A Functional CLT for Fields of Commuting Transformations Via Martingale Approximation

HAL Id: hal-01261297

https://hal.archives-ouvertes.fr/hal-01261297

Submitted on 25 Jan 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Functional CLT for Fields of Commuting Transformations Via Martingale Approximation

Christophe Cuny, Jerome Dedecker, Dalibor Volny

To cite this version:

Christophe Cuny, Jerome Dedecker, Dalibor Volny. A Functional CLT for Fields of Commuting Transformations Via Martingale Approximation. Zapiski Nauchnyh Seminarov POMI, 2016, 441 (5), pp.239-262. �10.1007/s10958-016-3145-y�. �hal-01261297�

TRANSFORMATIONS VIA MARTINGALE APPROXIMATION

CHRISTOPHE CUNY, J ´ER ˆOME DEDECKER, AND DALIBOR VOLN ´Y Dedicated to the memory of Mikhail (Misha) Gordin

Abstract. We consider a field f ◦T₁ⁱ¹◦ · · · ◦T_d^id, where T1, . . . , Td are completely commuting transformations in the sense of Gordin. If one of these transformations is ergodic, we give sufficient conditions in the spirit of Hannan under which the partial sum process indexed by quadrants converges in distribution to a brownian sheet.

The proof combines a martingale approximation approach with a recent CLT for martingale random fields due to Voln´y. We apply our results to completely commuting endomorphisms of them-torus. In that case, the conditions can be expressed in terms of theL²-modulus of continuity off.

1. Introduction

Let (Ω,A, µ) be a probability space andT a (non-invertible) measure preserving map.

LetU be the associated Koopman operator (U f =f◦T for everyf ∈L¹(Ω,A, µ)) and U^∗ be the associated Perron-Frobenius operator. In 1978, Gordin and Lifˇsic [10] (see also [8]) observed that if f = (I−U^∗)g for some g ∈L²(µ) (i.e. f is a coboundary for U^∗), then one has a decomposition f = (g −U U^∗g) + (U −I)U^∗g into the sum of a reverse martingale difference plus a coboundary (forU that time). This allows to prove the central limit theorem (CLT) and the weak invariance principle (WIP) for f from the corresponding results for stationary reverse martingale differences.

This fruitful approach presents a part of the martingale-approximation method, known also as Gordin’s method and started with the seminal paper [8] from 1969.

It has been further developped in many papers (to go beyond the coboundaries for U^∗). Let us mention the following references where optimal or sharp results have been obtained concerning the CLT as well as other limit theorems: Hannan [13, 14], Heyde [15], Maxwell and Woodroofe [17], Peligrad and Utev [18], Gordin and Peligrad [11], Cuny [4].

Consider now a family of commuting measure preserving transformationsT₁, . . . , T_d. In 2009, Gordin [9] proved a decomposition analogous to the above one (see Section 4), when the transformations are completely commuting (see the next section for the definition) and f = (I −U₁^∗)· · ·(I −U_d^∗)g. However, probably by lack of a CLT for

1991Mathematics Subject Classification. Primary: 60J05, 60J55, 47A35, 28D05, 37A05; Secondary:

47B38.

Key words and phrases. random fields, reverse martingales, endomorphisms of the torus.

1

“multi-dimensional” reverse martingale differences, he did not derive any CLT from that decomposition.

Very recently, the third author [19] proved such a CLT. Actually, he worked in the setting of martingale differences but, as shown in Section 3, its proof applies equally in the reverse martingale case and yields also the weak invariance principle. In this paper, we provide a suitable reverse-martingale approximation under a condition in the spirit of Hannan, from which the CLT and the WIP follow. Note that the WIP under Hannan’s condition has been recently obtained by Voln´y and Wang [20] in the case where the random field can be expressed as a function of an iid random field. In the one-dimensional setting this condition is known to be sharp (see for instance Dedecker [6]). The results of Voln´y and Wang can, using [20], be extended to random fields which are not Bernoulli.

We apply these results to prove a CLT and a WIP in the case where the transforma- tions are commutingdilating endomorphisms of them-dimensional torus. Note that, for such commuting endomorphisms (not necessarily dilating), the CLT has been obtained recently by mean of completely different technics under slightly stronger conditions, see Section 5 for a deeper discussion.

2. Setting of the paper Let us describe our setting.

Let (Ω,A, µ) be a probability space. Consider a family {T₁, . . . , T_d} of measure preserving transformations on Ω.

Denote by U₁, . . . , U_d the corresponding Koopman operators and by U₁^∗, . . . , U_d^∗ the associated adjoint operators, also known in that context as the Perron-Frobenius oper- ators. Recall that those operators are characterized as follows

U_if =f◦T_i; Z

Ω

Uif g dµ= Z

Ω

f U_i^∗g dµ ,

for every positive measurable functions f, g and everyi∈ {1, . . . , d}.

