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

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

Jérôme Dedecker, Florence Merlevède, Françoise Pene

To cite this version:

Jérôme Dedecker, Florence Merlevède, Françoise Pene. Rates in the strong invariance principle for

ergodic automorphisms of the torus. Stochastics and Dynamics, World Scientific Publishing, 2014, 14

(2). �hal-00709413�

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

J´erˆome Dedecker^a, Florence Merlev`ede<sup>b</sup> and Fran¸coise P`ene^{c 1}

a Universit´e Paris Descartes, Sorbonne Paris Cit´e, Laboratoire MAP5 and CNRS UMR 8145. Email:

jerome.dedecker@parisdescartes.fr

b Universit´e Paris Est, LAMA and CNRS UMR 8050.

E-mail: florence.merlevede@univ-mlv.fr

c Universit´e de Brest, Laboratoire de Math´ematiques de Bretagne Atlantique UMR CNRS 6205.

E-mail: francoise.pene@univ-brest.fr

Key words: Invariance principles, strong approximations, ergodic automorphisms of the torus.

Mathematical Subject Classification (2010): 37D30, 60F17.

Abstract

LetT be an ergodic automorphism of thed-dimensional torusT^d. In the spirit of Le Borgne [10], we give conditions on the Fourier coefficients of a functionf fromT^d toRunder which the partial sums f◦T+f◦T²+· · ·+f◦Tⁿsatisfies a strong invariance principle. Next, reinforcing the condition on the Fourier coefficients in a natural way, we obtain explicit rates of convergence in the strong invariance principle, up ton^1/4logn.

1 Introduction

We endow the d-dimensional torus T^d = R^d/Z^d with the Lebesgue measure ¯λ, and we denote by E(·) the expectation with respect to ¯λ. As usual, the L^p norm of a f from T^d to Ris denoted by kfk^p= (E(|f|^p))^1/p.

For d≥2, let T be an ergodic automorphism of T^d, and let f be a function from T^d to Rsuch thatE(f²)<∞andE(f) = 0. In [10], Le Borgne has proved that if the Fourier coefficients (c_k)_k∈Z^d off are such that, forθ >2 and every integerb >1,

X

|k|≥b

|c_k|²≤Rlog^−θ(b), where|k|= max

1≤i≤d|ki|, (1.1)

then the partial sums process

nX^[nt]

i=1

f◦Tⁱ, t∈[0,1]o

(1.2) properly normalized, satisfies both the weak and strong invariance principles. More precisely, Le Borgne has introduced in [10] an appropriate σ-field F0 such that F0 ⊆ T⁻¹(F0), for which the quantitieskE(f◦T^k|F0)k2andkf◦T^−k−E(f◦T^−k|F0)k2 can be controlled for any positive integer k. The weak and strong invariance principles follow then, by applying Gordin’s result (see [8]) and Heyde’s result (see [9]) respectively.

1Supported in part by the ANR project PERTURBATIONS

In Theorem 2.1 of this paper, we show that the weak and strong invariance principles still hold for functionsf satisfying (1.1) with θ >1 only, and we give a multivariate version of these results.

For the weak invariance principle, this follows from an improvement of Gordin’s criterion, which was already known in the univariate case (see [7]). For the strong invariance principle, this will follow from a new criterion for stationary sequences, presented in Theorem 4.1 of the appendix. Note that the condition (1.1) withθ >1 is satisfied if, for a positive constantA,

|c_k|²≤A

d

Y

i=1

1

(1 +|ki|) log^1+α(2 +|ki|) for some α >1, (1.3) improving on the condition α >2 given by Leonov in 1969 (see [11], Remark 1). Note that Leonov has also given a condition in terms of the modulus of continuity off in L².

The strong invariance principle means that, enlarging T^d if necessary, there exists a sequence of independent identically distributed (iid) Gaussian random variablesZi such that

sup

1≤k≤n

k

X

i=1

f ◦Tⁱ−

k

X

i=1

Zi

=o n^1/2(log logn)^1/2

almost surely, as n→ ∞. (1.4) It is also possible to exhibit rates of convergence in (1.4), provided that we reinforce the assumption (1.1). This has been done recently, thanks to a general result giving rates of convergence in the strong invariance principle for partial sums of stationary sequences. More precisely, letp∈]2,4] and q=p/(p−1). We have proved in Theorem 2.1 of [6] that if there exists R >0 such that for every integerb >1,

X

|k|≥b

|c_k|^q ≤Rlog^−θ(b) for someθ > p²−2

p(p−1), (1.5)

and

X

|k|≥b

|c_k|²≤Rlog^−β(b) for someβ > 3p−4

p , (1.6)

then the strong approximation (1.4) holds true with an error of ordero n^1/p(logn)^(t+1)/2

, fort >2/p.

A condition on theℓ^q norm of (c_k)_k∈Z^dseems appropriate in this context, since thisℓ^q-norm dominates theL^p norm off, which is required to be finite to get the rateo(n^1/p) in the iid situation.

