T 4 :

S :=     0 0 0 -1 1 0 0 2 0 1 0 0 0 0 1 2     .
When T is ergodic and non hyperbolic, the dynamical system (T d , T, λ) has no Markov partition. However, it is possible to construct some measurable partition [START_REF] Lind | Dynamical properties of quasihyperbolic toral automorphisms[END_REF], to prove a central limit theorem [START_REF] Leonov | Central limit theorem for ergofic endomorphisms of compact commutative groups[END_REF]. Moreover, in [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF], Le Borgne proved the functional central limit theorem and the Strassen strong invariance principle for (X k = f • T k ) k under weak hypotheses on f , thanks to Gordin's method and to the partitions studied by Lind in [START_REF] Lind | Dynamical properties of quasihyperbolic toral automorphisms[END_REF].

We give here rates of convergence in the strong invariance principle for

(X k = f • T k ) k under conditions on the Fourier coefficients of f : T d → R.
In what follows, for k ∈ Z d , we denote by |k| = max i∈{1,...,d} |k i |.

Theorem 2.1. Let T be an ergodic automorphism of T d with the notations as above. Let p ∈]2, 4] and q be its conjugate exponent. Let f : T d → R be a centered function with Fourier coefficients (c k ) k∈Z d satisfying, for any integer b ≥ 2,

|k|≥b |c k | q ≤ R log -θ (b) for some θ > p 2 -2 p(p -1) , (2.2)
and

|k|≥b |c k | 2 ≤ R log -β (b) for some β > 3p -4 p . (2.3)
Then the series

σ 2 = λ((f -λ(f )) 2 ) + 2 k>0 λ((f -λ(f ))f • T k )
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, for any t > 2/p,

sup 1≤k≤n k i=1 f • T i - k i=1 Z i = o n 1/p (log n) (t+1)/2 almost surely, as n → ∞.
(2.4)

Observe that (2.3) follows from (2.2) provided that θ > (3p-4)/(2p-2). Hence, (2.2) and (2.3) are both satisfied as soon as

|k|≥b |c k | q ≤ R log -θ (b) for some θ > 3p -4 2(p -1) . Example. Let p ∈]2, 4].
If we assume that the Fourier coefficients of f are such that

|c k | ≤ A d i=1 1 (1 + |k i |) 1/q log α (2 + |k i |) , (2.5)
for some positive constant A, then the conditions (2.2) and (2.3) are both satisfied provided that α > (2p 2 -p -2)/p 2 .

Let us now compare our hypotheses on Fourier coefficients with those appearing in other works. In [START_REF] Leonov | Central limit theorem for ergofic endomorphisms of compact commutative groups[END_REF], Leonov proved a central limit theorem (possibly degenerated) when

|c k | ≤ A d i=1 1 (1 + |k i |) 1/2 log α (2 + |k i |)
for some α > 3/2.

(2.6)

In [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF], Le Borgne proved the functional central limit theorem and the Strassen strong invariance principle when (2.3) holds true with β > 2 (and when f is not a coboundary), which is a weaker condition than (2.6). Observe that, as p converges to 2, (p 2 -2)/(p(p -1)) and (3p -4)/p both converge to 1.

Probabilistic results

In the rest of the paper, we shall use the following notations: E k (X) = E(X|F k ), and a n ≪ b n means that there exists a numerical constant C not depending on n such that a n ≤ Cb n , for all positive integers n.

In this section, we give rates of convergence in the strong invariance principle under projective criteria for stationary sequences that are non necessarily adapted to F i . Theorem 3.1. Let 2 < p < 4 and t > 2/p. Assume that X 0 belongs to L p , that

n≥2 n p-1 n 2/p (log n) (t-1)p/2 E 0 (X n ) p p + X -n -E 0 (X -n ) p p < ∞ , (3.1)
and that

n≥2 n 3p/4 n 2 (log n) (t-1)p/2 E 0 (X n ) p/2 2 + X -n -E 0 (X -n ) p/2 2 < ∞ . (3.2)
Assume in addition that there exists a positive integer m such that

n≥2 1 n 2 (log n) (t-1)p/2 E -nm (S 2 n ) -E(S 2 n ) p/2 p/2 < ∞ . (3.3) Then n -1 E(S 2 n ) converges to σ 2 = k∈Z Cov(X 0 , X k )
and, enlarging Ω 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 S k - k i=1 Z i = o n 1/p (log n) (t+1)/2
almost surely, as n → ∞.

(3.4)

Theorem 3.2. Let t > 1/2.
Assume that X 0 belongs to L 4 and that the conditions (3.1) and (3.3) hold with p = 4. Assume in addition that

n≥2 n(log n) 4-2t E 0 (X n ) 2 2 + X -n -E 0 (X -n ) 2 2 < ∞ . (3.5)
Then the conclusion of Theorem 3.1 holds with p = 4.

Of course, if (X i ) i∈Z is a sequence of iid random variables in L p , then, taking F i = σ(X k , k ≤ i), all the conditions (3.1), (3.2), (3.3) and (3.5) are satisfied. In that particular case, we obtain an extra power of log(n) compared to the optimal rate n 1/p . The conditions (3.1), (3.2) and (3.5) are similar to the conditions given by Gordin [START_REF] Gordin | The central limit theorem for stationary processes[END_REF] when p = 2 for the central limit theorem, and hence Theorem 3.1 and 3.2 have the same range of applicability as Gordin's result.

For the proof of Theorems 3.1 and 3.2, we shall use the approximating martingale M n introcuded by Gordin [START_REF] Gordin | The central limit theorem for stationary processes[END_REF], and we shall give an appropriate upper bound on R n = S n -M n . The next step is to get a strong approximation result for the martingale M n . This will be done by applying Proposition 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF], which itself is based on the Skorohod embedding for martingales, as in [START_REF] Shao | Almost sure invariance principles for mixing sequences of random variables[END_REF].

3.1. Proofs of Theorems 3.1 and 3.2 Proof. We first notice that since p > 2, (3.1) implies that

n>0 n -1/p E 0 (X n ) p < ∞ and n>0 n -1/p X -n -E 0 (X -n ) p < ∞
(apply Hölder's inequality to see this). Let P k (X) = E k (X)-E k-1 (X). Using Lemma 5.1 of the appendix with q = 1, we infer that k∈Z P 0 (X k ) p < ∞ .

(3.6)

In addition the condition (3.6) implies that n -1 E(S 2 n ) converges to the quantity σ 2 = k∈Z Cov(X 0 , X k ).

Let now d 0 := j∈Z P 0 (X j ). Then d 0 belongs to L p and E(d

0 |F -1 ) = 0. Let d i := d 0 • T i for all i ∈ Z. Then (d i ) i∈Z is a stationary sequence of martingale differences in L p . Let M n := n i=1 d i and R n := S n -M n .
The theorems will be proven if we can show that

R n = o n 1/p (log n) (t+1)/2 almost surely as n → ∞, (3.7)
and that (3.4) holds true with

M k replacing S k . Since E(d 2 0 ) = σ 2 and t > p/2, according to Proposition 5.1 in [2] (applied with ψ(n) := n 2/p (log n) t ), to prove that (3.4) holds true with M k replacing S k , it suffices to prove that n≥2 1 n 2 (log n) (t-1)p/2 E 0 (M 2 n ) -E(M 2 n ) p/2 p/2 < ∞ . (3.8)
By standard arguments, (3.7) will be satisfied if we can show that

r>0 max 1≤ℓ≤2 r |R ℓ | p p 2 r r (t+1)p/2 < ∞ . (3.9)
Now, by stationarity, max 1≤ℓ≤2 r |R ℓ | p ≪ 2 r/p r k=0 2 -k/p R 2 k p (see for instance inequality [START_REF] Field | Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions[END_REF] in [START_REF] Wu | Strong invariance principles for dependent random variables[END_REF]) and for all i, j ≥ 0, R i+j q ≤ R i q + R j q . Applying then Item 1 of Lemma 37 in [START_REF] Merlevède | Rosenthal inequalities for martingales and stationary sequences and examples[END_REF], we derive that for any integer n in

[2 r , 2 r+1 [, max 1≤ℓ≤2 r |R ℓ | p ≪ n 1/p n k=1 k -(1+1/p) R k p .
(3.10) Therefore using (3.10) followed by an application of Hölder's inequality, we get that for any α < 1,

r>0 max 1≤ℓ≤2 r |R ℓ | p p 2 r r (t+1)p/2 ≪ n≥2 1 n (log n) (t+1)p/2 n k=1 k -(1+1/p) R k p p ≪ n≥2 (log n) (p-1)(1-α) n (log n) (t+1)p/2 n k=1 k -2 (log k) α(p-1) R k p p .
Hence taking α ∈]1 -p/(2(p -1)), 1[ and changing the order of summation, we infer that (3.9) and then (3.7) hold provided that

n≥1 R n p p n 2 (log n) (t-1)p/2 < ∞ . (3.11)
On an other hand, we shall prove that condition (3.8) is implied by: there exists a positive finite integer m such that

n≥2 1 n 2 (log n) (t-1)p/2 E -nm (M 2 n ) -E(M 2 n ) p/2 p/2 < ∞ . (3.12)
For any nonnegative integer i, we set

V i := E 0 (M 2 i ) -E(M 2 i ) p/2 .
Using that M n is a martingale, we infer that, for any nonnegative integers i and j,

V i+j ≤ V i + V j . (3.13) Let now n ∈ [2 k , 2 k+1 -1]
∩ N, and write its binary expansion:

n = k ℓ=0 2 ℓ b ℓ where b k = 1 and b j ∈ {0, 1} for j = 0, . . . , k -1 .
Inequality (3.13) combined with Hölder's inequality implies that, for any η > 0,