Definition 1. We say that the family{T₁, . . . , T_d}(or the family {U₁, . . . , U_d}) is com- pletely commuting if it is commuting and if moreover

(1) U_iU_j^∗ =U_j^∗U_i for any i, j ∈ {1, . . . , d}, i6=j.

Notice that (as already observed by Gordin) this definition is slightly abusive since that property depends onµ. Sinceµwill be fixed in the sequel, we will not worry about that fact.

We consider now the natural filtrations associated with our transformations. For every i∈ {1, . . . , d} and every n∈N, denote Fn⁽ⁱ⁾ :=T_i⁻ⁿ(A).

Then, it is well-known (and not hard to prove) that, for every i ∈ {1, . . . , d} and every n ∈N, we have

(2) E(f|F_n⁽ⁱ⁾) = U_iⁿ(U_i^∗)ⁿf for any f ∈L¹(Ω,A, µ).

On the other hand, since the U_i’s are clearly isometries (of any L^p, 1 ≤p≤ ∞), we also have for every i∈ {1, . . . , d}

(3) U_i^∗U_if =f for any f ∈L¹(Ω,A, µ).

The relevance of the property of complete commutation lies in the fact that for every i, j ∈ {1, . . . , d} with i 6= j, and every n ∈ N, the operator U_i and the operator of conditional expectation with respect to Fn^(j) are commuting.

3. An invariance principle for stationary d-fields of reverse martingale differences

In all of this section we suppose given a completely commuting family {T₁, . . . , T_d} of transformations on the probability space (Ω,A, µ) and we make use of the previous notations.

We shall use the notationn to specify that n is a vector. Then, ifn= (n₁, . . . , n_d)∈ N^d (with N={0,1. . .}, and later N^∗ ={1,2. . .}), we shall use the notation

Uⁿf =U₁ⁿ¹· · ·U_d^ndf for any f ∈L¹(Ω,A, µ).

Definition 2. We shall say that (f_n)_n∈Nd is a commuting stationary d-field of reverse martingale differences if there exists f ∈ L¹(Ω,A, µ) with E(f|F₁⁽ⁱ⁾) = 0 for every i∈ {1, . . . , d} such that f_n=Uⁿf.

Let {e₁, . . . , e_d} be the canonical basis in R^d. If (f_n)_n∈Nd is a commuting stationary d-field of reverse martingale differences, then, for everyi∈ {1, . . . , d}, we have (withf as in the definition), U_i^∗f = 0 and

(4)

E(f_n|F_n⁽ⁱ⁾_i+m) = U_i^ni+m(U_i^∗)^ni+mUⁿf =U^n+mei(U_i^∗)^kf = 0 for any m∈N^∗, n∈N^d. For every k, h∈N^d, we shall write k h if for every i∈ {1, . . . , d} we have k_i ≤h_i. For every n ∈N^d and every t ∈[0,1]^d, we shall write [nt] := ([n₁t₁], . . . ,[n_dt_d]), where [·] stands for the integer part.

Let (Uⁿf)_n∈N^d be a random field on (Ω,A, µ). For every n = (n₁, . . . , n_d) ∈ N^d and every t = (t1, . . . , td)∈[0,1]^d set

S_n,t(f) := X

0k[nt]

d

Y

i=1

(k_i∧(n_it_i−1)−k_i+ 1)U^kf , and T_n,t(f) := S_n,t(f)

(Qd

i=1n_i)^1/2 .

Theorem 1. Let(Uⁿf)_n∈N^d be a commuting stationaryd-field of reverse martingale dif- ferences with f ∈L²(Ω,A, µ). Assume that one of theU_i’s is ergodic. Then, the process (T_n,t(f))_t∈[0,1]d)_n∈Nd converges in law in C([0,1]^d) to (kfk₂W_t)_t∈[0,1]d, where (W_t)_t∈[0,1]d

is the standard d-dimensional brownian sheet.

Remark. Recall that (W_t)_t∈[0,1]^d is the centered gaussian process characterized by E(W_sW_t) = Qd

i=1(s_i∧t_i). The ergodicity will be needed at the very end of the proof, in order to prove Lemma 4 below.

As usual we shall prove that result in two steps. The first one consists in proving tightness. The second one consists in proving convergence of the finite dimensional distributions.

3.1. Proof of the tightness. The tightness has been proved by Voln´y and Wang [20]

in the case of martingale differences rather than reverse martingale differences. Their argument carry on to our setting provided that we have a maximal inequality of Cairoli’s type for reverse martingale fields.