If we assume that the Fourier coefficients off are such that,

|c_k|^q ≤A

d

Y

i=1

1

(1 +|ki|) log^1+α(2 +|ki|), (1.7) then the conditions (1.5) and (1.6) are both satisfied provided that α > (p² −2)/(p²−p). Now, considering (1.3), one can wonder ifα >1 in (1.7) is enough to get an approximation error of order o(n^1/pL(n)) in (1.4), whereL(n) is a slowly varying function. The main result of this paper, Theorem 2.2 below, shows that the answer is positive.

2 Invariance principles for ergodic automorphisms of the torus

Let us first recall some probabilistic notations. A measurable functionf :T^d →R^m(with coordinates f1, ..., fm) is said to be centered if everyfi is integrable and centered. Such a function f is said to be square integrable if every fi is square integrable. Now, for every centered and square integrable

functionsf, g:T^d→R^m (withf = (f1, ..., fm) andg= (g1, ..., gm)), we define the covariance matrix Cov(f, g) off andg and the variance matrix Var(f) by

Cov(f, g) = (E(figj))i,j=1,...,m, and Var(f) = Cov(f, f).

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

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

An automorphism T of T^d is said to be hyperbolic if the matrix S associated to T admits no eigenvalue of modulus one. With the preceding characterization of ergodic automorphisms ofT^d, it is clear that every hyperbolic automorphism ofT^dis ergodic. Ergodic automorphisms ofT^d are partially hyperbolic but not necessarily hyperbolic (an example of a non-hyperbolic ergodic automorphism of T^d can be found in [10]).

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

Theorem 2.1. LetT be an ergodic automorphism of T^d. For anyj∈ {1, . . . , m}, let fj:T^d→Rbe a centered function and assume that its Fourier coefficients(c_k,j)_k∈Z^d satisfy the following condition:

there exists a positive constantR such that for every integerb >1, X

|k|≥b

|c_k,j|²≤Rlog^−θ(b) for someθ >1. (2.1) Letf = (f1, . . . , fm) :T^d→R^m. Then the seriesΣ =P

k∈ZCov(f, f◦T^k)converges, and

n→∞lim 1

nVarXⁿ

i=1

f◦Tⁱ

= Σ. (2.2)

In addition,

1. The process {n^−1/2P[nt]

i=1f◦Tⁱ, t∈[0,1]}converges in D([0,1],R^m)equipped with the uniform topology to a Wiener process{W(t), t∈[0,1]} with variance matrixVar(W(1)) = Σ.

2. Enlarging T^d if necessary, there exists a sequence (Zi)i≥1 of iid R^m-valued Gaussian random variables with zero mean and variance matrix Var(Z1) = Σsuch that

sup

1≤k≤n

k

X

i=1

f ◦Tⁱ−

k

X

i=1

Zi

=o n^1/2(log logn)^1/2

almost surely, asn→ ∞.

Whenm= 1, it is also possible to exhibit rates of convergence in (1.4) provided that we reinforce Condition (2.1).

Theorem 2.2. Let T be an ergodic automorphism of T^d. Let p ∈]2,4] and q := p/(p−1). Let f :T^d→Rbe a centered function with Fourier coefficients(c_k)_k∈Z^dsatisfying the following conditions:

there exists a positive constantR such that for every integerb >1, X

|k|≥b

|c_k|^q ≤Rlog^−θ(b) for someθ >1, (2.3)

and

X

|k|≥b

|c_k|²≤R b^−ζ for someζ >0. (2.4) Then the series

σ²=X

k∈Z

E(f.f◦T^k) (2.5)

converges absolutely and, enlarging T^d if necessary, there exists a sequence (Zi)i≥1 of iid Gaussian random variables with zero mean and variance σ² such that

sup

1≤k≤n

k

X

i=1

f ◦Tⁱ−

k

X

i=1

Zi

=o n^1/plogn

almost surely, asn→ ∞. (2.6) Observe that if (1.7) holds with α >1 then (2.3) and (2.4) are both satisfied, so that the strong approximation (2.6) holds. However Theorem 2.1 in [6] and Theorem 2.2 above have different ranges of applicability. Indeed, let γ > 1, and define c_k = ℓ^−γ/q if k = (2^ℓ,0, . . . ,0), c_k = −ℓ^−γ/q if k= (−2^ℓ,0, . . . ,0) forℓ∈N, and c_k = 0 otherwise. Let now b and rbe positive integers such that 2^r−1< b≤2^r. Since

X

|k|≥b

|c_k|^q = 2X

ℓ≥r

1 ℓ^γ, it follows thatλ1(logb)^1−γ ≤P

|k|≥b|c_k|^q ≤λ2(logb)^1−γ(whereλ1andλ2are two positive constants).

