Rates of convergence in the strong invariance principle for non adapted sequences. Application to ergodic automorphisms of the torus

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for non adapted sequences. Application to ergodic automorphisms of the torus

Jérôme Dedecker, Florence Merlevède, Françoise Pene

To cite this version:

Jérôme Dedecker, Florence Merlevède, Françoise Pene. Rates of convergence in the strong invari-

ance principle for non adapted sequences. Application to ergodic automorphisms of the torus. High

dimensional probability 6, Oct 2011, Banff, Canada. pp.113-138. �hal-00713797�

ance principle for non adapted sequences.

Application to ergodic automorphisms of the torus

J´ erˆ ome Dedecker, Florence Merlev` ede and Fran¸coise P` ene

Abstract. In this paper, we give rates of convergence in the strong invari- ance principle for non-adapted sequences satisfying projective criteria.

The results apply to the iterates of ergodic automorphismsT of thed- dimensional torusT^d, even in the non hyperbolic case. In this context, we give a large class of unbounded functionf fromT^dtoR, for which the partial sumf◦T+f◦T²+· · ·+f◦Tⁿsatisfies a strong invariance principle with an explicit rate of convergence.

Mathematics Subject Classification (2010).60F17; 37D30.

Keywords.almost sure invariance principle, strong approximations, non adapted sequences, ergodic automorphisms of the torus.

1. Introduction and notations

Let (Ω,A,P) be a probability space, andT : Ω7→Ω be a bijective bimeasur- able transformation preserving the probability P. For a σ-algebra F0 sat- isfying F0 ⊆ T⁻¹(F0), we define the nondecreasing filtration (Fi)i∈Z by Fi=T⁻ⁱ(F0). TheL^p norm of a random variableX is denoted bykXkp = (E(|X|^p))^1/p.

Let X0 be a real-valued and square integrable random variable such thatE(X0) = 0, and define the stationary sequence (Xi)i∈ZbyXi=X0◦Tⁱ. Define then the partial sum by Sn = X1+X2+· · ·+Xn. According to the Birkhoff-Khinchine theorem,Sn satisfies a strong law of large numbers.

One can go further in the study of the statistical properties ofSn. 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. P`ene is partially supported by the French ANR projects MEMEMO2 and PERTURBATIONS.

independent identically distributed (iid) Gaussian random variables (Zi)i≥1

such that sup

1≤k≤n

k

X

i=1

(Xi−Zi)

=o(n^1/pL(n)) almost surely, (1.1) forp∈]2,4] and Lan explicit slowly varying function. Let us recall that, in the iid case, Koml´os, Major and Tusn´ady [13] and Major [18] obtained an ASIP with the optimal rateo(n^1/p) in (1.1) as soon as the random variables admit a moment of orderp.

Since the seminal paper by Philipp and Stout [25], many authors have considered this problem in a dependent context, but most of the papers deal with the adapted case, when X0 is F⁰ measurable (for instance, F⁰ is the pastσ-algebraσ(Xi, 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 typekE(Xn|F⁰)k^p.

In this paper, we shall not assume thatX0 is F⁰-measurable, and we shall give conditions on the quantities kE(Xn|F0)kp, kX₋n−E(X₋n|F0)kp

andkE(S_n²|F−n)−E(S_n²)kp/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 [7] forp= 2 to get the usual central limit theorem. Our proof is based on the approximation

n

X

i=1

Xi=Mn+Rn

by the martingaleMn=d1+d2+· · ·+dn, wherediis the martingale difference di=X

k∈Z

(E(Xk|Fⁱ)−E(Xk|Fⁱ−1))

introduced by Gordin [7] and Heyde [10]. In the adapted case, similar condi- tions are given in the recent paper [2], 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 torusT^d, and P is the Lebesgue measure on T^d. In this context, we use the σ-algebras Fi

considered by Le Borgne [14]. As a consequence of Theorem 2.1, we obtain that (1.1) holds forp= 4 andXi=f◦Tⁱ, wheref :T^d→R, as soon as the Fourier coefficients (c_k)_k∈Z^d off are such that

|c_k| ≤A

d

Y

i=1

1

(1 +|ki|)^3/4log^α(2 +|ki|) for someα >13/8.