We shall just state the appropriate version of Cairoli’s maximal inequality needed and refer to [20] for the proof of the tightness. We state it in the stationary case, but it holds in a more general setting.

Proposition 2 (Cairoli, [1]). Let (Uⁿf)_n∈N^d be a commuting stationary d-field of re- verse martingale differences with f ∈ L^p(Ω,A, µ), p > 1. Then, for every n ∈ N^d we have

(5) E



max

1kn

X

0ik

Uⁱf

p

≤2^dp p

p−1 d

E

X

0in

Uⁱf

p!

Proof. We explain how to derive the result from Cairoli’s original result when d= 2.

Let n ∈ N². For every 0 k n write g_k := U^n−kf. Then, (P

0ikg_i)0kn is a so-called orthomartingale. Moreover, we see that for every 0k n,

X

0ik

Uⁱf = X

0in

Uⁱf − X

(k1,0)in

Uⁱf − X

(0,k2)in

Uⁱf+ X

(k1,k2)in

Uⁱf

= X

0in

g_i− X

0i(n1−k1,0)

g_i− X

0i(0,n2k2)

g_i+ X

0i(n1−k1,n2−k2)

g_i.

3.2. Convergence of the finite dimensional distributions. We have to prove that for every (t_k)0≤k≤L with t_k ∈[0,1]^d,

(6) Tn,t₀(f), . . . , Tn,t_L(f)

⇒ Wt₀(f), . . . Wt_L(f) as n1, . . . , nd→+∞.

Note first that, if

Se_n,t(f) := X

0k[nt]−1

U^kf and Te_n,t(f) := Sen,t(f) (Qd

i=1n_i)^1/2,

where 1 = (1, . . . ,1), then kT_n,t(f)−Te_n,t(f)k₂ →0 as n₁, . . . , n_d → +∞. This follows easily from the fact that, using the reverse-martingale property,

kS_n,t(f)−Se_n,t(f)k²₂ ≤ kfk²₂





d

X

i=1

Y

j∈{1,...,d},j6=i

n_j



.

Hence, is suffices to prove (6) with Te_n,ti instead of T_n,ti. Let (a_k)0≤k≤L be L+ 1 real numbers. By the Cramer-Wold device, it suffices to prove that

L

X

k=0

akTen,t_k(f)⇒

L

X

k=0

akWt_k(f). As a second simple remark, note that the sum PL

k=0a_kTe_n,t

k(f) can be written as a weighted sum over disjoint and adjacent rectangles. Hence, it suffices to prove the convergence in distribution for such rectangles.

We shall make the proof when d = 2, the general case can be proved by induction.

Let t₀ = 0 < t₁ <· · · < t_K ≤ 1 and s₀ = 0< s₁ <· · · < s_K ≤1. Let also (a<sub>k,`</sub>)1≤k,`≤K

be real numbers. From the remarks above, it suffices to prove that V_n1,n2 = 1

√n₁n₂

K

X

k=1 K

X

`=1

a_k,`

[nt_k]−1

X

i=[n1tk−1]

[ns_`]−1

X

j=[n2s`−1]

U₁ⁱU₂^j(f)

⇒

K

X

k=1 K

X

`=1

a_{k,`</sub> W<sub>(tk,s`)}(f) +W_{(tk−1,s`−1)</sub>(f)−W<sub>(tk,s`−1)}(f)−W_(tk−1,s`)(f) . asn₁, n₂ → ∞. Notice that the random variable on right hand is distributed according toN(0,Γ), with

Γ =kfk²₂

K

X

k=1 K

X

`=1

a²_{k,`</sub>(t<sub>k</sub>−tk−1)(s<sub>`}−s`−1).

Clearly, it suffices to prove the desired convergence in distribution whenn₁, n₂ →+∞

along any sequence (m_r, n_r)r≥1. Hence, let us fix a sequence (m_r, n_r)r≥1 such that m_r, n_r→+∞ asr →+∞. It remains to prove that

(7) V_r= 1

√m_rn_r

K

X

k=1 K

X

`=1

a_k,`

[mrt_k]−1

X

i=[mrtk−1]

[nrs_`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)⇒ N(0,Γ)

Proof of (7). Since one of the U_i⁰s is assumed to be ergodic, let us assume that U2 is.

We will apply the following result of McLeish as stated in Hall and Heyde [12] (see Theorem 3.6 p. 77). This theorem is stated for an array of martingale differences, but a simple change of time gives the next proposition. We first mention what we mean by an array of reverse martingale differences.