Similarlyλ1(logb)^1−2γ/q≤P

|k|≥b|c_k|²≤λ2(logb)^1−2γ/q. In this situation, the conditions (1.5) and (1.6) are both satisfied provided thatγ >1 + (p²−2)/(p²−p) whereas condition (2.4) fails.

To prove Theorem 2.2, we shall still use martingale approximations as done in [6], but with the following modifications: Condition (2.4) allows us to consider a non stationary sequenceX_ℓ^∗=fℓ◦T^ℓ, where the functionsfℓ are defined through a truncated series of the Fourier coefficients of f. For the partial sums associated to this non stationary sequence, the approximation error by a non stationary martingale can be suitably handled with the help of Condition (2.3).

3 Proofs of Theorems 2.1 and 2.2

As in [6], we consider the filtration as defined in [12, 10] that enables to suitably approximate the partial sums Pn

i=1f ◦Tⁱ by a martingale. To be more precise, given a finite partition P of T^d, we define the measurable partitionP0^∞by :

∀x¯∈T^d, P0^∞(¯x) := \

k≥0

T^kP(T^−k(¯x)) and, for every integern, theσ-algebraFn generated by

∀x¯∈T^d, P−n^∞(¯x) := \

k≥−n

T^kP(T^−k(¯x)) =T⁻ⁿ(P0^∞(Tⁿ(¯x)).

These definitions coincide with the ones of [10] applied to the ergodic toral automorphismT⁻¹. We obviously haveFn⊆ Fn+1=T⁻¹Fn. Note that the sequence (f ◦Tⁱ)i≥1is non adapted to (Fi)i≥1.

In what follows, we use the notationE_n(f) =E(f|Fn).

3.1 Proof of Theorem 2.1

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

X

n≥3

logn

n^1/2(log logn)^1/2kE₀(fj◦Tⁿ)k²<∞and X

n≥3

logn

n^1/2(log logn)^1/2kfj−E_n(fj)k²<∞. (3.1) But, according to the proof of Propositions 4.2 and 4.3 of [6] (see also [10]), for anyfj satisfying (2.1),

kE_−n(fj)k2+kf−E_n(fj)k2≪n^−θ/2. Sinceθ >1, (3.1) is satisfied.

3.2 Proof of Theorem 2.2

Let f : T^d → R be a centered function with Fourier coefficients (c_k)_k∈Z^d. For every nonnegative integerm, we write

fm:= X

|k|≤m

cke^2iπhk,·i. (3.2)

Notice that iff satisfies (2.4), then

kf −fmk²≤Rm^−ζ/2, (3.3)

and iff satisfies (2.3), then

kf−fmk^p≤R(log(m))^{−θ(p−1)/p}. (3.4) According to the proofs of Propositions 4.2, 4.3 and 4.4 of [6], there existc≥1 andγ, λ∈(0,1) such that, settingb(n) := [γ⁻ⁿ], we have

sup

m≤b(n)

(kE_−n(fm)kp+kfm−E_n(fm)kp)≪λⁿ (3.5) (according to (4.50), (4.51) and (4.53) of [6]), and

sup

N≥cn

sup

m≤b(n)

sup

ℓ∈{0,...,n}kE_−N(fmfm◦T^ℓ)−E(fmfm◦T^ℓ)kp/2≪λⁿ (3.6) (according to (4.61) and (4.62) of [6]). Moreover, according to the proof of Propositions 4.2 and 4.3 of [6], we have, for anyf satisfying (2.3),

sup

m≥1

(kE_−n(fm)kp+kfm−E_n(fm)kp)≪n^{−θ(p−1)/p}, (3.7) and

kE_−n(f)kp+kf−E_n(f)kp≪n^{−θ(p−1)/p}. (3.8) In addition, according to the proof of Proposition 4.4 of [6], there exists a positive integerc, such that for anyf satisfying (2.3) and (2.4),

1≤k≤nmax kE_−nc(S_k²(f))−E(S_k²(f))kp/2≪n2−2θ(p−1)/p. (3.9) For anyf satisfying (2.4), using the arguments developed in the proofs of Propositions 4.2 and 4.3 of [6], we infer that there existsβ∈(0,1) such that

sup

m≥1

(kE_−n(fm)k²+kfm−E_n(fm)k²)≪βⁿ, (3.10)

and

kE_−n(f)k2+kf −E_n(f)k2≪βⁿ. (3.11) Let us writePℓ(·) =E_ℓ(·)−E_ℓ−1(·). Now, letαbe a positive real such thatαζ ≥3−2/p. We then define

d^∗₁:=X

k∈Z

P1 f1◦T^k

, X₁^∗:=f1◦T , and, for everyj≥0 and everyℓ∈ {2^j+ 1, ...,2^j+1},

d^∗_ℓ :=X

k∈Z

Pℓ f[2^αj]◦T^k

, X_ℓ^∗:=f[2^αj]◦T^ℓ. For every positive integern, we define

M_n^∗(f) :=

n

X

ℓ=1

d^∗_ℓ and S_n^∗(f) :=

n

X

ℓ=1

X_ℓ^∗.