We also get that there exists a positiveεsuch that sup

1≤k≤n

k

X

i=1

(Xi−Zi)

=o(n^1/2−ε) almost surely, as soon as

|c_k| ≤A

d

Y

i=1

1

(1 +|ki|)^δ for some δ >1/2.

These rates of convergence in the almost sure invariance principle complement the results by Leonov [16] and Le Borgne [14] for the central limit theorem and the almost sure invariance principle respectively. Let us mention that Dolgopyat [5] established an ASIP with the rateo(n^1/2−ε) (for someε >0) valid for ergodic automorphisms of the torus and f a H¨older continuous function. Thanks to the decorrelation estimates obtained in [15], the rate for H¨older observables can be improved by applying the general result of Gou¨ezel in [8] to get the rateo(n^1/4+ε) for everyε >0, and by applying the results of the present paper to get the rateo(n^1/4L(n)). To our knowledge, the present work gives the first strong approximation results for such partially hyperbolic transformationsT for unbounded (and then non continuous) functionsf.

To conclude, let us mention some previous works in the context of dy- namical systems: several results have been established with the rateo(n^1/2−ε) for someε >0 (see [11, 4, 5, 24, 19]). Results giving a rate in o(n^1/4+ε) for every ε > 0 can be found in [21, 6, 20, 8]. Most of these results hold for bounded functionsf.

Let us reiterate that we can reach the rate o(n^1/4L(n)) instead of o(n^1/4+ε) for everyε >0. Moreover, our conditions giving the rateo(n^1/pL(n)) are related to moments of orderpoff. Such results are not very common in the context of dynamical systems (let us mention [8] in the particular case of Gibbs-Markov maps, and [3, 23] for generalized Pommeau-Manneville maps).

2. 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 measurableP-preserving transformationT : Ω→Ω. Such a dynamical system is said to be ergodic if the onlyA∈ Fsuch thatT⁻¹A=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≥1starts with the Birkhoff-Khintchine theorem [1, 12]. This theorem ensures that, for every integrable functionf : Ω→R, the sequencen⁻¹Pn

k=1f ◦T^k converges almost surely to E(f). This means that (f◦T^k)k≥1satisfies 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 functionsf, with a rate depending on the rate of convergence to

zero of the Fourier coefficients off. In this context, we use theσ-algebrasFi

considered by Le Borgne [14]. The stationary sequence (Tⁱ)i∈Zis non adapted to this filtration (Fⁱ)i∈Z, in the sense thatTⁱis notFⁱmeasurable. 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 everyx∈R^d, we write ¯xits class inT^d. We recall thatT is the quotient map of a linear map ˜T :R^d→R^d given by ˜T(x) =S·x, whereS is a d×d-matrix with integer entries and with determinant 1 or -1. The map x7→S·xpreserves the infinite Lebesgue measure λonR^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 ofS has modulus one) but is much weaker. Indeed, as mentioned in [14], the following matrix gives an example of an ergodic non hyperbolic automorphism ofT⁴:

S :=









0 0 0 −1

1 0 0 2

0 1 0 0

0 0 1 2







 .

WhenT 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 [17], to prove a central limit theorem [16]. Moreover, in [14], Le Borgne proved the functional central limit theorem and the Strassen strong invariance principle for (Xk=f◦T^k)k under weak hypotheses onf, thanks to Gordin’s method and to the partitions studied by Lind in [17].

We give here rates of convergence in the strong invariance principle for (Xk =f ◦T^k)k under conditions on the Fourier coefficients of f :T^d →R. In what follows, fork∈Z^d, we denote by|k|= maxi∈{1,...,d}|ki|.

