STUDY OF ALMOST EVERYWHERE CONVERGENCE OF SERIES BY MEANS OF MARTINGALE METHODS

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STUDY OF ALMOST EVERYWHERE

CONVERGENCE OF SERIES BY MEANS OF MARTINGALE METHODS

Christophe Cuny, Fan Ai-Hua

To cite this version:

Christophe Cuny, Fan Ai-Hua. STUDY OF ALMOST EVERYWHERE CONVERGENCE OF SE- RIES BY MEANS OF MARTINGALE METHODS. 2016. �hal-01261284�

SERIES BY MEANS OF MARTINGALE METHODS

CHRISTOPHE CUNY AND AI HUA FAN

Abstract. Martingale methods are used to study the almost everywhere con- vergence of general function series. Applications are given to ergodic series, which improves recent results of Fan [9], and to dilated series, including Dav- enport series, which completes results of Gaposhkin [12] (see also [13]). Appli- cation is also given to the almost everywhere convergence with respect to Riesz products of lacunary series.

1. Introduction

Let us first state two problems which motivate the investigation in this paper.

One comes from the classical analysis and the other from the ergodic theory.

Given a function f ∈ L¹(R/Z) (or equivalently a locally integrable 1-periodic function on the real line R) such that R1

0 f(x)dx = 0, an increasing sequence of positive integers (n_k)k≥0 ⊂ N and a sequence of complex numbers (a_k)k≥0 ⊂ C, one would like to investigate the convergence (almost everywhere convergence or L¹-convergence etc) of the following series, called dilated series,

(1)

∞

X

k=0

a_kf(n_kx).

If the series converges almost everywhere (a.e.) whence (a_k) ∈ `², we say that {f(n_kx)} is a convergence system. The famous Carleson theorem states that both {sinnx} and {cosnx} are convergence systems. In general, one should find suitable conditions on f and on (n_k) for {f(n_kx)} to be a convergence system.

This is a long standing problem. One may consult the survey by Berkes and Weber [2] and, for recent progresses, Weber [22] and the references therein.

The other problem is the almost everywhere convergence of the so-calledergodic series

(2)

∞

X

k=0

a_kf(T^kx)

where T is a measure-preserving map on a probability space (X,B, µ) and f ∈ L¹(µ) such that R

f dµ= 0. To be more precise, we want to find sufficient condi- tions on the dynamical system (X,B, µ, T) and/or on f, such that the series (2)

1991Mathematics Subject Classification. Primary: ; Secondary:

1

converges forany sequence (an)n∈N∈`^p, where 1 ≤p≤2 is fixed (our main inter- est is in the casep= 2). Sufficient conditions for the a.e. convergence with specific (regular) sequences (a_n)_n∈N may be found for instance in [4]. The fact that condi- tions have to be imposed to ensure the a.e. convergence of (2) may be illustrated by the following result of Dowker and Erd¨os [6]: if µ is an non-atomic ergodic measure, for every positive sequence (a_n)n∈N such thatP

k∈Na_k=∞, there exists f ∈L^∞(µ) withR

f dµ= 0 such thatP

k∈Na_kf(T^kx) diverges a.e. Some sufficient conditions are recently found in [9] for {f◦Tⁿ} be to be a convergence system.

In this paper, we will study series in a general setting. The above situations are special examples. Let (Ω,A,P) be a probability space. Let (Z_n)_n≥0 ⊂L^p(Ω,A,P), p ≥ 1, be a sequence of L^p-integrable random variables with EZ_n = 0. We shall study the almost sure convergence of the random series

(3)

∞

X

n=0

Z_n(ω).

Suppose that we are given an increasing filtration (A_n)_n∈N such that A₀ = {∅,Ω} and A_∞ = A where A_∞ := W∞

n=0A_n. We will analyze random variables using this filtration. For everyn ∈N∪ {∞}, denoteEⁿ :=E(·|A_n) and introduce the operator

D_n =Eⁿ⁺¹−Eⁿ.

If Z ∈L^p(Ω,A,P) with EZ = 0 (p≥1), we have the decomposition Z =

∞

X

n=0

D_nZ ,

where the convergence takes place a.e. and in L^p. This is what we mean by the analysis based on the filtration (An)n∈N and it is a simple consequence of Doob’s convergence theorem of martingales.

change

One of our main results is the following theorem, which is a special case of Theorem 5 corresponding to p= 2.

Theorem A. Let (Z_n)_n∈N ⊂ L²(Ω,A,P) be such that EZ_n = 0 for every n ∈N. Then the series P

n∈NZ_n converges P-a.s. and in L²(Ω,A,P)under the following set of conditions

(4)

∞

X

k=0

X^∞

n=0

kD_n+kZ_nk²₂1/2

<∞

and (5)

∞

X

k=1

X^∞

n=0

kD_nZ_n+kk²₂1/2

<∞.

One of ingredients in the proof of Theorem A is the Doob maximal inequality of martingales. Assume that {Zn} ⊂ L²(Ω,A, P) are independent and EZn = 0

for all n. It follows from Theorem A that P

E|Zn|² < ∞ implies the almost sure convergence of P

Z_n. This is a trivial application of Theorem A, because this known result of Kolmogorov is actually covered by the Doob convergence theorem of martingales.

We can also make an analysis using a decreasing filtration. Suppose that we are given a decreasing filtration (B_n)n∈N such that B₀ =A. Let B∞ :=T∞

n=0B_n. For every n∈N∪ {∞}, denote En =E(·|B_n) and introduce

d_n=En−En+1.