The conclusion of Theorem 2.2 comes from the three following lemmas.

Lemma 3.1. We have|Sn(f)−S_n^∗(f)|=o(n^1/p(logn))almost surely.

Lemma 3.2. We have|S_n^∗(f)−M_n^∗(f)|=o(n^1/p(logn))almost surely.

Lemma 3.3. The conclusion of Theorem 2.2 holds withM_n^∗(f)replacingSn(f).

Proof of Lemma 3.1. For any nonnegative integerj, let Dj := sup

1≤k≤2^j

k+2^j

X

ℓ=2^j+1

(Xℓ−X_ℓ^∗) . LetN ∈N^∗ and letk∈]1,2^N]. We first notice thatDj ≥ |P2^j+1

ℓ=2^j+1(Xℓ−X_ℓ^∗)|, so ifK is the integer such that 2^K−1< k≤2^K, then

Sk−S_k^∗

≤ |X1−X₁^∗|+

K−1

X

j=0

Dj.

Consequently, sinceK≤N,

1≤k≤2max^N|Sk−S_k^∗| ≤ |X1−X₁^∗|+

N−1

X

j=0

Dj. (3.12)

Therefore, by standard arguments, Lemma 3.1 will follow if we can prove thatDj=o j2^j/p almost surely. This will hold true as soon as

X

j≥1

kDjk^qq

2^jq/pj^q <∞ for someq∈[1, p]. (3.13) We shall verify (3.13) forq= 2. Notice that

kDjk2= sup

1≤k≤2^j

k+2^j

X

ℓ=2^j+1

(f−f_[2^αj_])◦T^ℓ

₂≤2^jkf−f_[2^αj_]k2.

Hence, by using (2.4), kDjk²2 ≪ 2^2j2^−ζαj, which together with the fact that αζ ≥2−2/p implies (3.13) withq= 2, and then Lemma 3.1.

Proof of Lemma 3.2. Without loss of generality, we assume thatθ <(p²−2)/(p(p−1)). Following the beginning of the proof of Lemma 3.1, Lemma 3.2 will be proven if (3.13) holds withDj defined by

Dj = sup

1≤k≤2^j

k+2^j

X

ℓ=2^j+1

(X_ℓ^∗−d^∗_ℓ)

. (3.14)

With this aim, setting, for everyk∈ {1, ...,2^j}, Rj,k=

k

X

ℓ=1

f[2^αj]◦T^ℓ−X

m∈Z

Pℓ f[2^αj]◦T^m

=

k+2^j

X

ℓ=2^j+1

(X_ℓ^∗−d^∗_ℓ)◦T^−2j, we first observe that

kDjk^p= sup

1≤k≤2^j|Rj,k|

_p≪2^j/p

j

X

k=0

2^−k/pkRj,2^kk^p , (3.15) (where for the inequality we have used inequality (6) in [15]). Now, according to the proof of Propo- sition 5.1 in [6] with Xℓ=f[2^αj]◦T^ℓ and using again stationarity, we get that for any integerk≥0 and any integerN ≥2^k,

1≤m≤2max^kkRj,mk^p≪

N

X

ℓ=1

kE_−ℓ(f[2^αj])k^p+

N−1

X

ℓ=0

kf[2^αj]−E_ℓ(f[2^αj])k^p

+ ²

k

X

m=1

X

ℓ≥m+N

P−ℓ(f[2^αj])◦T^ℓ

2 p

1/2

+ ²

k

X

m=1

X

ℓ≥m+N

Pℓ(f[2^αj])◦T^−ℓ

2 p

1/2

. (3.16) Let us first consider the case where [2^αj]≤b(2^k). Starting from (3.16) withN = 2^k and using the fact thatkP−ℓ(f[2^αj])kp≤2kE_−ℓ(f[2^αj])kp and thatkPℓ(f[2^αj])kp≤ kf[2^αj]−E_ℓ−1(f[2^αj])kp, we get that

1≤m≤2max^kkRj,mkp≪

2^k

X

ℓ=1

kE_−ℓ(f[2^αj])kp+

2^k−1

X

ℓ=0

kf[2^αj]−E_ℓ(f[2^αj])kp

+ 2^k/2 X

ℓ≥2^k+1

kE_−ℓ(f_[2^αj_])kp+ 2^k/2 X

ℓ≥2^k

kf_[2^αj_]−E_ℓ(f_[2^αj_])kp. Therefore, taking into account the upper bound (3.7) for the two first terms in the right hand side, and the upper bound (3.5) to handle the two last terms (since [2^αj]≤b(2^k)), we derive that

1≤m≤2max^kkRj,mk^p≪2k(1−θ(p−1)/p) (3.17) (recall that θ(p−1)/p < 1). On the other hand, starting from (3.16) with N = 2[2^kp/2] and using Lemma 5.1 in [6], we get that

1≤m≤2max^kkRj,mkp≪

2[2^kp/2]

X

ℓ=1

kE_−ℓ(f[2^αj])kp+

2[2^kp/2]

X

ℓ=0

kf[2^αj]−E_ℓ(f[2^αj])kp

+ 2^k/2 X

ℓ≥[2^kp/2]

kE_−ℓ(f[2^αj])k^p

ℓ^1/p + 2^k/2 X

ℓ≥[2^kp/2]

kf[2^αj]−E_ℓ(f[2^αj])k^p

ℓ^1/p .