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 →Rbe a centered function with Fourier coefficients(c_k)_k∈Z^dsatisfying, for any integer b≥2,

X

|k|≥b

|c_k|^q ≤Rlog^−θ(b) for someθ > p²−2

p(p−1), (2.2) and

X

|k|≥b

|c_k|²≤Rlog^−β(b) for someβ > 3p−4

p . (2.3)

Then the series

σ²= ¯λ((f−λ(f¯ ))²) + 2X

k>0

λ((f¯ −λ(f))f¯ ◦T^k)

converges absolutely and, enlarging T^d if necessary, there exists a sequence (Zi)i≥1of iid gaussian random variables with zero mean and varianceσ²such that, for anyt >2/p,

sup

1≤k≤n

k

X

i=1

f ◦Tⁱ−

k

X

i=1

Zi

=o n^1/p(logn)^(t+1)/2

almost surely, asn→ ∞. (2.4) Observe that (2.3) follows from (2.2) provided thatθ >(3p−4)/(2p−2).

Hence, (2.2) and (2.3) are both satisfied as soon as X

|k|≥b

|c_k|^q ≤Rlog^−θ(b) for someθ > 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

Y

i=1

1

(1 +|ki|)^1/qlog^α(2 +|ki|), (2.5) for some positive constant A, then the conditions (2.2) and (2.3) are both satisfied provided thatα >(2p²−p−2)/p².

Let us now compare our hypotheses on Fourier coefficients with those appearing in other works. In [16], Leonov proved a central limit theorem (possibly degenerated) when

|c_k| ≤A

d

Y

i=1

1

(1 +|ki|)^1/2log^α(2 +|ki|) for some α >3/2. (2.6) In [14], Le Borgne proved the functional central limit theorem and the Strassen strong invariance principle when (2.3) holds true withβ >2 (and whenf is not a coboundary), which is a weaker condition than (2.6). Observe that, as pconverges to 2, (p²−2)/(p(p−1)) and (3p−4)/pboth converge to 1.

3. Probabilistic results

In the rest of the paper, we shall use the following notations: E_k(X) = E(X|Fk), and an ≪bn means that there exists a numerical constant C not depending onnsuch thatan ≤Cbn, for all positive integersn.

In this section, we give rates of convergence in the strong invariance principle under projective criteria for stationary sequences that are non nec- essarily adapted toFⁱ.

Theorem 3.1. Let 2 < p < 4 and t > 2/p. Assume that X0 belongs to L^p, that

X

n≥2

n^p−1

n^2/p(logn)^(t−1)p/2 kE₀(Xn)k^pp+kX₋n−E₀(X₋n)k^pp

<∞, (3.1) and that

X

n≥2

n^3p/4

n²(logn)^(t−1)p/2 kE₀(Xn)k^p/22 +kX₋n−E₀(X₋n)k^p/22

<∞. (3.2) Assume in addition that there exists a positive integerm such that

X

n≥2

1 n²(logn)^(t−1)p/2

E_−nm(S_n²)−E(S_n²)

p/2

p/2<∞. (3.3) Then n⁻¹E(S_n²) converges to σ² = P

k∈ZCov(X0, Xk) and, enlarging Ω if necessary, there exists a sequence (Zi)i≥1 of iid Gaussian random variables with zero mean and varianceσ² such that

sup

1≤k≤n

Sk−

k

X

i=1

Zi

=o n^1/p(logn)^(t+1)/2

almost surely, as n→ ∞. (3.4) Theorem 3.2. Let t >1/2. Assume thatX0 belongs to L⁴ and that the con- ditions (3.1)and (3.3)hold with p= 4. Assume in addition that

X

n≥2

n(logn)^4−2t kE₀(Xn)k²2+kX₋n−E₀(X₋n)k²2

<∞. (3.5) Then the conclusion of Theorem3.1 holds withp= 4.

Of course, if (Xi)i∈Z is a sequence of iid random variables inL^p, then, takingFⁱ =σ(Xk, 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 raten^1/p.

The conditions (3.1), (3.2) and (3.5) are similar to the conditions given by Gordin [7] whenp= 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 Mn introcuded by Gordin [7], and we shall give an appropriate upper bound onRn=Sn−Mn. The next step is to get a strong approximation result for the martingaleMn. This will be done by applying Proposition 5.1 in [2], which itself is based on the Skorohod embedding for martingales, as in [26].

