Empirical central limit theorems for ergodic automorphisms of the torus

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Empirical central limit theorems for ergodic automorphisms of the torus

Jérôme Dedecker, Florence Merlevède, Françoise Pene

To cite this version:

Jérôme Dedecker, Florence Merlevède, Françoise Pene. Empirical central limit theorems for ergodic

automorphisms of the torus. ALEA : Latin American Journal of Probability and Mathematical Statis-

tics, Instituto Nacional de Matemática Pura e Aplicada, 2013, 10 (2), pp.731-766. �hal-00741466�

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Empirical central limit theorems for ergodic automorphisms of the torus

Jérôme Dedecker

^a

, Florence Merlevède

^b

and Françoise Pène

^c

a

Université Paris Descartes, Sorbonne Paris Cité, Laboratoire MAP5 and CNRS UMR 8145, 45 rue des Saints-Pères, F-75270 Paris cedex 06, France. E-mail: jerome.dedecker@parisdescartes.fr

b

Université Paris-Est, LAMA (UMR 8050), UPEMLV, CNRS, UPEC, F-77454 Marne-La-Vallée, FRANCE. E-mail: orence.merlevede@univ-mlv.fr

c

Université de Brest, Laboratoire de Mathématiques de Bretagne Atlantique UMR CNRS 6205. Sup- ported by the french ANR project Perturbations (ANR 10-BLAN 0106). E-mail: francoise.pene@univ- brest.fr

Key words: Empirical distribution function, Kiefer process, Ergodic automorphisms of the torus, Moment inequalities.

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

Abstract

Let T be an ergodic automorphism of the d -dimensional torus T

^d

, and f be a continuous function from T

^d

to R

^`

. On the probability space T

^d

equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f ◦ T

ⁱ

)

_i≥1

under some condition on the modulus of continuity of f . The proofs are based on new limit theorems and new inequalities for non-adapted sequences, and on new estimates of the conditional expectations of f with respect to a natural ltration.

1 Introduction

Let d ≥ 2 and T

^d

= R

^d

/ Z

^d

be the d -dimensional torus. For every x ∈ R

^d

, we write x ¯ its class in T

^d

. We denote by λ the Lebesgue measure on R

^d

, and by λ ¯ the Lebesgue measure on T

^d

.

On the probability space ( T

^d

, λ) ¯ , we consider a group automorphism T of 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 T ˜ preserves the innite Lebesgue measure λ on R

^d

and T preserves the probability Lebesgue measure λ ¯ .

We assume that T is ergodic, which is equivalent to the fact that no eigenvalue of S is a root of the unity. 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 mentionned in [9], 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 but non-hyperbolic, the dynamical system ( T

^d

, T, λ) ¯ has no Markov partition. How-

ever, it is possible to construct some measurable partition (see [11]), and to prove some decorrelation

properties for regular functions (see [11, 10]).

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Let ` be some positive integer, and let f = (f

1

, . . . f

`

) be a function from T

^d

to R

^`

. On the probability space ( T

^d

, λ) ¯ , the sequence (f ◦ T

^k

)

_k∈Z

is a stationary sequence of R

^`

-valued random variables. When ` = 1 and f is square integrable, Le Borgne [9] proved the functional central limit theorem and the Strassen strong invariance principle for the partial sums

n

X

i=1

(f ◦ T

ⁱ

− λ(f)) ¯

under weak hypotheses on the Fourier coecients of f , thanks to Gordin's method and to the partitions studied by Lind in [11]. In the recent paper [4], we slightly improve on Le Borgne's conditions, and we show how to obtain rates of convergence in the strong invariance principle up to n

^1/4

log(n) , by reinforcing the conditions on the Fourier coecients of f .

Now, for any s ∈ R

^`

, dene the partial sum S

n

(s) =

n

X

k=1

(1

_f◦Tk≤s

− F (s)) , (1.1)

where as usual 1

_f◦Tk≤s

= 1

_f₁_◦Tk≤s1

× · · · × 1

_f_`_◦Tk≤s`

, and F (s) = ¯ λ(1

_f◦Tk≤s

) is the multivariate distribution function of f .

In this paper, we give some conditions on the modulus of continuity of f for the weak convergence to a Gaussian process of the sequential empirical process

n S

_[nt]

(s)

√ n , t ∈ [0, 1], s ∈ R

^`

o

. (1.2)

The paper is organized as follows. Our main results are given in Section 2 and proved in Section 5. The proofs require new probabilistic results established in Section 3 combined with a key estimate for toral automorphisms which is given in Section 4. Let us give now an overview of our results.

In Section 2.1, we consider the case where ` = 1 and S

n

is viewed as an L

^p

-valued random variable for some p ∈ [2, ∞[ (this is possible because R

|S

n

(s)|

^p

ds < ∞ for any p ∈ [2, ∞[ ), so that the sequential empirical process is an element of D

_L^p

([0, 1]) , the space of L

^p

-valued càdlàg functions. We prove the weak convergence on D

_L^p

([0, 1]) equipped with the uniform metric to a L

^p

-valued Wiener process, and we give the covariance operator of this Wiener process. The proof is based on a new central limit theorem for dependent sequences with values in smooth Banach spaces, which is given in Section 3.1.1.

In Section 2.2, we state the convergence of the sequential empirical process (1.2) in the space

`

^∞

([0, 1] × R

^`

) of bounded functions from [0, 1] × R

^`

to R equipped with the uniform metric. In that case, the limiting Gaussian process is a generalization of the Kiefer process introduced by Kiefer in [8] for the sequential empirical process of independent and identically distributed random variables.

The proof is based on a new Rosenthal inequality for dependent sequences, which is given in Section 3.1.2. The weak convergence of the empirical process {n

^−1/2

S

n

(s), s ∈ R

^`

} has also been treated in [7] and [6]. We shall be more precise on these two papers in Section 2.2.

To prove these results, we shall use a control of the conditional expectations of continuous observables with respect to the ltration introduced by Lind [11], involving the modulus of continuity of the observables (See Theorem 18 of Section 4). As far as we know, such controls were known for Hölder observables only (see [10]). The inequalities given in Theorem 18 can be used in many other situations. Let us give two examples of applications. Let f be a continuous function from T

^d

to R with modulus of continuity ω(f, ·) (see Section 4, equation (4.1), for the denition).

1. Weak invariance principle. If

Z

1/2 0

ω(f, t)

t| log t|

^1/2

dt < ∞ ,

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then the series

σ

²

(f ) = ¯ λ((f − ¯ λ(f ))

²

) + 2 X

k>0

¯ λ((f − λ(f ¯ )) · f ◦ T

^k

)

converges absolutely, and the process n 1

√ n

[nt]

X

k=1

f ◦ T

^k

, t ∈ [0, 1] o

converges to a Wiener process with variance σ

²

(f ) in the space D([0, 1]) of càdlàg function equipped with the uniform metric .