Therefore, it follows from (3.7) that

1≤m≤2max^kkRj,mkp≪2(kp(1−θ(p−1)/p))/2. (3.18) Let

C= [α⁻¹(log(γ⁻¹))/(log 2)] andj0= (log 2)⁻¹(logj−logC). (3.19) Clearly, if j0 ≤k then [2^αj] ≤b(2^k). Therefore using the upper bound (3.18) whenk < j0 and the upper bound (3.17) whenk≥j0, we get that for any positive integerj

j

X

k=0

2^−k/pkRj,2^kk^p≪ j^p/2 j^1/pj^θ(p−1)/2,

sinceθ <(p²−2)/(p(p−1)). Now, sinceθ >1, it follows that X

j≥1

j^−pX^j

k=0

2^−k/pkRj,2^kk^pp

<∞.

From (3.15), this implies that (3.13) holds withDj defined by (3.14) andq=p. This ends the proof of lemma 3.2.

Proof of Lemma 3.3. Let Mn =Pn

ℓ=1dℓ where dℓ =d0◦T^ℓ with d0 =P

i∈ZP0(f ◦Tⁱ). Notice that the upper bound (3.8) and the fact thatθ >1 imply in particular that

X

n>1

n^−1/pkE_−n(f)kp<∞and X

n>1

n^−1/pkf −E_n(f)kp<∞, and then that P

k∈ZkP0(Xk)k^p < ∞ (use for instance Lemma 5.1 in [6] to see this). Therefore P

k∈ZkP0(Xk)k²<∞. Using (4.2) of Lemma 4.1, we get that kSn(f)−Mnk2=o(√

n). (3.20)

From (2.2) of Theorem 2.1, we know thatn⁻¹kSn(f)k²2 converges toσ². It follows from (3.20) that σ²=n⁻¹E(M_n²) =E(d²₀).

We shall prove now that

kM_n^∗−Mnk²=O n^1/p

. (3.21)

LetN be the positive integer such that 2^N−1 < n≤2^N. Since M_n^∗−Mn, is a martingale, we have that

kM_n^∗−Mnk²2=

n

X

ℓ=1

E((d^∗_ℓ−dℓ)²)≤E((d^∗₁−d1)²) +

N−1

X

j=0 2^j+1

X

ℓ=2^j+1

E((d^∗_ℓ−dℓ)²). (3.22) By stationarity, for anyℓ∈[2^j+ 1,2^j+1]∩Nwe get that

kd^∗_ℓ−dℓk²=kX

i∈Z

P0((f−f[2^αj])◦Tⁱ)k²

≤2^j+3kf−f[2^αj]k²+ X

i≥2^j+1

kP−i(f−f[2^αj])k²+ X

i≥2^j+1

kPi(f−f[2^αj])k². According to (3.3)

kf−f_[2^αj_]k2≤R2^−ζαj/2. (3.23)

On the other hand, by Lemma 5.1 in [6], X

i≥2^j+1

kP−i(f−f[2^αj])k²≪ X

k≥2^j

k^−1/2kE_−k(f−f[2^αj])k²

and X

i≥2^j+1

kPi(f −f[2^αj])k²≪ X

k≥2^j

k^−1/2kf−f[2^αj]+E_k(f −f[2^αj])k². Using the estimate (3.10) and (3.11), it follows that

X

i≥2^j+1

kP−i(f−f[2^αj])k²+kP−i(f−f[2^αj])k²

=O(β^2j). (3.24)

Combining the upper bounds (3.23) and (3.24) with the fact thatαζ ≥3−2/p, it follows that kd^∗_ℓ−dℓk2≪2−j(p−2)/(2p).

Using this estimate in (3.22), we obtain thatkM_n^∗−Mnk²2≪n^2/p, proving (3.21).