3.1. Proofs of Theorems3.1and3.2

Proof. We first notice that sincep >2, (3.1) implies that X

n>0

n^−1/pkE₀(Xn)kp<∞ and X

n>0

n^−1/pkX₋n−E₀(X₋n)kp<∞

(apply H¨older’s inequality to see this). LetPk(X) =E_k(X)−E_k−1(X). Using Lemma 5.1 of the appendix withq= 1, we infer that

X

k∈Z

kP0(Xk)k^p<∞. (3.6) In addition the condition (3.6) implies thatn⁻¹E(S_n²) converges to the quan- tityσ²=P

k∈ZCov(X0, Xk).

Let nowd0:=P

j∈ZP0(Xj). Thend0belongs toL^pandE(d0|F−1) = 0.

Let di := d0◦Tⁱ for all i ∈ Z. Then (di)i∈Z is a stationary sequence of martingale differences inL^p. Let

Mn:=

n

X

i=1

di and Rn:=Sn−Mn. The theorems will be proven if we can show that

Rn =o n^1/p(logn)^(t+1)/2

almost surely as n→ ∞, (3.7) and that (3.4) holds true withMkreplacingSk. SinceE(d²₀) =σ²andt > p/2, according to Proposition 5.1 in [2] (applied with ψ(n) := n^2/p(logn)^t), to prove that (3.4) holds true withMk replacingSk, it suffices to prove that

X

n≥2

1 n²(logn)^(t−1)p/2

E₀(M_n²)−E(M_n²)

p/2

p/2<∞. (3.8) By standard arguments, (3.7) will be satisfied if we can show that

X

r>0

kmax1≤ℓ≤2^r|Rℓ|k^pp

2^rr^(t+1)p/2 <∞. (3.9)

Now, by stationarity,kmax1≤ℓ≤2^r|Rℓ|kp ≪2^r/pPr

k=02^−k/pkR₂^kkp (see for instance inequality (6) in [27]) and for alli, j≥0,kRi+jkq ≤ kRikq+kRjkq. Applying then Item 1 of Lemma 37 in [22], we derive that for any integern in [2^r,2^r+1[,

max

1≤ℓ≤2^r|Rℓ| p

≪n^1/p

n

X

k=1

k^−(1+1/p)kRkk^p. (3.10) Therefore using (3.10) followed by an application of H¨older’s inequality, we get that for anyα <1,

X

r>0

kmax1≤ℓ≤2^r|Rℓ|k^pp

2^rr^(t+1)p/2 ≪X

n≥2

1 n(logn)^(t+1)p/2

Xⁿ

k=1

k^−(1+1/p)kRkkp

p

≪X

n≥2

(logn)^{(p−1)(1−α)} n(logn)^(t+1)p/2

n

X

k=1

k⁻²(logk)^α(p−1)kRkk^pp.

Hence takingα∈]1−p/(2(p−1)),1[ and changing the order of summation, we infer that (3.9) and then (3.7) hold provided that

X

n≥1

kRnk^pp

n²(logn)^(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 integermsuch that X

n≥2

1 n²(logn)^(t−1)p/2

E_−nm(M_n²)−E(M_n²)

p/2

p/2<∞. (3.12) For any nonnegative integer i, we set Vi := kE₀(M_i²)−E(M_i²)kp/2. Using thatMn is a martingale, we infer that, for any nonnegative integersiandj,

Vi+j≤Vi+Vj. (3.13)

Let nown∈[2^k,2^k+1−1]∩N, and write its binary expansion:

n=

k

X

ℓ=0

2^ℓbℓ wherebk= 1 and bj∈ {0,1}forj= 0, . . . , k−1. Inequality (3.13) combined with H¨older’s inequality implies that, for any η >0,

V_n^p/2≤X^k

ℓ=0

V2^ℓ

p/2

≪2^ηp(k+1)/2

k

X

ℓ=0

V2^ℓ

2^ηℓ p/2

. (3.14)

Therefore X

n≥2

1

n²(logn)^(t−1)p/2V_n^p/2≪X

k>0

2^ηp(k+1)/2 2^kk^(t−1)p/2

k

X

ℓ=0

V₂^ℓ 2^ηℓ

p/2

.