V p/2 n ≤ k ℓ=0 V 2 ℓ p/2 ≪ 2 ηp(k+1)/2 k ℓ=0 V 2 ℓ 2 ηℓ p/2 . (3.14) Therefore n≥2 1 n 2 (log n) (t-1)p/2 V p/2 n ≪ k>0 2 ηp(k+1)/2 2 k k (t-1)p/2 k ℓ=0 V 2 ℓ 2 ηℓ p/2 .
Changing the order of summation and taking η ∈]0, 2/p[, it follows that (3.8) is implied by

k≥1 1 2 k k (t-1)p/2 E 0 (M 2 2 k ) -E(M 2 2 k ) p/2 p/2 < ∞ (3.15)
(actually due to the subadditivity of the sequence (V i ) both conditions are equivalent, see the proof of item 1 of Lemma 37 in [START_REF] Merlevède | Rosenthal inequalities for martingales and stationary sequences and examples[END_REF] to prove that (3.8) entails (3.15)). Now, since (M n ) is a martingale,

E 0 (M 2 2 k ) -E(M 2 2 k ) = k j=1 E 0 ((M 2 j -M 2 j-1 ) 2 ) -E((M 2 j -M 2 j-1 ) 2 ) + E 0 (d 2 1 ) -E(d 2 1 ) ,
which implies by stationarity that

E 0 (M 2 2 k ) -E(M 2 2 k ) p/2 ≤ k-1 j=0 E -2 j (M 2 2 j ) -E(M 2 2 j ) p/2 + E 0 (d 2 1 ) -E(d 2 1 ) p/2 .
Therefore by using Hölder's inequality as done in (3.14) with η ∈]0, 2/p[, we infer that (3.15) is implied by

k≥1 1 2 k k (t-1)p/2 E -2 k (M 2 2 k ) -E(M 2 2 k ) p/2 p/2 < ∞ . (3.16)
Notice now that the sequence (W n ) n>0 defined by

W n := E -n (M 2 n ) -E(M 2 n ) p/2
is subadditive. Indeed, for any non negative integers i and j, using that M n is a martingale together with the stationarity, we derive that

W i+j = E -(i+j) (M 2 i ) -E(M 2 i ) + E -(i+j) ((M i+j -M i ) 2 ) -E((M i+j -M i ) 2 ) p/2 ≤ E -i (M 2 i ) -E(M 2 i ) p/2 + E -j (M j ) 2 ) -E(M j ) 2 ) p/2 ≤ W i + W j . Therefore W p/2 i+j ≤ 2 p/2 W p/2 i + 2 p/2 W p/2 j
. This implies that, for any integer ℓ and any integer 0

≤ j ≤ ℓ, W p/2 ℓ ≤ 2 p/2 (W p/2 j + W p/2 ℓ-j ) in such a way that (ℓ + 1)W p/2 ℓ ≤ 2 1+p/2 ℓ j=1 W p/2 j .
(3.17)

Therefore using (3.17) with ℓ = 2 k , we infer that condition (3.16) is implied by

n≥2 1 n 2 (log n) (t-1)p/2 E -n (M 2 n ) -E(M 2 n ) p/2 p/2 < ∞ . (3.18)
It remains to prove that (3.12) implies (3.18). With this aim, we have, for any positive integer m,

M n = m k=1 M k[nm -1 ] -M (k-1)[nm -1 ] + M n -M m[nm -1 ] .
Using that M n is a martingale together with the stationarity, we then infer that

E -n (M 2 n ) -E(M 2 n ) p/2 p/2 ≤ 2 p/2 m p/2 E -n (M 2 [nm -1 ] ) -E(M 2 [nm -1 ] ) p/2 p/2 + 2 p/2 E -n (M 2 n-m[nm -1 ] ) -E(M 2 n-m[nm -1 ] ) p/2 p/2 ,
which, together with the fact that n -m[nm -1 ] < m, implies that

E -n (M 2 n ) -E(M 2 n ) p/2 p/2 ≤ 2 p/2 m p/2 2 p/2 d 0 p p + E -n (M 2 [nm -1 ] ) -E(M 2 [nm -1 ] ) p/2 p/2 ≤ 2 p/2 m p/2 2 p/2 d 0 p p + E -m[nm -1 ] (M 2 [nm -1 ] ) -E(M 2 [nm -1 ] ) p/2 p/2 , (3.19)
where for the last line we have used the fact that n ≥ m[nm -1 ]. We notice now that due to the martingale property of (M n ) and to stationarity, the sequence (U i ) i≥0 defined for any non negative integer i by

U i := E -mi (M 2 i ) -E(M 2 i ) p/2 p/2
satisfies, for any positive integers i and j,

U i+j ≤ E -m(i+j) (M 2 i ) -E(M 2 i ) p/2 + E -m(i+j) ((M i+j -M i ) 2 ) -E((M i+j -M i ) 2 ) p/2 p/2 ≤ 2 p/2 U i + 2 p/2 U j .
Hence by (3.17) applied with

W p/2 i = U i , U [nm -1 ] ≤ 2 1+p/2 ([nm -1 ] + 1) -1 [nm -1 ] k=1 U k ≤ 2 1+p/2 [nm -1 ] k=1 U k k . (3.20)
Therefore starting from (3.19), considering (3.20) and changing the order of summation, we infer that (3.18) (and so (3.8)) holds provided that (3.12) does. To end the proof, it remains to show that under the conditions of Theorems 3.1 and 3.2, the conditions (3.11) and (3.12) are satisfied. This is achieved by using the two following lemmas.

R ℓ p ≪ max k=ℓ,N E 0 (S k ) p + max k=ℓ,N S k -E k (S k ) p + ℓ 1/2 |j|≥N P 0 (X j ) p .
Next, applying Lemma 5.1 given in the appendix with q = 1, and using the fact that by stationarity, for any positive integer k,

E 0 (S k ) p ≤ k ℓ=1 E 0 (X ℓ ) p and S k -E k (S k ) p ≤ k-1 ℓ=0 X -ℓ -E 0 (X -ℓ ) p , (3.21
) we derive that for, any positive integers N ≥ n,

max 1≤ℓ≤n R ℓ p ≪ N k=1 E 0 (X k ) p + N -1 k=0 X -k -E 0 (X -k ) p + n 1/2 k≥[N/2] E 0 (X k ) p k 1/p + n 1/2 k≥[N/2] X -k -E 0 (X -k ) p k 1/p . (3.22)
The lemma follows from (3.22) with N = [n p/2 ] by using Hölder's inequality (see the computations in the proof of Proposition 2.2 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF]). Proof. Let m be a positive integer such that (3.3) is satisfied. We first write that

E -nm (M 2 n ) -E(M 2 n ) p/2 ≤ E -nm (S 2 n ) -E(S 2 n ) p/2 + 2 E -nm (S n R n ) -E(S n R n ) p/2 + 2 R n 2 p .
By using Lemma 3.1, and since (3.3) holds, Lemma 3.2 will follow if we can prove that

n≥1 1 n 2 (log n) (t-1)p/2 E -nm (S n R n ) p/2 p/2 < ∞ . (3.23)
With this aim we shall prove the following inequality. For any non negative integer r and any positive integer u n such that u n ≤ n, we have that

E -r (S n R n ) p/2 ≪ √ u n E 0 (S n ) 2 + S n -E n (S n ) 2 + max k={n,n-un} R k 2 p + √ n E -un (S n ) 2 + S n -E n+un (S n ) 2 + max k={n,un} E -r (S 2 k ) -E(S 2 k ) p/2 + √ n n k=1 |j|≥k+n P 0 (X j ) 2 2 1/2 . (3.24)
Let us show how, thanks to (3.24), the convergence (3.23) can be proven. Let us first consider the case where 2 < p < 4. Notice that the following elementary claim is valid:

Claim 3.1. If F and G are two σ-algebras such that G ⊂ F, then for any random variable X in L q for q ≥ 1, X -E(X|F) q ≤ 2 X -E(X|G) q .

Starting from (3.24) with r = nm and u n = n, and using Claim 3.1, we derive that

E -nm (S n R n ) p/2 ≪ E -nm (S 2 n ) -E(S 2 n ) p/2 + √ n E 0 (S n ) 2 + S n -E n (S n ) 2 + R n 2 p + n |j|≥n P 0 (X j ) 2 .
This last inequality combined with condition (3.3) and Lemma 3.1 shows that (3.23) will be satisfied if we can prove that

n≥1 n p/4 n 2 (log n) (t-1)p/2 E 0 (S n ) 2 + S n -E n (S n ) 2 p/2 < ∞ , (3.25)
and

n≥1 n p/2 n 2 (log n) (t-1)p/2 |j|≥n P 0 (X j ) 2 p/2 < ∞ . (3.26)
To prove (3.25), we use the inequalities (3.21) with p = 2. Hence setting

a ℓ = E 0 (X ℓ ) 2 + X -ℓ+1 -E 0 (X -ℓ+1 ) 2 , (3.27)
and using Hölder's inequality, we derive that for any α < 1,

n≥1 n p/4 n 2 (log n) (t-1)p/2 E 0 (S n ) 2 + S n -E n (S n ) 2 p/2 ≪ n≥1 n p/4 n 2 (log n) (t-1)p/2 n ℓ=1 a ℓ p/2 ≪ n≥1 n p/4 n (1-α)(p/2-1) n 2 (log n) (t-1)p/2 n ℓ=1 ℓ α(p/2-1) a p/2 ℓ .
Taking α ∈](3p -8)/(2p -4), 1[ (this is possible since p < 4) and changing the order of summation, we infer that (3.25) holds provided that (3.2) does. It remains to show that (3.26) is satisfied. Using Lemma 5.1 and the notation (3.27), we first observe that

n≥1 n p/2 n 2 (log n) (t-1)p/2 |j|≥n P 0 (X j ) 2 p/2 ≪ n≥1 n p/2 n 2 (log n) (t-1)p/2 ℓ≥[n/2] ℓ -1/2 a ℓ p/2 .
Therefore by Hölder's inequality, it follows that for any α < 1,

n≥1 n p/2 n 2 (log n) (t-1)p/2 |j|≥n P 0 (X j ) 2 p/2 ≪ n≥1 n p/2 n (1-α)(p/2-1) n 2 (log n) (t-1)p/2 ℓ≥[n/2] ℓ α(p/2-
1 n(log n) 2(t-1) E -[ √ n] (S n ) 2 + S n -E n+[ √ n] (S n ) 2 2 < ∞ , (3.28) and n≥2 1 n 2 (log n) 2(t-1) E -nm (S 2 [ √ n] ) -E(S 2 [ √ n] ) 2 2 < ∞ . (3.29)
We start by proving (3.28). With this aim, using the notation (3.27), we first write that