Let p ≥ 1. For Z ∈ L^p(Ω,A,P) with E(Z|B∞) = 0 which implies EZ = 0, we have the decomposition

Z =

∞

X

n=0

d_nZ , where the series converges in L^p and P-a.s.

Theorem B. Let (Zn)n∈N ⊂ L²(Ω,A,P) be such that E^∞(Zn) = 0 for every n ∈ N. Then the series P

n∈Ng_n converges P-a.s. and in L²(Ω,A,P) under the following conditions

(6)

∞

X

k=0

X^∞

n=0

kd_n+kZ_nk²₂1/2

<∞

and (7)

∞

X

k=1

X^∞

n=0

kd_nZ_n+kk²₂1/2

<∞.

As an application of Theorem A, we have the following theorem, in which the condition (8) is sharp (see Proposition 16).

Recall first that a sequence of positive integers (n_k)_k∈N is said to be Hadamard lacunaryif inf_kn_k+1/n_k≥q >1 for someq and that themodulus ofL²-continuity of f ∈L²(R/Z) is defined by ω₂(δ, f) = sup_0≤h≤δkf(·+h)−f(·)k₂.

Theorem C. Suppose that f ∈L²(R/Z) satisfies R

f(x)dx= 0 and

(8) X

n∈N

ω₂(2⁻ⁿ, f)

√n <∞

and that (n_k)k∈N is Hadamard lacunary. Then{f(n_kx)} is a convergence system.

Let us look at a very interesting special system{f(n_kx)}wheref is a Davenport function. Let λ >0. The function

f_λ(x) =

∞

X

m=1

sin 2πmx m^λ

is well defined because the series converges everywhere. It is called Davenport function. Whenλ >1/2, it isL²-integrable and{f_λ(nx)}_n≥1 is a complete system in L²([0,1]) ([23]). When λ > 1, {f_λ(nx)}_n≥1 is even a Riesz basis ([15, 18]).

When 1/2 < λ ≤ 1, {f_λ(nx)}_n≥1 is not a Riesz basis, but Br´emont proved that {f_λ(n_kx)}_nk≥1 is a Riesz sequence (see Definition 1) for any Hadamard lacunary sequence {n_k}.

Theorem D Let λ > 1/2. Suppose that {n_k} ⊂ N is lacunary in the sense of Hadamard. Then the so-called Davenport seriesP∞

k=1a_kf_λ(n_kx)converges almost everywhere if ond only if P∞

k=1|a_k|² <∞.

In this special case of Davenport series, the condition P∞

k=1|a_k|² < ∞ is also proved necessary for the almost everywhere convergence. Both the sufficiency and necessity are new.

Now let us give an application of Theorem B. Consider a measure-theoretic dy- namical system (X,B, µT). LetLbe the associated transfer (or Perron-Frobenius) operator, which is defined by

(9)

Z

f·Lgdµ= Z

f ◦T ·gdµ (∀f ∈L^∞(µ),∀g ∈L¹(µ)).

As an application of Theorem B, we have the following theorem, in which the condition (H2) is sharp to some extent (see Proposition 18).

Theorem EAssume that(X,B, T, µ)is an ergodic measure-preserving dynamical system and f ∈L²(µ). Suppose

(H1) lim_m→∞E(f|T^−mB) = 0 a.e., which impliesR

f dµ = 0;

(H2) P∞ n=1

kL√ⁿfk2

n <∞

Then {f(Tⁿx)} is a convergence system.

This theorem improves a result in [9], where the normk · k∞ was used instead of the norm k · k₂.

The paper is organized as follows. In Section 2 we performs an analysis us- ing increasing filtrations. Theorem A will be proved there, together with some more general results. Conditions in Theorem A will be converted into some more practical conditions and divergence will also discussed. Section 3 is parallel to Section 2, as Theorem B is parallel to Theorem A. There an analysis is made using decreasing filtration. But details are omitted and some details can also be found in [9]. Application of Theorem A to ergodic series is discussed in Section 4 and Theorem E will be proved there. Section 5 is devoted to dilated series (The- orem C and Theorem D) and Section 6 is devoted to lacunary series and their almost everywhere convergence with respect to Riesz product and to more general inhomogeneous equilibriums.

2. Analysis using increasing filtration

Let (Ω,A,P) be a probability space and p≥1. Let (Z_n)n≥0 ⊂L^p(Ω,A,P) be a sequence random variables such thatEZ_n= 0 for all n. We shall study the almost sure convergence of the random seriesP∞

n=0Z_n(ω). Forn ≥0, denote the partial sum

Sn=

n

X

k=0

Zk.

Then define the maximal functions S^∗(ω) = sup

n∈N

|S_n(ω)|, S_N^∗(ω) = max

0≤n≤N|S_n(ω)| (∀N ≥0).

In this section, we give an analysis of the series by using an increasing filtration.

In the next section, we do the same by using an decreasing filtration. Notice that there will be minor differences between two analysis. But applications will show that there is a question of how to choose a filtration. For example, for the dilated series, we will introduce an increasing filtration and for the ergodic series, there is a natural decreasing filtration.

2.1. Decomposition relative to an increasing filtration. Suppose that we are given an increasing filtration (A_n)_n∈N. Assume thatA₀ ={∅,Ω}andA_∞ =A where A∞:=W∞

n=0A_n. For every n∈N∪ {∞}, denote Eⁿ :=E(·|A_n) and D_n =Eⁿ⁺¹−Eⁿ.