Definition 3. Let ((X_r,k)0≤k≤p_r−1)r≥1 be an array of variables and ((G_r,k)0≤k≤p_r−1)r≥1

be an array of σ-algebras, such that for every r ≥ 1, (G_r,k)0≤k≤p_r−1 is decreasing. We say that ((X_r,k,(G_r,k)0≤k≤p_r−1)r≥1 is an array of reverse martingale differences if X_r,k is G_r,k-measurable for every r ≥ 1 and 1 ≤ k ≤ p_r and if E(X_r,k|G_r,k+1) = 0 for every r≥1 and every 0≤k≤p_r−2.

Proposition 3 (McLeish). Let ((X_r,k,(G_r,k)0≤k≤pr−1)r≥1 be an array of reverse martin- gale differences in L². Assume that

(i) sup_{0≤k≤pr−1}|Xr,k|→^P 0; (ii) sup_{0≤k≤pr−1}E(X_r,k² )<∞ ; (iii) Ppr−1

k=0 X_r,k² →^P V for someV ≥0;

Then, P

0≤k≤pr−1X_r,k converges in distribution to N(0, V).

To apply this proposition, we write

√ 1 m_rn_r

mr−1

X

i=0 K

X

k=1

1{[mrtk−1]≤i≤[mrtk]−1}





K

X

`=1

ak,`

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)



:=

mr−1

X

i=0

Zr,i.

Note that Z_r,i is a reverse martingale difference with respect to G_r,i =F_i⁽¹⁾. We shall now prove that (Z_r,i)0≤i≤m_r−1 satisfies (i), (ii) and (iii) of Proposition 3 (with V = Γ for (iii)).

Proof of (i) and (ii). We have sup

0≤i≤mr−1

|Z_r,i| ≤ sup

1≤k≤K K

X

`=1

|a_k,`|

! sup

1≤`≤K

sup

0≤i≤mr−1

√ 1 mrnr

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f) .

Hence we just have to prove that for every ε >0, and every 1≤`≤K, γ<sub>`,ε,r</sub> :=µ



 sup

0≤i≤m_r−1

√ 1 mrnr

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)

≥ε



 −→

r→+∞0.

Using the stationarity and Markov inequality, we obtain that γ_j,ε,r ≤m_rµ





√ 1 m_rn_r

[nrs`]−1

X

j=[nrs`−1]

U₂^j(f)

≥ε





≤ 1 ε²E









√1 n_r

[nrs`]−1

X

j=[nrs`−1]

U₂^j(f)





2

1

P[nr s`]−1 j=[nr s`−1]U₂^j(f)

>√ mrnrε



 .

Notice that (U₂ⁱf)0≤ı≤mr−1 is a stationary sequence of reverse martingale differences.

Now, it is well-known (using stationarity, truncation and Burkholder inequality) that the family

(Y_r,`)r≥1 :=









√1 nr

[nrs`]−1

X

j=[nrs`−1]

U₂^j(f)





2



r≥1

is uniformly integrable, and (i) easily follows.

In the same way, (ii) can be proved by using the stationarity and the fact that (Y_r,`)r≥1 is bounded inL¹.

Proof of (iii). This is the difficult part. It suffices to prove that

r→∞lim

mr−1

X

i=0

Z_r,i² −Γ 1

= 0. Now

Z_r,i² =

K

X

k=1

1 mr

1_{{[mrtk−1]≤i≤[mrtk]−1}}





K

X

`=1

a_k,` 1

√n_r

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)





2

.

Hence, it suffices to prove that

r→∞lim

K

X

k=1

1 mr

[mrtk]−1

X

i=[mrtk−1]





K

X

`=1

a_k,` 1

√n_r

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)





2

−γ_k 1

= 0 where γ_k = kfk²₂PK

`=1a<sup>2</sup><sub>k,`</sub>(t_k −tk−1)(s<sub>`</sub> −s`−1). By stationarity, it suffices to prove that, for each k ∈ {1, . . . , K},

r→∞lim

1 m_r

[mrt_k]−[mrtk−1]−1

X

i=0





K

X

`=1

a_k,` 1

√n_r

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)





2

−γ_k 1

= 0. Settingu_r = [m_rt_k]−[m_rt_k−1], this is equivalent to

(8) lim

r→∞

1 u_r

ur−1

X

i=0





K

X

`=1

ak,`

√1 n_r

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)





2

− kfk²₂

K

X

`=1

a²<sub>k,`</sub>(s`−s`−1) 1

= 0. In order to prove (8), let us admit the following lemma for a while.

Lemma 4. Let ε > 0. If U₂ is ergodic, there exist integers v ≥ 1 (large enough) and p(v) (large enough), such that for every n≥p(v)

(9)

1 v

v−1

X

i=0





√1 n

K

X

`=1

a_k,`

[ns`]−1

X

j=[ns`−1]

U₁ⁱU₂^j(f)





2

−∆_k 1

< ε ,

where ∆k :=kfk²₂PK

`=1a<sup>2</sup><sub>k,`</sub>(s`−s`−1).