E -[ √ n] (S n ) 2 + S n -E n+[ √ n] (S n ) 2 ≤ n+[ √ n] k=[ √ n]+1 a k .
Therefore by Cauchy-Schwarz's inequality

n≥1 1 n(log n) 2(t-1) E -[ √ n] (S n ) 2 + S n -E n+[ √ n] (S n ) 2 2 ≪ n≥1 log n n(log n) 2(t-1) n+[ √ n] k=[ √ n]+1 ka 2 k ≪ n≥1 1 n n+[ √ n] k=[ √ n]+1 k log k (log k) 2(t-1) a 2 k .
Changing the order of summation, this proves that (3.28) holds provided that (3.5) does. It remains to prove (3.29). With this aim, we set for any positive real x,

h([x]) = E -m[x] (S 2 [x] ) -E(S 2 [x] ) 2 2 ,
and we notice that, for any integer n ≥ 0,

E -nm (S 2 [ √ n] ) -E(S 2 [ √ n] ) 2 2 ≤ h([ √ n]) . In addition, if x ∈ [n, n + 1[, then [ √ n] = [ √ x] or [ √ n] = [ √ x] -1. Therefore n≥3 1 n 2 (log n) (t-1)p/2 h([ √ n]) ≪ n≥3 h([ √ n]) [n,n+1[ 1 x 2 (log x) (t-1)p/2 dx ≪ ∞ 3 1 x 2 (log x) (t-1)p/2 h([ √ x])dx + ∞ 3 1 x 2 (log x) (t-1)p/2 h([ √ x] -1)dx ≪ ∞ 2 1 y 3 (log y) (t-1)p/2 h([y])dy ≪ n≥2 1 n 3 (log n) (t-1)p/2 h(n)dy .
For the last inequality, we have used that if y ∈ [n, n + 1[, then [y] = n. Therefore condition (3.3) implies (3.29). This ends the proof of (3.23) when p = 4.

It remains to prove (3.24). With this aim, we start with the decomposition of R n given in Proposition 5.1 of the appendix with N = n. Therefore setting

A n := n k=1 j≥2n+1 P k (X j ) + n k=1 j≥n P k (X -j ) ,
we write that

R n = E 0 (S n ) -E 0 (S n ) • T n + E -n (S n ) • T n + S n -E n (S n ) -(E 2n (S n -E n (S n )) • T -n -A n . (3.30)
Starting from (3.30) and noticing that

E -r (S n (E -n (S n ) • T n ) p/2 ≤ E 0 (S n (E -n (S n ) • T n ) p/2 ≤ E 0 (S n ) p E 0 (S 2n -S n ) p ,
and that

E -r (S n (S n -E n (S n )) = E -r ((S n -E n (S n ))
2 ), we first get

E -r (S n R n ) p/2 ≤2 E 0 (S n ) 2 p + S n -E n (S n ) 2 p + E -r (S n E n (S 2n -S n )) p/2 + E -r (S n E n (S n • T -n -E 0 (S n • T -n ))) p/2 + E -r (S n A n ) p/2 . (3.31)
Next, we use the following fact: if X and Y are two variables in L p with p ∈ [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF][START_REF] Denker | Approximation by Brownian motion for Gibbs measures and flows under a function[END_REF], then for any integer u,

E u (XY ) p/2 ≤ E u (X 2 ) -E(X 2 ) p/2 + Y 2 p + E(X 2 ) Y 2 . (3.32) Indeed, it suffices to write that E u (XY ) p/2 ≤ E 1/2 u (X 2 )E 1/2 u (Y 2 ) p/2 ≤ |E u (X 2 ) -E(X 2 )| 1/2 E 1/2 u (Y 2 ) p/2 + (E(X 2 )) 1/2 E 1/2 u (Y 2 ) p/2 ≤ E u (X 2 ) -E(X 2 ) p/2 + Y 2 p + (E(X 2 )) 1/2 E 1/2 u (Y 2 ) p/2 ,
and to notice that, since p ∈ [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF][START_REF] Denker | Approximation by Brownian motion for Gibbs measures and flows under a function[END_REF],

E 1/2 u (Y 2 ) p/2 ≤ E 1/2 u (Y 2 ) 2 = Y 2 .
Therefore, starting from (3.31) and using (3.32) together with E(S 2 n ) ≪ n, we infer that

E -r (S n R n ) p/2 ≪ E 0 (S n ) 2 p + S n -E n (S n ) 2 p + E -r (S n E n (S 2n -S n )) p/2 + E -r (S n E n (S n • T -n -E 0 (S n • T -n ))) p/2 + E -r (S 2 n ) -E(S 2 n ) p/2 + A n 2 p + n 1/2 A n 2 , and since E 0 (S n ) p ≤ R n p , S n -E n (S n ) p ≤ 2 R n p and A n p ≤ 8 R n p , we have overall that E -r (S n R n ) p/2 ≪ R n 2 p + E -r (S n E n (S 2n -S n )) p/2 + E -r (S n E n (S n • T -n -E 0 (S n • T -n ))) p/2 + E -r (S 2 n ) -E(S 2 n ) p/2 + n 1/2 A n 2 . (3.33)
By orthogonality and by stationarity,

A n 2 ≤ n k=1 j≥2n+1 P k (X j ) 2 2 1/2 + n k=1 j≥n P k (X -j ) 2 2 1/2 ≤ n k=1 ℓ≥k+n P 0 (X ℓ ) 2 2 1/2 + n k=1 ℓ≥k+n P 0 (X -ℓ ) 2 2 1/2 . (3.34)
Now for any integer u n such that u n ≤ n,

E -r (S n E n (S 2n -S n )) p/2 ≤ E -r ((S n -S n-un )E n (S 2n -S n )) p/2 + E -r (S n-un E n (S 2n -S n )) p/2 ≪ E -r (S 2 un ) -E(S 2 un ) p/2 + E 0 (S n ) 2 p + √ u n E 0 (S n ) 2 + E -r (S n-un E n (S 2n -S n )) p/2 , (3.35)
where for the last inequality we have used (3.32) together with E(S 2 un ) ≪ u n . Next, we write that

E -r (S n-un E n (S 2n -S n )) p/2 ≤ E -r ((S n-un -E n-un (S n-un ))E n (S 2n -S n )) p/2 + E -r (E n-un (S n-un )E n (S 2n -S n )) p/2 ≤ S n-un -E n-un (S n-un ) p E 0 (S n ) p + E -r (E n-un (S n-un )E n-un (S 2n -S n )) p/2 ≤ S n-un -E n-un (S n-un ) 2 p + E 0 (S n ) 2 p + E -r (S n E n-un (S 2n -S n )) p/2 + E -r ((S n -S n-un )E n-un (S 2n -S n )) p/2 .
Therefore using (3.32), we infer that

E -r (S n-un E n (S 2n -S n )) p/2 ≪ max k={n,n-un} R k 2 p + √ n E -un (S n ) 2 + max k={n,un} E -r (S 2 k ) -E(S 2 k ) p/2 . (3.36)
We deal now with the third term in the right-hand side of (3.33). With this aim, we first write that

E -r (S n E n (S n • T -n -E 0 (S n • T -n ))) p/2 ≤ E -r (S n E n (S n • T -n -E un (S n • T -n ))) p/2 + E -r (S n E un (S n • T -n -E 0 (S n • T -n ))) p/2 . (3.37)
By using (3.32) together with E(S 2 un ) ≪ n, stationarity and the fact that

S n -E n+un (S n ) 2 ≤ 2 R n p , we infer that E -r (S n E n (S n • T -n -E un (S n • T -n ))) p/2 ≪ E -r (S 2 n ) -E(S 2 n ) p/2 + R n 2 p + √ n S n -E n+un (S n ) 2 . (3.38)
On the other hand,

E -r (S n E un (S n • T -n -E 0 (S n • T -n ))) p/2 ≤ E -r (S un E un (S n • T -n -E 0 (S n • T -n ))) p/2 + E -r (E un (S n -S un )E un (S n • T -n -E 0 (S n • T -n )) p/2 .
We apply (3.32) to the first term of the right hand side together with the fact that E(S 2 un ) ≪ n. Hence by stationarity and since S n -E n (S n ) p ≤ 2 R n p , we derive that

E -r (S un E un (S n • T -n -E 0 (S n • T -n ))) p/2 ≪ E -r (S 2 un ) -E(S 2 un ) p/2 + R n 2 p + √ u n S n -E n (S n ) 2 .
On the other hand, by stationarity,

E -r (E un (S n -S un )E un (S n • T -n -E 0 (S n • T -n ))) p/2 ≤ E un (S n -S un ) p E un (S n • T -n -E 0 (S n • T -n )) p ≤ E 0 (S n-un ) p S n -E n (S n )) p . ≤ E 0 (S n-un ) 2 p + R n 2 p .
Therefore we get overall that

E -r (S n E un (S n • T -n -E 0 (S n • T -n ))) p/2 ≪ R n 2 p + E 0 (S n-un ) 2 p + E -r (S 2 un ) -E(S 2 un ) p/2 + √ u n S n -E n (S n ) 2 . (3.39)
Starting from (3.37) and taking into account (3.38) and (3.39), we get that 4. Proof of Theorem 2.1

E -r (S n E n (S n • T -n -E 0 (S n • T -n ))) p/2 ≪ √ u n S n -E n (S n ) 2 + √ n S n -E n+un (S n ) 2 + max k={n,un} E -r (S 2 k ) -E(S 2 k ) p/2 + max k={n,n-un}

Preparatory material

Let us denote by E u , E e and E s the S-stable vector spaces associated to the eigenvalues of S of modulus respectively larger than one, equal to one and smaller than one. Let d u , d e and d s be their respective dimensions. Let v 1 , ..., v d be a basis of R d in which S is represented by a real Jordan matrix. Suppose that v 1 , ..., v du are in E u , v du+1 , ..., v du+de are in E e and v du+de+1 , ..., v d are in E s . We suppose moreover that det(v Let r u be the spectral radius of S -1 |Eu . For every ρ u ∈ (r u , 1), there exists K > 0 such that, for every integer n ≥ 0, we have The properties satisfied by the filtration considered 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] and enabling the use of a martingale approximation method à la Gordin will be crucial here. Given a finite partition P of T d , we define the measurable partition P ∞ 0 by :

1 |v 2 | • • • |v d ) = 1. Let us write || • || the norm on R d given by d i=1 x i v i = max
∀h u ∈ E u , ||S n h u || ≥ Kρ -n u ||h u || (
∀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 torus automorphism T -1 . We obviously have F n ⊆ F n+1 = T -1 F n . Let r 0 > 0 be such that (h u , h e , h s ) → h u + h e + h s defines a diffeomorphism from B u (r 0 ) × B e (r 0 ) × B s (r 0 ) on its image in T d . Observe that, for every x ∈ T d , on the set x + B u (r 0 ) + B e (r 0 ) + B s (r 0 ), we have d λ(x + h u + h e + h s ) = dm u (h u )dm e (h e )dm s (h s ).