2. Rates of convergence in the strong invariance principle. Let p ∈]2, 4] , and assume that ω(f, x) ≤ C| log(x)|

^−a

in a neighborhood of 0 for some a > 1 + p

1 + 4p(p − 2)

2p + 1 − 2

p . Then, enlarging T

^d

if necessary, there exists a sequence (Z

i

)

_i≥1

of independent and identically distributed Gaussian random variables with mean zero and variance σ

²

(f ) such that, for any t > 2/p ,

sup

1≤k≤n

k

X

i=1

f ◦ T

ⁱ

+

k

X

i=1

Z

i

= o n

^1/p

(log(n))

^(t+1)/2

almost surely as n → ∞ .

In particular, we obtain the rate of convergence n

^1/4

log(n) as soon as a ≥ 3/2 . This follows from Theorem 3.1 in [3].

2 Empirical central limit theorems

2.1 Empirical central limit theorem in L

^p

In this section, L

^p

is the space of Borel-measurable functions g from R to R such that λ(|g|

^p

) < ∞ , λ being the Lebesgue measure on R. If f is a bounded function, then, for any any p ∈ [2, ∞[ , the random variable S

n

dened in (1.1) is an L

^p

-valued random variable, and the process {n

^−1/2

S

[nt]

, t ∈ [0, 1]}

is a random variable with values in D

_L^p

([0, 1]) , the space of L

^p

-valued càdlàg functions. In the next theorem, we give a condition on the modulus of continuity ω(f, ·) of f under which the process {n

^−1/2

S

_[nt]

, t ∈ [0, 1]} converges in distribution to an L

^p

-valued Wiener process, in the space D

_Lp

([0, 1]) equipped with the uniform metric. We refer to Section 4, equation (4.1), for the precise denition of ω(f, ·) .

By an L

^p

-valued Wiener process with covariance operator Λ

_p

, we mean a centered Gaussian process W = {W

_t

, t ∈ [0, 1]} such that E (kW

_t

k

²

L^p

) < ∞ for all t ∈ [0, 1] and, for any g, h in L

^q

( q being the conjugate exponent of p ),

Cov Z

R

g(u)W

t

(u)du, Z

R

h(u)W

s

(u)du

= min(t, s)Λ

p

(g, h) .

Theorem 1. Let f : T

^d

→ R be a continuous function, with modulus of continuity ω(f, ·) . Let p ∈ [2, ∞[ , and let q be its conjugate exponent. Assume that

Z

1/2 0

ω(f, t)

^1/p

t| log t|

^1/p

dt < ∞ .

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Then the process {n

^−1/2

S

_[nt]

, t ∈ [0, 1]} converges in distribution in the space D

_Lp

([0, 1]) to an L

^p

- valued Wiener process W , with covariance operator Λ

p

dened by

Λ

p

(g, h) = X

k∈Z

Cov Z

R

g(s)1

_f≤s

ds, Z

R

h(s)1

_f◦Tk≤s

ds

, for any g, h in L

^q

. (2.1) The proof of Theorem 1 is based on results of Sections 3 and 4 and is postponed to Section 5.

Remark 2. In particular, if f is Hölder continuous, then the conclusion of Theorem 1 holds for any p ∈ [2, ∞[ .

Let us give an application of this theorem to the Kantorovich-Rubinstein distance between the empirical measure of (f ◦ T

ⁱ

)

_1≤i≤n

and the distribution µ of f . Let

µ

_n

= 1 n

n

X

i=1

δ

_f◦Ti

and µ

_n,k

= 1 n

(n − k)µ +

k

X

i=1

δ

_f◦Ti

.

The Kantorovich distance between two probability measures ν

₁

and ν

₂

is dened as K(ν

1

, ν

2

) = inf n Z

|x − y|λ(dx, dy), λ ∈ M(ν

1

, ν

2

) o , where M(ν

1

, ν

2

) is the set of probability measures with margins ν

1

and ν

2

.

Corollary 3. Let f : T

^d

→ R be a continuous function, with modulus of continuity ω(f, ·) . Assume that

Z

1/2 0

p ω(f, t) t p

| log t| dt < ∞ . Then √

nK(µ

_n

, µ) converges in distribution to kW

₁

k

_L1

, and sup

_1≤k≤n

√

nK(µ

_n,k

, µ) converges in distribution to sup

_t∈[0,1]

kW

_t

k

_L1

, where W is the L

²

-valued Wiener process with covariance operator Λ

₂

dened by (2.1) .

Proof of Corollary 3. Applying Theorem 1 with p = 2 , we know that {n

^−1/2

S

_[nt]

, t ∈ [0, 1]} converges in distribution in the space D

_L2

([0, 1]) to an L

²

-valued Wiener process W , with covariance operator Λ

₂

dened by (2.1). Since f is continuous on T

^d

, it follows that |f | ≤ M for some positive constant M , so that S

_[nt]

(s) = 0 and W

t

(s) = 0 for any t ∈ [0, 1] and any |s| > M . Since k·k

_L1

is a continuous function on the space of functions in L

²

with support in [−M, M ] , it follows that n

^−1/2

kS

_n

k

_L₁

converges in distribution to kW

1

k

_L1

, and that sup

_t∈[0,1]

n

^−1/2

kS

_[nt]

k

_L₁

converges in distribution to sup

_t∈[0,1]

kW

t

k

_L1

. Now, if ν

1

and ν

2

are probabity measures on the real line, with distribution functions F

ν₁

and F

ν₂

respectively,

K(ν

1

, ν

2

) = Z

R

|F

ν₁

(t) − F

ν₂

(t)|dt .

Hence nK(µ

_n

, µ) = kS

n

k

_L₁

and sup

_1≤k≤n

nK(µ

_n,k

, µ) = sup

_t∈[0,1]

kS

[nt]

k

_L₁

, and the result follows.

2.2 Weak convergence to the Kiefer process

Let ` be a positive integer. Let f = (f

1

, . . . , f

`

) be a continuous function from T

^d

to R

^`

. The modulus of continuity ω(f, ·) of f is dened by

ω(f, x) = sup

1≤i≤`

ω(f

_i

, x) ,

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where we recall that ω(f

i

, x) is dened by equation (4.1).

As usual, we denote by `

^∞

([0, 1]× R

^`

) the space of bounded functions from [0, 1]× R

^`

to R equipped with the uniform norm. For details on weak convergence on the non separable space `

^∞

([0, 1] × R

^`

) , we refer to [17] (in particular, we shall not discuss any measurability problems, which can be handled by using the outer probability).

For any positive integer ` and any α ∈]0, 1] , let a(`, α) = min

p≥max(`+2,2`)

g

`,α

(p), where g

`,α

(p) = max p

α(p − 2`) , (p − 1)(2α + p) pα

. (2.2) Note that this minimum is reached at p

1

= max(3, p

0

) , where p

0

is the unique solution in ]2`, 4`[ of the equation

p

(p − 2`) = (p − 1)(p + 2α)

p (2.3)

(in particular, p

1

= p

0

if ` > 1 ).