Let

F_r,i =

K

X

`=1

a_k,` 1

√n_r

[nrs`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f).

Letr≥1, such thatur ≥v and writeur =vqr+tr, withqr ≥1 andtr ∈ {0, . . . , v−1}.

We have that

(10) 1

u_r

ur−1

X

i=0

F_r,i² −∆_k = v u_r

qr−1

X

k=0



 1 v

(k+1)v−1

X

i=kv

F_r,i² −∆_k



+ 1 u_r

ur−1

X

i=vqr

F_r,i² −t_r∆_k u_r

By stationarity, and Lemma 4, we see that, for n_r ≥p(v)

(11)

v u_r

qr−1

X

k=0



 1 v

(k+1)v−1

X

i=kv

F_r,i² −∆_k



 1

≤ 1 v

v−1

X

i=0





√1 n_r

K

X

`=1

a_k,`

[nrs_`]−1

X

j=[nrs`−1]

U₁ⁱU₂^j(f)





2

−∆_k 1

< ε .

On another hand, for a fixed v, one has

(12) lim

r→∞

1 u_r

ur−1

X

i=vqr

F_r,i² − t_r∆_k u_r

1

= 0.

From (10), (11) and (12), we see that (8) holds. This completes the proof of (iii).

Proof of Lemma 4 The proof relies on the following convergence in law.

Lemma 5. Let v ≥1. The sequence of random vectors





√1 n

K

X

`=1

a_k,`

[ns`]−1

X

j=[ns`−1]

U₁ⁱU₂^j(f)





1≤i≤v

converges in distribution to (N_i)_1≤i≤v, where the N_i’s are iid with common distribution N(0,kfk²₂PK

`=1a<sup>2</sup><sub>k,`</sub>(s`−s`−1)).

Lemma 4 follows easily from Lemma 5, a truncation argument and the law of large numbers in L¹ for (N_i²)1≤i≤v (with v →+∞).

Lemma 5 can be proved by applying Proposition 3, but it is shorter to notice that it is a consequence of the WIP for stationary and ergodicR^v-valued reverse martingale differences (note that it is the only place where we use the ergodicity of U2). Indeed,

letting V(f) = (U₁¹(f), . . . , U₁^v(f))⁰, it follows from the WIP that





√1 n

[ns1]−1

X

j=0

V(U₂^j(f)), . . . , 1

√n

[nsK]−1

X

j=[nsK−1]

V(U₂^j(f))





converges in distribution to (G₁, . . . , G_K), where the G_{`</sub>’s are independent Gaussian random vectors with respective covariance matrix (s<sub>`}−s_`−1)E(V(f)V(f)⁰). Now since E(U₁ⁱ(f)U₁^j(f)) = 0 if i 6= j, we see that E(V(f)V(f)⁰) = kfk²₂Id_v, where Id_v is the identity v×v matrix. Lemma 5 follows straightforwardly.

4. Reverse martingale approximation

We shall consider again a completely commuting family of (non invertible) measure preserving transformations T₁, . . . , T_d.

In all that section we assume the following property k(U_i^∗)ⁿfk₂ −→

n→+∞0 for any i∈ {1, . . . , d}, and any f ∈L²(Ω,A, µ) with E(f) = 0. This property is equivalent to the fact that each T_i is exact (see Definition 4.14 in Walters [21]).

We shall now prove a (reverse) martingale approximation result under a condition in the spirit of Hannan. For f ∈L²(µ) set

kfkX₂ : = X

n1,...,nd∈N

d

Y

i=1

(U_iⁿⁱ(U_i^∗)ⁿⁱ−U_iⁿⁱ⁺¹(U_i^∗)ⁿⁱ⁺¹)f 2

(13)

= X

n1,...,nd∈N

d

Y

i=1

((U_i^∗)ⁿⁱ−U_i(U_i^∗)ⁿⁱ⁺¹)f 2

, and

X2 :={f ∈L²(µ) : kfkX2 <∞}.

It is not hard to prove as in the one-dimensional case (see for instance the proof of Proposition 12 in [5]) that a sufficient condition for f to be in X₂ is that

(14) X

n1,...,nd∈N

k(U₁^∗)ⁿ¹· · ·(U_d^∗)^ndfk2

(n₁· · ·n_d)^1/2 <∞.

We first prove a maximal inequality. Its statement, as well as its proof, are analogous to Lemma 5.2 of [20].

Lemma 6. Let f ∈ X₂. Then,

(15) E



max

1kn

X

0ik

Uⁱf

2

≤2^3d(n₁· · ·n_d)kfk²_X2.