Proposition 4.1 ([17, 14] applied to T -1 ). There exist some Q > 0, K 0 > 0, α ∈ (0, 1) and some finite partition P of T d whose elements are of the form d i=1 I i v i where the I i are intervals with diameter smaller than min(r 0 , K) such that, for almost every x ∈ T d , 1. the local leaf P ∞ 0 (x) of P ∞ 0 containing x is a bounded convex set x + F (x), with 0 ∈ F (x) ⊆ E u , F (x) having non-empty interior in E u , 2. we have

E n (f )(x) = 1 m u (S -n F (T n x)) S -n F (T n x) f (x + h u ) dm u (h u ), (4.43)
3. for every γ > 0, we have

E -n (e 2iπ k,• )(x) ≤ K 0 m u (F (T -n (x)))
|k| de+ds α n , (4.45)

5. for every β ∈ (0, 1),

∃L > 0, ∀n ≥ 0, λ(m u (F (•)) < β n ) ≤ Lβ n/du . (4.46)
Proof. The first item comes from Proposition II.1 of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]. Item 2 comes from the formula given after Lemma II.2 of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF]. Item 3 follows from Lemma III.1 of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] and from the fact that the numbers a(P ∞ 0 (•)) considered in [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] are uniformly bounded. Item 4 comes from Proposition III.3 of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF] and from the uniform boundedness of a(P ∞ 0 (•)). Item 5 comes from the proof of Proposition II.1 of [START_REF] Borgne | Limit theorems for non-hyperbolic automorphisms of the torus[END_REF].

According to the first item of Proposition 4.1 and to (4.41), there exists c u > 0 such that, for almost every x ∈ T d and every n ≥ 1, we have sup

hu∈S -n F (T n (x)) |h u | ≤ c u ρ n u . ( 4
Then E 0 (f • T n ) p = E -n (f ) p = O(n -θ(p-1)/p ) .
Proof. Recall first that E 0 (f

•T n ) = E -n (f )•T n .
Let us consider α satisfying (4.45). Let β := α 1/2 , γ := max(α p/2 , β 1/du ) and

V n := x ∈ T d : m u (F (T -n (x)) ≥ β n .
Let b(n) := γ -n/(2p(d+de+ds)) . Let us write

f = f 1,n +f 2,n where f 1,n := |k|<b(n) c k e 2iπ k,• and f 2,n := |k|≥b(n) c k e 2iπ k,• . (4.49) We have Vn |E -n (f 1,n )| p d λ ≤ esssup x∈Vn |k|≤b(n) |c k | E -n (e 2iπ k,• )(x) p ≤ |k|≤b(n) |c k |K 0 β -n |k| de+ds α n p ,
according to (4.45) and thanks to the definition of V n . Now, since β = α 1/2 , we get

Vn |E -n (f 1,n )| p d λ ≤ 3 dp ||f || p 1 K p 0 α np 2 (b(n)) p(d+de+ds) . Hence Vn |E -n (f 1,n )| p d λ = O(γ n (b(n)) p(d+de+ds) ) = O(γ n/2 ). ( 4

.50)

Moreover, thanks to (4.46), we have

V c n |E -n (f 1,n )| p d λ ≤ λ(V c n ) |k|≤b(n) |c k | p = O((b(n)) dp β n/du ) = O((b(n)) dp γ n ) = O(γ n/2 ). ( 4

.51)

Since p ≥ 2 and since p/q = p -1, thanks to (4.48), we have

E -n (f 2,n ) p p ≤ f 2,n p p ≤ |k|≥b(n) |c k | q p/q ≤ R p-1 (log(b(n))) -θ(p-1) ≪ n -θ(p-1
E 0 (f • T -n ) -f p = E n (f ) -f p = O(n -θ(p-1)/p ) .
Proof. We consider the decomposition (4.49) with b(n) defined by b(n) = ρ

-n/(2(d+1)) u

. We have

E n (f 1,n ) -f 1,n p ≤ E n (f 1,n ) -f 1,n ∞ ≤ |k|≤b(n) |c k | E n (e 2iπ k,• ) -e 2iπ k,• ∞ ≤ |k|≤b(n) |c k |2π|k|c u ρ n u ,
according to (4.43) and to (4.47). Therefore

E n (f 1,n ) -f 1,n p ≪ (b(n)) d+1 ρ n u ≪ ρ n/2 u . (4.53)
Moreover, thanks to (4.48), we have where r is the spectral radius of S. Then First, we note that

E n (f 2,n ) -f 2,n p p ≤ 2 p f 2,n p p ≤ 2 p |k|≥b(n) |c k | q p/q ≤ 2 p R p-1 (log(b(n))) -θ(p-1) ≪ n -θ(p-1) . ( 4
E -nm (S 2 n (f )) -E(S 2 n (f )) p/2 ≪ n 2-2θ(p-1)/p + n (3-η)/2 . Proof. Let β := α 1/2 , V nm := x ∈ T d : m u (F (T -nm (x)) ≥ β nm
E -nm (S 2 n (f ))-E(S 2 n (f )) p/2 ≤ E -nm (S 2 1,n (f )) -E(S 2 1,n (f )) p/2 + E -nm (S 2 2,n (f )) -E(S 2 2,n (f )) p/2 + 2 E -nm (S 1,n (f )S 2,n (f )) -E(S 1,n (f )S 2,n (f )) p/2 ≤ E -nm (S 2 1,n (f )) -E(S 2 1,n (f )) p/2 + 2 S 2,n (f ) 2 p + 4 E -nm (S 1,n (f )S 2,n (f )) p/2 .
Next using (3.32), we get that

E -nm (S 1,n (f )S 2,n (f )) p/2 ≤ E -nm (S 2 1,n (f )) -E(S 2 1,n (f )) p/2 + S 2,n (f ) 2 p + S 1,n (f ) 2 S 2,n (f ) 2 ≤ E -nm (S 2 1,n (f )) -E(S 2 1,n (f )) p/2 + 2 S 2,n (f ) 2 p + S n (f ) 2 S 2,n (f ) 2 .
By Propositions 4.2 and 4.3, (4.55) implies that n>0

E -n (f ) 2 n 1/2 < ∞ and n>0 f -E n (f ) 2 n 1/2 < ∞ ,
which yields (3.6) with p = 2, and then S n (f ) 2 ≪ √ n. Therefore, we get overall that

E -nm (S 2 n (f )) -E(S 2 n (f )) p/2 ≪ E -nm (S 2 1,n (f )) -E(S 2 1,n (f )) p/2 + S 2,n (f ) 2 p + √ n S 2,n (f ) 2 . (4.58)
Since p ≥ 2 and p/q = p -1, (4.48) implies that

S 2,n (f ) p ≤ n f 2,n p ≤ n |k|≥b(n) |c k | q 1/q ≤ nR (p-1)/p (log(b(n))) -θ(p-1)/p ≪ n 1-θ(p-
E -nm (f 1,n • T i f 1,n • T i+j ) -E(f 1,n • T i f 1,n • T i+j ) p/2
≤ n 2 sup ℓ∈{0,...,n}

E -nm (f 1,n f 1,n • T ℓ ) -E(f 1,n f 1,n

Appendix

As in Section 3, let P k (X) = E k (X) -E k-1 (X).

Lemma 5.1. Let p ∈ [2, ∞[. Then, for any real 1 ≤ q ≤ p and any positive integer n, k≥2n P 0 (X k ) q p ≪ k≥n E 0 (X k ) q p k q/p and k≥2n P 0 (X -k ) q p ≪ k≥n X -k -E 0 (X -k ) q p k q/p .

Proof. The first inequality is Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF]. To prove the second one, we first consider the case p > q and we follow the lines of the proof Lemma 5.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF] with P k (X 0 ) replacing P -k (X 0 ). We get that k≥2n P 0 (X -k ) q p ≪ k≥n+1 k -q p ℓ≥k P 0 (X -ℓ ) p p q/p . Now, we notice that, by the Rosenthal's inequality given in Theorem 2.12 of [START_REF] Hall | Martingale limit theory and its application[END_REF], there exists a constant c p depending only on p such that Now when p = q, inequality (5.1) together with the fact that by Claim 3.1, for any integer k in [n + 1, 2n], X 0 -E 2n (X 0 ) p p ≤ 2 p X 0 -E k (X 0 ) p p imply the result. Indeed we have

k≥2n P 0 (X -ℓ ) p p ≤ c p X 0 -E 2n (X 0 ) p p ≪ 2n k=n+1 k -1 X 0 -E k (X 0 ) p p .
The proof is complete.

