Paul Doukhan
email: paul.doukhan@ensae.fr

Michael H Neumann
email: mi.neumann@tu-bs.de

Probability and moment inequalities for sums of weakly dependent random variables, with applications

Keywords: Bernstein inequality, cumulants, Rosenthal inequality, weak dependence 1991 MSC: 60E15, 62E99

introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random variables. We employ such cumulant estimates to derive inequalities of Bernstein and Rosenthal type which both improve on previous results. Furthermore, we consider several classes of processes and show that they fulfill appropriate weak dependence conditions. We also sketch applications of our inequalities in probability and statistics.

Introduction

For a long time mixing conditions have been the dominating type of conditions for imposing a restriction on the dependence between time series data. They are considered to be useful since they are fulfilled for many classes of processes and since they allow to derive tools similar to those in the independent case. On the other hand, it turns out that certain classes of processes which are of interest in statistics are not mixing although a successive forgetting of past states takes place. Typical examples are processes driven by discrete observations as they appear, for example, with model-based time series bootstrap methods. In 1999, Doukhan and Louhichi proposed a new concept of restricting dependence which focuses on covariances rather than the total variation distance between joint distributions and the product of corresponding marginals. It has been shown that this concept is more general than mixing and includes, under natural conditions on the process parameters, essentially all classes of processes of interest in statistics; see for example Ango Nze, Bühlmann and [START_REF] Ango Nze | Nonparametric regression estimation under weak dependence beyond mixing and association[END_REF] for an overview. It became readily apparent that the concept of weak dependence allows in many instances similar tools to be used as in the independent or mixing case. For example, versions of a central limit theorem are derived in [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] for sequences of random variables, in Coulon-Prieur and [START_REF] Coulon-Prieur | A triangular CLT for weakly dependent sequences[END_REF] for situations as they appear with nonparametric curve estimators, and in [START_REF] Neumann | Goodness-of-fit tests for Markovian time series models[END_REF] for general triangular schemes. A first exponential inequality was obtained in [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF], a Bennett inequality in [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF], and a Bernstein-type inequality in [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF].

The concept of weak dependence is particularly suitable for deriving upper estimates for the cumulants of sums of random variables. Such cumulant estimates can serve as a starting point for deriving rather precise approximations of distributions as well as rather tight probability inequalities. The main contributions in this paper are inequalities of Bernstein and Rosenthal type for sums of weakly dependent random variables. In the case of mixing, Bernstein-type inequalities can be easily derived from the well-known Bernstein inequality in the independent case by using coupling arguments; see for example [START_REF] Doukhan | Mixing: Properties and Examples[END_REF] and [START_REF] Rio | Théorie asymptotique pour des processus aléatoires faiblement dépendants[END_REF]. In the case of weakly dependent random variables, [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] proved a first exponential inequality via a combinatorial technique. Unfortunately, rather than the rate of t 2 in the exponent as in the independent case, only a rate of √ t was obtained by this approach. Using a new coupling result, [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] proved a Bennett inequality which possibly implies a Bernstein-type inequality with t 2 in the exponent. Using cumulant techniques [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF] derived a Bernstein-type inequality with a leading term of -t 2 /(2 var(X 1 + • • • + X n )) in the exponent, under weak dependence conditions tailor-made for causal processes and with an exponential decay of the coefficients of weak dependence. In this paper, we extend this result to more general conditions of weak dependence, including also noncausal processes and allowing a subexponential decay of the weak dependence coefficients. In Section 4, we discuss several statistical applications of this result. It turns out that certain purposes such as a law of iterated logarithm and a precise asymptotics for nonparametric curve estimators do actually require an exponential inequality with a tight leading term in the exponent.

A second major result is a Rosenthal-type inequality which in particular improves a previous inequality given in [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF]. Using again cumulant techniques we derive such an inequality with an asymptotically dominating term equal to p!/(2 p/2 (p/2)!)(var(X 1 + • • • + X n )) p/2 . Such an inequality allows for example to derive a central limit theorem via the method of moments; see again Section 4.

We present the main results, a Bernstein-type and a Rosenthal-type inequality in the next section. In Section 3, [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] concept of weak dependence is recalled and it is shown that the particular conditions used for our inequalities do actually follow from usual conditions of weak dependence in many instances. Section 4 contains typical applications of our probability inequalities in probability and statistics. A long list of examples of processes satisfying weak dependence conditions is presented in Section 5. Finally, the proofs of the main theorems and of some auxiliary results of general interest are given in Section 6.

A Bernstein-type and a Rosenthal-type inequality for sums of weakly dependent random variables

In this section we will be concerned with probability and moment inequalities for

S n = X 1 + • • • + X n ,
where X 1 , . . . , X n are zero mean random variables which fulfill appropriate weak dependence conditions. Throughout the paper, we denote by σ 2 n the variance of S n .

Our first result is a Bernstein-type inequality which generalizes and improves previous inequalities of [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] and [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF].

Theorem 1 Suppose that X 1 , . . . , X n are real-valued random variables defined on a probability space (Ω, A, P) with zero mean. Let Ψ : N 2 → N be one of the following functions:

(a) Ψ(u, v) = 2v, (b) Ψ(u, v) = u + v, (c)
Ψ(u, v) = uv, (d) Ψ(u, v) = α(u + v) + (1 -α)uv,
for some α ∈ (0, 1).