Changing the order of summation and takingη∈]0,2/p[, it follows that (3.8) is implied by

X

k≥1

1 2^kk^(t−1)p/2

E₀(M₂²k)−E(M₂²k)

p/2

p/2<∞ (3.15) (actually due to the subadditivity of the sequence (Vi) both conditions are equivalent, see the proof of item 1 of Lemma 37 in [22] to prove that (3.8) entails (3.15)). Now, since (Mn) is a martingale,

E₀(M₂²k)−E(M₂²k) =

k

X

j=1

E₀((M2^j −M2^j−1)²)−E((M2^j −M2^j−1)²) +E₀(d²₁)−E(d²₁),

which implies by stationarity that

E₀(M₂²k)−E(M₂²k) p/2≤

k−1

X

j=0

E

−2^j(M₂²j)−E(M₂²j) p/2

+

E₀(d²₁)−E(d²₁) p/2. Therefore by using H¨older’s inequality as done in (3.14) withη∈]0,2/p[, we infer that (3.15) is implied by

X

k≥1

1 2^kk^(t−1)p/2

E₋₂k(M₂²k)−E(M₂²k)

p/2

p/2<∞. (3.16) Notice now that the sequence (Wn)n>0 defined by

Wn :=

E_−n(M_n²)−E(M_n²) p/2

is subadditive. Indeed, for any non negative integersiand j, using thatMn

is a martingale together with the stationarity, we derive that Wi+j=

E_−(i+j)(M_i²)−E(M_i²) +E_−(i+j)((Mi+j−Mi)²)

−E((Mi+j−Mi)²) p/2

≤

E_−i(M_i²)−E(M_i²) _p/2+

E_−j(Mj)²)−E(Mj)²) _p/2

≤Wi+Wj.

ThereforeW_i+j^p/2≤2^p/2W_i^p/2+ 2^p/2W_j^p/2. This implies that, for any integerℓ and any integer 0≤j≤ℓ,W_ℓ^p/2≤2^p/2(W_j^p/2+W_ℓ^p/2_−j) in such a way that

(ℓ+ 1)W_ℓ^p/2≤2^1+p/2

ℓ

X

j=1

W_j^p/2. (3.17)

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

X

n≥2

1 n²(logn)^(t−1)p/2

E_−n(M_n²)−E(M_n²)

p/2

p/2<∞. (3.18) It remains to prove that (3.12) implies (3.18). With this aim, we have, for any positive integerm,

Mn=

m

X

k=1

M_k[nm−1]−M_(k−1)[nm−1]

+Mn−M_m[nm−1].

Using thatMn is a martingale together with the stationarity, we then infer that

E_−n(M_n²)−E(M_n²)

p/2

p/2≤2^p/2m^p/2

E_−n(M_[nm² −1])−E(M_[nm² −1])

p/2 p/2

+ 2^p/2

E_−n(M_n²_−m[nm−1])−E(M_n²_−m[nm−1])

p/2 p/2,

which, together with the fact thatn−m[nm⁻¹]< m, implies that

E_−n(M_n²)−E(M_n²)

p/2 p/2

≤2^p/2m^p/2

2^p/2kd0k^pp+

E_−n(M_[nm² −1])−E(M_[nm² −1])

p/2 p/2

≤2^p/2m^p/2

2^p/2kd0k^pp+

E_−m[nm−1](M_[nm² −1])−E(M_[nm² −1])

p/2 p/2

, (3.19) where for the last line we have used the fact thatn≥m[nm⁻¹]. We notice now that due to the martingale property of (Mn) and to stationarity, the sequence (Ui)i≥0 defined for any non negative integeriby

Ui:=

E_−mi(M_i²)−E(M_i²)

p/2 p/2

satisfies, for any positive integersiandj, Ui+j ≤

E_−m(i+j)(M_i²)−E(M_i²) _p/2 +

E_−m(i+j)((Mi+j−Mi)²)−E((Mi+j−Mi)²) _p/2

p/2

≤2^p/2Ui+ 2^p/2Uj. Hence by (3.17) applied withW_i^p/2=Ui,

U_[nm−1] ≤2^1+p/2([nm⁻¹] + 1)⁻¹

[nm⁻¹]

X

k=1

Uk≤2^1+p/2

[nm⁻¹]

X

k=1

Uk

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.