We are now in position to state the main result of this section.

Theorem 4. Let f = (f

₁

, . . . , f

_`

) : T

^d

→ R

^`

be a continuous function, with modulus of continuity ω(f, ·) . Assume that the distribution functions of the f

i

's are Hölder continuous of order α ∈]0, 1] . If

ω(f, x) ≤ C| log(x)|

^−a

for some a > a(`, α) ,

then the process {n

^−1/2

S

_[nt]

(s), t ∈ [0, 1], s ∈ R

^`

} converges in distribution in the space `

^∞

([0, 1] × R

^`

) to a Gaussian process K with covariance function Γ dened by: for any (t, t

⁰

) ∈ [0, 1]

²

and any (s, s

⁰

) ∈ R

^`

× R

^`

,

Γ(t, t

⁰

, s, s

⁰

) = min(t, t

⁰

)Λ(s, s

⁰

) with Λ(s, s

⁰

) = X

k∈Z

Cov(1

_f◦T_≤s

, 1

_f◦Tk≤s⁰

) .

The proof of Theorem 4 is given in Section 5. It uses results of Sections 3 and 4.

Remark 5. Using the Cardan formulas (see the appendix) to solve (2.3), we get

p

0

= 2 ` + 1 − α

3 + 2

r

− p

⁰

3 cos 1

3 arccos − q 2

s 27

−(p

⁰

)

³

!!

, with

p

⁰

:= −4α` + 2` − 2α − 1

3 (−2` + 2α − 2)

²

< 0 and

q := 1

27 (−2` + 2α − 2)(2(−2` + 2α − 2)

²

+ 36α` − 18` + 18α) + 4α` . For example, for α = ` = 1 , we get p

0

∼ 2.903211927 and nally a(1, 1) = 10/3 .

Recall that, by Theorem 1, if ` = 1 and p ∈]2, ∞[ , the weak invariance principle holds in D

_L^p

([0, 1]) as soon as a > p − 1 without any condition on the distribution function of f .

The weak convergence of the (non sequential) empirical process {n

^−1/2

S

_n

(s), s ∈ R

^`

} has been studied in [7] and [6]. When ` = 1 , a consequence of the main result of the paper [7] is that the empirical process converges weakly to a Gaussian process for any Hölder continuous function f having an Hölder continuous distribution function. In the paper [6] this result is extended to any dimension

` , under the assumption that the moduli of continuity of the distribution functions of the f

i

's are

smaller than C| log(x)|

^−a

in a neighborhood of 0 , for some a > 1 .

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Note that, in our case, one cannot apply Theorem 1 of [6]. Indeed, one cannot prove the multiple mixing for the sequence (f ◦ T

ⁱ

)

_i∈Z

by assuming only that ω(f, x) ≤ C| log(x)|

^−a

in a neighborhood of zero (in that case one can only prove that |Cov(f, f ◦ T

ⁿ

)| is O(n

^−a

) ). However, even if our condition on the regularity of f is much weaker than in [6], our result cannot be directly compared to that of [6], because we assume that the distribution functions of the f

i

's are Hölder continuous of order α , which is a stronger assumption than the corresponding one in [6].

3 Probabilistic results

In this section, C is a positive constant which may vary from lines to lines, and the notation a

_n

b

_n

means that there exists a numerical constant C not depending on n such that an a

_n

≤ Cb

_n

, for all positive integers n .

3.1 Limit theorems and inequalities for stationary sequences

Let (Ω, A, P ) be a probability space, and T : Ω 7→ Ω be a bijective bimeasurable transformation preserving the probability P. For a σ -algebra F

0

satisfying F

0

⊆ T

⁻¹

(F

0

) , we dene the nondecreasing ltration (F

i

)

_i∈Z

by F

i

= T

⁻ⁱ

(F

0

) . Let F

_−∞

= T

k∈Z

F

k

and F

_∞

= W

k∈Z

F

k

. Let I be the σ -algebra of T -invariant sets. As usual, we say that (T, P ) is ergodic if each element A of I is such that P (A) = 0 or 1.

Let ( B , | · |

_B

) be a separable Banach space. For a random variable X with values in B, let kX k

p

= ( E (|X |

^p_B

))

^1/p

and L

^p

( B ) be the space of B-valued random variables such that kXk

p

< ∞ . For X ∈ L

¹

( B ) , we shall use the notations E

k

(X) = E (X|F

k

) , E

∞

(X) = E (X|F

∞

) , E

−∞

(X) = E (X |F

−∞

) , and P

k

(X) = E

k

(X ) − E

k−1

(X ) . Recall that E (X |F

n

) ◦ T

^m

= E (X ◦ T

^m

|F

n+m

) .

Let X

0

be a random variable with values in B. Dene the stationary sequence (X

i

)

i∈Z

by X

i

= X

₀

◦ T

ⁱ

, and the partial sum S

_n

by S

_n

= X

₁

+ X

₂

+ · · · + X

_n

.

3.1.1 Weak invariance principle in smooth Banach spaces

Following Pisier [16], we say that a Banach space ( B , | · |

_B

) is 2 -smooth if there exists an equivalent norm k · k such that

sup

t>0

n 1

t

²

sup{kx + tyk + kx − tyk − 2 : kxk = kyk = 1} o

< ∞ .

From [16], we know that if B is 2 -smooth and separable, then there exists a constant K such that, for any sequence of B-valued martingale dierences (D

i

)

_i≥1

,

E (|D

₁

+ · · · + D

_n

|

²_B

) ≤ K

n

X

i=1

E (|D

_i

|

²_B

) . (3.1)

From [16], we see that 2 -smooth Banach spaces play the same role for martingales as space of type 2 do for sums of independent variables. Note that, for any measure space (T, A, ν) , L

^p

(T, A, ν) is 2 -smooth with K = p −1 for any p ≥ 2 , and that any separable Hilbert space is 2 -smooth with K = 2 . Let D

_B

([0, 1]) be the space of B-valued càdlàg functions. In the next theorem, we give a condition under which the process {n

^−1/2

S

_[nt]

, t ∈ [0, 1]} converges in distribution to an B-valued Wiener process, in the space D

_B

([0, 1]) equipped with the uniform metric.

By an B-valued Wiener process with covariance operator Λ

_B

, we mean a centered Gaussian process W = {W

t

, t ∈ [0, 1]} such that E (|W

t

|

²_B

) < ∞ for all t ∈ [0, 1] and, for any g, h in the dual space B

^∗

,

Cov(g(W

_t

), h(W

_s

)) = min(t, s)Λ

_B

(g, h) .