The operators Dn have the following remarkable properties. The proofs, which are easy, are left to the reader.

Lemma 1. Assume h ∈ L¹(Ω,A,P) and f, g ∈ L²(Ω,A,P). The operators D_n have the following properties.

(1) For any n≥0, D_n is A_n+1-measurable and EⁿD_nf = 0.

(2) For any distinct integers n and m, D_nf and D_mg are orthogonal.

(3) For any N₁ < N₂ we have

N2

X

n=N1

kDnfk²₂ =kE^N2+1f −E^N1fk²₂.

The first assertion implies that for any sequence (f_n)∈ L¹(Ω,A,P), (D_nf_n) is a sequence of martingale differences. The second assertions will be referred to as the orthogonality of the martingale difference.

For any integral random variable of zero mean, we may decompose it into martingale as follows. In the following lemma, we include an inequality of Bahr- Esseen-Rio and an inequality of Burkholder.

Lemma 2. Let Z ∈L^p(Ω,A,P), p≥1 with EZ = 0. We have the decomposition

(10) Z =

∞

X

n=0

DnZ ,

where the series converges in L^p and P-a.s. Moreover we have

(11) kZk_p ≤max(1,p

p−1)X^∞

n=0

kD_nZk^p_p⁰1/p⁰

,

where p⁰ := min(2, p) and, if p > 1, (12)

X^∞

n=0

|D_nZ|²1/2

p ≤C_pkZk_p.

Proof. By assumptions, (EⁿZ)n∈Nis a uniformly integrable martingale converging inL^p and P-a.s. toE^∞Z =Z. Hence, (10) follows from the equality

N

X

n=1

D_nZ =E^N+1Z−E⁰Z

where E⁰Z = E(Z|A₀) = EZ = 0, for A₀ ={∅,Ω}. Then, (11) follows from von Bahr-Esseen [1] when 1 ≤p≤ 2 and from Rio [21] when p≥2, while (12) is the Burkholder inequality.

Remark. The inequality (12) is nothing but the (reverse) Burkholder inequality.

In particular, we have a converse inequality but we shall only need (12) in the sequel.

We callD_nZ the n-th order detail ofZ with respect to (A_n).

For any Z_n, we have the following decomposition

Z_n =X_n+Y_n with X_n=Z_n−EⁿZ_n, Y_n=EⁿZ_n.

Notice that EⁿX_n= 0 i.e. X_n is conditionally centered, and Y_n isA_n-measurable with E(Y_n) = 0. Our random series is thus decomposed into two random series rajout

∞

X

n=0

Z_n=

∞

X

n=0

X_n+

∞

X

n=0

Y_n.

The convergence of P

Z_n is reduced to those of P

X_n and P

Y_n. In the sequel, we separately study these two series.

2.2. Conditionally centered series P

Xn. The maximal functions associated to the series P

X_n will be denoted by S_N,X^∗ and S_X^∗. Recall that we write p⁰ = min(2, p) forp≥1.

Proposition 3. Let (Xn)n∈N ⊂ L^p(Ω,A,P) be such that Eⁿ(Xn) = 0 for every n∈N (p >1). Then, for everyN ∈N, we have the following maximal inequality

(13)

S_N,X^∗

p ≤ p

p−1max(1,p p−1)

∞

X

k=0

X^N

`=0

kD_{`+k</sub>X<sub>`}k^p_p⁰1/p⁰

. Consequently, the series P

n∈NX_n converges P-a.s. and in L^p(Ω,A,P) under the following condition:

(14)

∞

X

k=0

X^∞

n=0

kD_n+kX_nk^p_p⁰1/p⁰

<∞.

Moreover, when (14) holds, we have S_X^∗ ∈L^p(Ω,A,P).

Proof. By the decomposition in Lemma 2 applied to each X<sub>`</sub> and using the fact E<sup>`</sup>(X<sub>`</sub>) = 0 which implies that, for every k≤`,E^kX<sub>`</sub> =E<sup>k</sup>(E<sup>`</sup>X_{`</sub>) = 0, hence that D<sub>k</sub>X<sub>`} = 0 for k < `, we have

X` =X

k≥`

DkX` =

∞

X

k=0

Dk+`X`.

LetN ∈N and 0≤n≤N. We obtain that

|S_n|=

n

X

`=0

X_`

=

n

X

`=0

∞

X

k=0

D_{`+k</sub>X<sub>`}

≤

∞

X

k=0

0≤m≤Nmax

m

X

`=0

D_{`+k</sub>X<sub>`} . Now, for every k ∈ N fixed, the sequence (Pm

`=0D<sub>`+k</sub>X_`)m∈N is a martingale.

Hence, by Doob’s maximal inequality and the von Bahr-Esseen-Rio inequality (11) in Lemma 2, we have

kS_N^∗ k_p ≤ p

p−1max(1,p p−1)

∞

X

k=0

X^N

`=0

kD_{`+k</sub>X<sub>`}k^p_p⁰1/p⁰

.

Thus we have proved the maximal inequality. Consequently, for any N⁰ ≤N⁰⁰, setting K_p := _p−1^p max(1,√

p−1), we have k max

N⁰≤p,q≤N⁰⁰|S_p−S_q|k_p ≤2k max

N⁰≤n≤N⁰⁰|S_n−S_N⁰|k_p ≤2K_p

∞

X

k=0

^N

00

X

`=N⁰+1

kD_{`+k</sub>X<sub>`}k^p_p⁰1/p⁰

.