X

  i = M n + R n by the martingale M n = d 1 +d 2 +• • •+d n , where d i is the martingale difference

Lemma 3 . 1 .max 1≤ℓ≤n R ℓ p p n 2 (

 312 Let p ∈[START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF][START_REF] Denker | Approximation by Brownian motion for Gibbs measures and flows under a function[END_REF]. Assume that (3.1) holds. Thenn≥1 log n) (t-1)p/2 < ∞ ,and (3.11) holds. Proof. Since (3.1) implies (3.6), Item 2 of Proposition 5.1 given in the appendix implies that, for any positive integers ℓ and N ,

Lemma 3 . 2 .

 32 Let p ∈[START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF][START_REF] Denker | Approximation by Brownian motion for Gibbs measures and flows under a function[END_REF] and assume that (3.1) and (3.3) are satisfied. Assume in addition that (3.2) holds when 2 < p < 4 and (3.5) does when p = 4 . Then (3.12) is satisfied.

R k 2 p

 2 . (3.40) Finally, starting from (3.33) and considering (3.34), (3.35), (3.36) and (3.40), we conclude that (3.24) holds.

  i=1,...,d |x i | and d 0 (•, •) the metric induced by || • || on R d . Let also d 1 be the metric induced by d 0 on T d . We define now B u (δ) := {y ∈ E u : ||y|| ≤ δ}, B e (δ) := {y ∈ E e : ||y|| ≤ δ} and B s (δ) = {y ∈ E s : ||y|| ≤ δ}. Let | • | be the usual euclidean norm on R d .

m

  u (∂(F (x))(γ)) ≤ Qγ, (4.44) where ∂F (β) := {y ∈ F : d(y, ∂F ) ≤ β}, 4. for every k ∈ Z d \ {0}, for every integer n ≥ 0,

Proposition 4 . 4 .

 44 .54) Considering (4.53) and (4.54), the proposition follows. Let p ∈ [2, 4] and set S n (f ) := n k=1 f • T k with f : T d → R be a centered function with Fourier coefficients satisfying (4.48) with θ > 0 and |k|≥b |c k | 2 ≤ R log -η (b) for some η > 1 . (4.55) Set m := -4(d e + d s ) log(r) log(α) + 1 . (4.56)

  , γ := max(α p/8 , β 1/du ) and b(n) := γ n m/(p(2d+de+ds)) . (4.57) We consider the decomposition (4.49) with b(n) defined by (4.57) and we set S 1,n (f ) := n k=1 f 1,n • T k and S 2,n (f ) := n k=1 f 2,n • T k .

f 2 1 K≪ 2 ≪

 12 0 β -nm (|k| + r ℓ |m|) de+ds α nm p/2 ≪ α nmp 4 r pℓ(de+ds)/2 (b(n)) p(2d+de+ds)/2 . Hence, since γ ≥ α p/8 , m ≥ 4(d e + d s ) log(r)/log(1/α), and according to the definition of b(n), we have sup ℓ∈{0,...,n} VnmE -nm (f 1,n .f 1,n • T ℓ ) -E(f 1,n .f 1,n • T ℓ ) α 3nmp/16 r pn(de+ds)/2 ≪ γ nm/2 . (4.61) Moreover, for any non negative integer ℓ,V c nm E -nm (f 1,n .f 1,n • T ℓ ) p/2 d λ ≤ λ(V c nm ) |k|,|m|≤b(n) |c k ||c m | p/(b(n)) dp β nm/du ≪ (b(n)) dp γ nm ≪ γ nm/2 , (4.62) according to (4.46) and to the definition of b(n) and of γ. Combining (4.61) and (4.62), we then derive that E -nm (S 2 1,n (f )) -E(S 2 1,n (f ))

ℓ≥k P 0

 0 (X -ℓ ) p p = ℓ≥k P ℓ (X 0 ) p p ≤ c p ℓ≥k P ℓ (X 0 ) p p = c p X 0 -E k (X 0 ) p p = c p X -k -E 0 (X -k ) p p . (5.1)

Proposition 5 . 1 .

 51 Let p ∈ [1, ∞[ and assume that the series d 0 = i∈Z P 0 (X i ) converges in L p .(5.2)

.

  1) ℓ -p/4 a Therefore taking α ∈]1, 2[ and changing the order of summation, we infer that (3.25) holds provided that (3.2) does. This ends the proof of (3.23) when p ∈]2, 4[. Now, we prove (3.23) when p = 4. With this aim we start from (3.24) with r = nm and u n = [

	√ n]. This inequality combined with condition (3.3),
	Lemma 3.1 and the arguments developed to prove (3.25) and (3.26) shows
	that (3.23) will be satisfied for p = 4 if we can prove that
	n≥1
	p/2
	ℓ

  , h s ) ∈ E e × E s , ||S n (h e + h s )|| ≤ K(1 + n) de ||h e + h s ||. (4.42)Let ρ u ∈ (r u , 1) and K satisfying (4.41) and (4.42). Let us denote by m u , m e , m s the Lebesgue measure on E u (in the basis v 1 , ..., v du ), E e (in the basis v du+1 , ..., v du+de ) and E s (in the basis v du+de+1 , ..., v d ) respectively. Observe that dλ(h u + h e + h s ) = dm u (h u )dm e (h e )dm s (h s ).

	4.41)
	and
	∀(h e

  .47) Proposition 4.2. Let p ≥ 2 and q be its conjugate exponent. Let θ > 0 and f : T d → R be a centered function with Fourier coefficients (c k ) k∈Z d satisfying |k|≥b |c k | q ≤ R log -θ (b) .

	(4.48)

  Proposition 4.3. Under the assumptions of Proposition 4.2,

) . (4.52) Combining (4.50), (4.51) and (4.52), the proposition follows.

  We deal now with the first term in the right hand side of(4.58). With this aim, we first observe that, for any non negative integer ℓ, e 2iπ k,T ℓ (•) = e 2iπ t S ℓ k,• , where t S ℓ is the transposed matrix of S ℓ . Therefore,Vnm E -nm (f 1,n .f 1,n • T ℓ ) -E(f 1,n .f 1,n • T ℓ ) p/2 d λ ≤ esssup x∈Vnm |k|,|m|≤b(n):k+ t S ℓ m =0 |c k ||c m | E -nm (e 2iπ k+ t S ℓ m,• )(x) |c k ||c m |K 0 β -nm |k + t Sℓ m| de+ds α nm p/2 , according to (4.45) and to the definition of V nm . It follows that Vnm E -nm (f 1,n .f 1,n • T ℓ ) -E(f 1,n .f 1,n • T ℓ )

			p/2
		≤	|k|,|m|≤b(n)
			p/2 d	λ
	≤	|k|,|m|≤b(n)
			1)/p .	(4.59)
	Similarly using (4.55), we get that

S 2,n (f ) 2 ≤ n f 2,n 2 ≪ n 1-η/2 .

(4.60)

  • T ℓ ) p/2 ≪ n 2 γ nm/p . Considering (4.59), (4.60) and (4.63) in (4.58), the proposition follows. 4.2. End of the proof of Theorem 2.1 Proof. Propositions 4.2 and 4.3 give (3.1) provided (2.2) is satisfied. Propositions 4.2 and 4.3 give (3.2) (when p ∈]2, 4[) and (3.5) (when p = 4), provided (2.3) is satisfied. Finally, Proposition 4.4 gives (3.3) provided (2.2) and (2.3) are satisfied. The proof follows now from Theorem 3.1 when p ∈]2, 4[ and from Theorem 3.2 when p = 4.

	(4.63)

Let M n := n i=1 d 0 • T i and R n := S n -M n . Then, for any positive integers n and N ,

where p ′ = min(2, p).

Proof. Notice first that the following decomposition is valid: for any positive integer n,

where

(5.4) Let N be a positive integer. According to item 1 of Proposition 2.1 in [START_REF] Dedecker | Rates of convergence in the strong invariance principle under projective criteria[END_REF],

On an other hand, we write that

Starting from (5.3) and considering (5.5) and (5.6), the first part follows. We turn now to the second part of the proposition. Applying Burkholder's inequality and using stationarity, we obtain that there exists a positive constant c p such that, for any positive integer n, The second part of the proposition follows from item 1 by taking into account the stationarity and by considering the bounds (5.7) and (5.8).


Jonathan Benchimol
email: jonathan.benchimol@essec.edu.

Laurent Clerc

Risk aversion in the Euro area

Keywords: Risk aversion, Output, Money, Euro area, New Keynesian DSGE models, Bayesian estimation. JEL Classi…cation Number: E23, E31, E51

We propose a New Keynesian Dynamic Stochastic General Equilibrium (DSGE) model where a risk aversion shock enters a separable utility function. We analyze …ve periods, each one lasting twenty years, to follow over time the dynamics of several parameters (such as the risk aversion parameter), the Taylor rule coe¢cients and the role of this risk aversion shock on output and real money balances in the Eurozone. Our analysis suggests that risk aversion was a more important component of output and real money balance dynamics between 2006 and 2011 than it had been between 1971 and 2006, at least in the short run.

Introduction

The New Keynesian model, as developed by [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF] and [START_REF] Walsh | Monetary theory and policy[END_REF], brings three equations together to characterize the dynamic behavior of three macroeconomic key variables: output, in ‡ation, and the nominal interest rate. The resulting output equation corresponds to the log-linearization of an optimizing household's Euler equation, linking consumption and output growth to the in ‡ation-adjusted return on nominal bonds, that is, to the real interest rate. The in ‡ation equation describes the optimizing behavior of monopolistically competitive …rms that either set prices in a randomly staggered fashion, as suggested by [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF], or face explicit costs of nominal price adjustment, as suggested by [START_REF] Rotemberg | Monopolistic Price Adjustment and Aggregate Output[END_REF]. The nominal interest rate equation, a monetary policy rule of the kind proposed by [START_REF] Taylor | Discretion versus Policy Rules in Practice[END_REF], dictates that the central bank should adjust the short-term nominal interest rate in response to a trade-o¤ between changes in in ‡ation and output and changes in the past interest rate.