Now, let us recall Theorem 2.1 in [13] (used withan=n^2/p(logn)): if there exists a finite constant Ksuch that

sup

k≥1kd^∗_kk^p≤K , (3.25)

and if

n

X

i=1

E((d^∗_i)²|Fⁱ⁻¹)−E((d^∗_i)²)

=o n^2/p(logn)

a.s. , (3.26)

then, sinceE((M_n^∗)²)∼nσ², enlargingT^d if necessary, there exists a sequence (Z_ℓ^∗)ℓ≥1of independent Gaussian random variables with zero mean and varianceE(Z_ℓ^∗)²=E(d^∗_ℓ)²= (σ_ℓ^∗)²such that

sup

1≤k≤n

M_k^∗−

k

X

ℓ=1

Z_ℓ^∗

=o n^1/p(logn)

almost surely, as n→ ∞. (3.27) Let (δk)k≥1 be a sequence of iid Gaussian random variables with mean zero and varianceσ², inde- pendent of the sequence (Z_ℓ^∗)ℓ≥1. We now construct a sequence (Zℓ)ℓ≥1 as follows. Ifσ^∗_ℓ = 0, then Zℓ=δℓ, elseZℓ= (σ/σ_ℓ^∗)Z_ℓ^∗. By construction, theZℓ’s are iid Gaussian random variables with mean zero and varianceσ². LetGℓ=Zℓ−Z_ℓ^∗ and note that (Gℓ)ℓ≥1is a sequence of independent Gaussian random variables with mean zero and variances Var(Gℓ) = (σ−σ_ℓ^∗)². Notice now that

v²_n= VarXⁿ

i=1

Gi

=

n

X

i=1

kdik2− kd^∗_ik2²

≤ kMn−M_n^∗k²2.

From the basic inequality P

1≤k≤nmax

k

X

i=1

Gi

> x

≤2 exp

− x² 2v_n²

,

and the fact that by (3.21),v²_n≪n^2/p, it follows that for anyε >0, X

n>1

n⁻¹P

1≤k≤nmax

k

X

i=1

Gi

> εn^1/p(logn)

<∞,

showing that max1≤k≤n|Pk

i=1Gi|=o(n^1/p(logn)) almost surely. Therefore starting from (3.27), we conclude that if (3.25) and (3.26) hold then Lemma 3.3 does; namely, enlargingT^dif necessary, there exists a sequence (Zℓ)ℓ≥1 of iid Gaussian random variables with zero mean and varianceσ²such that

sup

1≤k≤n

M_k^∗−

k

X

ℓ=1

Zℓ

=o(n^1/p(logn)) almost surely, asn→ ∞.

It remains to show that (3.25) and (3.26) are satisfied. We start with (3.25). Notice that kd^∗₁kp ≤ P

k∈ZkP−k(f1)kp and that, for everyj≥0 and everyℓ∈ {2^j+ 1, ...,2^j+1}, kd^∗_ℓk^p≤X

k∈Z

kP−k(f[2^αj])k^p. By Lemma 5.1 in [6],

X

k∈Z

kP−k(g)kp≪X

k≥1

k^−1/pkE_−k(g)kp+X

k≥1

k^−1/pkg−E_k(g)kp,

with constants non depending ong. Hence, using the estimate (3.7) and the fact thatθ >1, we get that, for everyj ≥0 and every ℓ∈ {2^j+ 1, ...,2^j+1}, there exists a constant K non depending onj such thatkd^∗_ℓkp≤K. This ends the proof of (3.25).

To prove (3.26), we proceed as follows. Following the beginning of the proof of Lemma 3.1, we infer that (3.26) will be proven if we can show that

Dj:= sup

1≤ℓ≤2^j

2^j+ℓ

X

i=2^j+1

(E((d^∗_i)²|Fⁱ⁻¹)−E((d^∗_i)²))

=o j2^2j/p

a.s. . (3.28)

This will hold true as soon as

X

j≥1

kDjk^p/2_p/2

2^jj^p/2 <∞. (3.29)

For anyj fixed and anyi∈Z, letdj,i=P

k∈ZPi f[2^αj]◦T^k

. By stationarity kDjkp/2:=

sup

1≤ℓ≤2^j

ℓ

X

i=1

(E(d²_j,i|Fi−1)−E(d²_j,i)) _p/2.

Observe now that, for anyj fixed, (dj,i)_i∈Z is a stationary sequence of martingale differences in L^p. Let

Mj,n:=

n

X

i=1

dj,i.

Applying Theorem 3 in [16] (since 1< p/2≤2) and using the martingale property of the sequence (Mj,n)n≥1, we get that

E sup

1≤ℓ≤2^j

ℓ

X

i=1

(E(d²_j,i|Fi−1)−E(d²_j,i))

p/2

≪2^jkd²_j,1k^p/2_p/2+2^j

j−1

X

k=0

kE(M_j,2² k|F0)−E(M_j,2² k)kp/2

2^2k/p

!^p/2 . Using the fact that kd²_j,1kp/2=kdj,1k²p ≤K where K does not depend onj, the convergence (3.29) will be then proven if we can show that

X

j≥1

1 j^p/2

^j−1X

k=0

2^−2k/pkE₀(M_j,2² k)−E(M_j,2² k)kp/2

p/2

<∞.