We assume that there exist constants K, M, L 1 , L 2 < ∞, µ, ν ≥ 0, and a nonincreasing sequence of real coefficients (ρ(n)) n≥0 such that, for all u-tuples (s 1 , . . . , s u ) and all v-tuples (t 1 , . . . , t

v ) with 1 ≤ s 1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v ≤ n the following inequalities are fulfilled: |cov (X s 1 • • • X su , X t 1 • • • X tv )| ≤ K 2 M u+v-2 Ψ(u, v) ρ(t 1 -s u ), (1)
where

∞ s=0 (s + 1) k ρ(s) ≤ L 1 L k 2 (k!) µ ∀k ≥ 0, (2)
and

E|X t | k ≤ (k!) ν M k ∀k ≥ 0. (3)
Then, for all t ≥ 0,

P (S n ≥ t) ≤ exp - t 2 /2 A n + B 1/(µ+ν+2) n t (2µ+2ν+3)/(µ+ν+2) , (4)
where A n can be chosen as any number greater than or equal to σ 2 n and

B n = 2 (K ∨ M ) L 2 2 4+µ+ν n K 2 L 1 A n ∨ 1 .
Remark 2 (i) Inequality (4) resembles the classical Bernstein inequality for independent random variables. Asymptotically, σ 2 n is usually of order O(n) and A n can be chosen equal to σ 2 n while B n is usually O(1) and hence negligible. In cases where σ 2 n is very small or where knowledge of the value of A n is required for some statistical procedure, it might, however, be better to choose A n larger than σ 2 n . It follows from ( 1) and ( 2) that a rough bound for σ 2 n is given by

σ 2 n ≤ 2 n K 2 Ψ(1, 1) L 1 . (5)
Hence, taking A n = 2nK 2 Ψ(1, 1)L 1 we obtain from (4) that

P (S n ≥ t) ≤ exp - t 2 C 1 n + C 2 t (2µ+2ν+3)/(µ+ν+2) , (6)
where

C 1 = 4K 2 Ψ(1, 1)L 1 and C 2 = 2B 1/(µ+ν+2) n with B n = 2(K ∨ M )L 2 ( 2 3+µ+ν Ψ(1, 1) ∨ 1).
Inequality ( 6) is then more of Hoeffding-type.

(ii) Based on a Rosenthal-type inequality, [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] also proved an exponential inequality for S n , however, with √ t instead of t 2 in the exponent.

(iii) [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] proved a Bennett inequality for weakly dependent random variables. This also implies a Bernstein-type inequality, however, with different constants. In particular, the leading term in the denominator of the exponent differs from σ 2 n . This is a consequence of their method of proof which consists of replacing weakly dependent blocks of random variables by independent ones according to some coupling device (an analogous argument is used in [START_REF] Doukhan | Mixing: Properties and Examples[END_REF] for the case of absolute regularity). Let M denote a sub σ-algebra of A and X ∈ R d be any random variable on the probability space (Ω, A, P). We recall from [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF] that

φ(M, X) = sup{ E(g(X)|M) -Eg(X) ∞ , Lip g ≤ 1} Set φk (r) = max l≤k 1 l sup i+r≤j 1 <j 2 <•••<j l φ(σ({X j , j ≤ i}), (X j 1 , . . . , X j l )). The pro- cess (X t ) t∈Z is called φ-dependent if φ(r) = sup k>0 φk (r) tends to 0 as r → ∞.
For an integrable function h and a Lipschitz function k this entails, for

s 1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v , that |Cov(h(X s 1 , . . . , X su ), k(X t 1 , . . . , X tv ))| ≤ vE|h (X s 1 , . . . , X su ) |Lip k φ(t 1 -s u )
Let us consider n φ-weakly dependent observations X 1 , . . . , X n . We assume here only that E|X t | 2 ≤ 1 and

P(|X t | ≤ √ n) = 1 for all 1 ≤ t ≤ n. If φ(r) ≤ exp(-ar b )
for any a > 0, b ∈ (0, 1), then the assumptions of Theorem 1 hold with µ = 1/b and it is easy to check that there exists a constant C such that the parameter B n is smaller than C √ n. The notion of φ-weak dependence is adapted to expanding dynamical systems; see [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF] for more details.

(iv) A Bernstein-type inequality with σ 2 n as a possible leading term in the denominator of the exponent has been derived in [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF] under a weak dependence condition which is tailor-made for causal processes with an exponential decay of the coefficients of weak dependence. The result above is more general and is also applicable to interesting classes of processes where [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF] inequality does not apply; see Section 5 for a thorough discussion of examples.

(v) Condition (1) in conjunction with (2) may be interpreted as a weak dependence condition in the sense that the covariances on the left-hand side tend to zero as the time gap between the two blocks of observations increases. Note that the supremum of expression (1) for all u-tuples (s 1 , . . . , s u ) and all v-tuples (t 1 , . . . , t [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] initial paper. Conditions (1) and (2) are typically fulfilled for truncated versions of random variables from many time series models; see also Proposition 8 below. The constant K in (1) is included to possibly take advantage of a sparsity of data as it appears, for example, in nonparametric curve estimation.

v ) with 1 ≤ s 1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v < ∞ such that t 1 -s u = r is denoted by C u+v (r) in
(vi) For unbounded random variables, the coefficients C p (r) may still be bounded by an explicit function of the index p under a weak dependence assumption; see Lemma 10 below. For example, assume that E exp(A|X t | a ) ≤ L holds for some constants A, a > 0, L < ∞. Since the inequality u p ≤ p!e u (p ∈ N, u ≥ 0) implies that

u m = (Aa) -m/a (Aau a ) m/a ≤ (Aa) -m/a (m!) 1/a e Au a ∀m ∈ N we obtain that E|X t | m ≤ L(m!) 1/a (Aa) -m/a
holds for all m ∈ N. Lemma 10 below provides then appropriate estimates for C p (r). These bounds may also be used to derive an exponential inequality of Bernstein-type; see for example Theorem 4.24 on page 102 in [START_REF] Saulis | Limit Theorems for Large Deviations, Mathematics and Its Applications[END_REF] for such a result in the case of α-mixing random variables.

