Christophe Cuny

ON THE RITT PROPERTY AND WEAK TYPE MAXIMAL INEQUALITIES FOR CONVOLUTION POWERS ON 1 (Z)

Keywords: MSC 2010 subject classification, 47A35, 37A99; Secondary, 60B15

In this paper we study the behaviour of convolution powers of probability measures µ on Z, such that (µ(n)) n∈N is completely monotone or such that ν is centered with a second moment. In particular we exhibit many new examples of probability measures on Z having the so called Ritt property and whose convolution powers satisfy weak type maximal inequalities in 1 (Z).

Introduction

Let µ be a probability on Z. Given an invertible bi-measurable transformation τ on a measure space (S, S, λ) we define a positive contraction of every L p (λ), 1 ≤ p ≤ ∞, by setting

P µ (τ )(f ) := k∈Z µ(k)f • τ k ∀f ∈ L p (m) .
Several authors, see for instance [START_REF] Bellow | Almost everywhere convergence of powers, Almost everywhere convergence[END_REF], [START_REF] Bellow | Almost everywhere convergence of weighted averages[END_REF], [START_REF] Reinhold-Larsson | Almost everywhere convergence of convolution powers in L1(X)[END_REF], [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF], [START_REF] Jones | Oscillation and variation inequalities for convolution powers[END_REF], [START_REF] Bellow | A weak-type inequality for convolution products[END_REF], [START_REF] Losert | The strong sweeping out property for convolution powers, Ergodic Theory Dynam[END_REF], [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF], [START_REF] Reinhold | Almost everywhere convergence of convolution measures[END_REF], [START_REF] Reinhold | Variation and oscillation inequalities for convolution products[END_REF] or [START_REF] Cohen | Almost everywhere convergence of powers of some positive Lp contractions[END_REF], studied the almost everywhere behaviour of the iterates of µ(τ ), i.e. of (µ * n (τ )) n≥1 , acting on L p (λ), 1 ≤ p < ∞.

When p > 1, the almost everywhere behaviour has been characterized thanks to the so called bounded angular ratio property, introduced in [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF] and which is equivalent to the so called Ritt property on p (Z), p > 1. Let us recall the definition of those properties. Definition 1.1. Let µ be a probability measure on Z. We say that µ is strictly aperiodic if |μ(θ)| < 1 for every θ ∈ (0, 2π). We say that µ has bounded angular ratio (BAR) if moreover [START_REF] Akcoglu | The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters[END_REF] sup

θ∈(0,2π) |1 -μ(θ)| 1 -|μ(θ)| < ∞ .
The strict aperiodicity is equivalent to the fact that the support of µ is not contained in a coset of a proper subgroup of Z. In particular, it holds whenever the support of µ contains two consecutive integers.

I am very grateful to Alexander Gomilko and, more particularly, to Yuri Tomilov who both noticed several inaccuracies in a previous version. The paper substantially benefited from our discussions. Definition 1.2. We say that a probability measure µ on Z is Ritt on p (Z), for some p ≥ 1, if sup n≥1 n µ * n -µ * (n+1) p (Z) < ∞ .

When p = 1 we say simply that µ is Ritt, because then, it is Ritt on all r (Z), r ≥ 1.

Denote by R the set of Ritt probability measures on Z.

A version of the next Theorem may be found for instance in Cohen, Cuny and Lin [START_REF] Cohen | Almost everywhere convergence of powers of some positive Lp contractions[END_REF]Theorem 4.3]. Their Theorem 4.3 is not formulated exactly as below but the proof of Theorem 1.1 may be done similarly. The equivalence the item (vi) with the other items follow from their Proposition 6. [START_REF] Bellow | Almost everywhere convergence of powers, Almost everywhere convergence[END_REF]. In all the paper we use the notation N := {0, 1, 2 . . .}.

Theorem 1.1. Let µ be a strictly aperiodic probability on Z. The following are equivalent: i) µ has BAR; ii) There exist p > 1 and C p > 0 such that for every invertible bi-measurable transformation τ on a measure space (S, S, λ),

(P µ (τ )) n f | p,λ ≤ C p f p,λ ∀f ∈ L p (λ) ; (2) sup n≥1 |
(iii) There exists p > 1 such that for every invertible bi-measurable transformation τ on a probability space (S, S, λ) and every f ∈ L p (λ), ((P µ (τ )) n f ) n∈N converges λ-a.e. (iv) There exist p > 1 and C p > 0 such that

(3) sup n≥1 |(P µ (R)) n f | p (Z) ≤ C p f p (Z) f ∈ p (Z) ,
where R is the right shift on Z; v) There exists p > 1 such that µ is Ritt on p (Z). (vi) There exists p > 1 such that for every m ∈ N, there exists C m,p > 0 such that

sup n≥1 n m |(I -P µ ) m (P µ (R)) n f | p (Z) ≤ C m,p f p (Z) f ∈ p (Z) ,
Actually, if any of the above properties holds, then the conclusion of (ii), (iii), (iv) or (v) holds for all p > 1.

The proof of the above theorem follows from recent works of Le Merdy and Xu, [START_REF] Merdy | Maximal theorems and square functions for analytic operators on Lpspaces[END_REF] and [START_REF] Merdy | Strong q-variation inequalities for analytic semigroups[END_REF], who studied positive Ritt contractions T of L p (S, S, m) (p being fixed). Recall that a contraction T on a Banach space X is Ritt if sup n∈N n T n -T n+1 X < ∞, which is compatible with our definition 1.2 which just says that the operator of convolution by µ is Ritt on X = p (Z).

Le Merdy and Xu proved that any positive Ritt contraction satisfies maximal inequalities in spirit of [START_REF] Bellow | Transference principles in ergodic theory[END_REF]. They also obtained square function estimates, oscillation inequalities and variation inequalities. See also [START_REF] Cohen | Almost everywhere convergence of powers of some positive Lp contractions[END_REF] for related results.

In this paper we are concerned with the case when p = 1, and we address the following two questions.

-Question 1: For what probability measures µ on Z does one have a weak type (1, 1)-maximal inequality: [START_REF] Bellow | Almost everywhere convergence of powers, Almost everywhere convergence[END_REF] #{k ∈ Z : sup

n≥1 |µ * n * f | > λ} ≤ C λ f 1 (Z) ∀λ ≥ 0 .
More generally, given m ∈ N, does there exist C m > 0 such that (with the convention (δ 0 -µ) * 0 = δ 0 )

(5) sup

λ>0 λ#{k ∈ Z : sup n≥1 n m |µ * n * (δ 0 -µ) * m * f (k)| ≥ λ} ≤ C m f 1 (Z) ∀f ∈ 1 (Z) .
-Question 2: For what probability measures µ on Z does one have the Ritt property in 1 (Z): [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF] sup

n≥1 n µ * n -µ * (n+1) 1 (Z) < ∞ .
Notice that if µ satisfies (4) then, by the Marcinkiewicz interpolation theorem (between weak L 1 and L ∞ ), it does satisfy [START_REF] Bellow | Transference principles in ergodic theory[END_REF], hence µ has BAR. Notice also that if µ satisfies [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF] then, by Theorem 1.1 has also BAR. Hence, the questions we intend to answer are: what extra conditions, in addition to the BAR property, are sufficient to have (4), [START_REF] Bellow | Almost everywhere convergence of weighted averages[END_REF] or [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF] ?

Let us discuss the known results concerning those questions, before presenting our results. As far as we know, when m ≥ 1, (5) has not been investigated before.

The simplest examples of probability measures having BAR are the symmetric ones. Bellow, Jones and Rosenblatt [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF] proved that if µ is symmetric such that (µ(n)) n≥0 is non increasing then (4) holds. We do not know whether (6) holds as well, in this case, but we provide sufficient conditions in Section 5.

Another case where (4) holds is when k∈Z k 2 µ(k) < ∞ (µ has a second moment) and k∈Z kµ(k) = 0. This has been proved by Bellow and Calderón [START_REF] Bellow | A weak-type inequality for convolution products[END_REF]. Again the Ritt property is not known in that case. The proof of Bellow and Calderón is based on general intermediary results that have been extended recently by Wedrychowicz [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF]. Wedrychowicz proved that (4) holds for centered probability measures (hence with a first moment) having BAR and satisfying some extra conditions. Examples without second moment are presented in [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF].

Several examples of probabilities having the Ritt property in 1 (Z) may be found in Dungey [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF], see sections 4 and 5 there.

Let us now present our results. As mentionned above, the method of Bellow and Calderón is fairly general. Actually, if one follows carefully their paper, one realizes that the following definition comes somewhat naturally into play. Definition 1.3. We say that a probability measure µ on Z satisfies the hypothesis (H) if μ is twice continuously differentiable on [-π, π]-{0} and if there exists an even and continuous function ψ on [-π, π], vanishing at 0 and continuously differentiable on [-π, π]-{0}, and some constants c, C > 0 such that for every θ ∈ (0, π]

(i) |μ(θ)| ≤ 1 -cψ(θ); (ii) |θ μ (θ)| ≤ Cψ(θ); (iii) |μ (θ)| ≤ Cψ (θ); (iv) |θ μ (θ)| ≤ Cψ (θ).
Let us denote by H the set of probability measures satisfying hypothesis (H).

The relevance of the hypothesis (H) lies in the following, where we also give stability properties of H as well as of R. We say that a set of probability measures on Z is stable by symmetrization if whenever µ = (µ(n)) n∈Z belongs to that set so does μ = (µ(-n)) n∈Z .

Theorem 1.2.

(i) The set H is convex and stable by convolution and by symmetrization.

(ii) The set R is convex and stable by convolution and by symmetrization.

(iii) Let µ ∈ H. Then, µ satisfies (4).

(iv) Let µ ∈ H ∩ R. Then, for every m ∈ N, there exists C m > 0 such that µ satisfies (5).

Theorem 1.2 follows from several results: item (i) follows from Proposition 2.4, item (ii) may be proved as Proposition 3.10, items (iii) and (iv) follow from Proposition 2.3.

Then, our goal is to provide many examples of elements of H ∩ R.