Proof. Let f ∈ X2. Using that for every i ∈ {1, . . . , d}, k(U_i^∗)ⁿfk2 −→ 0 as n → ∞, we obtain the following orthogonal decomposition

(16) f = X

m1,...,m_d∈N d

Y

i=1

U_i^mi(U_i^∗)^mi −U_i^mi+1(U_i^∗)^mi+1

f := X

m∈N^d

f_m.

Then, clearly

1knmax

X

0ik

Uⁱf

≤ X

m∈N^d 1knmax

X

0ik

Uⁱf_m

Now, for every m = (m₁, . . . , m_d) ∈ N^d, (Uⁱf_m)_i∈N^d is a commuting stationary d- field of reverse martingale differences associated with fm ∈L²(Ω,Am, µ), where Am = T₁^−m1 ◦ · · · ◦T_d^−md(A). Hence, the result follows from Proposition 2.

We shall also need the following lemma.

Lemma 7. Let T₁ be a measure preserving transformation andU₁ the associated Koop- man operator. Let f, g∈L²(Ω,A, µ) be such that for h∈ {f, g},

(17) X

n≥0

(U₁^∗)ⁿ−U₁(U₁^∗)ⁿ⁺¹ h

2 <∞.

Then,P

k∈Z|E(U₁^k+f U₁^k−g)|<∞, where k⁺ = max{0, k}andk⁻= max{0,−k}. More- over, writing f˜:=P

k≥0((U₁^∗)^k−U₁(U₁^∗)^k+1)f and g˜:=P

k≥0((U₁^∗)^k−U₁(U₁^∗)^k+1)g,

(18) E

f˜g˜

=X

k∈Z

E

U₁^k+f U₁^k−g .

Proof. The absolute convergence of the series will follow from the proof. Hence, we only prove (18). For everyf ∈L²(Ω,A, µ) such thatE(f) = 0, using thatk(U₁^∗)ⁿfk2 −→

n→+∞0, we have

f =X

n∈N

U₁ⁿ(U₁^∗)ⁿ−U₁ⁿ⁺¹(U₁^∗)ⁿ⁺¹ f , (19)

where the summands are orthogonal.

Let

A_k,`(f, g) = E (U₁^∗)^k−U₁(U₁^∗)^k+1 f

(U₁^∗)^{`</sup>−U<sub>1</sub>(U<sub>1</sub><sup>∗</sup>)<sup>`+1} g Using (17) to permuteE and P

, we have (with absolute convergence) E

f˜˜g

= X

k,`∈N

A_k,`(f, g) = X

k,`:k>`

A_k,`(f, g) + X

k,`:k<`

A_k,`(f, g) + X

k,`:k=`

A_k,`(f, g). The first two sums on right hand are symmetric one from the other, hence we shall deal only with the second one. Since E U₁^∗)^k−U₁(U₁^∗)^k+1

f

U₁(U₁^∗)^`+1g

= 0 for

`≥k, X

k∈N

X

`>k

A_k,`(f, g) = X

k∈N

X

m∈N^∗

E U₁^k(U₁^∗)^k−U₁^k+1(U₁^∗)^k+1 f

(U₁^∗)^mg

= X

m∈N^∗

E(U₁^mf g) , where we have used (19). In the same way

X

k∈N

A_k,k(f, g) = X

k∈N

E U₁^k(U₁^∗)^k−U₁^k+1(U₁^∗)^k+1 f

g

=E(f g).

Theorem 8. Let f ∈ X₂. Then, there exists a commuting stationary d-field of reverse martingale differences (Uⁿ(d))_n∈N^d with d∈L²(µ) such that

(20) E



max

0kn

X

0ik

Uⁱf −Uⁱd

2

=o(n₁· · ·n_d), as n₁, . . . , n_d→+∞, and kdk²₂ = P

n∈N^dE(Uⁿ⁺f Uⁿ⁻f), where we use the notations n⁺ = (n⁺₁, . . . , n⁺_d) and n⁻ = (n⁻₁, . . . , n⁻_d).

Proof. By (19) we see that k · k_X2 is definite on X₂ hence that it is a norm. Moreover, (X₂,k · k_X2) is a Banach space.

Let i ∈ {1, . . . , d}. We easily see that k(U_i^∗)ⁿfk_X2 −→ 0 as n → +∞. Hence, U_i^∗ is mean ergodic on X₂ (with no fixed points), that is

X₂ = (I−U_i^∗)X₂^X2. Then, it follows that

(21) X₂ = Y

1≤i≤d

(I−U_i^∗)X₂

X2

.