A first Rosenthal-type inequality for weakly dependent random variables was derived by [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] via direct expansions of the moments of even order. Unfortunately, the variance of the sum did not explicitly show up in their bound. Instead, a rough bound for this expression based on upper estimates was used. Using cumulant bounds in conjunction with Leonov and Shiryaev's formula we are able to obtain a tighter moment inequality which resembles the Rosenthal inequality in the independent case (see [START_REF] Rosenthal | On the subspaces of L p (p > 2) spanned by sequences of independent random variables[END_REF] and [START_REF] Johnson | Best constants in moment inequalities for linear combinations of independent and exchangeable random variables[END_REF] in the independent case, and Theorem 2.12 in [START_REF] Hall | Martingale Limit Theory and Its Applications[END_REF] in the case of martingales).

Theorem 3 Suppose that X 1 , . . . , X n are real-valued random variables on a probability space (Ω, A, P) with zero mean and let p be a positive integer. We assume that there exist constants K, M < ∞, and a nonincreasing sequence of real coefficients (ρ(n)) n≥0 such that, for all u-tuples (s 1 , . . . , s u ) and all v-

tuples (t 1 , . . . , t v ) with 1 ≤ s 1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v ≤ n and u + v ≤ p, condition (
1) is fulfilled. Furthermore, we assume that

E|X i | p-2 ≤ M p-2 .
Then, with Z ∼ N (0, 1),

|ES p n -σ p n EZ p | ≤ B p,n 1≤u<p/2 A u,p K 2u (M ∨ K) p-2u n u , where B p,n = (p!) 2 2 p max 2≤k≤p {ρ p/k k,n }, ρ k,n = n-1 s=0 (s + 1) k-2 ρ(s) and A u,p = 1 u! k 1 +•••+ku=p, k i ≥2 ∀i p! k 1 ! • • • k u ! .
Remark 4 (i) For even p, the above result implies that

ES p n ≤ (p -1)(p -3) • • • 1 σ p n + B p,n 1≤u<p/2 A u,p K 2u (M ∨ K) p-2u n u ,
which resembles the classical Rosenthal inequality from the independent case. If sup n B p,n < ∞ and σ 2 n n, then the first term on the right-hand side is asymptotically dominating, as n → ∞. This term is equal to the p-th moment of a Gaussian random variable with mean 0 and variance σ 2 n .

(ii) [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] also obtained a Rosenthal-type inequality, however, essentially with n ∞ k=-∞ |EX 0 X k | instead of var(S n ) in the first term.

Remark 5 The inequality from Theorem 3 is well suited for proving a central limit theorem via the method of moments. Assume first that the random variables X t are uniformly bounded, centred and satisfy condition (1) with lim s→∞ ρ(s)/s p = 0, for all p > 0. Then

lim n→∞ σ 2 n n = σ 2 = ∞ k=-∞ EX 0 X k
is a convergent series, and thus the method of moments implies the central limit theorem,

1 √ n S n d -→ n→∞ σZ.
The next section introduces the suitable frame of weak dependence in order to apply those results.

Weak Dependence

A large class of examples for the assumptions in Theorem 1 to hold is provided by [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] with weakly dependent processes. Consider a stationary process (X t ) t∈Z with values in R d . Then any real-valued, Lipschitz and bounded function Y t = f (X t ) will be proved to satisfy the assumptions in the previous theorems if it is weakly dependent. We first recall this notion. Consider a process with values in R d endowed with some norm • . Let h : (R d ) u → R be an arbitrary function. We set

Lip h = sup | h(x 1 , . . . , x u ) -h(y 1 , . . . , y u ) | x 1 -y 1 + • • • + x u -y u : (x 1 , . . . , x u ) = (y 1 , . . . , y u ) .
Moreover, Λ denotes the set of functions h : R u → R, for some u ∈ N, such that Lip h < ∞, and Λ (1) = {h ∈ Λ : h ∞ ≤ 1}. For each u ≥ 1, we identify the sets R d u and R du .

Definition 6 [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] The sequence (X t ) t∈Z is called (Λ (1) , ψ, )-weakly dependent if there exists a function ψ : R 2 + × N 2 → R + and a sequence = ( r ) r∈N decreasing to zero at infinity such that, for any

g 1 , g 2 ∈ Λ (1) with g 1 : R du → R, g 2 : R dv → R (u, v ∈ N) and for any u-tuple (s 1 , . . . , s u ) and any v-tuple (t 1 , . . . , t v ) with s 1 ≤ • • • ≤ s u ≤ s u + r ≤ t 1 ≤ • • • ≤ t v , the following inequality is fulfilled: cov g 1 (X s 1 , . . . , X su ), g 2 (X t 1 , . . . , X tv ) ≤ ψ(Lip g 1 , Lip g 2 , u, v) r .
Important examples of processes correspond to the following choices of the function ψ: [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] and [START_REF] Dedecker | A new covariance inequality and applications[END_REF]) and we shall always denote r = θ r .