Letting N⁰⁰→+∞, we infer that

(15) k sup

p,q≥N⁰

|S_p −S_q|k_p ≤2K_p

∞

X

k=0

X

`≥N⁰+1

kD_{`+k</sub>X<sub>`}k^p_p⁰1/p⁰

<∞,

by (14). By the Lebesgue dominated convergence theorem on N, applied to the counting measure, we deduce that

k sup

p,q≥N⁰

|S_p−S_q|k_p −→

N⁰→+∞0. By the Fatou lemma we infer that almost surely

sup

p,q≥N⁰

|S_p−S_q| −→

N⁰→+∞0,

which finishes the proof.

Remark. If p ≥ 2, then p⁰ = 2 and the condition (14) with p ≥ 2 is stronger than the condition (14) with p= 2. Hence, the relevance of the proposition when p ≥ 2 lies in the integrability of the maximal function S_X^∗. While if one is only concerned with the a.s. convergence it is better to apply the proposition with p= 2. However, as we shall see, the control of S^∗ inL^p with p >2 will allows us to study the divergence of the series P

X_n. 2.3. Adapted seriesP

Yn. The maximal functions associated to the seriesP Yn

will be denoted by S_N,Y^∗ and S_Y^∗.

Proposition 4. Let (Y_n)n≥1 ⊂L^p(Ω,A,P) be such that Y_n is A_n-measurable and EY_n = 0 for every n≥1 ( p >1). Then, for every N ≥1,

(16) kS_N,Y^∗ k_p ≤K_p

N

X

k=1

X^N

`=k

kD`−kY<sub>`</sub>k^p_p⁰1/p⁰

Consequently, the series P

n∈NY_n converges P-a.s. and in L^p(Ω,A₀,P) under the following condition

(17)

∞

X

k=1

X^∞

n=0

kD_nY_n+kk^p_p⁰1/p⁰

<∞.

Moreover, when (17) holds, we have S_Y^∗ ∈L^p(Ω,A,P).

Proof. The proof is similar to that of Proposition 3. By assumption, we infer that for every ` ≥1,

Y_` =

`−1

X

k=0

D_kY_` =

`

X

k=1

D`−kY<sub>`</sub>. Hence,

|

n

X

`=1

Y`|=|

n

X

`=1

`

X

k=1

D`−kY`| ≤

N

X

k=1

k≤m≤Nmax |

m

X

`=k

D`−kY`|. Since, for every fixedk ∈N, (Pm

`=kD`−kY_`)m≥1 is a martingale, then (16) follows from the Doob maximal inequality and the Bahr-Esseen-Rio inequality (11).

The proof of the P-a.s. and L^p-convergence may be done as for Proposition 3.

2.4. Convergence and integrability of general series P

Zn. Combining Proposition 3 and Proposition 4 and using the decomposition Z_n = X_n +Y_n, we derive the following theorem.

Theorem 5. Let p > 1. Let (Z_n)n∈N ⊂ L^p(Ω,A,P) be such that EZ_n = 0 for every n ∈N. Then, for every N ∈N,

(18) kS_N,Z^∗ k_p ≤K_pX^∞

k=0

X^N

`=0

kD_{`+k</sub>Z<sub>`}k^p_p⁰1/p⁰

+

N

X

k=1

X^N

`=k

kD`−kZ<sub>`</sub>k^p_p⁰1/p⁰ . Consequently, the series P

n∈NZ_n converges P-a.s. and in L^p(Ω,A,P) under the following set of conditions

(19)

∞

X

k=0

X^∞

n=0

kD_n+kZ_nk^p_p⁰1/p⁰

<∞

and (20)

∞

X

k=1

X^∞

n=0

kD_nZ_n+kk^p_p⁰1/p⁰

<∞.

Moreover, if both conditions (19) and (20) are satisfied, then S_Z^∗ ∈L^p(Ω,A,P).

Remark. If (Z_n) is adapted to (A_n), then the condition (19) is trivially satisfied.

Proof. Recall that X_n = Z_n −EⁿZ_n and Y_n = EⁿZ_n. The theorem follows immediately from the decompositionZn=Xn+Yn, Proposition 3 and Proposition 4 and the following simple facts

S_N^∗ ≤S_N,X^∗ +S_N,Y^∗ ,

D`+kX`=E<sup>`+k+1</sup>(Z`−E<sup>`</sup>Z`)−E<sup>`+k</sup>(Z`−E`</sup>Z`) =D`+kZ`, D`−kY<sub>`</sub> =E`−k+1</sup>(E<sup>`Z`</sub>)−E<sup>`−k(E<sup>`</sup>Z<sub>`) = D`−kZ<sub>`</sub>.

We call (19) the condition on higher order details and (20) the condition on lower order details.

2.5. Practical criteria. In order to apply Theorem 5, it is sometimes more con- venient to use the sufficient conditions in the next lemma. The proof of the lemma will use the Burkholder inequality.

The condition (21) suggests that we need some order of approximation of Z_k byE(Z_k|A_n) asn tends to infinity and the condition (22) suggests that the condi- tional expectationE(·|A_n)) would contract in some order on the spaceL^p₀(Ω,A, P) consisting ofp-integrable variables with zero mean; sometimes it is really the case (see Lemma 21, see also Theorem 2 in [8] and Lemma 2 in [10]).