Define a linear operatorD onX₂ by setting Df := X

n∈N^d d

Y

i=1

(U_i^∗)ⁿⁱ−U_i(U_i^∗)ⁿⁱ⁺¹ f .

Let us observe that iff =Q

1≤i≤d(I−U_i^∗)g withg ∈ X₂, thenDf =Q

1≤i≤d(I−U_iU_i^∗)g.

Obviously,

(22) kD(f)k₂ ≤ kfkX₂.

Let us prove (20) with d=D(f). Let us admit for a while that (20) holds whenever f belongs to Q

1≤i≤d(I −U_i^∗)X₂. Let us show then that (20) holds for everyf ∈ X₂.

Letf ∈ X2. Let ε >0. By (21), there existsg ∈ X2 such that

f− Y

1≤i≤d

(I−U_i^∗)g X2

< ε .

For every n∈N^d, we have, setting ˜g :=Q

1≤i≤d(I−U_i^∗)g,

X

0kn

U^kf −U^kDf

≤

X

0kn

U^kf−U^k˜g

+

X

0kn

U^kh−U^kD˜g

+

X

0kn

U^kD(f−g)˜ .

Using (15) to deal with the first term above and (22) and (5) to deal with the third term, and since we admit for the moment that (20) holds for ˜g, we infer that

lim sup

n1,...,nd→+∞

1 n₁· · ·n_dE



max

1kn

X

0ik

Uⁱf−Uⁱd

2

≤Cε ,

and (20) follows by letting ε→0.

It remains to deal with the case where f =Q

1≤i≤d(I−U_i^∗)g, for someg ∈ X₂. To do so we use the following simple identity (see also Gordin [9], Proposition 1):

for i∈ {1, . . . , d}, I −U_i^∗ =I−U_iU_i^∗+ (U_i−I)U_i^∗. Letf =Q

1≤i≤d(I−U_i^∗)g with g ∈ X₂. We have

f =Df+ X

E⊂{1,...,d},E6=∅

Y

i∈E^c

(I−U_iU_i^∗)Y

j∈E

(U_j−I)U_j^∗g : =Df+h .

(23)

The proof relies on the fact that the remainder in (23) (i.e. h) behaves like a cobound- ary in some “directions” and like a sum of reverse martingale differences in the other

“directions”. For the sake of simplicity, we only prove the results for d = 2, but the general case can be handled similarly.

We have

f− Df = (I−U₁U₁^∗)(U₂−I)U₂^∗g+ (I−U₂U₂^∗)(U₁−I)U₁^∗g+ (U₁−I)(U₂−I)U₁^∗U₂^∗g . For every 0≤k₁ ≤n₁ and every 0≤k₂ ≤n₂, we have

X

0≤i₁≤k₁

U₁ⁱ¹ X

0≤i₂≤k₂

U₂ⁱ²(I−U2U₂^∗)(U1−I)U₁^∗g = (U₁^k1 −I) X

0≤i₂≤k₂

U₁ⁱ¹(I−U2U₂^∗)U₁^∗g .

DefineZ_k1,n2 := max1≤k2≤n2|P

0≤i₂≤k₂U₂ⁱ²(I−U₂U₂^∗)U₁^k1U₁^∗g|. For everyA >0, we have

1≤kmax1≤n1

1≤kmax2≤n2

X

0≤i1≤k1

U₁ⁱ¹ X

0≤i2≤k2

U₂ⁱ²(I−U₂U₂^∗)(U₁−I)U₁^∗g

2

≤2|Z0,n2|²+ 2n2A²+ 2 X

0≤k₁≤n₁

Zk1,k21_{|Zk

1,n2|≥A√ n₂}

2

,

and, by stationarity E



 max

1≤k2≤n2

1≤kmax1≤n1

X

0≤i₁≤k₁

U₁ⁱ¹ X

0≤i₂≤k₂

U₂ⁱ²(I−U₂U₂^∗)(U₁−I)U₁^∗g

2



≤2E |Z_0,n2|²

+ 2n₂A²+ 2n₁E

Z_0,n21_{|Z0,n

2|≥A√ n₂}

2 . Since, (Z_0,n²

2/n₂)_n2≥1 is uniformly integrable, it follows that 1

n₁n₂E



 max

1≤k₂≤n₂ max

1≤k₁≤n₁

X

0≤i₁≤k₁

U₁ⁱ¹ X

0≤i₂≤k₂

U₂ⁱ²(I −U2U₂^∗)(U1−I)U₁^∗g

2

 −→

n1,n2→+∞0. We may deal similarly with the sum associated with the term (I−U₁U₁^∗)(U₂−I)U₂^∗g.

To deal with the sum associated with the term (U₁−I)−U₂ −I)U₁^∗)U₂^∗g is somehow easier.