According to the arguments developed in the proof of Theorems 3.1 and 3.2 in [6] (see (3.19) and (3.20) of [6]), since (Mj,k)k≥1is a sequence of martingales, we infer that this last convergence will be satisfied as soon as there exists a positive integerc such that

X

j≥1

1 j^p/2

X^j

k=0

2^−2k/p

E_−c2k(M_j,2² k)−E(M_j,2² k) _p/2

p/2

<∞. (3.30)

We shall prove in what follows that this convergence holds as soon ascis chosen in such a way that (3.9) holds true.

For any positive integern, let Sj,n=

n

X

ℓ=1

f_[2^αj_]◦T^ℓ and Rj,n=Sj,n−Mj,n. We first write that

kE_−c2k(M_j,2^{2 k})−E(M_j,2^{2 k})kp/2≤ kE_−c2k(S_j,2^{2 k})−E(S_j,2^{2 k})kp/2

+ 4kE_−c2k(Sj,2^kRj,2^k)kp/2+ 2kRj,2^kk²p. (3.31) Letj0be defined as in (3.19). Using the upper bound (3.18) whenk < j0and the upper bound (3.17) whenk≥j0, we get that for any positive integerj,

j

X

k=0

2^−2k/p max

1≤m≤2^kkRj,mk²p≪ j^p j^2/pj^θ(p−1),

since we can assume without loss of generality that θ < (p²−1)/(p(p−1)). Now, since θ > 1, it follows that

X

j≥1

1 j^p/2

X^j

k=0

2^−2k/p max

1≤m≤2^kkRj,mk²p

p/2

<∞. (3.32)

On an other hand,c being chosen such that (3.9) holds true, the upper bound in (3.9) together with the fact thatθ >1, implies that

j

X

k=0

2^−2k/p max

1≤m≤2^kkE_−c2k(S_j,m² )−E(S_j,m² ) p/2≪

j

X

k=0

2^−2k/p22k(1−θ(p−1)/p)=O(1). Therefore,

X

j≥1

1 j^p/2

X^j

k=0

2^−2k/p max

1≤m≤2^kkE_−c2k(S_j,m² )−E(S_j,m² ) _p/2

p/2

<∞. (3.33) Starting from (3.31), and taking into account (3.32) and (3.33), we then infer that (3.30) will hold true if we can show that

X

j≥1

1 j^p/2

X^j

k=0

2^−2k/pkE_−c2k(S_j,2^kR_j,2^k) _p/2

p/2

<∞. (3.34)

With this aim, we use Inequality (3.24) in [6] (takingn= 2^k,un= [2^k/2],r=c2^k). Therefore, kE_−c2k(Sj,2^kRj,2^k)

_p/2≪2^k/4 kE₀(Sj,2^k)k²+kSj,2^k−E₂k(Sj,2^k)k²

+ max

m={2^k,2^k−[2^k/2]}kRj,mk²p+ 2^k/2 kE_−[2k/2](Sj,2^k)k²+kSj,2^k−E₂k+[2^k/2](Sj,2^k)k²

+ max

m={2^k,[2^k/2]}kE_−c2k(S_j,m² )−E(S_j,m² )kp/2+ 2^k X

|ℓ|≥2^k

kP0(f[2^jα]◦T^ℓ)k2. (3.35)

By stationarity,

kE₀(Sj,2^k)k²+kSj,2^k−E₂k(Sj,2^k)k²≤

2^k

X

ℓ=1

kE_−ℓ(f[2^jα])k²+

2^k−1

X

ℓ=0

kf[2^jα]−E_ℓ(f[2^jα])k². Hence by using (3.10),

2^k/4 kE₀(S_j,2^k)k2+kS_j,2^k−E₂k(S_j,2^k)k2

≪2^k/4. (3.36)

Using again (3.10) and the stationarity, we get that kE_−[2k/2](S_j,2^k)k2+kS_j,2^k−E₂k+[2^k/2](S_j,2^k)k2

≤

2^k

X

ℓ=1

kE_−([2k/2]+ℓ)(f[2^jα])k2+

2^k−1

X

ℓ=0

kf[2^jα]−E_[2k/2]+ℓ(f[2^jα])k2≪β^2k/2. Therefore

2^k/2 kE_−[2k/2](Sj,2^k)k²+kSj,2^k−E_2k+[2k/2](Sj,2^k)k²

=O(1). (3.37)

On an other hand, using again (3.10) and the stationarity, X

|ℓ|≥2^k

kP0(f[2^jα]◦T^ℓ)k2≤ X

ℓ≥2^k

kE_−ℓ(f[2^jα])k2+kf[2^jα]−E_ℓ−1(f[2^jα])k2

≪β^2k, which implies that

2^k X

|ℓ|≥2^k

kP0(f[2^jα]◦T^ℓ)k2=O(1). (3.38) Starting from (3.35) and taking into account the convergence (3.32) and (3.33), and the upper bounds (3.36), (3.37) and (3.38), we then derive that (3.34) holds. This ends the proof of (3.26) and therefore of Lemma 3.3.