Our first examples are the ones already considered by Bellow and Calderón, in particular the fact that µ as in the next theorem satisfies (4) is not new, while the Ritt property is new. The proof of Theorem 1.3 is done in Section 2.3. Theorem 1.3. Let µ be a centered and strictly aperiodic probability measure on Z with finite second moment. Then µ ∈ H ∩ R.

Then, we shall consider probability measures µ, such that (µ(n)) n≥0 is completely monotone (see the next section for the definition). In this context we are able to characterize the BAR property. The idea of considering completely monotone sequences was motivated by Gomilko-Haase-Tomilov [START_REF] Gomilko | On rates in mean ergodic theorems[END_REF] and Cohen-Cuny-Lin [START_REF] Cohen | Almost everywhere convergence of powers of some positive Lp contractions[END_REF].

Theorem 1.4. Let µ be a probability measure on Z supported on N, such that (µ(n)) n∈N is completely monotone. Then (i) µ has BAR if and only if there exists C > 0 such that

(7) n k=1 kµ(k) ≤ Cn k≥n µ(k) ∀n ≥ 1 .
(ii) Assume that µ has BAR. Let σ be a probability measure on

Z such that n∈Z n 2 σ(n) < ∞. Then, µ * σ ∈ H ∩ R and for every α ∈ (0, 1] αµ + (1 -α)σ ∈ H ∩ R. In particular (take σ = δ 0 ), µ ∈ H ∩ R.
Remarks. Notice that we do not assume σ to be centered. The conclusion of item (ii) actually holds for σ such that σ is twice continuously differentiable on [-π, π] -{0} with σ and θ → θσ (θ) bounded. Moreover, see Proposition 3.10, it is possible to relax the conditions on σ if one is only concerned with the Ritt Property. We were not able to provide a perturbation result in the spirit of Theorem 1.5.

Item (i) follows from Propositions 3.3 and 3.5. Item (ii) is proved in sections 3.2 and 3.3.

The proof of the Ritt property in Theorem 1.4 is based on a recent of Gomilko and Tomilov [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF]. The fact that when µ has BAR δ 1 * µ is Ritt has been proven by Gomilko and Tomilov [START_REF] Gomilko | On discrete subordination of power bounded and Ritt operators[END_REF], see their Theorem 7.1. Their proof is also based on [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF].

Theorem 1.5. Let µ be a centered probability measure on Z supported on {-1} ∪ Z, such that (µ(n)) n∈N is completely monotone. Then (i) µ has BAR if and only if there exists C > 0 such that

(8) n k≥n kµ(k) ≤ C n k=1 k 2 µ(k) ∀n ≥ 1 .
(ii) Assume that µ has BAR. Let σ be a centered probability measure on Z, such that there exists c > 0

such that n∈Z n 2 |σ(n) -cµ(n)| < ∞. Then, σ ∈ H ∩ R. In particular (take c = 1 and σ = µ), µ ∈ H ∩ R.
Moreover (see section 5), we also study symmetric probability measures with completely monotone coefficients.

In the above theorems, we obtain weak type maximal inequalities in 1 (Z). Of course, by mean of transference principles (see e.g. [START_REF] Bellow | Transference principles in ergodic theory[END_REF] or [25, page 164], one may derive similar results for the operator P µ (τ ) in the spirit of (2) as well as some almost everywhere convergence results. We leave that "standard" task to the reader.

The paper is organized as follows. In Section 2, we prove Theorem 1.2 and prove the Ritt property under a slightly stronger assumption than hypothesis (H). In section 3, we consider probability measures as in Theorem 1.4 and prove Theorem 1.4 In section 4, we consider probability measures as in Theorem 1.5 and prove Theorem 1.5. In section 5 we consider symmetric probability measures. Finally, in section 6 we discuss several open questions on the topic.

Before going to the proofs, we would like to mention that the above theorems provide new situations to which the results of Cuny and Lin [START_REF] Cuny | Limit theorems for Markov chains by the symmetrization method[END_REF] apply, see examples 1 and 2 there.

General criteria for maximal inequalities and for the Ritt property

In this section we give general conditions ensuring weak type maximal inequalities associated with sequences of probabilities on Z and conditions ensuring the Ritt property.

In the case of weak type maximal inequalities, the obtained conditions are derived from slight modifications of known results, see e.g. [START_REF] Bellow | A weak-type inequality for convolution products[END_REF] and [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF].

2.1. Sufficient conditions for weak type maximal inequalities. We start with the following result of Bellow and Calderón [START_REF] Bellow | A weak-type inequality for convolution products[END_REF], see also Zo [START_REF] Zo | A note on the approximation of the identity[END_REF] for a related result. Actually, Bellow and Calderón considered only the case of probability measures, but their proof extends to the situation below.

Theorem 2.1 (Bellow-Calderón). Let (σ n ) n∈N be a sequence of finite signed measures on Z such that sup n∈N σ n 1 < ∞. Assume that there exists C > 0 such that k, ∈ Z with 0 < 2|k| ≤ , [START_REF] Cuny | Limit theorems for Markov chains by the symmetrization method[END_REF] |σ

n (k + ) -σ n ( )| ≤ C k 2 ∀n ∈ N .
Then, there exists C > 0 such that for every f ∈ 1 (Z),

sup λ>0 #{k ∈ Z : sup n∈N |σ n * f (k)| ≥ λ} ≤ C λ f 1 .
In order to apply Theorem 2.1 we shall need the following version of Corollary 3.4 of [START_REF] Bellow | A weak-type inequality for convolution products[END_REF]. Lemma 2.2. Let (σ n ) n∈N be a sequence of finite signed measures, such that for every n ∈ N, σn is twice continuously differentiable on R -2πZ. If moreover [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF] sup

n∈N π -π |θσ n (θ)|dθ < ∞ , and (11)
lim θ→0,θ =0 θσ n (θ) = 0 ∀n ∈ N , then (9) holds.
Remark. It follows from [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF] (and the continuity of σn at 0) that the limit in [START_REF] Farkas | Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces[END_REF] exists, hence condition (11) is just that the limit is 0. Proof. For every k ∈ Z-{0}, we have σ n (k) = π -π σn (θ)e -ikθ dθ. Let π > ε > 0. Performing two integration by parts as in [START_REF] Bellow | A weak-type inequality for convolution products[END_REF] to evaluate π ε σn (θ)e -ikθ dθ and -ε -π σn (θ)e -ikθ dθ, using our assumptions and letting ε → 0, we see that

σ n (k) = π -π σ n (θ) 1 -e -ikθ k 2 dθ .
Then, we conclude as in [START_REF] Bellow | A weak-type inequality for convolution products[END_REF].

Proposition 2.3. Let µ be a probability measure on Z satisfying hypothesis (H). Then, for every m ∈ N,

(12) sup n≥1 n m π -π |θ||(μ n (1 -μ) m ) (θ)|dθ < ∞ , and lim θ→0,θ =0 θ(μ n (1 -μ) m ) (θ) = 0 ∀n ∈ N ,
In particular, there exists C > 0 such that for every f ∈ 1 (Z), [START_REF] Gomilko | Bernstein functions and rates in mean ergodic theorems for operator semigroups[END_REF] sup

λ>0 λ#{k ∈ Z : sup n≥1 |µ * n * f (k)| ≥ λ} ≤ C f 1 .
If moreover µ is Ritt then, for every m ≥ 1, there exists C m > 0 such that for every f ∈ 1 (Z), ( 14)

sup λ>0 λ#{k ∈ Z : sup n≥1 n m |µ * n * (δ 0 -µ) * m * f (k)| ≥ λ} ≤ C m f 1 .
Remarks. The proposition is related to Theorem 2.10 of Wedrychowicz [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF]. Notice that, by (ii), ψ is non-negative and by (iii) it is non-decreasing. We shall see in proposition 2.5 that if there exists C > 0 such that for every θ ∈ (0, π], ψ(θ) ≤ Cθψ (θ), then µ is automatically Ritt.

Proof. If µ = δ 0 the result is trivial. Hence we assume that µ = δ 0 . In particular, by (ii), ψ cannot vanish in a neighbourghood of 0, hence is positive on (0, π]. Then |μ| < 1 on (0, π] (hence µ is strictly aperiodic).

We have, on (0, π].

n m |(μ n (1 -μ) m ) | ≤ n m+2 |μ| n-2 |μ | 2 |1 -μ| m + 2mn m+1 |μ| n-1 |μ | 2 |1 -μ| m-1 (15) +n m+1 |μ| n-1 |μ ||1 -μ| m-1 + mn m |μ| n |μ | |1 -μ| m-1 + m(m -1)n m |μ| n |μ | 2 |1 -μ| m-2 .
Using (i) and the fact that ψ is continuous with ψ(0) = 0, there exist η > 0 and c > 0, such that for every θ ∈ [0, η], [START_REF] Jones | Oscillation and variation inequalities for convolution powers[END_REF] |μ(θ)| ≤ e -cψ(θ) .

Since, sup θ∈[η,π] |μ(θ)| < 1, taking c smaller if necessary, we may assume that (16) holds for every θ ∈ [0, π]. Using (iii) and that ψ is continuous at 0, we see that ψ and μ are in L 1 , hence that

(17) |1 -μ(θ)| ≤ ψ(θ) ∀θ ∈ (0, π] .
Combining ( 16) and ( 26) with (ii), (iii) and (iv) (taking care with the cases m = 0 and m = 1), we see that there exists C m > 0 such that for every θ ∈ (0, π] and every n ∈ N,

|θ(μ n ) (θ)| ≤ C m n 2 ψ m+1 (θ) + nψ m (θ) + (m -1)ψ m-1 (θ) e -c(n-2)ψ(θ) n m ψ (θ) .
Using that the integrand below is even and the change of variable u = (n -2)ψ(θ), we see that

sup n∈N π -π |θ||(μ n (1 -μ) m ) (θ)|dθ ≤ Cm +∞ 0 (u m+1 + u m + (m -1)u m-1 )e -cu du < ∞,
and (12) holds.

The fact that (11) holds follows from item (ii), using that ψ is continuous at 0, with ψ(0) = 0, and that μ is bounded.