Lemma 6. Let (Zn)n∈N ⊂ L^p(Ω,A,P), p > 1, with EZn = 0 for all n. The condition (19) on the higher order details is satisfied if

(21)

∞

X

`=0

2^`(1−1/p)X^∞

k=0

kZ_k−E²

`+k−1Z_kk^p_p⁰1/p⁰

<∞.

The condition (20) on the lower order details is satisfied if (22)

∞

X

`=0

2^`(1−1/p)X^∞

k=2^`

kE^k+1−2

`Z_kk^p_p⁰1/p⁰

<∞.

Proof. We make the proof when p≥2, then p⁰ = 2. The proof when 1< p < 2 may be done similarly. Look first at the condition (19) on the higher order of details. Cutting the sum over k into dyadic blocks and applying Cauchy-Schwarz inequality to each block we get

∞

X

k=0

X^∞

n=0

kD_n+kZ_nk²_p1/2

≤

∞

X

`=0

2^`/22

`+1−2

X

k=2^`−1

∞

X

n=0

kD_n+kZ_nk²_p1/2

.

Now, using successively the H¨older inequality (notice that p/2 ≥ 1), the trivial inequalityk · k_{`</sub><sup>p</sup> ≤ k · k<sub>`}² and the reverse Burkholder inequality (12), we get that

2^`+1−2

X

k=2^`−1

kD_n+kZ_nk²_p ≤ 2^`(1−2/p)2

`+1−2

X

k=2^`−1

kD_n+kZ_nk^p_p2/p

= 2^`(1−2/p) E

2^`+1−2

X

k=2^`−1

(D_n+kZ_n)^p2/p

≤ 2^`(1−2/p)

X

k≥2^`−1

(D_n+kZ_n)²1/2

2 p

≤ C_p2^`(1−2/p)kZ_n−Eⁿ⁺²

`−1

Z_nk²_p. Thus the first assertion follows. Similarly, we have

∞

X

k=1

X^∞

n=0

kD_nZ_n+kk²_p1/2

=

∞

X

k=1

X^∞

n=k

D_n−kZ_nk²_p1/2

≤

∞

X

`=0

2^`/22

`+1−1

X

k=2^`

∞

X

n=k

Dn−kZ_nk²_p1/2

.

Now, we first change the order of summation to get

2^`+1−1

X

k=2^`

∞

X

n=k

kDn−kZ_nk²_p =

∞

X

n=2^`

min(n,2^`+1−1)

X

k=2^`

kDn−kZ_nk²_p.

Then, using the same arguments as above, we have

min(n,2^`+1−1)

X

k=2^`

kDn−kZ_nk²_p ≤ C_p2^`(1−2/p)kEⁿ⁺¹⁻²

`Z_n−E⁰Z_nk²_p

= C_p2^`(1−2/p)kEⁿ⁺¹⁻²

`Z_nk²_p. Finally we get

∞

X

k=1

X^∞

n=0

D_nZ_n+kk²_p1/2

≤C_p

∞

X

`=0

2^`(1−1/p)X^∞

k=2^`

kE^k+1−2

`Z_kk²_p1/2

.

In view of the above results and of Lemma 5 of [9], one expects to have better integrability properties forS_Z^∗ when (Z_n)⊂L^∞(Ω,A,P).

Theorem 7. Let (Z_n)n∈N ⊂L^∞(Ω,A,P) with EZ_n= 0 for all n. Assume that

(23) ∆₁ :=

∞

X

`=0

X^∞

k=0

kZ_k−E^`+kZ_kk²_∞1/2

<∞.

and

(24) ∆₂ :=

∞

X

`=0

X^∞

k=`

kE^k+1−`Z_kk²_∞1/2

<∞. Then P

n∈NZ_n converges P-a.e. and in any L^p with p≥1. Moreover, we have E(e^β(SZ∗)2)<∞,

for every β < _4e(∆¹

1+∆2)².

Proof. We follow a standard strategy to prove the exponential integrability S_Z^∗ by estimating the p-th moments of S_Z^∗. To estimate the p-th moments we shall not use Lemma 6 which yields badly behaving constants C_p, as p → ∞, due to the use of Burkholder’s inequality. Hence, we shall use Theorem 5 instead.

First remark that kS_Z^∗k_p (p ≥ 2) is bounded by K_p times the sum of the two terms in (19) and (20). Notice that

kD_{`+k</sub>Z<sub>`}k_p ≤ kE^{`+k+1</sup>Z<sub>`</sub>−Z<sub>`</sub>k<sub>p</sub>+kE<sup>`+k}Z_{`</sub>−Z<sub>`}k_p.

Then, using Minkowski’s inequality in `² we obtain that for every N ∈N,

∞

X

k=0

X^∞

`=0

kD_{`+k</sub>Z<sub>`}k²_p1/2

≤2

∞

X

k=0

X^∞

`=0

kZ<sub>`</sub>−E<sup>`+k</sup>Z_`k²_∞1/2

= 2∆₁. On the other hand, using

kDnZn+kkp ≤2kEⁿ⁺¹Zn+kkp,

we have

∞

X

k=1

X^∞

n=0

kD_nZ_n+kk²_p1/2

≤2

∞

X

k=1

X^∞

n=0

kEⁿ⁺¹Z_n+kk²_∞1/2

= 2∆₂.

Thus we have proved kS_Z^∗k_p ≤ 2K_p∆ with ∆ = ∆₁ + ∆₂ for p ≥ 2. Let β > 0.

We have

E(e^β(SZ∗)2) =

∞

X

p=0

β^pkS_Z^∗k^2p_2p

p! ≤1 +

∞

X

p=1

β^p(2K_2p∆)^2p

p! .