(a) If ψ(Lip g 1 , Lip g 2 , u, v) = vLip g 2 , then the sequence is called θ-dependent (see
(b) If ψ(Lip g 1 , Lip g 2 , u, v) = uLip g 1 + vLip g 2 , then the sequence is called η- dependent and we shall always denote r = η r . (c) If ψ(Lip g 1 , Lip g 2 , u, v) = uvLip g 1 Lip g 2 , then the sequence is called κ- dependent and we shall always denote r = κ r . (d) If ψ(Lip g 1 , Lip g 2 , u, v) = uLip g 1 + vLip g 2 + uvLip g 1 Lip g 2 , then the se-
quence is called λ-dependent and we shall always denote r = λ r , as in [START_REF] Doukhan | A central limit theorem under non causal weak dependence conditions and sharp moment assumptions[END_REF]. This case includes the two previous ones.

Remark 7 (i) Assume that (X t ) t∈Z is an R d -valued and stationary process which is (Λ (1) , ψ, )-weakly dependent. Then, for any Lipschitz-continuous 1) , ψ, )-weakly dependent.

function F : R d → R with F ∞ = M < ∞ and Lip F ≤ 1, the process Y t = F (X t ) is real valued, stationary, and Y t ∞ ≤ M . Moreover, it is also (Λ (
(ii) In the more general case when Lip F possibly exceeds 1 (e.g., if the function F depends on the sample size in a statistical context), then weak dependence still holds where only ψ(a, b, u, v) has to be replaced by ψ Y (a, b, u, v) = ψ(aLip F, bLip F, u, v). For the special cases of η, κ and λ weak dependence conditions, one may re-formulate this as (Y t ) t∈Z is still an η, κ or λ-weakly dependent sequence but now we have to respectively consider the coefficients

η Y r = Lip F η r , κ Y r = Lip 2 F κ r , λ Y r = max Lip F, Lip 2 F λ r .
Now we relate the condition in Definition 6 to condition (1) which was also considered in [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF]. Suppose that (X t ) t∈Z is a stationary sequence of real-valued random variables with X t ∞ ≤ M which satisfies the condition in Definition 6. To see the connection to condition (1), we consider functions g 1 and g 2 with

g 1 (x 1 , . . . , x u ) = u i=1 f (x i /M ) and g 2 (x 1 , . . . , x v ) = v i=1 f (x i /M ), where f (u) = u ∨ (-1) ∧ 1. (These functions satisfy Lip g i ≤ 1/M and g i ∞ ≤ 1.)
The covariance in Definition 6 can be expressed as in inequality (1), up to a factor M u+v since g

1 (X s 1 , . . . , X su ) = X s 1 • • • X su /M u and g 2 (X t 1 , . . . , X tv ) = X t 1 • • • X tv /M v . Proposition 8 Assume that X t ∞ ≤ M and that the real-valued sequence (X t ) t∈Z is (Λ (1) , ψ, )-weakly dependent. Then |cov (X s 1 • • • X su , X t 1 • • • X tv )| ≤ M u+v ψ(M -1 , M -1 , u, v) t 1 -su . (7)
Moreover, if r = exp(-ar), for some a > 0, then we may choose in inequality (2) µ = 1 and

L 1 = L 2 = 1/(1 -e -a
).

If r = exp(-ar b ), for some a > 0, b ∈ (0, 1), then we may choose µ = 1/b and L 1 , L 2 appropriately as only depending on a and b.

Remark 9 (i) According to Proposition 8, (Λ (1) , ψ, )-dependence implies condition (1) to be fulfilled with

(a) Ψ(u, v) = 2v, K 2 = M and ρ(r) = θ r /2, under θ-dependence, (b) Ψ(u, v) = u + v, K 2 = M and ρ(r) = η r , under η-dependence, (c) Ψ(u, v) = uv, K = 1 and ρ(r) = κ r , under κ-dependence, (d) Ψ(u, v) = (u+v+uv)/2, K 2 = M ∨1 and ρ(r) = 2λ r , under λ-dependence.
(ii) If a vector-valued process (X t ) t∈Z is an η, κ or λ-weakly dependent sequence and

F : R d → R is a Lipschitz function with F ∞ = M < ∞, then the process Y t = F (X t ) is real-valued and the relation (1) holds with (a) Ψ(u, v) = 2v, K 2 = M Lip F and ρ(r) = θ r /2, under θ-dependence, (b) Ψ(u, v) = u + v, K 2 = M Lip F and ρ(r) = η r , under η-dependence, (c) Ψ(u, v) = uv, K = Lip F and ρ(r) = κ r , under κ-dependence, (d) Ψ(u, v) = (u + v + uv)/2, K 2 = (M ∨ 1)(Lip 2 F ∨ Lip F ) and ρ(r) = 2λ r , under λ-dependence.
Inequality (7) together with the specifications of Ψ(u, v) given in Remark 9 allow to use the Bernstein-type inequality in Theorem 1 for sums of functions of weakly dependent sequences. Finally, we intend to determine sharp bounds for the coefficients C p (r) in the case of not necessarily bounded random variables.