Let m ∈ N and set σ n = σ n,m := n m µ * n (δ 0 -µ) * m for every n ∈ N. It follows from Theorem 2.1 and Lemma 2.2, that (13) holds. When m ≥ 1, it follows from Theorem 2.1 and Lemma 2.2, that ( 14) holds provided that [START_REF] Merdy | Strong q-variation inequalities for analytic semigroups[END_REF] sup

n∈N σ n,m 1 (Z) < ∞ .
When m = 1, ( 18) is just the definition of the Ritt property. Let m ≥ 2, write n = m + k, with ∈ N and 0 ≤ k ≤ m -1. We have

n m µ * n * (δ 0 -µ) * m 1 (Z) ≤ m m ( + 1)µ * * (δ 0 -µ) m 1 (Z)
, and the latter is bounded uniformly with respect to ∈ N, since µ is Ritt.

To conclude this subsection we shall study stability properties of set of probabilities satisfying the weak type maximal inequalities.

It is well-known, see e.g. Proposition 3.2 of [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF], that the set of Ritt probability measures on Z is convex and stable by convolution. Actually, [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF] deals with probability measures supported by N, but the proof is the same.

Let p > 1. It is not difficult to see that the set of probability measures µ on Z, such that there exists C p > 0 such that (3) holds is also convex and stable by convolution.

However it is unclear (and probably not true) whether the set of probability measures µ on Z satisfying [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF] for every m ∈ N (or for some m ∈ N) is also convex and stable by convolution. Nevertheless, we have the following. Proposition 2.4. Let µ 1 and µ 2 be probability measures satisfying hypothesis (H). Let α ∈ (0, 1). Then, μ1 , µ 1 * µ 2 and αµ 1 + (1 -α)µ 2 satisfy hypothesis (H).

Remark. Recall that μ1 is the probability measure defined by μ1 (n) = µ 1 (-n) for every n ∈ Z. Proof. The fact that μ1 satisfies hypothesis (H) is obvious.

Let ψ i , c i , C i be the terms associated with µ i (i ∈ {0, 1}) such that the items (i) -(iv) of hypothesis (H) be satisfied.

Define

µ := µ 1 * µ 2 and ψ := c 1 ψ 1 + c 2 ψ 2 . Let θ ∈ (0, π]. We have |μ(θ)| = |μ 1 (θ)| |μ 2 (θ)| ≤ 1 -ψ(θ) + c 1 c 2 ψ 1 (θ)ψ 2 (θ).
Since ψ 1 and ψ 2 are continuous with ψ 1 (0) = ψ 2 (0) = 0, there exist c ∈ (0, 1) and

η ∈ (0, π) such that c 1 c 2 ψ 1 (θ)ψ 2 (θ) ≤ (1 -c)ψ(θ) for every θ ∈ (0, η).
Hence, |μ| ≤ 1 -cψ on (0, η). Arguing as in the previous proof, we see that taking c smaller if necessary, the inequality holds on (0, π] either.

Using that μ = μ 1 μ2 + μ1 μ 2 , we infer that items (ii) and (iii) of hypothesis (H) hold.

We have μ = μ 1 μ2 + 2μ 1 μ 2 + μ1 μ 2 . Hence, for every θ ∈ (0, π]

|θ μ (θ)| ≤ C 1 ψ 1 (θ) + 2C 1 ψ 1 (θ)C 2 ψ 2 (θ) + C 2 ψ 2 (θ) ,
and we see that item (iv) holds, since ψ 1 is bounded.

Let α ∈ (0, 1). Let µ := αµ 1 + (1 -α)µ 2 . One can see that items (i) -(iv) of hypothesis (H) hold with ψ := αψ 1 + (1 -α)ψ 2 .

2.2.

A sufficient condition for the Ritt property. In this subsection we derive a condition ensuring that a probability measure is Ritt. This condition will be used for centered probability measure with either a second moment, or a first moment and completly monotone coefficients. For non centered probability measure another argument will be needed.

We start with a general result.

Proposition 2.5. Let (σ n ) n∈N be a sequence of finite signed measures on Z, such that for every n ∈ N, σn is twice differentiable on R -2πZ. Assume moreover the following

(i) sup n∈N π -π |σn(θ)| |θ| dθ < ∞; (ii) sup n∈N π -π |σ n (θ)|dθ < ∞; (iii) sup n∈N π -π |θ| |σ n (θ)|dθ < ∞ . Then, sup n∈N σ n 1 (Z) < ∞.
Proof. We first notice that, by (i),

sup n∈N |σ n (0)| ≤ sup n∈N π -π |σ n (θ)|dθ < ∞ Let k ∈ Z -{0}. We have σ n (k) = π -π σn (θ)e -ikθ dθ = π/|k| -π/|k| σn (θ)e -ikθ dθ + [-π,π]-[-π/|k|,π/|k|] σn (θ)e -ikθ dθ .
Integrating by part and using that σn is 2π-periodic, we have

[-π,π]-[-π/|k|,π/|k|] σn (θ)e -ikθ dθ = - [-π,π]-[-π/|k|,π/|k|] σ n (θ) e -ikθ -ik dθ + σ n (-π/|k|) -σ n (π/|k|) -ik ,
and

[-π,π]-[-π/|k|,π/|k|] σ n (θ) e -ikθ -ik dθ = - [-π,π]-[-π/|k|,π/|k|] σ n (θ) e -ikθ -k 2 dθ + σ n (-π/|k|) -σ n (π/|k|) -k 2 Now, |k|≥1 | π/|k| -π/|k| σn (θ)e -ikθ dθ| ≤ π -π |σ n (θ)| 1≤|k|≤π/|θ| 1 dθ ≤ 2π π -π |σ n (θ)| |θ| dθ ,
and |k|≥1 | [-π,π]-[-π/|k|,π/|k|] σ n (θ) e -ikθ -k 2 dθ| ≤ π -π |σ n (θ)| |k|≥π/|θ| 1 k 2 ≤ C π -π |θ| |σ n (θ)|dθ .
Hence, it remains to show that sup n∈N |k|≥1

|σn(π/k)| |k| < ∞ and sup n∈N |k|≥1 |σ n (π/k)| k 2 < ∞.
Let f n (θ) := θσ n (θ), for every θ ∈ R -2πZ. Then f n is differentiable on R -2πZ and, by (i) and (ii), σ n ∈ L 1 ([0, 2π]), f n ∈ L 1 ([0, 2π]). Hence, σn and f n can be continuously extended to R with f n (0) = 0. Then, for every k ≥ 1,

π k |σ n (π/k)| = π/k 0 f n (θ)dθ ≤ π/k 0 |σ n (θ)|dθ + π/k 0 θ|σ n (θ)|dθ .
Dealing similarly with k ≤ -1 we infer that

|k|≥1 |σ n (π/k)| |k| ≤ k≥1 π/k -π/k |σ n (θ)|dθ + π/k -π/k θ|σ n (θ)|dθ ≤ π π -π |σ n (θ)| |θ| dθ + π π -π |σ n (θ)|dθ ,
which is bounded uniformly with respect to n.

Proceeding as above with g n (θ) := θ 2 σ n (θ) in place of f n (θ) we see that, by (ii) and (iii), sup n∈N |k|≥1

|σ n (π/k)| k 2 < ∞.
Let µ be a probability measure on Z. We say that µ satisfies hypothesis ( H) if it satisfies hypothesis (H) with a function ψ such that there exists D > 0 such that for every θ ∈ (0, π], [START_REF] Losert | A remark on almost everywhere convergence of convolution powers[END_REF] ψ(θ) ≤ Dθψ (θ) .

Proposition 2.6. Let µ be a probability measure on Z satisfying hypothesis ( H). Then (σ n ) n∈N := (n(µ * n -µ * (n+1) )) n∈N satisfies to items (i), (ii) and (iii) of Proposition 2.5.

In particular, sup n∈N n µ * n -µ * (n+1) 1 (Z) < ∞, i.e. µ is Ritt.

Proof. By Proposition 2.3 we already know that (iii) holds. It follows from the proof of Proposition 2.3 and from ( 19) that there exist C, c > 0 such that for every θ ∈ (0, π],

|σ n (θ)|/θ ≤ Cne -cnψ(θ) ψ(θ)/θ ≤ CDne -cnψ(θ) ψ (θ) |σ n (θ)| ≤ Cne -cnψ(θ) ψ (θ)(nψ(θ) + 1) .
then, we conclude as in the proof of Proposition 2.3.

We now provide a sufficient condition on sequence of finite signed measure on Z to be bounded in 1 (Z), that will be needed in the sequel. Proposition 2.7. Let (σ n ) n∈N be a sequence of finite signed measures on Z such that for every n ∈ N, σn is continuously differentiable on

[-π, π] -{0}. Assume that (i) sup n∈N n π -π |σ n (θ)|dθ < ∞; (ii) sup n∈N π -π |σ n (θ)| 2 n+1 dθ < ∞. Then, sup n∈N σ n 1 (Z) < ∞. Proof. Let n ≥ 1. Let k ∈ Z. We have σ n (k) = π -π σn (θ)e -ikθ dθ , (20)
and if k = 0,

σ n (k) = π -π σ n (θ) e -ikθ ik dθ , (21)
Using [START_REF] Losert | The strong sweeping out property for convolution powers, Ergodic Theory Dynam[END_REF], we infer that 0≤|k|≤n |σ n (k)| ≤ (2n + 1) π -π |σ n (θ)|dθ. Using [START_REF] Miller | Completely monotonic functions, Integral Transform[END_REF], Cauchy-Schwarz and Parseval, we infer that

( |k|>n |σ n (k)|) 2 ≤ π -π |σ n (θ)| 2 dθ |k|>n 1 k 2 ≤ C n π -π |σ n (θ)| 2 dθ .
Then, we conclude thanks to (i) and (ii).

2.3.

Centered probability measures with a second moment. It is known, see [START_REF] Bellow | A weak-type inequality for convolution products[END_REF], that a centered and strictly aperiodic probability measure µ on Z with a second moment satisfies [START_REF] Gomilko | Bernstein functions and rates in mean ergodic theorems for operator semigroups[END_REF]. As an application of the previous subsections we add here that µ is moreover Ritt and satisfies [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF]. Indeed, we shall prove Theorem 1.3.