In this framework, even if money or real balances are included in the utility (MIU) or in the central bank's reaction function, real or nominal monetary aggregates generally become an irrelevant variable and are then neglected, at least for the US, as in [START_REF] Woodford | Foundations of a Theory of Monetary Policy[END_REF] and [START_REF] Ireland | Money's role in the monetary business cycle[END_REF]. Additionally, for the Eurozone, even if a money variable appears in the monetary policy reaction function, as in [START_REF] Andrés | Money in an Estimated Business Cycle Model of the Euro Area[END_REF] or in [START_REF] Barthélemy | A two-pillar DSGE monetary policy model for the euro area[END_REF], money plays no role in the dynamics.

However, [START_REF] Benchimol | Money and risk in a DSGE framework: A Bayesian application to the Eurozone[END_REF] show that the role of money in the business cycle is dependent on the risk aversion level, at least in the Eurozone. They estimate a New Keynesian DSGE model with non-separable household preferences between consumption and money, as in the [START_REF] Ireland | Money's role in the monetary business cycle[END_REF] or [START_REF] Andrés | Money and the natural rate of interest: Structural estimates for the United States and the euro area[END_REF] to analyze the role of money in the dynamics of the variables under a high level of risk aversion. In this context, they establish a signi…cant link between money, output and risk by showing that real money has a signi…cant role with regard to output and ‡exible-price output dynamics in the short term only if the relative risk aversion level is su¢ciently high (twice the standard value). They study the role of the level of risk aversion in a non-standard MIU function case and do not include a standard case or a study of a micro-founded risk aversion shock for the Eurozone. They only consider standard micro-founded shocks (price-markup, monetary policy and technology) and a money demand shock.

Finally, as in other studies, we also consider a price-markup shock, a monetary policy shock and a technology shock. To analyze the role of risk aversion in the dynamics of other variables, we do not consider a money shock that has no role in the dynamics of this framework (because of the separa-bility assumption between consumption and money), as shown by [START_REF] Smets | An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area[END_REF], but we do consider a money equation to take account of the behaviors of national central banks (before 1999) and the European Central Bank (after 1999) and to close the model as much with historical variables as with exogenous shocks.

In contrast, as relative risk aversion measures the willingness to substitute consumption over di¤erent periods, the lower the level of risk aversion, the more households substitute consumption over time. [START_REF] Wachter | A consumption-based model of the term structure of interest rates[END_REF] and [START_REF] Bekaert | Stock and bond returns with Moody Investors[END_REF] show that an increase in risk aversion involve an increase in equity and bond premiums and may increase the real interest rate through a consumption smoothing e¤ect or decrease it through a precautionary savings e¤ect. [START_REF] Bommier | Comparative Risk Aversion: A Formal Approach with Applications to Saving Behaviors[END_REF] also show that risk aversion enhances precautionary savings. These studies con…rm the potential link between money holdings, output and risk aversion.

However, few studies quantify this link, or even consider risk aversion as a shock, in a New Keynesian DSGE framework. Moreover, no studies use Bayesian techniques as [START_REF] Fernández-Villaverde | The econometrics of DSGE models[END_REF] does to analyze the role of the risk aversion shock in output and money dynamics in the Eurozone. In the nearly same technical and theoretical context, [START_REF] Alpanda | Identifying the Role of Risk Shocks in the Business Cycle Using Stock Price Data[END_REF] highlights the important role played by risk aversion shocks in US output between 2006 and 2011.

Accordingly, this article contributes to the literature in several ways. First, we analyze the role of a micro-founded risk aversion shock in the dynamics of a New Keynesian DSGE model. Second, the development of a completely micro-founded model with a risk aversion shock is original in terms of …ndings as well as in terms of estimation techniques. Mainly inspired by [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF] and [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF], our model explores the role of risk aversion in in ‡ation, output, interest rate and real money balances, as well as in ‡exible-price output..

A speci…c emphasis will be placed on how the risk aversion shock impacts the dynamics of these key variables through time. We use Bayesian techniques, as in [START_REF] An | Bayesian Analysis of DSGE Models[END_REF], to estimate …ve subsamples of the Eurozone between 1971 and 2011, each one lasting twenty years. This original focus on the last forty years will show that risk aversion shocks have had stronger e¤ects on output and real money balances in recent years than in the more distant past.

Last but not least, our framework allows us to analyze successively the informational content of the last two crises (subprimes and sovereign debts) in comparison with other crises that occurred between 1971 and 2006 in the Euro area.

Bayesian estimations and dynamic analyses of the model, with impulse response functions and short-and long-run variance decompositions following structural shocks, yield di¤erent relationships between risk aversion and other structural variables. This approach sheds light on the importance of risk aversion and its impact on output and real money balances during the last …ve years (2006 to 2011). It also shows that the role of monetary policy as regards output in the short run has decreased in the recent years in comparison to the more distant past.

Finally, this study explores with modern theoretical and empirical tools a fundamental question about the role of the perception of economic risks, e.g., the ability for households to consume now or later, in the dynamics of the main economic variables for the Eurozone.

Section 2 describes the theoretical set up. In Section 3, the model is calibrated and estimated with Euro area data and impulse response functions and variance decomposition are analyzed. Interpretation of the results is provided in Section 4. Section 5 concludes, and the Appendix presents additional theoretical and empirical results.

The model

The model consists of households that supply labor, purchase goods for consumption and hold money and bonds, and of …rms that hire labor and produce and sell di¤erentiated products in monopolistically competitive goods markets. Each …rm sets the price of the good it produces, but not all …rms reset their price during each period. Households and …rms behave optimally: households maximize the expected present value of utility, and …rms maximize pro…ts. There is also a central bank that controls the nominal rate of interest. This model is essentially inspired by [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF] and [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF].

Households

We assume a representative in…nitely lived household, seeking to maximize

E t " 1 X k=0 k U t+k # (1)
where U t is the period utility function and < 1 is the discount factor. The household decides how to allocate its consumption expenditures among the di¤erent goods. This requires that the consumption index C t be maximized for any given level of expenditures, as in [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF]. Furthermore, and conditional on such optimal behavior, the period budget constraint takes the form

P t C t + M t + Q t B t B t 1 + W t N t + M t 1 (2)
where t = 0; 1; 2:::, P t is an aggregate price index, M t is the quantity of money holdings at time t, B t is the quantity of one-period nominally riskless discount bonds purchased in period t and maturing in period t+1 (each bond pays one unit of money at maturity and its price is Q t where i t = log Q t is the short term nominal rate), W t is the nominal wage, and N t is hours of work (or the measure of household members employed). The above sequence of period budget constraints is supplemented with a solvency condition 1 . Preferences are measured with a common time-separable utility function (MIU). Under the assumption of a period utility given by

U t = C 1 t t 1 t + 1 M t P t 1 N 1+ t 1 + (3)
consumption, money demand, labor supply, and bond holdings are chosen to maximize (1) subject to (2) and the solvency condition. This MIU utility function depends positively on the consumption of goods, C t , positively on real money balances, Mt Pt , and negatively on labor N t . t = + " r t is the timevarying coe¢cient of the relative risk aversion of households (or the inverse of the intertemporal elasticity of substitution), where " r t is a risk aversion shock. is the inverse of the elasticity of money holdings with respect to the interest rate, and is the inverse of the elasticity of work e¤ort with respect to the real wage. and are positive scale parameters.

This setting leads to the following conditions 2 , which, in addition to the budget constraint, must hold in equilibrium. The resulting log-linear version of the …rst-order condition corresponding to the demand for contingent bonds implies that

c t = E t [c t+1 ] 1 t (i t E t [ t+1 ] c ) (4)
where

c t = ln (C t ) is the logarithm of the aggregate consumption, i t is the nominal interest rate, E t [ t+1 ]
is the expected in ‡ation rate in period t + 1 with knowledge of the information in period t, and c = ln ( ).

The demand for cash that follows from the household's optimization problem is given by

t c t mp t m = a 2 i t (5) 1 Such as 8t lim n !1 E t [B n ] 0, in order to avoid Ponzi-type schemes. 2 See Appendix 6.A
where mp t = m t p t are the log linearized real money balances, m = ln ( ) + a 1 , and a 1 and a 2 are resulting terms of the …rst-order Taylor approximation of log (1

Q t ) = a 1 + a 2 i t .
Real cash holdings depend positively on consumption with an elasticity equal to t and negatively on the nominal interest rate 3 . In what follows, we take the nominal interest rate as the central bank's policy instrument.

The resulting log-linear version of the …rst-order condition corresponding to the optimal consumption-leisure arbitrage implies that

w t p t = t c t + n t n (6)
where w t p t corresponds to the log of the real wage, n t denotes the log of hours of work, and n = ln ( ).

Finally, these equations represent the Euler condition for the optimal intratemporal allocation of consumption (Eq. ( 4)), the intertemporal optimality condition setting the marginal rate of substitution between money and consumption equal to the opportunity cost of holding money (Eq. ( 5)), and the intratemporal optimality condition setting the marginal rate of substitution between leisure and consumption equal to the real wage (Eq. ( 6)). [START_REF] Backus | International Real Business Cycles[END_REF] have shown that capital appears to play a rather minor role in the business cycle. To simplify the analysis and focus on the role of risk, we do not include a capital accumulation process in this model, as in [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF].

Firms

We assume a continuum of …rms indexed by i 2 [0; 1]. Each …rm produces a di¤erentiated good, but they all use an identical technology, represented by the following production function

Y t (i) = A t N t (i) 1 (7)
where A t = exp (" a t ) represents the level of technology, assumed to be common to all …rms and to evolve exogenously over time, and " a t is a technology shock.

All …rms face an identical isoelastic demand schedule and take the aggregate price level P t and aggregate consumption index C t as given. As in the standard [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] model, our generalization features monopolistic competition and staggered price setting. At any time t, only a fraction 1 of …rms, with 0 < < 1, can reset their prices optimally, while the remaining …rms index their prices to lagged in ‡ation.