SinceK_2p^2p ≤ _2p−1^2p 2p

p^p ∼ep^p and, by Stirling’s formula,p!∼ ^p_ep√

2πp, we infer that E(e^β(SZ∗)2)<∞ as soon as β <(4e∆²)⁻¹. 2.6. Series of the form P∞

n=0a_nW_n. In our applications, we shall be concerned with the situation where

Z_n=a_nW_n

with (an)n∈N a deterministic sequence and (Wn)n∈N a stationary sequence or at least a sequence behaving somehow closely to a stationary one. When p ≥ 2 (resp. when 1< p <2), we are going to control theL^p-moment ofP

a_nW_nby the

`<sup>2</sup>-moment (resp. the `^p-moment) of (a_n). Before stating the result, let us state Cauchy’s condensation principle whose proof is easy.

Lemma 8. Let (u_n)n∈N be positive numbers such that u_n+m ≤ Ku_n for every n, m∈ N and for some K > 0. Then the series P

`∈Nu<sub>`</sub> converges if and only if the series P

`∈N2<sup>`</sup>u₂^` converges.

Theorem 9. Let 1< p≤ ∞. Let (W_n)n∈N⊂L^p(Ω,A,P) be such that (25) ∆e_p :=

∞

X

`=0

1 (`+ 1)^1/p

sup

k∈N

kW_k−E^k+`−1W_kk_p+ sup

m∈N

kE^m+1W_m+`k_p

<∞. Then the series P

n∈Na_nW_n converges P-a.e. for every (a_n)_n∈N ∈`^p0. Moreover, when 1 < p < ∞, there exists C_p > 0 (independent from (a_n)_n∈N and (W_n)_n∈N) such that

(26) k sup

N∈N

|

N

X

n=0

a_nW_n| k_p ≤C_p∆e_pk(a_n)n∈Nk_`p0 ; and when p=∞, we have

Eexp βsup

N∈N

|

N

X

n=0

anWn|²

<∞

for every β <(4e∆e∞k(an)n∈Nk_`²)⁻¹.

Proof. Let 1< p <∞. By Theorem 5 and Lemma 6, we only have to check that (27)

∞

X

`=0

2^`(1−1/p) sup

k∈N

kW_k−E^k+2

`−1

W_kk_p+ sup

m∈N

kE^m+1W_m+2^`k_p

<∞. This is actually equivalent to the condition (25), by the Cauchy condensation principle. Let us check the condition in the Cauchy condensation principle. Notice that

sup

m∈N

kE^m+1W_m+`+1k_p ≤ sup

m∈N

kE^m+2W_m+`+1k_p ≤ sup

m∈N

kE^m+1W_m+`k_p,

and that, by the Burkholder inequality (12), for everyk, `∈N and every m≥1 C<sub>p</sub>kW<sub>k</sub>−E<sup>k+`−1</sup>W_kk_p ≥ k W_k−E^{k+`+m−1</sup>W<sub>k</sub>)<sup>2</sup> + (E<sup>k+`+m−1}W_k−E^k+`−1W_k)²1/2

k_p

≥ kW_k−E^k+`+m−1W_kk_p

The case p=∞can be proved similarly basing on Theorem 7.

To conclude this section we shall prove that, when p≥ 2, there are situations where the condition (a_n)n∈N ∈ `² in the above theorem is also necessary for the P-a.e. convergence of P

n∈NanWn.

Definition 1. We say that a sequence (Wn)n∈N ⊂ L²(Ω,A,P) is a Riesz system if there exists C >0 such that for every (b_n)_n∈N ∈`²,

C⁻¹k(b_n)_n∈Nk_`² ≤ kX

n∈N

b_nW_nk₂ ≤Ck(b_n)_n∈Nk_`².

If moreover (W_n)n∈N is complete in L²(Ω,A,P), we say that is a Riesz basis.

Theorem 10. Let 2< p≤ ∞. Suppose that (W_n)_n∈N⊂L^p(Ω,A,P) such that (28)

∞

X

`=0

1 (`+ 1)^1/p

sup

k∈N

kW_k−E^k+`−1W_kk_p+ sup

m∈N

kE^m+1W_m+`k_p

<∞ and that(W_n)n∈Nis is a Riesz sequence inL²(Ω,A,P). Then the seriesP

n∈Na_nW_n does not converge P-a.e. for every (a_n)_n∈N such that P

n∈N|a_n|² =∞.

Remark. We do not know whether the series is P-a.e. divergent under the above conditions.

Proof. We proceed as in the proof of Theorem 2.6 of [9]. Recall first the following Paley-Zygmund inequality (see Kahane [16] for the case q = 2, the proof being the same for general q > 1): Let Z ∈ L^q(Ω,A,P) be non negative (q > 1). For any 0< λ <1, we have

P(Z ≥λEZ)≥

(1−λ) EZ kZkq

q/(q−1)

.

We apply the inequality with q=p/2 and Z =ZN = sup_0≤n≤N|Pn

k=0akWk|². By Theorem 9 and the hypothesis that (W_n)n∈Nis a Riesz sequence, we know that there existC, D > 0 such that

D v u u t

N

X

k=0

|a_k|² ≤EZ_N ≤ kZ_Nk_q≤C v u u t

N

X

k=0

|a_k|².

Hence, we infer that

P ZN ≥λD

N

X

k=0

|ak|²

≥

(1−λ)D C

q/(q−1)

. Since, P

n∈N|a_n|² =∞, the result follows.