By Proposition 2.3 and Proposition 2.6, it suffices to prove that a centered and strictly aperiodic probability measure µ with a second moment satisfies condition ( H) for some function ψ.

We shall take ψ(θ) = θ 2 , for every θ ∈ [-π, π]. Then ψ satisfies (19) hence we just have to prove that µ satisfies (H).

Since µ has a second moment and is centered, it is twice continuously differentiable on [-π, π] and we have lim

θ→0,θ =0 (1 -Re μ(θ))/θ 2 = μ (0)/2 > 0 ,
and lim

θ→0,θ =0 Im μ(θ))/θ 2 = 0 .
It follows that item (i) of hypothesis (H) is satisfied for θ close enough to 0. Then, taking c smaller if necessary, it holds on (0, π] by strict aperiodicity. Using again that µ has a second moment and is centered we see that for every θ ∈

[-π, π], |μ (θ)| ≤ μ ∞ |θ|.
Hence items (ii) and (iii) of hypothesis (H) hold. Similarly, item (iv) holds.

Probability measures without first moment

In this section as well as in sections 4 and 5, we shall consider probability measures µ on Z such that (µ(n)) n∈N is completely monotone sequence. Let us recall some definition and facts. Definition 3.1. Let ∆ be the operator defined for every sequence (t n ) n∈N of real numbers, by (∆t n ) n∈N = (t n -t n+1 ) n∈N . We say that a sequence (t n ) n≥0 is completely monotone if for every m ≥ 0 (with the convention ∆ 0 = Id, (∆ m t n ) n≥0 is non-negative. Definition 3.2. We say that an infinitely differentiable function f : [s, +∞) → [0, +∞) is completely monotone, if for every m ≥ 0, (-1) m f (m) ≥ 0.

The following characterization of completely monotone sequences is due to Hausdorff and may be found in Widder [START_REF] Widder | The Laplace Transform[END_REF], p.108. Proposition 3.1 (Hausdorff). A sequence (µ n ) n∈N is completely monotone if and only if there exists a finite positive measure ν on [0, 1], such that µ n = 1 0 t n ν(dt for every n ∈ N. A way to generate completely monotone sequences is the following, see [START_REF] Widder | The Laplace Transform[END_REF], Theorem 11d, p. 158. Proposition 3.2. Let f be a completely monotone function. Then (f (n + 1)) n∈N is a completely monotone sequence. Definition 3.3. We say that a probability measure µ on Z is CM if it is supported on N and if there exists a finite (positive) measure ν on [0, 1], such that

(22) 1 0 ν(dt) 1 -t = 1 . 1 and (23) µ(n) = 1 0 t n ν(dt) ∀n ∈ N .
To emphasize the measure ν we shall say that µ is a CM probability measure on Z with representative measure ν.

Notice that for µ as above, µ(n) > 0 for every n ∈ N, hence µ is strictly aperiodic.

3.1. Characterization of the BAR property. We first give an equivalent formulation of the BAR property that will be more convenient in the sequel.

Definition 3.4. We say that a subset of C is a Stolz region if it is the convex hull of 1 and a circle centered at 0, with radius 0 < r < 1.

It is known that µ is strictly aperiodic and has BAR if and only if the range of μ is included in a Stolz region. If µ is strictly aperiodic, for every ε ∈ (0, π), μ([ε, 2π -ε]) is included in a disk centered at 0 with radius strictly smaller than 1. Hence, a strictly aperiodic µ has BAR if and only if [START_REF] Reinhold | Almost everywhere convergence of convolution measures[END_REF] sup

θ∈(0,2π) |Im (μ(θ))| 1 -Re (μ(θ))| < ∞ .
We shall consider the following condition on ν: there exists L > 0 such that for every x ∈ [0, 1), ( 25)

x 0 t (1 -t) 2 ν(dt) ≤ L 1 -x 1 x t 1 -t ν(dt) .
Notice that this condition implies that

1 0 ν(dt
(1-t) 2 = +∞ or, equivalently, that n∈N na n = +∞, i.e. µ does not have first moment. Proposition 3.3. Let µ be a CM probability measure on Z with representative measure ν. Then, µ has BAR if and only if there exists L > 0 such that ν satisfies [START_REF] Weber | Dynamical Systems and Processes[END_REF]. Moreover, then

(26) 1 -Re μ(θ) |θ| -→ θ→0 +∞ .
We deduce the following corollary, in the spirit of Theorem 4.1 of Dungey [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF].

Corollary 3.4. Let µ be a CM probability measure on Z with representative measure ν satisfying (25) for some L > 0. Let τ be a probability measure on Z such that there exists c > 0 such that

n∈Z n|τ (n) -aµ(n)| < ∞ .
Then, τ has BAR.

Throughout the paper we will make use of the following easy inequalities.

| sin θ| ≤ |θ| , 1 -cos θ ≤ θ 2 2 ∀θ ∈ R , | sin θ| ≥ 2|θ|/π , 1 -cos θ ≥ θ 2 4 ∀θ ∈ [-1, 1] .
Proof of Proposition 3.3. Assume first that ν satisfies [START_REF] Weber | Dynamical Systems and Processes[END_REF]. Since ν is not null, the support of µ is N and µ is strictly aperiodic.

Hence, we just have to prove that there exists K > 0, such that

(27) |Im μ(θ)| ≤ K(1 -Re μ(θ)) ∀θ ∈ [-π, π] .
We have, for every θ ∈

[-π, π], μ(θ) = 1 0 ν(dt) 1-te iθ . Notice that |1 -te iθ | 2 = 1 + t 2 -2t cos θ = (1 -t) 2 + 2t(1 -cos θ) and that Re 1 1 -t - (1 -te -iθ ) |1 -te iθ | 2 = (1 -t) 2 + 2t(1 -cos θ) -(1 -t)(1 -t cos θ) (1 -t)|1 -te iθ | 2 = t(1 -cos θ) (1 -t)|1 -te iθ | 2 .
Hence, using [START_REF] Reinhold-Larsson | Almost everywhere convergence of convolution powers in L1(X)[END_REF], we have

1 -Re μ(θ) = 1 0 t(1 -cos θ) (1 -t)((1 -t) 2 + 2t(1 -cos θ)) ν(dt) . (28) Moreover, (29) Im μ(θ) = 1 0 t sin θ |1 -te iθ | 2 ν(dt) = 1 0 t sin θ (1 -t) 2 + 2t(1 -cos θ) ν(dt) .
Since, μ is continuous and 1 -Re μ vanishes only at 0, on [-π, π], it is enough to prove [START_REF] Widder | The Laplace Transform[END_REF]

for θ ∈ [-1/2, 1/2]. Moreover, (27) is clear for θ = 0. So, let θ ∈ [-1/2, 1/2] -{0}. Let us first estimate 1 -Re μ(θ). Using that (1 -t) 2 + 2t(1 -cos θ) ≤ (1 -t) 2 + θ 2 ≤ 2 max((1 -t) 2 , θ 2 ), we obtain 1 -Re μ(θ) ≥ 1 2 1-|θ| 0 t(1 -cos θ) (1 -t) 3 ν(dt) + 1 8 1 1-|θ| t 1 -t ν(dt) (30)
Now, we estimate Im μ. We have,

1 1-|θ| t| sin θ| (1 -t) 2 + 2t(1 -cos θ) ν(dt) ≤ 1 1-|θ| tθ 2 (1 -t)((1 -t) 2 + 2t(1 -cos θ)) ν(dt) ≤ 4 1 1-|θ| t(1 -cos θ) (1 -t)((1 -t) 2 + 2t(1 -cos θ)) ν(dt) ≤ 4(1 -Re μ(θ)) .
Now, using our assumption on ν and (30), we obtain

1-|θ| 0 t| sin θ| (1 -t) 2 + 2t(1 -cos θ) ν(dt) ≤ 1-|θ| 0 t| sin θ| (1 -t) 2 ν(dt) ≤ L |θ| 1 1-|θ| t 1 -t ν(dt) ≤ 8L(1 -Re μ(θ)) .
and we see that [START_REF] Widder | The Laplace Transform[END_REF] holds.

Let us prove the converse. Assume that (27) holds.

Let S ≥ 1 be fixed for the moment.

Let θ ∈ [-1/2S, 1/2S] -{0}. Using that |1 -te iθ | 2 ≤ (1 + 1/S 2 )(1 -t) 2 , whenever 0 ≤ t ≤ 1 -S|θ|, we see that (31) 1-S|θ| 0 ν(dt) (1 -t) 2 ≤ 1 + 1/S 2 2|θ|/π |Im μ(θ)| ≤ C 1 + 1/S 2 2|θ|/π (1 -Re μ(θ)) .
Now, we see that

1 -Re μ(θ)) ≤ 1 -cos θ S|θ| 1-S|θ| 0 tν(dt) (1 -t) 2 + 1 1-S|θ| tν(dt) 1 -t .
Hence, taking S large enough and using (31), we infer that there exists D > 0 such that

1-S|θ| 0 ν(dt) (1 -t) 2 ≤ D |θ| 1 1-S|θ| tν(dt) 1 -t ,
which prove that [START_REF] Weber | Dynamical Systems and Processes[END_REF] holds

It remains to prove [START_REF] Wedrychowicz | Almost everywhere convergence of convolution powers without finite second moment[END_REF]. Using [START_REF] Weber | Dynamical Systems and Processes[END_REF], we see that

1 -Re μ(θ) |θ| ≥ 1 -cos θ 2|θ| 3 1 1-|θ| t (1 -t) ν(dt) ≥ 1 -cos θ 2|θ| 2 1-|θ| 0 t (1 -t) 2 ν(dt) -→ θ→0 +∞ , hence the result.
Proof of Corollary 3.4. By assumption and Proposition 3.3, there exists

K > 0 such that n≥1 n|τ (n) -aµ(n)| ≤ K and for every θ ∈ [-π, π], |Im (μ(θ))| ≤ K(1 -Re (μ(θ)).
Let us prove that τ is strictly aperiodic. If τ were not strictly aperiodic, there would exists ≥ 2 and 0 ≤ k ≤ -1, such that the support of τ would be contained in k + Z. In particular τ (k + 1 + m) = 0 for every m ∈ Z. Hence, m∈Z |m|µ(k + 1 + m) < ∞ and (using that (µ(n)) n≥1 is non increasing) µ must have a first moment, contradicting [START_REF] Weber | Dynamical Systems and Processes[END_REF] (see the remark after (25)).