Lemma 10 Assume that the real-valued sequence (X t ) t∈Z is η, κ or λ-weakly dependent and that E|X t | m ≤ M m , for all m > p. Then, according to the type of the weak dependence condition:

C p (r) ≤ 2 p+3 p 2 M p-1 m-1 m η 1-p-1 m-1 r , (8)
≤ 2 p+3 p 4 M p-2 m-2 m κ 1-p-2 m-2 r , (9)
≤ 2 p+3 p 4 M p-1 m-1 m λ 1-p-1 m-1 r . ( 10
)
Remark 11 This lemma is the essential tool to provide a version of Theorem 3 which yields both a Rosenthal-type moment inequality and a rate of convergence for moments in the central limit theorem. We also note that this result does not involve the assumption that the involved random variables are a.s. bounded. In fact, even the use of Theorem 1 does not really require such a boundedness; see Remark 2-(vi) above.

Applications

In this section we present various applications of the previous results in probability and statistics. A first subsection addresses the basic case of the bounded LIL, while the others, specific to statistics, are concerned with empirical processes and nonparametric curve estimation.

Bounded LIL

Suppose that (X t ) t∈Z is a stationary process satisfying the assumptions of Theorem 1 and that

σ 2 = lim n→∞ σ 2 n /n > 0. Then lim sup n→∞ 1 σ √ 2n log log n |S n | ≤ 1 a.s. ( 11
)
To prove (11), we select a subsequence (n k ) k∈Z as n k = [a k ], for any a > 1.

We obtain from Theorem 1 that, for n k ≤ n < n k+1 and any fixed c,

P |S n | √ 2nσ 2 > c log log n k ≤ 2 exp -c 2 log log n k σ 2 n σ 2 n (1 + o(1)) = 2 exp -c 2 log log n k (1 + o(1)) = O k -c 2 (1+o(1)) .
This implies by the maximal inequality given in Theorem 2.2 in Móricz, Serfling and Stout (1982) that

P max n k ≤n<n k+1 |S n | √ 2nσ 2 > c log log n k ≤ C k -c , (12)
where c < c 2 can be chosen arbitrarily close to c 2 and C is an appropriate finite constant; see the remark following the proof of Theorem 2.2 in [START_REF] Móricz | Moment and probability bounds with quasi-superadditive structure for the maximum partial sum[END_REF]. Since lim k→∞ max n k ≤n<n k+1 log log n log log n k = 1 we conclude from ( 12) by the Borel-Cantelli lemma that lim sup

n→∞ 1 σ √ 2n log log n |S n | ≤ c a.s.,
for any c > 1, which in turn implies (11).

Kernel type density estimation in the supremum norm

Let (X t ) t∈Z be a stationary η or λ-weakly dependent R d -valued process. We denote K : R d → R a Lipschitz function, compactly supported with

R d K(x) dx = 1. Then, if h n → 0, f n (x) = 1 nh d n n t=1 K X t -x h n
is an estimator of the marginal density f of X 0 , provided this function exists. This section is devoted to derive asymptotic properties of

D n = sup x∈R d | f n (x) -f (x)|.
In the independent case, [START_REF] Giné | A law of the iterated logarithm for kernel density estimators in the presence of censoring[END_REF] proved the following tight asymptotic bound for the supremum deviation of f n from its expectation:

sup x∈R d | f n (x) -E f n (x)| = O log n/(nh d n ) (a.s.) (13)
We first analyze the bias. A Hölder class of β-regular probability densities is given by

F ∞ (β, L) = f : f is a density, f (b) (x + y) -f (b) (x) ≤ L|y| β-β , ∀x, y, ∀b = (b 1 , . . . , b d ) with b 1 + • • • + b d = β } ,
where β denotes the greatest integer strictly less than β (β > 0) and

f (b) = ∂ β f /∂ b 1 x 1 • • • ∂ b d x d is a partial derivative.
Assume that the density f belongs to F ∞ (β, L) and that K is a kernel of order β , i.e., P (x)K(x) dx = P (0) for each polynomial of degree less than or equal to β . Then it follows from Taylor's formula that, for each

x ∈ R d , |E f n (x) -f (x)| ≤ L R d u β |K(u)| du • h β n β ! .
Furthermore, it is easy to prove that if the joint densities f k of (X 0 , X k ) are bounded, uniformly with respect to k > 0, if in addition nh d n → ∞ and η r = O(r -a ) for some a > 3 or if λ r = O(r -a ) for a > 4 are fulfilled, then (see [START_REF] Doukhan | Functional estimation for weakly dependent stationary time series[END_REF])

var( f n (x)) ∼ f (x) K 2 (u)du/(nh d ).
This section has a double purpose, we first precise a uniform and almost sure convergence rate under weak dependence assumptions in a first subsection, while the second subsection provides precise constants in this asymptotics in the univariate case (d = 1). For this, we describe the necessary modifications of the arguments in [START_REF] Butucea | Exact asymptotics for estimating the marginal density of discretely observed diffusion processes[END_REF], who stated such a result under an absolute regularity assumption (β-mixing).