To finish the proof of the theorem, it remains to identifykDfk²₂. But, this follows by applying inductively Lemma 7, noticing that

X

n∈N^d d

Y

i=1

(U_i^∗)ⁿⁱ−U_i(U_i^∗)ⁿⁱ⁺¹

=

d

Y

i=1

X

ki∈N

(U_i^∗)^ki−U_i(U_i^∗)^ki+1

and using the fact that (U1, . . . , Ud) is completely commuting.

5. Expanding endomorphisms of the m-dimensional torus

LetA be a m×m (m ≥1) matrix with integer entries. We say thatA is expanding if all its eigenvalues have modulus strictly greater than 1.

A induces a transformation θ_A of the m-dimensional torus [0,1)^m, preserving the Lebesgue-Haar measure λ. We denote by U_A the corresponding Koopman operator, and by U_A^∗ the Perron-Frobenius operator.

Let us give a simple condition under whichθ_A and θ_B are completely commuting.

Lemma 9. LetAandB be two expanding m×m (m≥1) matrices with integer entries.

Assume that A and B commutes and that they have coprime determinants. Then, θA

and θB are completely commuting.

Proof. We have to prove that U_A^∗UB = UBU_A^∗. Let Γ be representatives of Z^m/AZ^m with distinct images inZ^m/AZ^m. By (34) (using that A⁻¹ andB commute), it suffices to prove that B induces a bijection (modulo Z^m) of the set (A⁻¹γ)_γ∈Γ.

Letγ, γ⁰ ∈Γ be such that there exists β ∈Z^m such that BA⁻¹γ =BA⁻¹γ⁰+β. Set δ := γ −γ⁰ ∈ Z^m. Writing A⁻¹ = (detA)⁻¹A, where ˜˜ A is the adjugate matrix of A (with integer entries) and similarly, B⁻¹ = (detB)⁻¹B˜, we see that

(detB) ˜Aδ = (detA) ˜Bβ .

Since detB ∧detA = 1, by Gauss lemma, we infer that detA divides all entries ˜Aδ, hence that A⁻¹δ ∈ Z^m and δ ∈ AZ^m. By definition of Γ, we see that γ = γ⁰ and the

lemma is proved.

The fact that A and B have coprime determinants is by no mean necessary for θ_A and θ_B to be completely commuting as one may see from the following basic example:

A=

2 0 0 3

and B =

3 0 0 2

.

The next proposition is an easy consequence of a result by Fan [7] (see Proposition 13 of the Appendix). Recall that the modulus of continuity in L² is given by

Ω_2,f(δ) := sup

0≤|x|≤δ

kf(·+x)−fk₂ , where |x| stands for the euclidean norm ofx∈[0,1)^m.

Proposition 10. Let A₁, . . . , A_dbe commuting expanding m×m matrices with integral entries. Let λ_min >1 be the infimum of the modulus of their eigenvalues. There exists C >0 such that, for every f ∈L²(λ) and every n₁, . . . , n_d∈N,

(24) k(U_A^∗

1)ⁿ¹· · ·(U_A^∗

d)^ndfk₂ ≤Ω_2,f

Cλ⁻⁽ⁿ_min^1+···+nd) . Proof. Let us first notice that (U_A^∗₁)ⁿ¹· · ·(U_A^∗

d)^nd is the Perron-Frobenius operator associated with θ_Adnd···A1n1. Now, since the matrices are commuting, it follows from standard linear algebra results, that the set of the eigenvalues of A_d^nd· · ·A₁ⁿ¹ is in- cluded in {λ_d^nd· · ·λ₁ⁿ¹ : λ_i is an eigenvalue ofA_i}. In particular, A_d^nd· · ·A₁ⁿ¹ is an expanding matrix. Then, the result follows from Proposition 13, noticing that

∆ A⁻ⁿ₁ ¹· · ·A⁻ⁿ_d ^d([0,1]^m)

≤Cλ⁻⁽ⁿ_min^1+···+nd), where ∆ A⁻ⁿ₁ ¹· · ·A⁻ⁿ_d ^d([0,1]^m)

is the diameter of A⁻ⁿ₁ ¹· · ·A⁻ⁿ_d ^d([0,1]^m).

We shall use the following notation: Uⁱ =U_Aⁱ¹

1· · ·U_A^id

d for every (i₁, . . . , i_d)∈N^d. Theorem 11. Let d≥1. Let f ∈L²([0,1)^m, λ) centered such that

(25)

Z 1 0

|log(t)|^(d−2)/2

t Ω_2,f(t)dt <∞.

Let A1, . . . , Ad be expanding m×m matrices with integral entries. Assume that they are commuting and that their determinants are pairwise coprime. Then, there exists