4 Appendix

Let (Ω,A,P) be a probability space, and T : Ω 7→ Ω be a bijective bimeasurable transformation preserving the probabilityP. Let us denote by | · |m the euclidean norm onR^m and by<·,·>mthe associated scalar product. For aσ-algebraF0satisfyingF0⊆T⁻¹(F0), we define the nondecreasing filtration (Fi)_i∈Z byFi=T⁻ⁱ(F0). LetF−∞=T

k∈ZFk andF∞=W

k∈ZFk. For a random variable X with values inR^m, we denote bykXkp,m= (E(|X|^pm))^1/p its norm in L^p(R^m).

In what followsX0is a random variable with values inR^m, and we define the stationary sequence (Xi)_i∈Z by Xi = X0◦ Tⁱ. We shall use the notations E_k(X) = E(X|Fk), E_∞(X) = E(X|F∞), E_−∞(X) =E(X|F−∞), andPk(X) =E_k(X)−E_k−1(X).

The aim of this section is to collect some results about invariance principles for stationary se- quences that are non necessarily adapted to the underlying filtration. We start with a martingale approximation result. The estimate (4.2) of Proposition 4.1 below is a generalization of Item 2 of Theorem 1 in [15] to the multidimensional case and to the case where the variables are non necessarily adapted to the filtration under consideration. The convergence (4.3) is new. Notice that the proof of the next lemma is based on algebraic computations and on Burkholder’s inequality. Burkholder’s inequality being also valid in Hilbert spaces (see [2]), the approximation lemma below is then also valid for variables taking values in a separable Hilbert space,H, by replacing the norm| · |m by the norm onH, let say| · |H.

Proposition 4.1. Let p∈[1,∞[ andp^′ = min(2, p). Assume thatE_−∞(X0) = 0 almost surely, that E_∞(X0) =X0 almost surely, and that

X

i∈Z

kP0(Xi)k^p,m<∞. (4.1)

Letd0=P

i∈ZP0(Xi),Mn:=Pn

i=1d0◦Tⁱ andRn:=Sn−Mn. For any positive integern, kRnk^pp,m^′ ≪

n

X

k=1

X

|ℓ|≥k

kPℓ(X0)k^p,mp^′

. (4.2)

In addition,

max

1≤k≤n|Rk|m

_p=o(n^1/p′) as n→ ∞. (4.3)

Remark 4.1. The constant appearing in (4.2) depends only on pand not on (Ω,A,P, T, X0,F0).

Proof of Proposition 4.1. It will be useful to note that Pi(Xj) =Pi−j(X0)◦T^j =P0(Xj−i)◦Tⁱ almost surely. The following decomposition is valid:

Rn =

n

X

k=1

Xk−

n

X

j=1

Pj(Xk)

−

n

X

k=1

X

j≥n+1

Pk(Xj)−

n

X

k=1

∞

X

j=0

Pk(X−j)

= E₀(Sn)−

n

X

k=1

X

j≥n+1

Pk(Xj) +Sn−E_n(Sn)−

n

X

k=1

∞

X

j=0

Pk(X−j). (4.4) Applying Burkholder’s inequality for multivariate martingales, and using the stationarity, we obtain that there exists a positive constantcp such that, for any positive integern,

n

X

k=1

X

j≥n+1

Pk(Xj)

p^′ p,m≤cp

n

X

k=1

X

j≥n+1

Pk(Xj)

p^′ p,m≤cp

n

X

k=1

X

j≥k

kP0(Xj)kp,m

p^′

, (4.5)

and

n

X

k=1

X

j≥0

Pk(X−j)

p^′ p,m ≤cp

n

X

k=1

X

j≥0

Pk(X−j)

p^′ p,m =cp

n

X

k=1

X

j≥k

kP0(X−j)kp,m

p^′

. (4.6)

On an other hand, since E_−∞(X0) = 0 almost surely, we have E₀(Sn) = P

k≥0P−k(Sn) almost surely. Hence by Burkholder’s inequality for multivariate martingales together with stationarity, there exists a positive constantcp depending only on psuch that

kE₀(Sn)k^pp,m^′ ≤cp

X

k≥0

kP−k(Sn)k^pp,m^′ ≤cp

X

k≥0

Xⁿ

ℓ=1

kP−k(Xℓ)kp,m

p^′

≤cp n−1

X

k=0

X

ℓ≥k+1

kP−ℓ(X0)k^p,mp^′

+cp

X

k≥n

^k+nX

ℓ=k+1

kP−ℓ(X0)k^p,mp^′

≤cp n−1

X

k=0

X

ℓ≥k+1

kP−ℓ(X0)k^p,mp^′

+cp

X

i≥n+1

kP−i(X0)k^p,mp^′−1X

k≥n k+n

X

ℓ=k+1

kP−ℓ(X0)k^p,m

≤cp n−1

X

k=0

X

ℓ≥k+1

kP−ℓ(X0)kp,m

p^′

+cpn X

ℓ≥n+1

kP−ℓ(X0)kp,m

p^′

.