Multivariate case, rate of convergence

Using the remark following Proposition 8 we obtain by Theorem 1 that the result (13) still holds if h n ≥ Cn -c for a constant C > 0 and for some c = c(d). We set

X n,t = 1 nh d n K X t -x h n -EK X t -x h n . Then f n (x)-E f n (x) = X n,1 +• • •+X n,n . We write X n,t = F n (X t ), for a function F n which satisfies F n ∞ ≤ 2 K ∞ /(nh d n ) and Lip F n ≤ Lip K/(nh d+1 n ).
As mentioned above, mild conditions on the weak dependence coefficients imply that σ

2 n = var(X 1,n + • • • + X n,n ) satisfies nh d n σ 2 n → n→∞ f (x) K 2 (u)
du. Now the Bernstein-type inequality from Theorem 1 writes here,

• in the case of η-dependent sequences with M = F n ∞ ∼ 1/(nh d n ), K 2 ∼ 1/(n 2 h 2d+1 ) and B n ∼ 1/(nh 2d+2 n ). Now, with t = C log n/(nh d n ), Theo- rem 1 implies | f n (x) -E f n (x)| = O log n/nh d n , (a.s.) for each x of R d if h n ≥ n -c
, for a c < 1/(3d + 4). Note that an optimal uniform window width h n ∼ (log n/n) 1/(2β+d) yields the tight uniform a.s. rate in the case of

β > d + 2. • in the case of λ-dependence, with M = F n ∞ ∼ 1/(nh d n ), K ∼ 1/(nh d+1 n ) and B n ∼ 1/(nh 2d+3 n ). Now, again with t = C log n/(nh d n )
, Theorem 1 implies the same as before if h n ≥ n -c , for a c < 1/(3d + 6). The choice of h n ∼ (log n/n) 1/(2β+d) yields the tight uniform a.s. rate if β > d + 3. Now we assume that the kernel K is compactly supported, i.e., K(x) = 0 if x ≥ C, for some C < ∞. Note that the exponential inequality used here allows to extend the above convergence rates to hold uniformly with respect to x ≤ n A /C, for an arbitrary norm on R d and any A > 0. To this end, we consider f n (x) -E f n (x) first on a sequence of increasingly fine grids X n for the set {x : x ≤ n/C} with a cardinality of order n γ , for some γ < ∞. It follows from our Bernstein-type inequality that

P max x∈Xn | f n (x) -E f n (x)| > C λ log n/(nh d n ) = O(n -λ )
holds for arbitrary λ < ∞ and some finite C λ . Moreover, it follows from the Lipschitz continuity of the kernel function K that

| f n (x) -f n (y)| ≤ Ch -d n x -y ,
for some C < ∞. Hence, with X n sufficiently dense, we can conclude that

P sup {x: x ≤n A /C} | f n (x) -E f n (x)| > C λ log n/(nh d n ) = O(n -λ ),
which implies by the Borel-Cantelli Lemma that sup {x:

x ≤n A /C} | f n (x) -E f n (x)| = O log n/(nh d n ) (a.s.)
In order to prove that convergence holds uniformly over R d , and then deduce (13), assume that E X 0 a < ∞ for some a > 0. Then, if A is large enough, sup x ≥n A /C f (x) → n→∞ 0 as fast as needed; indeed, the existence of a moment together with an Hölder assumption entail some Riemannian decay of f . The same occurs a.s. for f n . We have that

∞ n=1 P max 1≤t≤n X t > n A -Ch n ≤ ∞ n=1 1 ∧ E( X 1 a + • • • + X n a ) (n A -Ch n ) a < ∞,
for A sufficiently large. This implies by the Borel-Cantelli Lemma that sup x ≥n A | f n (x)| → n→∞ 0 a.s. as fast as needed if A is large enough.

Note that for lower order regularities our result does not apply but the Bernstein inequality from [START_REF] Ragache | Convergence rates for density estimates of weakly dependent time series[END_REF], following [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF], is still working and the a.s. convergence rate is now obtained only with the log n term replaced by log 2+2/µ n under our previous assumptions.

Univariate case, exact asymptotics

In [START_REF] Butucea | Exact asymptotics for estimating the marginal density of discretely observed diffusion processes[END_REF], it was shown that asymptotically exact minimax results can be obtained by appropriately tuned kernel density estimators in the case that the real-valued observations (d = 1) X 1 , . . . , X n are mixing, all joint densities p X 0 ,X k are uniformly (in k) bounded, the error is measured in the supremum norm and the density is assumed to belong to some Hölder class. Armed with the new Bernstein-type inequality, we can easily generalize these results to the case of weakly dependent observations.

Based on observations X 1 , . . . , X n , we consider the previous kernel estimator f n , where

h n = C(log n/n) 1/(2β+1) for f ∈ F ∞ (β, L).
Here the expectations are taken relatively to the parameter L which is the distribution of the process (X t ) t∈Z , where the marginal distributions of L has the density f and the process is associated with a certain decay rate of the dependence coefficients.

The supremum risk of f n can be decomposed in a bias and a stochastic term as

E f n -f ∞ ≤ E f n -f ∞ + E f n -E f n ∞ . Now it follows, for all f ∈ F ∞ (β, L), that E f n -f ∞ ≤ sup x 1 h n K u -x h n (f (u) -f (x)) du ≤ sup x K(u)(f (x + h n u) -f (x)) du ≤ L h β n B(β), (14)
where B(β) = sup g∈F∞(β,L) | K(u)(g(u) -g( 0)) du|.