Central bank

The central bank is assumed to set its nominal interest rate according to an augmented smoothed [START_REF] Taylor | Discretion versus Policy Rules in Practice[END_REF] rule such as:

i t = (1 i ) ( t c ) + x y t y f t + m (mp t mp c ) + i i t 1 + " i t (8)
where , x and m are policy coe¢cients re ‡ecting, respectively, the weight on in ‡ation, the output gap and real money; the parameter 0 < i < 1 captures the degree of interest rate smoothing; and " i t is an exogenous ad hoc shock accounting for ‡uctuations of the nominal interest rate. c is an in ‡ation target, and mp c is a money target, essentially included to account for changes in targeting policies of in ‡ation and monetary aggregates, as in, respectively, [START_REF] Svensson | In ‡ation targeting as a monetary policy rule[END_REF] and Fourçans and Vranceanu (2007) 4 .

m takes also into account the potential national central bank's money targeting before the creation of the European Central Bank (ECB, 1999). After 1999, the ECB follows an explicit money targeting until 2004 called the Two Pillars policy, as explained in [START_REF] Barthélemy | A two-pillar DSGE monetary policy model for the euro area[END_REF], and may even follow an implicit one after this date, as suggested by [START_REF] Kahn | The role of money in monetary policy: why do the Fed and ECB see it so di¤erently ?[END_REF].

Empirical results

DSGE model

Our macro model consists of …ve equations and …ve dependent variables: in ‡ation, nominal interest rate, output, real money balances, and ‡exibleprice output. Flexible-price output is completely determined by shocks.

y f t = 1 + t (1 ) + + " a t + (1 ) log (1 ) + n log " " 1 t (1 ) + + (9) t = E t [ t+1 ] + (1 ) (1 ) ( t (1 ) + + ) (1 + ") y t y f t ( 10
)
y t = E t [y t+1 ] 1 t (i t E t [ t+1 ] c ) (11) mp t = t y t a 2 i t m ( 12
)
4 Other studies introduce a relevant money variable in the Euro area Taylor rule: Andrés, López- [START_REF] Andrés | Money in an Estimated Business Cycle Model of the Euro Area[END_REF], [START_REF] Andrés | Money and the natural rate of interest: Structural estimates for the United States and the euro area[END_REF], [START_REF] Barthélemy | A two-pillar DSGE monetary policy model for the euro area[END_REF], [START_REF] Benchimol | Money and risk in a DSGE framework: A Bayesian application to the Eurozone[END_REF].

i t = (1 i ) ( t c ) + x y t y f t + m (mp t mp c ) + i i t 1 + " i t ( 13
)
where a 1 = log 1 e .

All structural shocks are assumed to follow a …rst-order autoregressive process with an i.i.d. normal error term, such as " k t = k " k t 1 + ! k;t , where " k;t N (0; k ) for k = fp; i; a; rg.

Euro area data

In our model of the Eurozone, t is the detrended in ‡ation rate measured as the yearly log di¤erence of the detrended GDP De ‡ator from one quarter to the same quarter of the previous year; y t is the detrended output per capita measured as the di¤erence between the log of the real GDP per capita and its trend; and i t is the short-term (3-month) detrended nominal interest rate. These data are extracted from the AWM database of [START_REF] Fagan | An area-wide model (AWM) for the Euro area[END_REF]. mp t is the detrended real money balances per capita measured as the di¤erence between the real money per capita and its trend, where real money per capita is measured as the log di¤erence between the money stock per capita and the GDP De ‡ator. We use the M 3 monetary aggregate from the Eurostat database.

Calibration

Following standard conventions, we calibrate beta distributions for parameters that fall between zero and one, inverted gamma distributions for parameters that need to be constrained to be greater than zero, and normal distributions in other cases.

The parameters of the utility function are assumed to be distributed as follows. Only the discount factor is …xed in the estimation procedure to 0:98. The intertemporal elasticity of substitution (i.e. the level of relative risk aversion) is set at 2, a mean between the calibrations of Rabanal and Rubio-Ramírez (2005) and [START_REF] Casares | Monetary Policy Rules in a New Keynesian Euro Area Model[END_REF], and consistent with the calibrated value used by [START_REF] Kollmann | The exchange rate in a dynamic-optimizing business cycle model with nominal rigidities: a quantitative investigation[END_REF] and the value estimated by [START_REF] Lindé | Monetary policy in an estimated open-economy model with imperfect pass-through[END_REF]. The inverse of the Frisch elasticity of labor supply is assumed to be approximately 1, as in [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF] and the scale parameters on money and labor are assumed to be approximately 0:2, as in [START_REF] Benchimol | Money and risk in a DSGE framework: A Bayesian application to the Eurozone[END_REF].

The calibration of , , and " comes from [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF], [START_REF] Casares | Monetary Policy Rules in a New Keynesian Euro Area Model[END_REF] and [START_REF] Galí | Monetary Policy, In ‡ation and the Business Cycle: An Introduction to the New Keynesian Framework[END_REF]. The smoothed Taylor rule ( i , , x and m ) priors are calibrated following [START_REF] Smets | An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area[END_REF], [START_REF] Andrés | Money and the natural rate of interest: Structural estimates for the United States and the euro area[END_REF], and [START_REF] Barthélemy | A two-pillar DSGE monetary policy model for the euro area[END_REF]. To observe both the behavior of the central bank and risk aversion, we assign a higher standard error (0:2) and a Normal prior law for the relative risk aversion level and for the Taylor rule's coe¢cients (including the in ‡ation and money targets), except for the smoothing parameter, which is restricted to be positive and less than one (Beta distribution). The in ‡ation target, c , is calibrated to 2%, and the money target, mp c , is assumed to be approximately 4%.

The calibration of the shock persistence parameters and the standard errors of the innovations follow [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF]. All of the standard errors of shocks are assumed to be distributed according to inverted Gamma distributions, with prior means of 0:01. The latter ensures that these parameters have positive support. The autoregressive parameters are all assumed to follow Beta distributions. All of these distributions are centered approximately 0:75, except for the autoregressive parameter of the monetary policy shock and the risk aversion shock, which are centered approximately 0:50, as in [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF]. We take a common standard error of 0:15 for the shock persistence parameters, which is a mean between that of [START_REF] Benchimol | Money and risk in a DSGE framework: A Bayesian application to the Eurozone[END_REF] and [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF].

Results

The model is estimated with 160 observations from 1971 (Q1) to 2011 (Q1) with Bayesian techniques, as in [START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF]. However, to capture di¤erent policies and risk perceptions in the Euro area between 1971 and 2011, and more speci…cally between 2006 and 2011, we divide this large sample into …ve subsamples, each one consisting of 80 observations (20 years).

This procedure allows us to analyze …ve di¤erent periods with a su¢ciently large sample, as speci…ed in Fernandez-Villaverde and Rubio-Ramirez ( 2004). Accordingly, we estimate our model over …ve di¤erent periods: from 1971Q1 to 1991Q1 (P1); from 1976Q1 to 1996Q1 (P2); from 1981Q1 to 2001Q1 (P3); from 1986Q1 to 2006Q1 (P4); and from 1991Q1 to 2011Q1 (P5). The estimation of the implied posterior distribution of the parameters over the …ve periods (Fig. 1) is performed using the Metropolis-Hastings algorithm (10 distinct chains, each of 100000 draws). The average acceptation rates per chain are included in the interval [0:19; 0:22] and the student's ttests are all above 1:96. To assess the model validation, we insure convergence of the proposed distribution to the target distribution for each period in Appendix 6.B. Priors and posteriors distributions are presented in Appendix 6.C.

Simulations

Impulse response functions

As in the literature, Appendix 6.D (Fig. 9) shows that a price-markup shock increases in ‡ation and the nominal interest rate and decreases output, the output gap, the real interest rate, real money balances and real money growth.

The response of output, real money balances and real money growth to a technology shock is positive (Fig. 9). Notice that the improvement in technology is partly accommodated by the central bank, which lowers the nominal and real interest rate, while increasing the quantity of money in circulation.

Fig. 9 also presents the response to an interest rate shock. In ‡ation, output and the output gap, real money balances and real money growth all fall. The real and nominal interest rate rise.

This case is very interesting because Fig. 9 shows that a risk aversion shock leads to a decrease in output and an increase in in ‡ation: it implies a tightening of monetary policy (because of the strong weight that the central banker places on in ‡ation), and its strength depends on the period (strong monetary policy tightening in P1 and low monetary policy tightening in P5). The risk aversion shock also implies an increase in real money balances and real money growth and a decrease in the output gap.

Household consumption is reduced (decreasing output), and companies increase their price (to face high risk aversion and possibly low consumption), which implies an increase in the in ‡ation rate, constrained by a tightening of monetary policy.

Variance decompositions

We analyze the forecast error variance decomposition of each variable following exogenous shocks. The analysis is conducted via an unconditional variance decomposition to analyze long-term variance decomposition (the grey bar in Fig. 2) and via a conditional variance decomposition, conditionally to the …rst period, to analyze short-run variance decomposition (the black bar in Fig. 2).

Fig. 2 shows that output is mainly explained by the technology shock in the long run (approximately 90%) and by the monetary policy shock (ap- proximately 35%) and the technology shock (approximately 50%) in the short run. The rest of the variance in output is explained by the risk aversion shock (approximately 5% from P1 to P4 and more than 15% for P5) in the short term, whereas risk aversion shock has a limited role in output variance in the long run.

Fig. 2 also shows that, in accordance with the literature, in ‡ation is mainly explained by the price-markup shock and that interest rate variance is mainly driven by monetary policy in the short run and by monetary policy and price-markup in the long run. Furthermore, most of the variance in real money balances is induced by the risk aversion shock (approximately 40%) and the monetary policy shock (approximately 25%) in the short run, whereas in the long run, real money balance variance is mainly driven by the technology shock. All of these results are in line with the literature.

Interpretation

Appendix 6.B shows that the estimation results are valid and that convergence is obtained for all estimations and all moments. Appendix 6.C shows that the maximum of the posterior distribution reaches the posterior mean of each estimated parameter. The estimation is relatively well identi…ed, and the data are quite informative for most of the estimated micro-parameters.