Lemma 3.1. Letp∈[2,4]. Assume that (3.1)holds. Then X

n≥1

max1≤ℓ≤nkRℓk^pp

n²(logn)^(t−1)p/2 <∞, and (3.11) holds.

Proof. Since (3.1) implies (3.6), Item 2 of Proposition 5.1 given in the ap- pendix implies that, for any positive integersℓandN,

kRℓkp≪ max

k=ℓ,NkE₀(Sk)kp+ max

k=ℓ,NkSk−E_k(Sk)kp+ℓ^1/2 X

|j|≥N

kP0(Xj)kp.

Next, applying Lemma 5.1 given in the appendix withq= 1, and using the fact that by stationarity, for any positive integerk,

kE₀(Sk)kp≤

k

X

ℓ=1

kE₀(Xℓ)kp and kSk−E_k(Sk)kp ≤

k−1

X

ℓ=0

kX₋ℓ−E₀(X₋ℓ)kp, (3.21) we derive that for, any positive integersN ≥n,

1max≤ℓ≤nkRℓkp≪

N

X

k=1

kE₀(Xk)kp+

N−1

X

k=0

kX₋k−E₀(X₋k)kp

+n^1/2 X

k≥[N/2]

kE₀(Xk)kp

k^1/p +n^1/2 X

k≥[N/2]

kX₋k−E₀(X₋k)kp

k^1/p . (3.22) The lemma follows from (3.22) withN = [n^p/2] by using H¨older’s inequality (see the computations in the proof of Proposition 2.2 in [2]).

Lemma 3.2. Let p ∈ [2,4] 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.

Proof. Letmbe a positive integer such that (3.3) is satisfied. We first write that

kE_−nm(M_n²)−E(M_n²)

p/2≤ kE_−nm(S_n²)−E(S_n²) p/2

+ 2kE_−nm(SnRn)−E(SnRn)kp/2+ 2kRnk²p. By using Lemma 3.1, and since (3.3) holds, Lemma 3.2 will follow if we can prove that

X

n≥1

1

n²(logn)^(t−1)p/2kE_−nm(SnRn)k^p/2_p/2<∞. (3.23) With this aim we shall prove the following inequality. For any non negative integerrand any positive integerun such thatun≤n, we have that

kE_−r(SnRn)kp/2≪√un kE₀(Sn)k2+kSn−E_n(Sn)k2

+ max

k={n,n−un}kRkk²p+√ n kE_−u

n(Sn)k2+kSn−E_n+u

n(Sn)k2

+ max

k={n,un}kE_−r(S_k²)−E(S_k²)kp/2+√ nXⁿ

k=1

X

|j|≥k+n

P0(Xj)

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 variableX inL^q forq≥1,kX−E(X|F)k^q ≤2kX−E(X|G)k^q.

Starting from (3.24) withr=nmandun=n, and using Claim 3.1, we derive that

kE_−nm(SnRn)kp/2≪ kE_−nm(S_n²)−E(S_n²)kp/2

+√

n kE₀(Sn)k²+kSn−E_n(Sn)k²

+kRnk²p+n X

|j|≥n

kP0(Xj)k². This last inequality combined with condition (3.3) and Lemma 3.1 shows that (3.23) will be satisfied if we can prove that

X

n≥1

n^p/4

n²(logn)^(t−1)p/2 kE₀(Sn)k²+kSn−E_n(Sn)k²p/2

<∞, (3.25) and

X

n≥1

n^p/2 n²(logn)^(t−1)p/2

X

|j|≥n

kP0(Xj)k²p/2

<∞. (3.26) To prove (3.25), we use the inequalities (3.21) withp= 2. Hence setting

aℓ=kE₀(Xℓ)k²+kX₋ℓ+1−E₀(X₋ℓ+1)k², (3.27) and using H¨older’s inequality, we derive that for anyα <1,