We first prove that there exists C > 0 such that for every θ ∈ [-π, π],

(32)

Re (1 -τ (θ)) ≥ C|θ| .

Since τ is strictly aperiodic, it is enough to prove the result for small enough θ's. By Proposition 3.3, there exists δ ∈ (0, π), such that for every θ ∈ [-δ, δ], |θ| ≤ a(1 -Re (μ(θ))/2K. Then, using that 1 -cos u ≤ |u| for every u ∈ R,

|θ| ≤ (1 -Re (τ (θ))/2K + 1 2K n≥1 |τ (n) -aµ(n)|(1 -cos(nθ)) ≤ (1 -Re (τ (θ))/2K + |θ|/2 ,
and (32) follows.

Let θ ∈ [-π, π]. We have, using that | sin u| ≤ u for every u ∈ R,

|Im (τ (θ))| ≤ a|Im (μ(θ))| + n≥1 |τ (n) -aµ(n)| | sin(nθ)| ≤ aK(1 -Re (μ(θ)) + K|θ| ≤ K(1 -Re (τ (θ)) + n≥1 |τ (n) -aµ(n)| |(1 -cos(nθ)| + K|θ| ≤ K(1 -Re (τ (θ)) + 2K|θ| ≤ K(1 + 2C)(1 -Re (τ (θ)) ,
and the corollary is proved.

From a pratical point of view it is better to have a condition on (µ(n)) n∈Z . Indeed we may consider completely monotone sequences given thanks to Proposition 3.2, in which case, we do not know ν. Proof. Assume [START_REF] Weber | Dynamical Systems and Processes[END_REF]. Let n ≥ 1. We have

n k=1 kµ(k) ≤ 1-1/n 0 t (1 -t) 2 ν(dt) + n 1 1-1/n t 1 -t ν(dt) ≤ (1 + L)n 1 1-1/n t 1 -t ν(dt) .
Using that k≥n µ(k) =

1 0 t n 1-t ν(dt) and that (1 -1/n) n -→ n→∞ e -1
, we see that (33) holds.

Assume now that (33) holds.

Let A ≥ 1 be a positive integer fixed for the moment. Let n ≥ 2.

Let 1 ≤ m ≤ n -1 be an integer and let t ∈ [1 -1/m, 1 -1/(m + 1)]. Using that the sequence ((1 -1/k) k-1 ) k≥1 decreases to 1/e, we obtain that (with the convention 0 0 = 1)

An k=1 kt k ≥ t m-1 k=0 (k + 1)(1 -1/m) m-1 ≥ tm(m + 1) 2e ≥ t e(1 -t) 2 .
Hence,

1-1/n 0 t (1 -t) 2 ν(dt) ≤ e An k=1 kµ(k) ≤ eDAn k≥An µ(k) = eDAn 1 0 k≥An t k ν(dt) . Now notice that for t ∈ [0, 1 -1/n], k≥An t k-1 ≤ 1 A 2 n 2 k≥An k(k + 1)t k-1 ≤ 1 A 2 n 2 (1 -t) 3 ≤ 1 A 2 n(1 -t) 2 ,
and that for t

∈ [1 -1/n, 1], k≥An t k-1 ≤ 1/(1 -t).
Hence, taking A large enough we infer that (25) holds.

Hypothesis (H)

for CM probability measures. We shall prove that the conditions imposed in the previous subsection guarantee hypothesis (H).

Proposition 3.6. Let µ be a CM probability measure on Z satisfying (33). Then, µ satisfies hypothesis (H).

Proof. To check the conditions (i) -(iv) of hypothesis (H) with a suitable function ψ we must first estimate μ and its derivatives.

Let us first compute the derivatives of μ. Recall that for every θ ∈ [-π, π],

1 -Re μ(θ) = 1 0 t(1 -cos θ) (1 -t) (1 -t) 2 + 2t(1 -cos θ) ν(dt) = 1 2 1 0 ν(dt) 1 -t - 1 2 1 0 (1 -t) (1 -t) 2 + 2t(1 -cos θ) ν(dt) ,
and

Im μ(θ) = 1 0 t sin θ |1 -te iθ | 2 ν(dt) = 1 0 t sin θ (1 -t) 2 + 2t(1 -cos θ) ν(dt) .
Hence, for every θ ∈ [-π, π] -{0},

Re μ (θ) = -sin θ 1 0 t(1 -t) (1 -t) 2 + 2t(1 -cos θ) 2 ν(dt) , (34) Re μ (θ) = -cos θ 1 0 t(1 -t) (1 -t) 2 + 2t(1 -cos θ) 2 ν(dt) (35) +2 sin 2 θ 1 0 t(1 -t) (1 -t) 2 + 2t(1 -cos θ) 3 ν(dt) Im μ (θ) = 1 0 t cos θ (1 -t) 2 + 2t(1 -cos θ) ν(dt) (36) - 1 0 2t 2 sin 2 θ (1 -t) 2 + 2t(1 -cos θ) 2 ν(dt) , Im μ (θ) = 1 0 -t sin θ (1 -t) 2 + 2t(1 -cos θ) ν(dt) (37) - 1 0 8t 2 sin θ cos θ (1 -t) 2 + 2t(1 -cos θ) 2 ν(dt) + 1 0 4t 3 sin 3 θ (1 -t) 2 + 2t(1 -cos θ) 3 ν(dt) . Define, for θ ∈ [-π, π], (38) ψ
(θ) = 1 0 t|θ| (1 -t)(1 -t + t|θ|) ν(dt) = 1 - 1 0 ν(dt) (1 -t + t|θ|) .
Hence, for every θ ∈ (0, π],

ψ (θ) = 1 0 tν(dt) (1 -t + tθ) 2 . ( 39
)
Notice that, for every θ ∈ (0, 1/2],

θ 2 1-θ 0 tν(dt) (1 -t) 2 + 1 2 1 1-θ tν(dt) 1 -t ≤ ψ(θ) ≤ θ 1-θ 0 tν(dt) (1 -t) 2 + 1 1-θ ν(dt) 1 -t ; (40) θ 4 1-θ 0 tν(dt) (1 -t) 2 + 1 4θ 1 1-θ tν(dt) ≤ θψ (θ) ≤ θ 1-θ 0 tν(dt) (1 -t) 2 + 2 |θ| 1 1-θ ν(

dt) . (41)

Claim 1. There exists C > 0, such that for every θ ∈ [0, π],

1 -Re μ(θ) ≥ Cψ(θ) .
Proof. It suffices to prove the claim for θ ∈ (0, 1/2]. Using (30), ( 25) and (40) we have

1 -Re μ(θ) ≥ 1 1-|θ| t 8(1 -t) ν(dt) ≥ 1 8(L + 2) ψ(θ) ,
and the claim follows.

Claim 2. There exists C > 0, such that for every θ ∈ (0, π], |μ (θ)| ≤ Cψ (θ) .

Proof. Again, we only consider the case when θ ∈ (0, 1/2]. We deal separately with the real and imaginary part of µ . We have, using (34) and ( 41)

|Re μ (θ)| ≤ 2θ 1-θ 0 tν(dt) (1 -t) 3 + 2 θ 3 1 1-θ t(1 -t)ν(dt) ≤ 8ψ (θ) , Similarly, |Im μ (θ)| ≤ 1-θ 0 t (1 -t) 2 + tθ 2 (1 -t) 4 ν(dt) + 1 1 -cos θ + 2θ 2 (1 -cos θ) 2 1 1-θ tν(dt) ≤ Cψ (θ) .
Claim 3. There exists C > 0, such that for every θ ∈ (0, π], |θ μ (θ)| ≤ Cψ(θ) .

Proof. Combine Claim 2 and (41).

Claim 4. There exists C > 0, such that for every θ ∈ (0, π], |θ μ (θ)| ≤ Cψ (θ) .

Proof. We assume that θ ∈ (0, 1/2]. By ( 35) and (41), we have

|Re μ (θ)| ≤ 1-θ 0 2t (1 -t) 3 + 4tθ 2 (1 -t) 5 ν(dt) + 2 (1 -cos θ) 2 1 1-θ t 1 -t ν(dt) + 4θ 2 (1 -cos θ) 3 1 1-θ t(1 -t)ν(dt) ≤ Cψ (θ)/θ.
|μ(θ)| 2 = |Im μ(θ)| 2 + 1 -2(1 -Re μ(θ)) + (1 -Re μ(θ)) 2 1 -(1 -Re μ(θ))(2 -(C + 1)(1 -Re μ(θ)))
Since (1 -Re μ(θ)) -→ θ→0 0, using Claim 1, we infer that for θ small enough

|μ(θ)| 2 ≤ 1 -Cψ(θ) .
Hence (i) holds for, say, θ ∈ [0, η], with η small enough. Since sup θ∈[η,π] |μ(θ)| < 1, wee see that (i) holds for every θ ∈ [0, π], taking c smaller if necessary.

Corollary 3.7. Let µ be a CM probability measure on Z. Let σ be a probability measure on Z such that σ is twice continuously differentiable on [-π, π] -{0} and such that σ and θ → θσ (θ) are bounded. Then, σ * µ satisfies hypothesis (H). Moreover, if µ satisfies hypothesis ( H), so does σ * µ.

Remark. The assumptions on ν holds, for instance, as soon as n∈Z n 2 σ(n) < ∞. Proof. Let ψ be the function defined in (38). Since

1 0 ν(dt)
(1-t) 2 = +∞, one easily infers from (39) that lim inf θ→0,θ>0 ψ (θ) = +∞. In particular, there exists K > 0 such that for every θ ∈ (0, π], ψ (θ) ≥ K and, consequently, ψ(θ) ≥ Kθ. Then, the fact that σ * µ satisfies hypothesis (H), with the same function ψ as µ, may be proved exactly as Proposition 2.4. Since we use the same function ψ for σ * µ and µ, then σ * µ satisfies hypothesis ( H) as soon as µ does.