3. Analysis using decreasing filtration

We can also use decreasing filtrations to analyze our series. Recall that (Ω,A,P) is a probability space and (Z_n)n≥0 ⊂L¹(Ω,A,P) is a sequence of random variables such that EZ_n= 0 for all n. Our object of study is the random series

(29)

∞

X

n=0

Z_n(ω).

Suppose that we are given an decreasing filtration (Bn)n∈N. Assume thatB0 =A.

LetB∞ :=T∞

n=0Bn. We will suppose that

(30) ∀n, E(Zn|B∞) = 0

which implies EZn= 0.

Forn ≥0, denote the partial sum S_n=

n

X

k=0

Z_k.

ForN ≥0, define the maximal function S_N^∗(ω) = max

0≤n≤N|S_n(ω)|.

The basic idea is to convert the random series into reverse martingales.

For every n∈N∪ {∞}, denote Eⁿ :=E(·|Bn) and dn=Eⁿ−Eⁿ⁺¹.

Let us state the useful properties of the operatorsdnin the following proposition.

The following lemma is the same as Lemma 1.

Lemma 11. Let h ∈ L¹(Ω,A,P) and f, g ∈ L²(Ω,A,P). The operators dn have the following properties:

(1) For andn ≥0, D_n is B_n-measurable and Eⁿ⁺¹D_nf = 0.

(2) For any distinct integers n and m, dⁿf and d^mg are orthogonal.

(3) For any N₁ < N₂ we have

N2

X

n=N1

kd_nfk²₂ =kE^N1f−E^N2+1fk²₂.

The first assertion implies that for any sequence (f_n)∈L¹(Ω,A,P), (d_nf_n) is a sequence of reverse martingale differences.

For any integral random variable such that E(Z|B∞) = 0, we may decompose it into martingale as follows. We have the following analogue of Lemma 2, whose proof is the same (it does not matter here that we are dealing with reverse mar- tingale differences rather that martingale differences).

Lemma 12. Let Z ∈ L^p(Ω,A,P), p ≥ 1 with E(Z|B∞) = 0. We have the decomposition

(31) Z =

∞

X

n=0

D_nZ ,

where the series converges in L^p and P-a.s. Moreover we have

(32) kZk_p ≤max(1,p

p−1)X^∞

n=0

kd_nZk^p_p⁰1/p⁰

,

where p⁰ := min(2, p) and, if p > 1, (33)

X^∞

n=0

|dnZ|²1/2

p ≤CpkZkp.

We call dnZ the n-th order detail of Z with respect to (B_n). Since B_n is de- creasing, we say that d_nZ is a detail of higher order than d_mZ when n < m.

Theorem 13. Let (Z_n)n∈N ⊂ L^p(Ω,A,P), p > 1, be such that E^∞(Z_n) = 0 for every n ∈N. Then, for every N ∈N,

(34) kS_N^∗k_p ≤2K_pX^N

k=1

X^N

`=k

kd`−kZ<sub>`</sub>k^p_p⁰0

1/p⁰

+

∞

X

k=0

X^N

`=0

kd_{`+k</sub>Z<sub>`}k^p_p⁰0

1/p⁰ . Consequently, the series P

n∈Ngn converges P-a.s. and in L^p(Ω,A,P) under the following conditions

(35)

∞

X

k=0

X^∞

n=0

kdn+kZnk^p_p⁰0

1/p⁰

<∞

and (36)

∞

X

k=1

X^∞

n=0

kd_nZ_n+kk^p_p⁰0

1/p⁰

<∞.

If(Z_n)is adapted to(B_n), the condition (36) on the higher order details is trivially satisfied.

The proof being similar to that of Theorem 5, we leave it to the reader. Some similar arguments can be found [9].

In order to apply Theorem 13, it will be convenient to use the sufficient condi- tions in the next lemma.

Lemma 14. Let (Z_n)n∈N ⊂ L^p(Ω,A,P), p > 1, with E^∞Z_n = 0 for all n. The condition (36) on the higher order details is satisfied if

(37)

∞

X

`=0

2^`(1−1/p)X^∞

n=2^`

kZ_n−E^n−2`+1Z_nk^p_p⁰0

1/p⁰

<∞.

The condition (35) on the lower order details is satisfied if (38)

∞

X

`=0

2^`(1−1/p)X^∞

n=0

kEn+2^`−1Z_nk^p_p⁰0

1/p⁰

<∞.

The proof is similar to that of Lemme 6 and we leave it to the reader.

Results similar to Theorems 7, 9 and 10 holds.

4. Convergence of ergodic series

Let (X,B, µ, T) be a measure-preserving dynamical system. By an ergodic series we mean a series of the form

(39)

∞

X

n=0

a_nf_n(Tⁿx)

where it is assumed that the f_n’s are integrable with Ef_n = 0 and that (a_n) is a sequence of numbers. The almost everywhere convergence of such series was studied in [9] where the martingale method was already used, and which is a motivation of our present study.

In this case, the natural filtration that we can use to analyze the series is the one defined by

Bn =T⁻ⁿB.

It is a decreasing filtration. Let L be the transfer operator associated to the dynamical system which can be defined by

Z

f·Lgdµ= Z

f ◦T ·gdµ (∀f ∈L^∞(µ),∀g ∈L¹(µ)).