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Proposition 6. Assume that B be a 2-smooth Banach space having a Schauder Basis, that (T, P ) is ergodic, that kX

0

k

2

< ∞ and that E (X

0

) = 0 . If

X

k∈Z

kP

0

(X

i

)k

2

< ∞ (3.2)

then the process {n

^−1/2

S

_[nt]

, t ∈ [0, 1]} converges in distribution in the space D

_B

([0, 1]) equipped with the uniform metric to a B-valued Wiener process W

Λ_B

, where Λ

_B

is the covariance operator dened by

for any g, h in B

^∗

, Λ

_B

(g, h) = X

k∈Z

Cov(g(X

0

), h(X

k

)) .

Proof of Proposition 6. Let us prove rst that the result holds if E (X

0

|F

−1

) = 0 , that is when (X

k

)

k∈Z

is a martingale dierence sequence. As usual, it suces to prove that:

1. for any 0 = t

₀

< t

₁

< · · · < t

_d

= 1

√ 1

n (S

[nt₁]

, S

[nt₂]

− S

[nt₁]

, · · · , S

[nt_d]

− S

[nt_d−1]

)

converges in distribution to a the Gaussian distribution µ on B

^d

dened by µ = µ

1

⊗ µ

2

· · · ⊗ µ

d

, where µ

_i

is the Gaussian distribution on B with covariance operator C

_i

:

for any g, h in B

^∗

, C

_i

(g, h) = (t

_i

− t

_i−1

)Cov(g(X

₀

), h(X

₀

)) . 2. For any ε > 0 ,

lim

δ→0

lim sup

n→∞

1 δ P

max

1≤k≤[nδ]

|S

k

|

_B

> 2 √ nε

= 0

The rst point can be proved exactly as in [18], who proved the result only for t

₁

= 1 . Let us prove the second point. For any positive number M , let

X

_i⁰

= X

_i

1

_|X_i_|

B≤M

− E (X

_i

1

_|X_i_|

B≤M

|F

_i−1

) and X

_i⁰⁰

= X

_i

− X

_i⁰

.

Let also S

_n⁰

= X

₁⁰

+ · · · + X

_n⁰

and S

_n⁰⁰

= X

₁⁰⁰

+ · · · + X

_n⁰⁰

. Since B is 2-smooth, Burkholder's inequality holds (see for instance [15]), in such a way that E (max

_1≤k≤n

|S

_k⁰

|

^q

B

) ≤ K

q

M

^q

n

^q/2

for any q ≥ 2 . Hence, applying Markov's inequality at order q > 2 ,

1 δ P

max

1≤k≤[nδ]

|S

_k⁰

|

_B

> √ nε

≤ K

q

M

^q

δ

^(q−2)/2

ε

^q

.

As a consequence, we get that

δ→0

lim lim sup

n→∞

1 δ P

max

1≤k≤[nδ]

|S

_k⁰

|

_B

> √ nε

= 0. (3.3)

In the same way, applying Markov's inequality at order 2 1

δ P

max

1≤k≤[nδ]

|S

_k⁰⁰

|

_B

> √ nε

≤ K

2

ε

²

E (|X

0

|

²_B

1

_|X₀_|

B>M

) . (3.4)

The term E (|X

0

|

²_B

1

_|X₀_|

B>M

) is as small as we wish by choosing M large enough. The point 2 follows

from (3.3) and (3.4).

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We now consider the general case. Since B is 2-smooth, Burkholder's inequality holds and so Proposition 4.1 in [4] applies: if (3.2) holds, then, setting d

k

= P

i∈Z

P

k

(X

i

) , we have

max

1≤k≤n

k

X

i=1

X

_i

−

k

X

i=1

d

_i

B

₂

= o( √

n). (3.5)

Since (d

i

)

_i∈Z

is a stationary martingale dierences sequence in L

²

( B ) , we have just proved that it satises the conclusion of Proposition 6. From (3.5) it follows that the conclusion of Proposition 6 is also true for (X

i

)

_i∈Z

with

Λ

_B

(g, h) = Cov(g(d

0

), h(d

0

)), for any g, h in B

^∗

.

It remains to see that this covariance function can also be written as in Proposition 6. Recall that, for any g and h in B

^∗

,

X

k∈Z

|Cov(g(X

0

), h(X

k

))| ≤ X

k∈Z

kP

0

(g(X

k

))k

2

X

k∈Z

kP

0

(h(X

k

))k

2

< ∞ .

(see the proof of item 1 of Theorem 4.1 in [4]). Hence, for any g in B

^∗

,

n→∞

lim 1 n E

X

ⁿ

k=1

g(X

k

)

²

= X

k∈Z

Cov(g(X

0

), g(X

k

)) . (3.6) Now, from (3.5), we also know that

n→∞

lim 1 n E

X

ⁿ

k=1

g(X

_k

)

²

= E ((g(d

₀

))

²

) . (3.7)

Applying (3.6) and (3.7) with g , h and g + h , we infer that Cov(g(d

₀

), h(d

₀

)) = X

k∈Z

Cov(g(X

₀

), h(X

_k

)) ,

which completes the proof.

3.1.2 A Rosenthal inequality for non adapted sequences

We begin with a maximal inequality that is useful to compare the moment of order p of the maximum of the partial sums of a non necessarily adapted process to the corresponding moment of the partial sum. The adapted version of this inequality has been proven in the adapted case (that is when X

0

is F

0

-measurable) in [12]. Notice that Proposition 2 of [12] is stated for real valued random variables, but it holds also for variables taking values in a separable Banach space ( B , | · |

_B

) .

Proposition 7. Let p > 1 be a real number and q be its conjugate exponent. Let X

₀

be a random variable in L

^p

( B ) and F

0

a σ -algebra satisfying F

0

⊆ T

⁻¹

(F

0

) . Then, for any integer r , the following inequality holds:

max

1≤m≤2^r

|S

m

|

_B

_p

≤ qkS

2^r

k

p

+ q2

^r/p

r−1

X

`=0

2

^−`/p

k E

0

(S

₂`

)k

p

+ (q + 1)2

^r/p

r

X

`=0

2

^−`/p

kS

₂`

− E

2^`

(S

₂`

)k

p

.

(3.8)

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Remark 8. If we do not assume stationarity, so if we consider a sequence (X

i

)

_i∈_Z

in L

^p

( B ) for a p > 1 , and an increasing ltration (F

i

)

_i∈Z

, our proof reveals that the following inequality holds true:

for any integer r ,

max

1≤m≤2^r

|S

m

|

_B

_p

≤ qkS

2^r

k

p

+ q

r−1

X

l=0

²

r−l−1

X

k=1

k E

k2^l

(S

_(k+1)2l

− S

_k2l

)k

^p_p

^1/p

+ (q + 1)

r

X

l=0

²

r−l

X

k=1

kS

_k2l

− S

_(k−1)2l

− E

k2^l

(S

_k2l

− S

_(k−1)2l

)k

^p_p

1/p

.