Corollary 3.8. Let τ be a probability measure on Z. Assume that there exists a CM probability measure µ and c > 0 such that n∈Z n 2 |τ (n) -cµ(n)| < ∞. Then, τ satisfies hypothesis (H). If moreover µ satisfies hypothesis ( H), so does τ .

Remark. It follows from the proof that we only need that σ be twice continuously differentiable on [-π, π] -{0} and that τ and θ → θτ (θ) be bounded. Proof. Define a signed measure by setting σ := τ -cµ. Then, σ is twice continuously differentiable on [-π, π], σ(0) = 1 -c and there exists C > 0 such that for every θ ∈

[0, π], |σ(θ) -(1 -c)| ≤ Cθ.
Then, the proof may be finished using the same arguments as in the proof of Corollary 3.7.

3.3.

The Ritt property on 1 (Z). In this section, we finish the proof of Theorem 1.4. We first prove the Ritt property of CM probability measures, which corresponds to the case where σ = δ 0 .

Let µ be a probability measure on Z. Notice that the fact µ is Ritt is equivalent to the fact that sup

n≥1 n π n µ -π n+1 µ 1 (Z) < ∞ ,
where π µ stands for the operator of convolution by µ.

Let Γ be the open unit disk in the complex plane. By Theorem 1.5 of Dungey, µ is Ritt if and only if the spectrum σ(π µ ) of π µ is contained in Γ ∪ {1} and the semi-group (e -t(I-πµ) ) t≥0 is bounded analytic. The fact that (e -t(I-πµ) ) t≥0 is bounded analytic means that sup t>0 e -t(δ 0 -µ)

1 (Z) + t (I -T )e -t(δ 0 -πµ) 1 (Z) < ∞ .
Remark. Notice that Theorem 1.5 of Dungey is valid for probabilities supported on N.

Proposition 3.9. Let µ be a CM probability measure on Z with representative measure ν satisfying [START_REF] Weber | Dynamical Systems and Processes[END_REF]. Then, µ is Ritt.

We already saw that ν satisfies [START_REF] Weber | Dynamical Systems and Processes[END_REF] if and only if µ has BAR. The fact that a CM probability measure on Z having BAR is Ritt has been proved very recently (see their Theorem 7.1) by Gomilko and Tomilov [START_REF] Gomilko | On discrete subordination of power bounded and Ritt operators[END_REF] as a consequence of another very recent result of their own [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF].

The latter paper deals with subordination semi-groups hence is written in a continuous setting.

For reader's convenience we explain below how to derive Proposition 3.9 from the work [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF].

First of all, by Theorem 2.1 of Dungey [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF], we have σ(π µ ) ⊂ μ([-π, π]) ⊂ Γ ∪ {1}, where the latter inclusion follows from the fact that µ has BAR. Hence, Proposition 3.9 will be proved if we can prove that (e -t(I-πµ) ) t≥0 is bounded analytic. Definition 3.5. An infinitely differentiable function f : (0, +∞) → [0, +∞) is called a Bernstein function if f is completely monotone. If lim x→0 + f (x) exists and if f admits an holomorphic extension to {z ∈ C : Im z > 0}, such that Im f (z) ≥ 0, then f is called complete Bernstein.

For every x ≥ 0, define χ(x) := 1 -

1 0 ν(dt) 1-t+tx = 1 0 ν(dt) 1-t - 1 0 ν(dt)
1-t+tx . Then χ is nondecreasing, with χ(0) = 0, hence it is non-negative. It is not hard to see that it is infinitely differentiable and that χ is completely monotone, hence χ is a Bernstein function and one can easily see that it is actually a complete Bernstein function.

Since χ is Bernstein, it is well known (see e.g. Theorem 1.2.4 of [START_REF] Farkas | Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces[END_REF]) that there exists a convolution semi-group (σ t ) t≥0 (of probability measures on [0, ∞)), such that for every x ≥ 0, and every t ≥ 0, ∞ 0 e -xy σ t (dy) = e -tχ(x) .

Following Dungey [10, p. 1734], we consider the Poisson semi-group (P s ) s≥0 acting by convolution on 1 (N), and defined by

P s := e -s(δ 0 -δ 1 ) = e -s k≥0 s k k! δ k ∀s ≥ 0 .
Consider now the associated subordinated semi-group (Q t ) s≥0 defined by

Q t := ∞ 0 P s σ t (ds) ∀t ≥ 0 .
Let t ≥ 0. Then, Q t is a probability measure on N, whose generating function is given (on [0, 1]) by (1-x) .

x → ∞ 0 e -s(1-x) σ t (ds) = e -tχ
Let G µ denote the generating function of µ, i.e.

G µ (x) = n≥0 µ(n)x n = 1 0 ν(dt 1 -tx = 1 -χ(1 -x) ,
for every x ∈ [0, 1]. Then, for every t ≥ 0, the generating function of the probability e -t(I-µ) = e -t k≥0

t k µ * k k! is given by e -t k≥0 t k G k µ k! = e -t(1-Gµ) .
In particular, we see that the semi-groups (e -t(I-πµ) ) t≥0 and (Q t ) t≥0 coïncide. Hence, to prove that (e -t(I-πµ) ) t≥0 is bounded analytic, it is enough to prove that any subordinated semi-group associated with (σ t ) t≥0 is bounded analytic (see the introduction of [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF] for more details). To prove the latter point, since χ is complete Bernstein, by Corollary 7.10 of [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF], it is enough to prove that χ sends the half-plane {z ∈ C : Re z ≥ 0} to a sector {z ∈ C : |Im z| ≤ CRe z}, for some C > 0.

Let z = a + ib such that a ≥ 0 and |z| 2 = a 2 + b 2 ≤ 1/4. We have, using [START_REF] Weber | Dynamical Systems and Processes[END_REF]

|Im χ(z)| = |b| 1 0 t (1 -t + at) 2 + t 2 b 2 ν(dt) ≤ |b| 1-|z| 0 t (1 -t) 2 ν(dt) + 4b |z| 2 1 1-|z| tν(dt) ≤ K 1 1-|z| t 1 -t ν(dt) .
On the other hand,

Re χ(z) = 1 0 at(1 -t) + |z| 2 t 2 (1 -t) (1 -t + at) 2 + t 2 b 2 ν(dt) ≥ 1 1-|z| |z| 2 t 2 5|z| 2 (1 -t) ν(dt) ≥ 1 10 1 1-|z| t 1 -t ν(dt) .
This gives the desired bound when |z| 2 ≤ 1/4. Assume now that |z| 2 ≥ 1/4. In particular, we have 4|z| ≥ 2. Hence,

|Im χ(z)| ≤ (4|z|) -1 0 t|z| (1 -t) 2 ν(dt) + |z| |z| 2 1 (4|z|) -1 t -1 ν(dt) ≤ 1 4 1/2 0 ν(dt) (1 -t) 2 + 4 1 0 ν(dt) < ∞ .
Moreover, using that the integrand below is non decreasing with respect to |z|, we have

Re χ(z) ≥ 1 0 |z| 2 t 2 2(1 -t) (1 -t) 2 + t 2 |z| 2 ν(dt) ≥ 1 8 1 0 t 2 (1 -t) 2 + t 2 /4 ν(dt) > 0 ,
which finishes the proof.

Proposition 3.10. Let µ be a CM probability measure on Z with representative measure ν satisfying [START_REF] Weber | Dynamical Systems and Processes[END_REF]. Let σ be a probability measure on Z such that σ is continuously differentiable on [-π, π] -{0} and such that σ is bounded on

[-π, π] -{0}. Then, (42) sup n∈N
n (δ 0 -σ) * µ * n 1 < ∞ .
In particular, σ * µ is Ritt and for every α ∈ (0, 1], αµ + (1 -α)σ is Ritt.

Proof. To prove (42), we check that (σ n ) n∈N := (n(δ 0 -σ) * µ * n ) n∈N satisfies items (i) and (ii) of Proposition 2.7.

By assumption there exists L > 0 such that |σ | ≤ L and it follows that

|1-σ(θ)| ≤ L|θ| for every θ ∈ [-π, π].
Let ψ be the function given in (38). Recall that there exists K > 0 such that for every θ ∈ [-π, π] -{0}, ψ(θ) ≥ Kθ and ψ (θ) ≥ K. Hence, for every n ∈ N and every

θ ∈ [-π, π] -{0}, n|σ n (θ)| ≤ Ln 2 K 2 ψ(θ)ψ (θ)(μ(θ)) n .
Hence, arguing as in the proof of Proposition 2.6, we see that (σ n ) n∈N satisfies item (i) in Proposition 2.7.

For every θ ∈ [-π, π] -{0}, we have

σ n (θ) = -nσ (θ)μ n (θ) + n 2 (1 -σ(θ))μ (θ)μ n-1 (θ) .
Then, we infer that

|σ n (θ)| 2 ≤ 2n 2 L 2 K ψ (θ)|μ 2n |(θ) + 2n 4 L 2 K 3 ψ 2 (θ)ψ (θ)|μ n-1 |(θ)
, Hence, arguing as in the proof of Proposition 2.6, we see that (σ n ) n∈N satisfies item (ii) in Proposition 2.7.

It remains to prove the second part of the Proposition. Let n ≥ 1. We have

(δ 0 -σ * µ) * (σ * µ) * n = σ * n * (δ 0 -µ) * µ * n + (δ 0 -σ) * µ * (n+1) * σ * n , which proves that σ * µ is Ritt.
Let α ∈ (0, 1] and n ≥ 1, and τ := αµ + (1 -α)σ. We have

(δ 0 -τ ) * τ * n = n k=0 n k α k (1 -α) n-k k + 1 α(k + 1)(δ 0 -µ) * µ * k + (1 -α)(k + 1)(δ 0 -σ) * µ * k * σ * (n-k) .
Hence,

(n + 1) (δ 0 -τ ) * τ * n 1 ≤ C α n k=0 n + 1 k + 1 α k+1 (1 -α) (n+1)-(k+1) ≤ C ,
and we see that τ is Ritt.

Examples.