To analyze the stochastic term, we choose an arbitrary > 0 and some appropriate < ∞. Then, with

σ n = A(β, L)/(nh n ) K 2 2 log(1/h n ), A(β, L) = max{g(0) : g ∈ F ∞ (β, L)}, E f n -E f n ∞ ≤ (1 + ) σ n + (1 + ) σ n P f n -E f n ∞ > (1 + )σ n + h -1 n K ∞ P f n -E f n ∞ > (1 + )σ n = T 1 + T 2 + T 3 , (15)
say. The term T 1 can be made arbitrarily close to the desired σ n . Since var( 1)) we can show, using chaining techniques in conjunction with the Bernstein-type inequality from Theorem 1, that

f n (x)) ≤ A(β, L)/(nh n ) K 2 2 (1 + o(
P f n -E f n ∞ > (1 + )σ n = o(1) (16)
and

P f n -E f n ∞ > (1 + )σ n = O(n -λ ), (17)
for arbitrary λ < ∞, provided is sufficiently large. Actually, [START_REF] Butucea | Exact asymptotics for estimating the marginal density of discretely observed diffusion processes[END_REF] used a Bernstein-type inequality for mixing random variables in the course of deriving results analogous to ( 16) and ( 17), in particular to derive their equations (5.43), (5.45) and (5.52). It can be seen that we can apply our Bernstein-type inequality in the same way which finally leads to ( 16) and ( 17); for details see the proof of Theorem 4.2 in [START_REF] Butucea | Exact asymptotics for estimating the marginal density of discretely observed diffusion processes[END_REF].

From ( 14) to ( 17) we can conclude that lim sup

n→∞ sup f ∈F∞(β,L) (n/ log n) β/(2β+1) E f n -f ∞ ≤ n log n β/(2β+1)    A(β, L) nh n K 2 2 log(1/h n ) + L h β n B(β)    . (18)
Choosing now the kernel function K and the constant C in the definition of h n in an optimum manner we can show that the right-hand side of (18) matches the known asymptotic minimax bound in the case of independent data (see [START_REF] Korostelev | The asymptotic minimax constant for sup-norm loss in nonparametric density estimation[END_REF]) which is also the minimax bound under mixing (see again [START_REF] Butucea | Exact asymptotics for estimating the marginal density of discretely observed diffusion processes[END_REF]) and, hence, also the asymptotic risk bound in the more general framework of weak dependence.

Further applications

Exponential inequalities are quite useful when a large number of random sums has to be simultaneously bounded. Further possible applications in statistics include nonparametric estimation in Barron's classes (see [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF]), adaptive estimation as in [START_REF] Gannaz | Adaptive density estimation with dependent observations[END_REF].

Possible adaptation of the generic chaining arguments in [START_REF] Talagrand | The Generic Chaining[END_REF] are very attractive but use of classes of Lipschitz functions makes it difficult to use directly. We think that such applications deserve a detailed study which is, however, well beyond the scope of this paper.

Examples

Following [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF] and [START_REF] Doukhan | A central limit theorem under non causal weak dependence conditions and sharp moment assumptions[END_REF] we describe here some models which satisfy the previous weak dependence conditions.

• Gaussian and associated processes are κ-dependent with

κ r = max t≥r | cov(X 0 , X t )|,
if they are stationary. Note that such covariances are nonnegative under association. • Bernoulli shifts. Let H : R Z → R be a measurable function. If the sequence (ξ t ) t∈Z is independent and identically distributed on the real line, a Bernoulli shift with innovation process (ξ t ) t∈Z is defined as

X t = H ((ξ t-i ) i∈Z ) , t ∈ Z.
This sequence is stationary. The simplest case of an infinitely dependent Bernoulli shift is a moving average process

X t = ∞ i=-∞ a i ξ t-i . ( 19
)
Bernoulli shifts are η-weakly dependent with η r ≤ 2δ [r/2] , where

δ s ≥ max 0≤t≤s E X 0 -X (t) 0 , X (t) 0 = H (ξ (t) i ) i∈Z , with ξ (t) i = ξ i if |i| < t and ξ (t) i = 0 if |i| ≥ t.
• Chaotic Volterra models. A Volterra process is a stationary process defined through a convergent Volterra expansion

X t = v 0 + ∞ k=1 V k;t , where V k;t = i 1 <...<i k a k;i 1 ,...,i k ξ t-i 1 • • • ξ t-i k .
Here v 0 is a constant and (a k;i 1 ,...,i k ) (i 1 ,...,i k )∈Z k are real numbers for each k ≥ 1. Let p ≥ 1. Then this expression converges in L p , provided that E|ξ 0 | p < ∞ and the weights satisfy

∞ k=1 i 1 <•••<i k |a k;i 1 ,...,i k | p < ∞.
Those processes are η-dependent since δ s from above is now the tail of the previous series. The forthcoming examples contain some particular models of this type.

• LARCH(∞) models. A vast literature is devoted to the study of conditionally heteroskedastic models; see [START_REF] Giraitis | Recent advances in ARCH modelling[END_REF]. It was shown in [START_REF] Doukhan | Vector valued LARCH infinity processes[END_REF] that a simple equation in terms of a vector-valued process allows a unified treatment of those models. Let (ξ t ) t∈Z be an iid sequence of random d × m matrices, (a j ) j∈N be a sequence of m × d matrices, and a be a vector in R m . A vector valued LARCH(∞) model is a solution of the recurrence equation

X t = ξ t   a + ∞ j=1 a j X t-j   . ( 20
)
Below we provide sufficient conditions for the following chaotic expansion

X t = ξ t   a + ∞ k=1 j 1 ,...,j k ≥1 a j 1 ξ t-j 1 a j 2 ξ t-j 1 -j 2 • • • a j k ξ t-j 1 -•••-j k a   . ( 21
)
Such vector-valued LARCH(∞) models include a large variety of models, for example

• Bilinear models, X t = ζ t α + ∞ j=1 α j X t-j + β + ∞ j=1 β j X t-j
, where the variables are real-valued and ζ t is the innovation. Expansion (21) coincides then with the chaotic expansion in [START_REF] Giraitis | Recent advances in ARCH modelling[END_REF].