Remark 9. Under the assumptions of Proposition 7, we also have that for any integer n ,

max

1≤k≤n

|S

k

|

_B

_p

≤ 2q max

1≤k≤n

kS

k

p

+ a

p

n

^1/p

n

X

`=1

k E

0

(S

`

)k

p

`

^1+1/p

+ b

p

n

^1/p

2n

X

`=1

kS

`

− E

`

(S

`

)k

p

`

^1+1/p

, (3.9) where

a

_p

= 2

^1+1/p

q

1 − 2

^−1−1/p

and b

_p

= 2(q + 1) 2

^1+1/p

1 − 2

^−1−1/p

. The proof of this remark will be done at the end of this section.

In the next results, we consider the case where ( B , | · |

_B

) = ( R , | · |) . The next inequality is the non adapted version of the Rosenthal type inequality given in [12] (see their Theorem 6).

Theorem 10. Let p > 2 be a real number and q be its conjugate exponent. Let X

0

be a real-valed random variable in L

^p

and F

0

a σ -algebra satisfying F

0

⊆ T

⁻¹

(F

0

) . Then, for any positive integer r , the following inequality holds:

E

1≤j≤2

max

^r

|S

j

|

^p

2

^r

E (|X

0

|)

^p

+ 2

^r

r−1

X

k=0

k E

0

(S

₂k

)k

p

2

^k/p

!

^p

+ 2

^r

r

X

k=0

kS

₂k

− E

2^k

(S

₂k

)k

p

2

^k/p

!

^p

+ 2

^r

r−1

X

k=0

k E

0

(S

₂²k

)k

^δ_p/2

2

^2δk/p

!

^p/(2δ)

, (3.10) where δ = min(1, 1/(p − 2)) .

Remark 11. The inequality in the above theorem implies that for any positive integer n ,

E

1≤j≤n

max |S

j

|

^p

n E (|X

1

|)

^p

+ n

n

X

k=1

1 k

^1+1/p

k E

0

(S

k

)k

p

!

^p

+ n

2n

X

k=1

1 k

^1+1/p

kS

k

− E

k

(S

k

)k

p

!

^p

+ n

n

X

k=1

1 k

^1+2δ/p

k E

0

(S

_k²

)k

^δ_p/2

!

^p/(2δ)

. To see this, it suces to use the arguments developed in the proof of Remark 9 together with the following additional subadditivity property: for any integers i and j , and any δ ∈]0, 1] :

k E

0

(S

_i+j²

)k

^δ_p/2

≤ 2

^δ

k E

0

(S

_i²

)k

_p/2

+ 2

^δ

k E

0

(S

_j²

)k

_p/2

. So, according to the rst item of Lemma 37 of [12], for any integer n ∈]2

^r−1

, 2

^r

] ,

r−1

X

k=0

k E

0

(S

₂²_k

)k

^δ_p/2

2

^2δk/p

n

X

k=1

1 k

^1+2δ/p

k E

0

(S

_k²

)k

^δ_p/2

.

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Remark 12. Theorem 10 has been stated in the real case. Notice that if we assume X

0

to be in L

^p

( B ) where ( B , | · |

_B

) is a separable Banach space and p is a real in ]2, ∞[ , then a Rosenthal-type inequality similar as (3.10) can be obtained but with a dierent δ for 2 < p < 4 . To be more precise, we get that

E

1≤j≤2

max

^r

|S

_j

|

^p

B

2

^r

E (|X

₀

|

_B

)

^p

+ 2

^r

r

X

k=0

kS

₂k

− E

2^k

(S

₂k

)k

p

2

^k/p

!

p

+ 2

^r

r−1

X

k=0

k E

0

(|S

₂k

|

²_B

)k

^δ_p/2

2

^2δk/p

!

^p/(2δ)

, (3.11) where δ = min(1/2, 1/(p − 2)) . The proof of this inequality is given later.

As a consequence of (3.10), one can prove the following inequality. This inequality will be used to prove the tightness of the sequential empirical process (1.2) in the space `

^∞

([0, 1] × R

^`

) (see the proof of Theorem 4, Section 5).

Proposition 13. Let p > 2 . Let X

0

be a real-valed random variable in L

^p

and F

0

a σ -algebra satisfying F

0

⊆ T

⁻¹

(F

0

) . For any j ≥ 1 , let

A(X, j) = max 2 sup

i≥0

k E

0

(X

_i

X

_j+i

)k

_p/2

, sup

0≤i≤j

k E

0

(X

_j

X

_j+i

) − E (X

_j

X

_j+i

)k

_p/2

. (3.12)

Then for every positive integer n ,

max

1≤j≤n

|S

j

|

_p

n

^1/2

ⁿ⁻¹

X

k=0

| E (X

0

X

k

)|

^1/2

+ n

^1/p

kX

1

k

p

+ n

^1/p

n

X

k=1

1 k

^1/p

k E

0

(X

k

)k

p

+ n

^1/p

2n

X

k=1

1 k

^1/p

kX

₀

− E

k

(X

₀

)k

_p

+ n

^1/p

X

ⁿ

k=1

1 k

^(2/p)−1

(log k)

^γ

A(X, k)

1/2

.

where γ can be taken γ = 0 for 2 < p ≤ 3 and γ > p − 3 for p > 3 . The constant that is implicitly involved in the notation depends on p and γ but it depends neither on n nor on the X

_i

's.

The proof of this proposition is left to the reader since it uses the same arguments as those developed for the proof of Proposition 20 in [12].

We would like also to pint out that Theorem 10 implies the following Burkholder-type inequality.

This has been already mentioned in the adapted case in [12, Corollary 13].

Corollary 14. Let p > 2 be a real number, X

0

be a real random variable in L

^p

and F

0

a σ -algebra satisfying F

0

⊆ T

⁻¹

(F

0

) . Then, for any integer r , the following inequality holds

E max

1≤j≤2^r

|S

j

|

^p

2

^rp/2

E (|X

0

|

^p

) + 2

^rp/2

^r−1

X

j=0

k E

0

(S

₂j

)k

p

2

^j/2

^p

+ 2

^rp/2

X

^r

j=1

kS

₂j

− E

2^j

(S

₂j

)k

p

2

^j/2

^p

.

The above corollary (up to constants) is then the non adapted version of [13, Theorem 1] when p > 2 . Let us give an application of it for the partial sums associated to continuous functions of the iterates of an ergodic automorphism of the torus. Let T be a ergodic automorphism of T

^d

as dened in the introduction. Let f be a continuous function from T

^d

to R with modulus of continuity ω(f, ·) (see Section 4, equation (4.1), for the denition). Using Corollary 14 together with Theorem 18 (see also Remark 19), we infer that if

Z

1/2 0

ω(f, t)

t| log t|

^1/2

dt < ∞ ,

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then for any p > 2 ,

max

1≤k≤n

k

X

i=1

(f ◦ T

ⁱ

− ¯ λ(f ))

_p

n

^1/2

.