To exhibit examples we will make use of Proposition 3.2. Hence we shall first exhibit completely monotone functions.

Lemma 3.11 ). Let f, g : (0, +∞) → (0, +∞) be infinitely differentiable functions functions such that g is completely monotone. (i) If f is completely monotone then f • g is completely monotone either;

(ii) If f is completely monotone then (f • g) is completely monotone either.

Proof. Item (i) is just Theorem 2 of [START_REF] Miller | Completely monotonic functions, Integral Transform[END_REF]. Let us prove item (ii). We have (f • g) = f • g × g . By (i), f • g is completely monotone. Then, (f • g) is completely monotone by Theorem 1 of [START_REF] Miller | Completely monotonic functions, Integral Transform[END_REF].

Define by induction L 1 (x) = L(x) := log(1 + x) and L k+1 (x) = L(L k (x)) for every x > 0.

Corollary 3.12. For every integer k ≥ 1 and every real numbers α 1 , . . . , α k ∈ [0, +∞) and α ∈ [0, +∞) the function given by

f α,α 1 ,...,α k (x) = 1 x α L 1 (x) α 1 . . . L α k k (x)
∀x ≥ 0 , is completely monotone.

Proof. Obviously, x → x -α is completely monotone. By (ii) of the previous lemma L k admits a completely monotone derivative and then L -α k k is completely monotone by (i). The fact that f α,α 1 ,...,α k is also completely monotone then follows from Theorem 1 of [START_REF] Miller | Completely monotonic functions, Integral Transform[END_REF].

Example 1. Let µ be a probability measure supported on N such that µ(n) = cf α,α 1 ,...,α k (n+ 1) for every n ∈ N, where α 1 , . . . , α k ∈ [0, +∞), α ∈ (1, 2) and c is a normalizing constant ensuring that we have a probability. Then, µ ∈ H ∩ R Of course one may take α = 1 and α 1 > 1, and so on... But for α = 2, µ does not even have BAR.

It is more difficult to produce examples allowing negative α k s. One way to handle the difficulty is to proceed as in the proof of Proposition 5.11 of [START_REF] Cohen | On the convergence of power series of L p -contractions[END_REF].

Example 2. Our next example is a basic example of Ritt probability measures already considered by Dungey [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF] and Gomilko and Tomilov [START_REF] Gomilko | On discrete subordination of power bounded and Ritt operators[END_REF]. Let γ ∈ (0, 1). We have a power series expansion 1 -(1 -t) γ = n≥1 a n (γ)t n , 0 ≤ t ≤ 1. Notice that n≥1 a n (γ) = 1 and a n (γ) ≥ 0 for every n ≥ 1. Define two probability measures τ and µ by setting for every n ∈ N, µ(n) = a n+1 (γ) = τ (n + 1). so that τ = δ 1 * µ. Then, see for instance example 3.10a of [START_REF] Gomilko | On discrete subordination of power bounded and Ritt operators[END_REF], τ is a CM probability measure which has BAR. In particular, τ ∈ H ∩ R and µ ∈ H ∩ R.

Probability measures with a first moment

When µ has a first moment, a necessary condition for the BAR property is that µ be centered, i.e. n∈Z nµ(n) = 0, see Proposition 1.9 of [START_REF] Bellow | Almost everywhere convergence of convolution powers, Ergodic Theory Dynam[END_REF].

Hence we cannot consider probability measures µ supported by N anymore. We shall consider the following situation. Definition 4.1. We say that a probability measure µ on Z is CCM if it is supported on {-1} ∪ N and if there exists a finite positive measure ν on [0, 1], such that

1 0 ν(dt) (1 -t) 2 = 1 . and µ(n) := 1 0 t n ν(dt) ∀n ∈ N; µ(-1) = 1 - 1 0 ν(dt) 1 -t = 1 0 tν(dt) (1 -t) 2 .
It is not hard to see that µ is indeed a probability measure and that it is centered.

4.1. Characterization of the BAR property. Let µ be a CCM probability measure on Z with representative measure ν.

For every θ ∈ [-π, π], we have

μ(θ) = 1 0 1 -2t + 2t 2 e -iθ -t 2 e -2iθ (1 -t) 2 ((1 -t) 2 + 2t(1 -cos θ)) ν(dt) .
In particular, (43) 1 -Re μ(θ) = (1 -cos θ)

1 0 2t(1 -t cos θ) (1 -t) 2 ((1 -t) 2 + 2t(1 -cos θ)) ν(dt) ,

and

(44) Im μ(θ) = 2 sin θ (1 -cos θ)

1 0 t 2 (1 -t) 2 ((1 -t) 2 + 2t(1 -cos θ)) ν(dt)
Consider the following condition on ν: there exists L > 0, such that for every x ∈ [0, 1),

(45) 1 1 -x 1 x tν(dt) (1 -t) 2 ≤ L x 0 tν(dt) (1 -t) 3 . Notice that if 1 0 ν(dt)
(1-t) 3 < ∞ (i.e. µ has a moment of order 2), condition (45) is automatically satisfied.

Proposition 4.1. Let µ be a CCM probability measure on Z with representative measure ν. Then, µ has BAR if and only if there exists L > 0 such that ν satisfies (45).

Proof. Assume (45). Let us prove that µ satisfies [START_REF] Reinhold | Almost everywhere convergence of convolution measures[END_REF]. As noticed previously, it is enough to consider θ ∈ [-1/2, 1/2]. We have

|Im μ(θ)| ≤ C |θ| 3 1-|θ| 0 t (1 -t) 4 ν(dt) + |θ| 1 1-|θ| tν(dt) (1 -t) 2 .
Using that 1 -t cos θ ≥ 1 -t, we see that Let θ ∈ [-1/2, 1/2] and α ∈ (0, 1]. We have

1 -Re μ(θ) ≥ Cθ 2 1-|θ| 0 tν(dt) (1 -t) 3 ,
|Im μ(θ)| ≥ |θ| 4(1 + α 2 ) 1 1-α|θ| tν(dt) (1 -t) 2 .
It is not hard to prove that there exists C α , D > 0 such that

1 -t cos θ ≤ C α (1 -t) ∀t ∈ [0, 1 -α|θ|] ; 1 -t cos θ ≤ Dα|θ| ∀t ∈ (1 -α|θ|, 1] .
Hence, using (46), we infer that

|θ| 4(1 + α 2 ) 1 1-α|θ| tν(dt) (1 -t) 2 ≤ C C α |θ| 2 1-|θ| 0 tν(dt) (1 -t) 3 + α|θ| 1 1-α|θ| tν(dt) (1 -t) 2 .
Taking α = 1/(8C) gives the desired result.

As before, we shall now characterize the BAR property in terms of the coefficients of µ. Proposition 4.2. Let µ be a CCM probability measure on Z with representative measure ν. Then, ν satisfies (45) if and only if there exists L > 0 such that

(47) n k≥n kµ(k) ≤ L n k=1 k 2 µ(k) ∀n ∈ N .
Proof. Assume (45). Let n ≥ 2, we have 3 , where we used that (1 -1/m) m-1 decreases to e -1 . Hence, (47) holds.

n k≥n kµ(k) ≤ 1-1/n 0 k≥n k 2 t k ν(dt) + n 1 1-1/n tν(dt) (1 -t) 2 ≤ (1 + L) 1-1/n 0 tν(dt) (1 -t) 3 . Now, for every 1 ≤ ≤ n -1 and every t ∈ [1 -1/ , 1 -1/( + 1)], we have n k=1 k 2 t k ≥ t k=1 k 2 e -1 ≥ Ct/(1 -t)
Assume that (47) holds. Let γ ∈ (0, 1] and n ≥ 2. For every t ∈

[1 -1/n, 1], since γ ≤ 1, we have k≥γn kt k = k≥0 (k + n)t k+n ≥ t(1 -1/n) n (1 -t) 2 ≥ t(2e) -1 (1 -t) 2 .
Hence

n 1 1-1/n tν(dt) (1 -t) 2 ≤ 2en k≥γn kµ(k) ≤ 2nLe [γn] [γn] k=1 k 2 µ(k) ≤ 2nLe [γn] 1-1/n 0 tν(dt) (1 -t) 3 + [γn] 2 1 1-1/n tν(dt) 1 -t ,
and we conclude by taking γ small enough Theorem 4.3. Let µ be a CCM probability measure on Z with representative measure ν.

Assume that (µ(n)) n∈N satisfies (47). Then, µ is Ritt and for every m ∈ N, there exists

C m > 0 such that for every f ∈ 1 (Z), sup λ>0 λ#{k ∈ Z : sup n≥1 n m |µ * n * (δ 0 -µ) * m * f (k)| ≥ λ} ≤ C m f 1 .
Proof. It suffices to check that µ satisfies hypothesis (H) and to apply Propositions 2.3 and 2.5.

To check the conditions we must estimate μ and its derivatives. Define (48)

ψ(θ) = θ 2 1 0 tν(dt) (1 -t)((1 -t) 2 + θ 2 ) = tν(dt) 1 -t - 1 0 t(1 -t)ν(dt) (1 -t) 2 + θ 2 . Im μ (θ) = -sin θ 1 0 t (1 -t) 2 ν(dt) + sin θ 1 0 t(1 + t 2 ) ((1 -t) 2 + 2t(1 -cos θ)) 2 ν(dt) (54) -4 sin θ 1 0 t(2t 2 -t(1 + t 2 ) cos θ) ((1 -t) 2 + 2t(1 -cos θ)) 3 ν(dt)
We now derive the necessary estimates on μ and its derivatives. Using (43), we infer that Claim 5. There exists

C > 0 such that 1 -Re μ(θ) ≥ Cθ 2 1-θ 0 t (1-t) 3 ν(dt), for every θ ∈ (0, 1/2].
Using (51), we infer that Claim 6. There exists Notice that there exists α > 0 such that for every t ∈ [0, 1] and every θ ∈ (0, 1/2],

C > 0 such that |Re μ (θ)| ≤ Cθ 1-θ 0 t (1-t) 3 ν(dt) + C θ 2 1 1-θ tν(dt), for every θ ∈ (0, 1/2].