Fig. 9 shows that from P1 to P5, the impact of the price-markup shock on in ‡ation and output is almost halved. It also shows that the risk aversion shock has a longer impact in P5 than it does in the other periods. This is due to the increase over the periods of the autoregressive parameter of the risk aversion shock, r , as shown in Fig. 1.

Fig. 2 shows that output and real money balances variances have an important component coming from the risk aversion shock. This …nding shows the leading role of relative risk aversion in the dynamics of output, as in [START_REF] Black | Risk aversion as a technology factor in the production function[END_REF], and of real money balances, as in [START_REF] Benchimol | Money and risk in a DSGE framework: A Bayesian application to the Eurozone[END_REF]. Although the in ‡ation rate, the nominal interest rate and the ‡exible-price output are strong components of output, risk aversion has a minor role to play in the variance of in ‡ation and interest rate, and it has also no role to play with regard to the ‡exible-price output (less than 0:2% in the short and long run), which is completely determined by the technology shock. It also shows that in ‡ation and interest rate variances are quasi una¤ected by the introduction of the risk aversion shock, letting these variables be mainly explained by, respectively, the price-markup shock and the monetary policy shock.

The leading role of the risk aversion shock in the dynamics of real money balances in the short run is another important …nding. Fig. 2 shows that real money balances are mainly explained by the technology shock (approximately 80%) in the long run, whereas in the short run, real money balances are mainly explained by the risk aversion shock and the monetary policy shock.

Fig. 2 shows that technology plays an increasingly important role in the short term for the in ‡ation rate and, thus, the interest rate through the selected periods. This …gure also shows that, in the short run, risk aversion has a more signi…cant role in output dynamics in the last period (P5) than in the other periods (P1 to P4). This …nding re ‡ects the increasing role assumed by risk aversion in more recent years (between 2006 and 2011) as compared to the past (between 1971 and 2006).

Finally, Fig. 2 shows that monetary policy has a lower role in the short run concerning output in the last period (P5), approximately 22%, than it had in the past, approximately 35%. It highlights the transfer from the monetary policy role to the risk aversion role during the recent years. This con…rms the declining in ‡uence of European monetary policy relative to the in ‡uence of risk aversion shocks.

Conclusion

Risk aversion is a concept in economics and …nance that is based on the behavior of consumers and investors who are exposed to uncertainty. It is the reluctance of a person to accept a bargain with an uncertain payo¤ rather than another bargain o¤ering a more certain, but possibly lower, expected payo¤.

This paper presents a standard New Keynesian DSGE model that includes a risk aversion shock. It shows the involvement of this risk aversion shock in the dynamics of the economy: it increases in ‡ation, decreases output (Fig. 9) and diminishes the impact of the central bank's actions on output variance, at least in the short run (Fig. 2). Risk aversion plays also an important role for output and real money balance dynamics. The negative role played by risk aversion in determining output is clearly identi…ed, whereas it increases real money balances and real money growth in the …rst periods (Fig. 9).

Moreover, while estimations are quite robust (Fig. 3 to Fig. 8), they show that the risk aversion shock has a stronger impact on output dynamics during the last twenty years (P5) as compared to other analyzed periods (P1 to P4). This result is explained by the inclusion in P5 of the subprime and sovereign debt crises from 2007 to 2011.

This enhanced baseline model shows the importance of such a parameter to the economy, and especially its impact on output, money, and monetary policy. It also serves to show how it is important to control shocks to the risk aversion of agents, by communication for example.

Appendix

A Solving the model Price dynamics

Let's assume a set of …rms not reoptimizing their posted prices in period t. Using the de…nition of the aggregate price level and the fact that all …rms resetting prices choose an identical price P t , leads to

P t = h P 1 t t 1 + (1 ) (P t ) 1 t i 1 1 t , where t = 1 + 1 1 " 1 +" p t
is the elasticity of substitution between consumption goods in period t, and t t 1 is the markup of prices over marginal costs (time varying). Dividing both sides by P t 1 and log-linearizing around P t = P t 1 yields t = ( 1) (p t p t 1 )

In this setup, we do not assume inertial dynamics of prices. In ‡ation results from the fact that …rms reoptimizing in any given period their price plans, choose a price that di¤ers from the economy's average price in the previous period.

Price setting

A …rm reoptimizing in period t chooses the price P t that maximizes the current market value of the pro…ts generated while that price remains e¤ective. This problem is solved and leads to a …rst-order Taylor expansion around the zero in ‡ation steady state: 1)

p t p t 1 = (
1 X k=0 ( ) k E t c mc t+kjt + (p t+k p t 1 ) (15)
where c mc t+kjt = mc t+kjt mc denotes the log deviation of marginal cost from its steady state value mc = , and = log " " 1 is the log of the desired gross markup.

Equilibrium

Market clearing in the goods market requires Y t (i) = C t (i) for all i 2 [0; 1] and all t. Aggregate output is de…ned as

Y t = R 1 0 Y t (i) 1 1 t di t t 1 ; it
follows that Y t = C t must hold for all t. One can combine the above goods market clearing condition with the consumer's Euler equation (4) to yield the equilibrium condition

y t = E t [y t+1 ] 1 t (i t E t [ t+1 ] c ) (16)
Market clearing in the labor market requires N t = R 1 0 N t (i) di. With the production function ( 7) and taking logs, one can write the following approximate relationship between aggregate output, employment and technology as

y t = " a t + (1 ) n t ( 17
)
An expression is derived for an individual …rm's marginal cost in terms of the economy's average real marginal cost:

mc t = (w t p t ) mpn t (18) = w t p t 1 1 (" a t y t ) log (1 )
for all t, where mpn t de…nes the economy's average marginal product of labor.

As mc t+kjt = (w t+k p t+k ) mpn t+kjt we have

mc t+kjt = mc t+k t 1 (p t p t+k ) (19)
where the second equality follows from the demand schedule combined with the market clearing condition c t = y t . Substituting ( 19) into (15) yields

p t p t 1 = (1 ) 1 X k=0 t+k ( ) k E t [ c mc t+k ] + 1 X k=0 ( ) k E t [ t+k ] ( 20
)
where t = 1 1 + t
1 is time varying to take into account the markup shock.

Finally, ( 14) and ( 20) yield the in ‡ation equation

t = E t [ t+1 ] + mct c mc t (21)
where , mct = (1 )(1 ) t . mct is strictly decreasing in the index of price stickiness , in the measure of decreasing returns , and in the demand elasticity t .

Next, a relationship is derived between the economy's real marginal cost and a measure of aggregate economic activity. From ( 6) and ( 17), the average real marginal cost can be expressed as

mc t = t + + 1 y t 1 + 1 " a t log (1 ) n ( 22
)
Under ‡exible prices, the real marginal cost is constant and equal to mc = . De…ning the natural level of output, denoted by y f t , as the equilibrium level of output under ‡exible prices leads to

mc = t + + 1 y f t 1 + 1 " a t log (1 ) n (23) thus implying y f t = a " a t + c ( 24
)
where a = 1+ t( 1)+ + and c =

(1 )(log(1 )+ n log( " " 1 ))

t( 1)+ +

. Subtracting (25) from (24) yields

c mc t = t + + 1 y t y f t (25)
where c mc t = mc t mc is the real marginal cost gap and y t y f t is the output gap. Combining the above equation with (23), we obtain

t = E t [ t+1 ] + x y t y f t (26) where x = (1 )(1 )( t(1 )+ + ) (1 + ")
and y t y f t is the output gap. The second key equation describing the equilibrium of the model is obtained by rewriting (19) to determine output

y t = E t [y t+1 ] 1 t (i t E t [ t+1 ] c ) (27)
Equation ( 27) thus a dynamic IS equation including the real money balances.

The third key equation describes the behavior of the real money balances. From (5), we obtain

mp t = t y t a 2 i t m (28)

B Model validation

The red and blue lines in Fig. 3 represent an aggregate measure based on the eigenvalues of the variance-covariance matrix of each parameter both within and between chains. Each graph represents speci…c convergence measures and has two distinct lines that represent the results within and between chains. Those measures are related to the analysis of the parameter's mean (…rst moment), variance (second moment) and third moment of the model in the considered period. Convergence requires that both lines for each of the three measures become relatively constant and converge to each other. The diagnoses concerning the numerical maximization of the posterior kernel indicate that the optimization procedure was able to obtain a robust maximum for the posterior kernel. A diagnosis of the overall convergence for the Metropolis-Hastings sampling algorithm is provided in Fig. 3.

Diagnoses for each individual parameter were also obtained, following the same structure as that of the overall. Most of the parameters do not seem to exhibit convergence problems, notwithstanding the fact that this evidence is stronger for some parameters than it is for others.

D Impulse response functions

The black line, blue line, red line, cyan line and magenta line, represent, respectively the P5, P4, P3, P2 and P1 impulse response functions.

Figure 1 :

 1 Figure 1: Bayesian estimation of parameters over the selected periods

Figure 2 :

 2 Figure 2: Long-term (grey) and short-term (black) variance decompositions

Figure 3 :Figure 4 :Figure 6 :Figure 7 :Figure 8 :

 34678 Figure 3: Multivariate Metropolis-Hastings convergence diagnosis

Figure 9 :

 9 Figure 9: Impulse response functions

Table 1 :

 1 Priors summary

		Law	Mean Std.		Law	Mean Std.
		beta	0.33 0.10	m	normal	1.00 0.20
		beta	0.66 0.10	c	normal	2.00 0.20
		normal 2.00 0.20	mp c	normal	4.00 0.20
	v normal 1.50 0.10	a	beta	0.75 0.15
	" normal 6.00 0.10	p	beta	0.75 0.15
		normal 1.00 0.10	i	beta	0.50 0.15
		beta	0.20 0.05	r	beta	0.50 0.15
		beta	0.20 0.05	a	invgamma 0.01 2.00
	i	beta	0.50 0.10	p	invgamma 0.01 2.00
		normal 3.00 0.20	i	invgamma 0.01 2.00
	x	normal 1.50 0.20	r	invgamma 0.01 2.00

Because 1 > 1, a 2 > 0.