Proof of Proposition 7. For any k ∈ {1, . . . , 2

^r

} , we have that

S

k

= S

k

− E

k

(S

k

) + E

k

(S

2^r

) − E

k

(S

2^r

− S

k

) . Consequently

max

1≤k≤2^r

|S

k

|

_B

_p

≤

max

1≤k≤2^r

| E

k

(S

2^r

)|

_B

_p

+

max

1≤m≤2^r−1

| E

2^r−m

(S

2^r

− S

2^r−m

)|

_B

_p

+ kS

2^r

− E

2^r

(S

₂r

)k

p

+

max

1≤m≤2^r−1

|S

m

− E

m

(S

_m

)|

_B

_p

. (3.13) Following the proof of Proposition 2 in [12], we get that

max

1≤k≤2^r

| E

k

(S

₂r

)|

_B

_p

+

max

1≤m≤2^r−1

| E

2^r−m

(S

₂r

− S

₂r−m

)|

_B

_p

≤ q k E (S

₂r

|F

2^r

)k

p

+ q

r−1

X

`=0

²

r−`−1

X

k=1

k E

k2^`

(S

_(k+1)2`

− S

_k2`

)k

^p_p

^1/p

.

So, by stationarity,

max

1≤k≤2^r

| E

k

(S

2^r

)|

_B

p

+ max

1≤m≤2^r−1

| E

2^r−m

(S

2^r

− S

2^r−m

)|

_B

p

≤ q k E (S

2^r

|F

2^r

)k

p

+ q2

^r/p

r−1

X

`=0

2

^−`/p

k E (S

₂`

|F

0

)k

p

. (3.14) We now bound the last term in the right hand side of (3.13). For any m ∈ {1, . . . , 2

^r

− 1} , we consider its binary expansion:

m =

r−1

X

i=0

b

i

(m)2

ⁱ

, where b

i

(m) = 0 or b

i

(m) = 1 . Set m

l

= P

r−1

i=l

b

i

(m)2

ⁱ

, and write that for any p ≥ 1 ,

|S

m

− E

m

(S

m

)|

_B

≤

r−1

X

l=0

|S

m_l

− S

m_l+1

− E

m

(S

m_l

− S

m_l+1

)|

_B

, (3.15) since S

₀

= 0 and m

_r

= 0 . Now, since for any l = 0, . . . , r − 1 , F

ml

⊆ F

m

, the following decomposition holds:

|S

m_l

− S

m_l+1

− E

m

(S

m_l

− S

m_l+1

)|

_B

≤ |S

m_l

− S

m_l+1

− E

m_l

(S

m_l

− S

m_l+1

)|

_B

+

E S

m_l

− S

m_l+1

− E

m_l

(S

m_l

− S

m_l+1

)|F

m

)

B

. Notice that m

_l

6= m

_l+1

only if m

_l

= k

_m,l

2

^l

with k

_m,l

odd. Then, setting

B

_r,l

= max

1≤k≤2^r−l,kodd

|S

_k2l

− S

_(k−1)2l

− E

k2^l

(S

_k2l

− S

_(k−1)2l

)|

_B

,

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it follows that

|S

m_l

− S

m_l+1

− E

m

(S

m_l

− S

m_l+1

)|

_B

≤ B

r,l

+ | E (B

r,l

|F

m

)| . Starting from (3.15), we then get that

max

1≤m≤2^r−1

|S

m

− E

m

(S

m

)|

_B

_p

≤

r−1

X

l=0

kB

r,l

k

p

+

r−1

X

l=0

max

1≤m≤2^r−1

| E (B

r,l

|F

m

)|

_p

.

Since ( E (B

r,l

|F

m

))

_m≥1

is a martingale, by using Doob's maximal inequality, we get that

max

1≤m≤2^r−1

| E (B

r,l

|F

m

)|

_p

≤ qk E (B

r,l

|F

2^r−1

)k

p

≤ qkB

r,l

k

p

, yielding to

max

1≤m≤2^r−1

|S

_m

− E

m

(S

_m

)|

_B

_p

≤ (q + 1)

r−1

X

l=0

kB

_r,l

k

_p

.

Since

B

_r,l

≤

2^r−l−1

X

k=1

|S

_k2l

− S

_(k−1)2l

− E

k2^l

(S

_k2l

− S

_(k−1)2l

)|

^p

B

!

1/p

,

we derive that

kS

2^r

− E

2^r

(S

2^r

)k

p

+ max

1≤m≤2^r−1

|S

m

− E

m

(S

m

)|

_B

_p

≤ (q + 1)

r−1

X

l=0

²

r−l−1

X

k=1

kS

_k2l

− S

_(k−1)2l

− E

k2^l

(S

_k2l

− S

_(k−1)2l

)k

^p_p

^1/p

.

So, by stationarity, max

1≤m≤2^r−1

|S

_m

− E

m

(S

_m

)|

_B

_q

≤ (q + 1)2

^r/p

r−1

X

l=0

2

^−l/p

kS

₂l

− E

2^l

(S

₂l

)k

_p

. (3.16) Starting from (3.13) and taking into account (3.14) and (3.16), the inequality (3.8) follows.

Proof of Theorem 10. Thanks to Proposition 7, it suces to prove that the inequality (3.10) is satised for E |S

2^r

|

^p

instead of E max

_1≤j≤2^r

|S

j

|

^p

. Let a

n

= kS

n

k

p

. According to the proof of Lemma 11 in [12], the theorem will follow if we can prove the following recurrence formula: for any positive integer n ,

a

^p_2n

≤ 2a

^p_n

+ c

1

a

^p−1_n

k E

0

(S

n

)k

p

+ kS

n

− E

n

(S

n

)k

p

+ c

2

a

^p−2δ_n

k E

0

(S

_n²

)k

^δ_p/2

. (3.17) where c

₁

and c

₂

are positive constants depending only on p . To prove (3.17), we denote by S ¯

_n

= X

_n+1

+ · · · + X

_2n

, and we write that

S

2n

= S

n

− E

n

(S

n

) + E

n

(S

n

) + ¯ S

n

.

Recall rst the following algebraic inequality: Let x and y be two positive real numbers and p ≥ 1 any real number. Then

(x + y)

^p

≤ x

^p

+ y

^p

+ 4

^p

(x

^p−1

y + xy

^p−1

) . (3.18)

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(see Inequality (87) in [12]). The above inequality with x = | E

ⁿ

(S

n

) + ¯ S

n

| and y = |S

n

− E

ⁿ

(S

n

)|

gives

a

^p_2n

≤ k E

n

(S

n

) + ¯ S

n

k

^p_p

+ kS

n

− E

n

(S

n

)k

^p_p

+ 4

^p

E | E

n

(S

_n

) + ¯ S

_n

|

^p−1

|S

n

− E

n

(S

_n

)|

+ 4

^p

E | E

n

(S

_n

) + ¯ S

_n

||S

n

− E

n

(S

_n

)|

^p−1

, which combined with Hölder's inequality and stationarity leads to

a

^p_2n

≤ k E

n

(S

_n

) + ¯ S

_n

k

^p_p

+ 2

^p−1

(1 + 2

^2p+1

)a

^p−1_n

kS

_n

− E

n

(S

_n

)k

_p

. (3.19) Starting from (3.19), (3.17) will follow if we can prove that there exist two positive constants c and c

₂

depending only on p such that

k E

n

(S

n

) + ¯ S

n

k

^p_p

≤ 2a

^p_n

+ c a

^p−1_n

k E

0

(S

n

)k

p

+ c

2

a

^p−2δ_n

k E

0

(S

_n²

)k

^δ_p/2

.