Using (53), we infer that

Claim 7. There exists C > 0 such that |Im μ (θ)| ≤ Cθ 2 1-θ 0 t (1-t) 4 ν(dt)+C 1 1-θ t (1-t) 2 ν(dt), for every θ ∈ (0, 1/2].

Using (52), we infer that

|2t 2 -t(1 + t 2 ) cos θ)| = t|(1 + t 2 )(1 -cos θ) -(1 -t) 2 | ≤ α max(θ 2 , (1 -t) 2 ) .
Combining this estimate with (54), we infer that Claim 9. There exists

C > 0 such that |Im μ (θ)| ≤ Cθ 1-θ 0 t (1-t) 4 ν(dt) + C θ 3 1 1-θ tν(dt), for every θ ∈ (0, 1/2].
We already saw that [START_REF] Losert | A remark on almost everywhere convergence of convolution powers[END_REF] ψ (θ)/θ = +∞ (θ → 0) .

It follows from the proof of Theorem 4.3 that there exists an even function ψ continuous on [-π, π] and continuously differentiable on (0, π], with ψ(0) = 0 such that µ and ψ satisfy the item (i) -(iv) of hypothesis (H), for some C, c > 0.

Since τ = (τ -aν) + aμ is clearly twice differentiable on (0, π], the proposition will be proved if we can show that the items (i) -(iv) of hypothesis (H) hold with τ in place of µ with the same ψ, but for possibly different C, c > 0.

We already saw that τ must be strictly aperiodic. Hence |μ| < 1 on (0, π]. In particular, to prove item (i) it suffices to consider θ ∈ (0, η] for some small enough η > 0.

For every θ ∈ (0, π], we have Hence, using Claim 5, we infer that there exists η > 0 such that for every θ ∈ (0, η], |τ (θ)| ≤ 1 -δψ(θ), for some δ > 0.

τ (θ) = n∈Z (τ (n) -aµ(n))(e inθ -
The proofs of item (ii) -(iii) are similar (but simpler) hence we leave them to the reader.

Example 3. Let α ∈ (2, +∞) and α 1 , . . . , α k ≥ 0. Let µ be a probability on Z such that n∈Z |n|µ(n) < ∞, n∈Z nµ(n) = 0 and n∈Z n 2 |µ(n) -af α,α 1 ,...,α k (n + 1)| < ∞, for some a > 0, where we extended f α,α 1 ,...,α k to Z -by setting, f α,α 1 ,...,α k (-n) = 0 for every n ∈ N.

Symmetric probability measures

In this section we consider symmetric probability measures. If µ is symmetric (i.e. μ = µ), then μ is real valued, hence has BAR. It is known that if moreover (µ(n)) n∈N is non-increasing then (13) holds. However we are not aware of any result concerning the Ritt property or [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF] with m ≥ 1.

We shall again investigate the situation where we have completely monotone coefficients. To be more precise we consider the following situation. Definition 5.1. We say that a probability measure µ on Z is SCM if it is symmetric and if there exists a finite positive measure on [0, 1] such that Then µ 1 is a probability measure, (µ 1 (n)) n∈N is completely monotone and µ = 1 2 (μ 1 + µ 1 ). In particular, it follows from Proposition 2.4 and Theorem 3.6, that µ satisfies hypothesis (H) as soon as µ satisfies (33). The fact that µ is Ritt when it satisfies (33) may be proved similarly (but more easily).

We could use a similar argument based on Theorem 4.3. However, doing so, we would miss some symmetric probability measures satisfying hypothesis (H).

Let us explain how to be more precise. Let µ be a SCM probability measure. It follows from previous computations that, for every θ ∈ R, 1 -μ(θ) = 1 -Re μ(θ) = Consider the following condition on ν: there exists L > 0 such that for every x ∈ [0, 1), (56)

1 x t 1 -t ν(dt) ≤ L(1 -x) 2
x 0 t (1 -t) 3 ν(dt) . This condition can be proved to be equivalent to the following one: there exists D > 0 such that for every n ≥ 1, (57)

n 2 k≥n µ(k) ≤ L n k=1 k 2 µ(k) .
One can prove that if (56) holds, then µ satisfies hypothesis ( H) with ψ given by

ψ(θ) = θ 2 1 0 t (1 -t)(1 -t + |θ|) 2 ν(dt)
∀θ ∈ [-π, π] -{0} .

Notice that ψ (θ) = 2θ 1 0 t (1-t+θ) 3 ν(dt), for every θ ∈ (0, π]. Then, one can prove that a SCM probability measure satisfying (57) is Ritt and satisfies [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF] for every m ∈ N and some C m > 0.

In particular, we have the following. Theorem 5.1. Let µ be a SCM probability measure such that (µ(n)) n∈N satisfies either (33) or (57). Then, µ is Ritt and satisfies [START_REF] Gomilko | On subordination of holomorphic semigroups[END_REF] for every m ∈ N and some C m > 0.

Example 4. Let α > 1 and α 1 , . . . , α k ≥ 0. Let µ be a symmetric probability measure defined by µ(0) = 2cf α,α 1 ,...,α k (1) and for every n ≥ 1 µ(n) = cf α,α 1 ,...,α k (n + 1), where c is a normalizing sequence ensuring that µ is a probability. Then, µ is a SCM probability measure for which the above theorem apply.

Discussion and open questions

-Most of the examples of (strictly aperiodic) probability measures on Z that have BAR are known to be Ritt. We do not believe that the BAR property and the Ritt property are equivalent, but one has to find a counterexample. This problem was also formulated by Dungey [START_REF] Dungey | Subordinated discrete semigroups of operators[END_REF] (see his remarks page 1729).

-One may wonder whether, in the symmetric case, the condition "(µ(n)) n∈N is nonincreasing" is sufficient for the Ritt property or for weak type maximal inequalities (5), since it is sufficient for the weak type maximal inequality (4). At least, for a SCM probability measure on Z, can one remove the conditions (33) and ( 57 . Assume that µ has BAR. When p > 1, it follows from the work of Le Merdy and Xu [START_REF] Merdy | Maximal theorems and square functions for analytic operators on Lpspaces[END_REF] that there exists C p > 0 such that for every f ∈ p (Z), s µ (f ) p ≤ C p f p , i.e. s(f ) satifies a strong p -p inequality. A natural question is whether s(f ) satisfies a weak 1 -1 inequality.

1

  All along the paper (for esthetical reasons) we shall adopt the convention b a ϕdν = [a,b] ϕdν. Hence, for non-negative ϕ we will have c a ϕdν ≤ b a ϕdν + c b ϕdν with equality if ν({b}) = 0

Proposition 3 . 5 .

 35 Let µ be a CM probability measure on Z with representative measure ν. Then, ν satisfies (25) if and only if there exists D > 0, such that for every n ≥ 1,

1 -

 1 Similar computations based on (34) and (41) yields |Im μ (θ)| ≤ Cψ (θ)/θ . Then, items ii), (iii) and (iv) of Proposition 2.3 follows from the combination of the claims 2, 3 and 4. Let us prove item (i) of Proposition 2.3. Let θ ∈ [0, 1/2]. Recall that |Im μ(θ)| ≤ C(Re μ(θ)). Hence,

and ( 24 )

 24 holds, by (45). Let us prove that if µ has BAR, then (45) holds. There exists C > 0 such that for every θ ∈ [-1/2, 1/2], (46) |Im μ(θ)| ≤ C(1 -Re μ(θ)) .

Claim 8 .

 8 There exists C > 0 such that |Re μ (θ)| ≤ C t(1t)ν(dt, for every θ ∈ (0, 1/2].

  1) + 1 -a + a n∈Z µ(n)e inθ := χ(θ) + φ(θ) . Using that n∈Z n 2 |τ (n) -aµ(n)| < ∞ and that n∈Z n(τ (n) -aµ(n)) = 0, we see that lim θ→0,θ =0 χ(θ)/θ 2 = χ (0) exists. In particular, lim θ→0,θ =0 χ(θ)/ψ(θ) = 0. Now,we have |φ(θ)| 2 = (1-a+aRe μ(θ)) 2 +a 2 (Im μ(θ)) 2 = 1-2a(1-Re μ(θ))+a 2 (1-Re μ(θ)) 2 +a 2 (Im μ(θ)) 2 .

1 . 1 (t

 11 Let µ be an SCM probability measure on Z with repesentative measure ν. Define another measure on Z, supported on N, by setting µ n ν(dt) ∀n ≥ 1 .

1 0t( 1 -

 11 cos θ) (1 -t)((1 -t) 2 + 2t(1 -cos θ))ν(dt) .

  ) from Theorem 5.1 ? -Let µ be a probability measure on Z. Let f ∈ p (Z), p ≥ 1. Consider the square function defined by s µ (f )(k) := n≥1 n (µ * n -µ * (n+1) ) * f (k)

  holds. Let us prove that items (i) -(iv) of Proposition 2.Let τ be a centered probability measure on Z such that n∈Z |n|τ (n) < ∞. Assume moreover that there exists a CCM probability measure µ satisfying (47) and that there exists a > 0 such that n∈Z n 2 |τ (n) -aµ(n)| < ∞. Then the conclusion of Theorem 4.3 holds for τ .Proof. We shall assume that n∈Z n 2 τ (n) = ∞, otherwise, the result holds by Theorem 1.3. In particular we must have n∈Z n 2 µ(n) = ∞ and by (48) and (49)

	hold. Item (i) follows from Claim 5 and (45) (see the proof of Theorem 3.6. Item (ii) follows from Claims 6 and 7 combined with (45) and (50). Item (iii) follows from item (ii) combined with (19). Item (iv) follows from Claims 8 and 9 combined with (45) and (50). lim inf ψ(θ)/θ 2 = +∞ and lim inf Proposition 4.4. (55) θ∈(0,π] θ∈(0,π]	3

Then, (49) ψ (θ) = 2θ

Hence for every θ ∈ [0, 1/2], we have

In particular, using (45), we see that [START_REF] Losert | A remark on almost everywhere convergence of convolution powers[END_REF] holds. Let us compute the derivatives of μ. We shall not give the full details here. Using (43), we infer that

Using (44), we infer that Im μ(θ) = sin θ