X

n≥1

n^p/4

n²(logn)^(t−1)p/2 kE₀(Sn)k2+kSn−E_n(Sn)k2p/2

≪X

n≥1

n^p/4 n²(logn)^(t−1)p/2

Xⁿ

ℓ=1

aℓ

p/2

≪X

n≥1

n^p/4n^{(1−α)(p/2−1)} n²(logn)^(t−1)p/2

n

X

ℓ=1

ℓ^α(p/2−1)a^p/2_ℓ . Takingα∈](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

X

n≥1

n^p/2 n²(logn)^(t−1)p/2

X

|j|≥n

kP0(Xj)k2

p/2

≪X

n≥1

n^p/2 n²(logn)^(t−1)p/2

X

ℓ≥[n/2]

ℓ^−1/2aℓ

p/2

.

Therefore by H¨older’s inequality, it follows that for anyα <1, X

n≥1

n^p/2 n²(logn)^(t−1)p/2

X

|j|≥n

kP0(Xj)k2

p/2

≪X

n≥1

n^p/2n^{(1−α)(p/2−1)} n²(logn)^(t−1)p/2

X

ℓ≥[n/2]

ℓ^α(p/2−1)ℓ^−p/4a^p/2_ℓ .

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) whenp∈]2,4[.

Now, we prove (3.23) whenp= 4. With this aim we start from (3.24) withr=nmandun= [√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 forp= 4 if we can prove that

X

n≥1

1

n(logn)^2(t−1) kE

−[√

n](Sn)k²+kSn−E_n+[√

n](Sn)k²2

<∞, (3.28)

and

X

n≥2

1 n²(logn)^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

kE_−[√n](Sn)k²+kSn−E_n+[√n](Sn)k²≤

n+[√n]

X

k=[√n]+1

ak.

Therefore by Cauchy-Schwarz’s inequality

X

n≥1

1

n(logn)^2(t−1) kE

−[√

n](Sn)k²+kSn−E_n+[√

n](Sn)k²2

≪X

n≥1

logn n(logn)^2(t−1)

n+[√ n]

X

k=[√ n]+1

ka²_k

≪X

n≥1

1 n

n+[√ n]

X

k=[√n]+1

klogk (logk)^2(t−1)a²_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 realx,

h([x]) = E

−m[x](S_[x]² )−E(S²_[x])

2 2, and we notice that, for any integern≥0,

E_−nm(S_[^2√_n])−E(S_[^2√_n])

2

2≤h([√ n]).

In addition, ifx∈[n, n+ 1[, then [√ n] = [√

x] or [√ n] = [√

x]−1. Therefore X

n≥3

1

n²(logn)^(t−1)p/2h([√

n])≪X

n≥3

h([√ n])

Z

[n,n+1[

1

x²(logx)^(t−1)p/2dx

≪ Z ∞

3

1

x²(logx)^(t−1)p/2h([√ x])dx+

Z ∞ 3

1

x²(logx)^(t−1)p/2h([√

x]−1)dx

≪ Z ∞

2

1

y³(logy)^(t−1)p/2h([y])dy≪X

n≥2

1

n³(logn)^(t−1)p/2h(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 decompo- sition ofRn given in Proposition 5.1 of the appendix withN =n. Therefore setting

An:=

n

X

k=1

X

j≥2n+1

Pk(Xj) +

n

X

k=1

X

j≥n

Pk(X₋j), we write that

Rn=E₀(Sn)−E₀(Sn)◦Tⁿ+E_−n(Sn)◦Tⁿ+Sn−E_n(Sn)

−(E_2n(Sn−E_n(Sn))◦T⁻ⁿ−An. (3.30) Starting from (3.30) and noticing that

kE_−r(Sn(E_−n(Sn)◦Tⁿ)kp/2≤ kE₀(Sn(E_−n(Sn)◦Tⁿ)kp/2

≤ kE₀(Sn)kpkE₀(S2n−Sn)kp, and thatE_−r(Sn(Sn−E_n(Sn)) =E_−r((Sn−E_n(Sn))²), we first get

kE_−r(SnRn)kp/2≤2kE₀(Sn)k²p+kSn−E_n(Sn)k²p

+kE_−r(SnE_n(S2n−Sn))kp/2

+kE_−r(SnE_n(Sn◦T⁻ⁿ−E₀(Sn◦T⁻ⁿ)))kp/2

+kE_−r(SnAn)

p/2. (3.31)