This inequality can be proven by following the lines of the end of the proof of Theorem 6 in [12].

Indeed, it suces to replace in their proof, x = S

n

by x = E

ⁿ

(S

n

) , and to use the following estimates (coming from the proof of their lemma 34 and the stationarity assumption): for any reals p and u such that 0 ≤ u ≤ p − 2 ,

E (| E

n

(S

_n

)|

^u

| S ¯

_n

|

^p−u

) ≤ a

^{p−2u/(p−2)}_n

k E

n

( ¯ S

_n²

)k

^u/(p−2)_p/2

, and

E (| E

n

(S

_n

)|

^p−1

| S ¯

_n

|) ≤ a

^p−1_n

k E

n

( ¯ S

_n²

)k

^1/2_p/2

.

Proof of Remark 12. As it is pointed out in the proof of Theorem 10, the remark will be proven with the help of Proposition 7, if we can show that

a

^p_2n

≤ 2a

^p_n

+ c

₁

a

^p−1_n

kS

_n

− E

n

(S

_n

)k

_p

+ c

₂

a

^p−2δ_n

k E

0

(|S

_n

|

²_B

)k

^δ_p/2

,

where a

^p_n

= E (|S

n

|

^p_B

) , c

1

and c

2

are positive constants depending only on p and δ = min(1/2, 1/(p−2)) . Indeed, the second term in the right-hand side of (3.8) can be bounded by the last term in the right- hand side of (3.11). To see this it suces to use Jensen's inequality and the fact that δ ≤ 1/2 .

Starting from (3.19) (by replacing the absolute values by the norm | · |

_B

), we see that to prove the above recurrence formula it suces to show that there exists a positive constant c depending only on p such that

k E

ⁿ

(S

n

) + ¯ S

n

k

^p_p

≤ 2a

^p_n

+ ca

^p−2δ_n

k E

⁰

(|S

n

|

²_B

)k

^δ_p/2

.

The dierence at this step with the proof of Theorem 10 is that the inequality (3.18) is used whatever p ∈]2, ∞[ (in Theorem 10, so in the real case, when p ∈]2, 4[ more precise inequalities can be used as done in the proof of Theorem 6 in [12]).

Proof of Corollary 14. To prove the corollary, it suces to show that for any 0 < δ ≤ 1 and any real p > 2 ,

2

^r

r−1

X

k=0

k E

0

(S

₂²_k

)k

^δ_p/2

2

^2δk/p

!

^p/(2δ)

2

^rp/2

k E

0

(X

₁²

)k

^p/2_p/2

+ 2

^rp/2

^r−1

X

j=0

k E

0

(S

₂j

)k

p

2

^j/2

^p

, (3.20)

and to apply Theorem 10. Following the proof of Lemma 12 in [12] and setting b

n

= k E

0

(S

_n²

)k

_p/2

, we infer that (3.20) will follow if we can prove that, for any integer n ,

b

2n

≤ 2b

n

+ 2b

^1/2_n

k E

0

(S

n

)k

p

+ 2b

^1/2_n

kS

n

− E

n

(S

n

)k

p

. (3.21)

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By using the notation S ¯

n

= X

n+1

+ · · · + X

n

and the fact that S

²_2n

= S

_n²

+ ¯ S

_n²

+ 2 E

ⁿ

(S

n

) ¯ S

n

+ 2(S

n

− E

n

(S

n

)) ¯ S

n

, we get, by stationarity, that

b

2n

≤ 2b

n

+ 2k E

0

E

n

(S

n

) E

n

( ¯ S

n

)

k

_p/2

+ 2k E

0

(S

n

− E

n

(S

n

)) ¯ S

n

k

_p/2

.

Hence the inequality (3.21) follows from the following upper bounds: Applying Cauchy-Schwarz inequality twice and using stationarity, we get

k E

0

E

n

(S

n

) E

n

( ¯ S

n

)

k

p/2

≤ k E

0

( E

²n

(S

n

))k

^1/2_p/2

× k E

0

( E

²n

( ¯ S

n

))k

^1/2_p/2

≤ k E

0

(S

_n²

))k

^1/2_p/2

× k E

²n

( ¯ S

n

)k

^1/2_p/2

≤ b

^1/2_n

k E

0

(S

n

)k

p

, and

k E

0

(S

_n

− E

n

(S

_n

)) ¯ S

_n

k

_p/2

≤ k E

0

(((S

_n

− E

n

(S

_n

))

²

)k

^1/2_p/2

k E

0

( ¯ S

_n²

)k

^1/2_p/2

≤ b

^1/2_n

kS

n

− E

n

(S

n

)k

p

.

Proof of Remark 9. Let n and r be integers such that 2

^r−1

≤ n < 2

^r

. Notice rst that

max

1≤k≤n

|S

_k

|

_B

_p

≤

max

1≤k≤2^r

|S

_m

|

_B

_p

and kS

₂r

k

_p

≤ 2kS

₂r−1

k

_p

≤ 2 max

1≤k≤n

kS

_k

k

_p

(3.22) (for the second inequality we use the stationarity). Now, setting V

m

= k E

⁰

(S

m

)k

p

, we have by stationarity that for all n, m ≥ 0 , V

n+m

≤ V

n

+ V

m

and then, according to the rst item of Lemma 37 of [12],

2

^r/p

r−1

X

`=0

2

^−`/p

k E

0

(S

₂`

)k

p

≤ n

^1/p

2

^1/p

2

^2+1/p

2

^1+1/p

− 1

n

X

k=1

k E

0

(S

k

)k

p

k

^1+1/p

≤ n

^1/p

2

^1+1/p

1 − 2

^−1/p−1

n

X

k=1

k E

0

(S

k

)k

p

k

^1+1/p

. (3.23)

On an other hand, let W

_m

= kS

_m

− E

m

(S

_m

)k

_p

, and note that the following claim is valid:

Claim 15. If F and G are σ -algebras such that G ⊂ F , then for any X in L

^p

( B ) where p ≥ 1 , kX − E (X|F )k

p

≤ 2kX − E (X |G)k

p

.

The above claim together with the stationarity imply that for all n, m ≥ 0 , W

n+m

≤ 2(W

n

+W

m

) . Therefore, using once again the rst item of Lemma 37 of [12], we get that

2

^r/p

r

X

`=0

2

^−`/p

kS

₂`

− E

2^`

(S

₂`

)k

p

≤ 2n

^1/p

2

^1+1/p

1 − 2

^−1/p−1

2n

X

`=1

kS

`

− E

`

(S

`

)k

p

`

^1+1/p

. (3.24)

The inequality (3.9) then follows from the inequality (3.8) by taking into account the upper bounds

(3.22), (3.23) and (3.24).