Theorem 15. Assume that(X,B, T, µ) is an ergodic measure-preserving dynam- ical system. Let (f_n)_n∈N⊂L^p(µ), 1< p ≤ ∞. Suppose

(H1) ∀n ∈N, lim_m→∞kE(f_n|T^−mB)k_p = 0;

(H2) P∞

i=02^{`(1−1/p)</sup>sup<sub>n≥0</sub>kL<sup>2`}f_nk_p <∞.

Then for any complex sequence (a_n)_n∈N ⊂ C such that P∞

n=0|a_n|^p0 < ∞, the ergodic series P∞

n=0a_nf_n(Tⁿx) converges a.e., and in L^p(X,B, T, µ) if p < ∞.

Moreover, when 1 < p < ∞, sup_N∈N|PN

n=0a_nf_n(Tⁿx)| ∈ L^p(X,B, T, µ) and when p=∞, E(exp(βsup_N∈

N|PN

n=0a_nf_n◦Tⁿ|²))<∞ for some β >0.

Proof. Since (f_n◦ Tⁿ) is adapted to (B_n), we can apply Theorem 13 by just checking the condition (38) on the higher order details. The checking is easy and is a direct consequence of the fact that for m≥n we have

E(fn◦Tⁿ|T^−mB) = (L^m−nf)◦T^m. Then, letting Z_n =a_nf ◦Tⁿ, it suffices to notice

kEn+2^{`</sup>−1Z<sub>n</sub>k<sup>p</sup><sub>p</sub><sup>0</sup> =|a<sub>n</sub>|<sup>p0</sup>k(L<sup>2`−1}f_n)◦T^mk^p_p⁰ =|a_n|^p0kL^2`−1f_nk^p_p⁰.

When p =∞, the condition (H2) reads P∞

i=02^{`</sup>sup<sub>n≥0</sub>kL<sup>2`}f_nk∞ <∞. Under this last condition, the above theorem was proved in [9] and the exponential in- tegrability of the maximal function was not mentioned in [9] but it follows from the obtained estimates there. If one is only interested in almost everywhere con- vergence but not in the exponential integrability, Theorem 15 relax the condition in [9] by requiring only L^p-integrability, 2 ≤p≤ ∞ and weakening the exponent of 2^` in (H2).

In the study of dynamical systems, the decay of Lⁿf (measured by L^∞-norm or the H¨older norm) was extensively studied for regular functions f like H¨older functions. The above theorem shows that for the almost everywhere convergence of the ergodic series, weaker regularity would be sufficient and that there is an interest to study the decay ofLⁿf measured byL²-norm. Several situations where such a decay is measured, for unbounded functionals, may be found for instance in [5], section 3.2 or in [8].

To conclude the section, let us show that the condition (H2) in Theorem 15 is sharp. We shall use the notation L₀(x) = 1, L₁(x) = max(1,logx) and L_m(x) = L₁◦ · ◦L₁(x) (where L₁ appears m times).

LetT be the measure preserving transformation on ([0,1),B([0,1)), λ) (λbeing the Lebesgue measure) defines by T x = 2x mod 1. Then kLⁿgk₁ → 0 for every g ∈L¹([0,1),B([0,1)), λ) such that R

g(x)dx= 0.

Proposition 16. Consider the dynamics ([0,1),B([0,1)), λ, T) where λ is the Lebesque measure andT x= 2x mod 1. For everym∈N, there exist (an)n∈N∈`² and f ∈ L^p([0,1),B([0,1)), λ) for every 1 ≤ p < ∞ with R

f(x)dx = 0, such

that kL²ⁿfk2 = O(_L^2−n/2

m(2ⁿ)) and that the series P

n∈Nanf ◦ Tⁿ diverges almost everywhere.

The proposition follows from Proposition 18 in the next section and the fact that for T x = {2x} and for f ∈ L²([0,1),B([0,1)), λ) with R

f(x)d(x)dx = 0, we have kLⁿfk₂ = ω₂(f,2⁻ⁿ) (Theorem 2 in [8]), where ω₂ is the L²-modulus of continuity.

5. Convergence of dilated series

We want to apply our general results in Section 2 to the study of the following series, called dilated series,

(40)

∞

X

k=0

a_kf(n_kx),

where we assumef ∈L^p([0,1],B, λ) for somep >1 (λbeing the Lebesgue measure and B the Borel σ-algebra), (a_k)k∈N ∈ `^p0(N) (p⁰ := min(2, p)), and (n_k)k∈N an increasing sequence of positive integers.

There is an extensive literature on the topic forp= 2, which is our main concern here. Let us mention the surveys [11, (1966)] by Gaposhkin and [2, (2009)] by Berkes and Weber. For more recent results, one may refer to Weber [22], see also the references therein.

5.1. Lacunary dilated series. Before stating our next result we need a defini- tion.

Definition 2. For every f ∈L^p([0,1],B, λ), we define its L^p-modulus of continu- ity ω_p(·, f) by

ωp(δ, f) := sup

0≤h≤δ

kτhf −fkp ∀δ ∈[0,1]

where τ_hf(x) :=f(x+h).

Theorem 17. Let f ∈L^p([0,1],B, λ), 1< p ≤ ∞, be such that

(41) X

n∈N

ω_p(2⁻ⁿ, f) n^1/p <∞.

Let (n_k)k∈N be a Hadamard lacunary sequence of positive integers. Then for every (a_k)k∈N∈`^p0(N), the series P

ka_kf(n_k·)converges λ-a.s. and in L^p. Moreover, if 1< p <∞, we have

sup

N∈N

|

N

X

k

akf(nk·)| ∈L^p([0,1],B, λ);