Next, we use the following fact: if X and Y are two variables in L^p with p∈[2,4], then for any integeru,

kE_u(XY)kp/2≤ kE_u(X²)−E(X²)kp/2+kYk²p+p

E(X²)kYk². (3.32) Indeed, it suffices to write that

kE_u(XY)kp/2≤ kE^1/2_u (X²)E^1/2_u (Y²)kp/2

≤ k|E_u(X²)−E(X²)|^1/2E^1/2

u (Y²)kp/2+ (E(X²))^1/2kE^1/2

u (Y²)kp/2

≤ kE_u(X²)−E(X²)kp/2+kYk²p+ (E(X²))^1/2kE^1/2_u (Y²)kp/2,

and to notice that, since p∈ [2,4], kE^1/2_u (Y²)kp/2 ≤ kE^1/2_u (Y²)k² =kYk². Therefore, starting from (3.31) and using (3.32) together with E(S_n²)≪ n, we infer that

kE_−r(SnRn)kp/2≪kE₀(Sn)k²p+kSn−E_n(Sn)k²p

+kE_−r(SnE_n(S2n−Sn))kp/2

+kE_−r(SnE_n(Sn◦T⁻ⁿ−E₀(Sn◦T⁻ⁿ)))kp/2

+kE_−r(S_n²)−E(S_n²)

p/2+kAnk²p+n^1/2kAnk2, and since kE₀(Sn)k^p ≤ kRnk^p, kSn −E_n(Sn)k^p ≤ 2kRnk^p and kAnk^p ≤ 8kRnkp, we have overall that

kE_−r(SnRn)kp/2≪kRnk²p+kE_−r(SnE_n(S2n−Sn))kp/2

+kE_−r(SnE_n(Sn◦T⁻ⁿ−E₀(Sn◦T⁻ⁿ)))kp/2

+kE_−r(S_n²)−E(S²_n)

p/2+n^1/2kAnk². (3.33) By orthogonality and by stationarity,

kAnk²≤Xⁿ

k=1

X

j≥2n+1

Pk(Xj)

2 2

1/2

+Xⁿ

k=1

X

j≥n

Pk(X₋j)

2 2

1/2

≤Xⁿ

k=1

X

ℓ≥k+n

P0(Xℓ)

2 2

1/2

+Xⁿ

k=1

X

ℓ≥k+n

P0(X₋ℓ)

2 2

1/2

. (3.34) Now for any integerun such thatun≤n,

kE_−r(SnE_n(S2n−Sn))kp/2≤kE_−r((Sn−Sn−un)E_n(S2n−Sn))kp/2

+kE_−r(Sn−unE_n(S2n−Sn))kp/2

≪kE_−r(S_u²_n)−E(S²_un) p/2

+kE₀(Sn)k²p+√unkE₀(Sn)k2

+kE_−r(Sn−unE_n(S2n−Sn))kp/2, (3.35) where for the last inequality we have used (3.32) together withE(S_u²_n)≪un. Next, we write that

kE_−r(Sn−unE_n(S2n−Sn))kp/2

≤kE_−r((Sn−un−E_n−u

n(Sn−un))E_n(S2n−Sn))kp/2

+kE_−r(E_n−u

n(Sn−un)E_n(S2n−Sn))kp/2

≤kSn−un−E_n−un(Sn−un)k^pkE₀(Sn)k^p +kE_−r(E_n−u

n(Sn−un)E_n−u

n(S2n−Sn))kp/2

≤kSn−un−E_n−u

n(Sn−un)k²p+kE₀(Sn)k²p

+kE_−r(SnE_n−u

n(S2n−Sn))kp/2

+kE_−r((Sn−Sn−un)E_n−un(S2n−Sn))kp/2.