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3.2 A tightness criterion

We begin with the denition of the number of brackets of a family of functions.

Denition 16. Let P be a probability measure on a measurable space X . For any measurable function f from X to R, let kf k

_P,1

= P(|f |) . If kf k

_P,1

is nite, one says that f belongs to L

¹_P

. Let F be some subset of L

¹_P

. The number of brackets N

P,1

(ε, F) is the smallest integer N for which there exist some functions f

₁⁻

≤ f

1

, . . . , f

_N⁻

≤ f

N

in F such that: for any integer 1 ≤ i ≤ N we have kf

i

− f

_i⁻

k

P,1

≤ ε , and for any function f in F there exists an integer 1 ≤ i ≤ N such that f

_i⁻

≤ f ≤ f

i

.

The rst step in the proof of Theorem 4 is the following proposition, whose proof is based on a decomposition given in [1] (see also [5]).

Proposition 17. Let (X

i

)

_i≥1

be a sequence of identically distributed random variables with values in a measurable space X , with common distribution P . Let P

n

be the empirical measure P

n

= n

⁻¹

P

n

i=1

δ

X_i

, and let S

n

be the empirical process S

n

= n(P

n

− P) . Let F be a class of functions from X to R and G = {f − l, (f, l) ∈ F × F } . Assume that there exist r ≥ 2 and p > 2 such that for any function g of G ∪ F , we have

max

1≤k≤n

|S

k

(g)|

_p

≤ C( √

nkgk

^1/r_P,1

+ n

^1/p

) , (3.25) where the constant C does not depend on g nor on n . If moreover

Z

1 0

x

^(1−r)/r

(N

P,1

(x, F ))

^1/p

dx < ∞ and lim

x→0

x

^p−2

N

P,1

(x, F ) = 0 , then

δ→0

lim lim sup

n→∞

E

1≤k≤n

max sup

g∈G,kgkP,1≤δ

n

^−p/2

|S

k

(g)|

^p

= 0 , (3.26)

and lim

δ→0

lim sup

n→∞

1 δ E

max

1≤k≤[nδ]

sup

f∈F

n

^−p/2

|S

k

(f )|

^p

= 0 . (3.27)

Proof of Proposition 17. It is almost the same as that of Proposition 6 in [5]. Let us only give the main steps.

For any positive integer k , denote by N

k

= N

P,1

(2

^−k

, F) and by F

k

a family of functions f

₁^k,−

≤ f

₁^k

, . . . , f

_N^k,−

k

≤ f

_N^k

k

in F such that kf

_i^k

− f

_i^k,−

k

P,1

≤ 2

^−k

, and for any f in F , there exists an integer 1 ≤ i ≤ N

k

such that f

_i^k,−

≤ f ≤ f

_i^k

.

We follow exactly the proof of Proposition 6 in [5]. One can prove that for any ε > 0 , there exist N(ε) and m = m(ε) such that: for any n ≥ N(ε) there exists f

_n,m

in F

m

for which

max

1≤k≤n

sup

f∈F

n

^−1/2

|S

k

(f) − S

k

(f

n,m

)|

_p

≤ ε . (3.28)

Now, (3.26) follows from (3.28) as in [1] (see the end of the proof of Proposition 6 in [5]).

Let us prove (3.27). We apply (3.28) with ε = 1 : for n ≥ δ

⁻¹

N (1) , we infer from (3.28) that there exists f

_[nδ],m

in F

m

for which

max

1≤k≤[nδ]

sup

f∈F

n

^−1/2

|S

k

(f ) − S

k

(f

_[nδ],m

)|

_p

≤ √

δ .

Hence

max

1≤k≤[nδ]

sup

f∈F

n

^−1/2

|S

k

(f )|

_p

≤ √

δ + max

1≤k≤[nδ]

sup

f∈F

n

^−1/2

|S

k

(f

_[nδ],m

)|

_p

. (3.29)

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Now, since F

m

contains 2N

m

functions (g

`

)

_`∈2N_m

(each g

`

being one of the functions f

_i^m

or f

_i^m,−

in N

m

), it follows that

max

1≤k≤[nδ]

sup

f∈F

n

^−1/2

|S

k

(f

[nδ],m

)|

_p

≤

2N_m

X

`=1

√ 1 n

max

1≤k≤[nδ]

|S

k

(g

`

)|

_p

.

Let K

m

= max

_f∈F_m

kfk

P,1

. Applying (3.25), we infer that

max

1≤k≤[nδ]

sup

f∈F

n

^−1/2

|S

k

(f

_[nδ],m

)|

_p

≤ 2CN

m

(K

_m^1/r

√

δ + n

^{−(p−2)/2p}

δ

^1/p

) . (3.30) Since m = m(1) is xed, (3.27) follows from (3.29) and (3.30) and the fact that p > 2 .

4 Inequalities for ergodic torus automorphisms

In this section, we keep the same notations as in the introduction. 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

such that v

₁

, ..., v

_d_u

are in E

_u

, v

_d_u₊₁

, ..., v

_d_u_+d_e

are in E

_e

and v

_d_u_+d_e₊₁

, ..., v

_d

are in E

_s

. We suppose moreover that det (v

₁

|v

2

| · · · |v

d

) = 1 . Let || · || be the norm on R

^d

given by

d

X

i=1

x

i

v

i

= max

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

namely, d

1

(¯ x, y) = inf ¯

z∈Z^d

d

0

(x + z, y) .

We dene 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} . For every f : T

^d

→ R, we consider the moduli of continuity dened, for every δ > 0 , by ω(f, δ) := sup

¯

x,¯y∈T^d:d₁(¯x,¯y)≤δ

|f (¯ x) − f (¯ y)| , (4.1)

ω

_(s,e)

(f, δ) = sup{|f (¯ x) − f (¯ x + h

s

+ h

e

)|, ¯ x ∈ T

^d

, h

s

∈ B

s

(δ), h

e

∈ B

e

(δ)}

and

ω

_(u)

(f, δ) = sup{|f (¯ x) − f (¯ x + h

u

)|, x ¯ ∈ T

^d

, h

u

∈ B

u

(δ)} . Let r

u

be the spectral radius of S

_|E⁻¹

u

. For every ρ

u

∈ (r

u

, 1) , there exists K > 0 such that, for every integer n ≥ 0 , we have

∀h

_u

∈ E

_u

, ||S

⁻ⁿ

h

_u

|| ≤ Kρ

ⁿ_u

||h

_u

|| (4.2) and

∀(h

e

, h

s

) ∈ E

e

× E

s

, ||S

ⁿ

(h

e

+ h

s

)|| ≤ Kn

^d^e

||h

e

+ h

s

|| . (4.3)

The following inequality is an extension to continuous functions of a result for Hölder functions

established in [10].