• GARCH(p, q) models,      r t = σ t ε t σ 2 t = p j=1 β j σ 2 t-j + γ 0 + q j=1 γ j r 2 t-j
where β i ≥ 0, γ => 0 and γ j ≥ 0 (j ≥ 1), and the variables ε t are centred at expectation; see [START_REF] Giraitis | Recent advances in ARCH modelling[END_REF].

• ARCH(∞) processes, given by

     r t = σ t ε t σ 2 t = β 0 + ∞ j=1 β j σ 2 t-j
Now we turn to the general case given by equation ( 20). Assume that λ = j≥1 a j E ξ 0 < 1. Then a stationary solution of equation ( 20) in L 1 is given as ( 21). With A(x) = j≥x a j , this solution is θ-weakly dependent with

θ r = E ξ 0 E ξ 0 r-1 k=1 kλ k-1 A r k + λ r 1 -λ a .
There exist some constants K > 0 and b, C > 0 such that

θ r ≤      K (log(r)) b∨1 r b , under Riemannian decay, A(x) ≤ Cx -b , K(q ∨ λ) √ r ,
under geometric decay, A(x) ≤ Cq x , q < 1.

• Non causal LARCH(∞) models (now a j is defined for j = 0) allow the same results of existence (only replace summation for j > 0 by summation for j = 0) and dependence is now of the η type with

η r = E ξ 0   E ξ 0 0≤2k<r kλ k-1 A r 2k + λ r/2 1 -λ   a , A(x) = |j|≥x a j .
• Stable Markov processes are even θ-weakly dependent.

We consider a stationary sequences satisfying a recurrence equation

X t = F (X t-1 , . . . , X t-d , ξ t ),
where the sequence (ξ t ) t∈Z is iid. In this case, Y t = (X t , . . . , X t-d+1 ) is a Markov chain such that

Y t = M (Y t-1 , ξ t ) with M (x 1 , . . . , x d , ξ) = (F (x 1 , . . . , x d , ξ), x 1 , . . . , x d-1 ). Then E F (x, ξ) -F (y, ξ) ≤ a x -y if a = ( d i=1 a i ) 1/d < 1, where a i ≥ 0 are such that E|M (x, ξ 0 )-M (y, ξ 0 )| ≤ d i=1 a i |x i -y i |, and (x 1 , . . . , x d ) = max 1≤i≤d a i-1 |x i | denotes a norm on R d .
In this setting, it is simple to derive that θ-dependence holds with θ r = O(a r ) for: Examples of such Markov processes are nonlinear AR(1) processes (case B ≡ 1), stochastic volatility models (case A ≡ 0), or classic ARCH(1) models (case A(u) = αu, B(u) = √ β + γu 2 with α, β, γ ≥ 0, but here we also need Eξ 0 = 0 and Eξ 2 0 < ∞). • Branching type models. Set ξ t = (ξ (1) t , . . . , ξ (D) t ) , d = 1, and D ≥ 2. Let now A 1 , . . . , A D be Lipschitz functions R → R, and:

• Functional AR models, X t = r(X t-1 , . . . , X t-d ) + ξ t , if E|ξ 0 | < ∞ and |r(u 1 , . . . , u d ) -r(v 1 , . . . , v d )| ≤ d i=1 a i |u i -v i |, for some a 1 , . . . , a d ≥ 0 with a = ( d i=1 a i ) 1/d < 1. • ARCH-type processes. With d = 1, let M (u, z) = A(u) + B(u)z,
M u, z (1) , . . . , z (D) = D j=1 A j (u)z (j) , (u, z (1) , . . . , z (D) ) ∈ R D+1 .
For such kernels, we also require a = D j=1 Lip (A j )E|ξ

(j) 0 | < 1.
• Compound processes may be λ-dependent. Instead of independence, assume that the sequence (ξ t ) t∈Z is stationary and η-dependent with coefficients (ρ ξ,r ) r≥0 . Then, for example, linear processes ( 22) are then η-weak dependent with η r = η ξ,r/2 + δ r/2 and δ r/2 according to the definition after equation ( 22). Such hereditary properties of weak dependence are unknown under mixing. We present here the results from [START_REF] Doukhan | A central limit theorem under non causal weak dependence conditions and sharp moment assumptions[END_REF] who sharpen analogue results from [START_REF] Borovkova | Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation[END_REF].

We now focus on specific examples of two sided linear sequences

X t = i∈Z b i Y t-i (22)
with dependent inputs (Y t ) t∈Z .

Let us now denote by (Y t ) t∈Z a weakly dependent innovation process. The coefficient λ is proved to be very useful to study Bernoulli shifts with such stationnary innovations as in Lemma 12 below. Let H : R (Z) → R be a measurable function on the space of finitely supported real valued sequences we shall define X t = H(Y t-i , i ∈ Z). Such models are proved to exhibit either λ or η-weak dependence properties. We set here x = sup i∈Z |x i |.

In order to study weak dependence properties of (X t ) t∈Z , we assume that

H : R (Z) → R is such that for each s ∈ Z, if x, y ∈ R (Z) satisfy x i = y i for each index i = s, |H(x) -H(y)| ≤ b s |x s -y s |, (23)
The following lemma proves both the existence and the weak dependence properties of such models. Lemma 12 [START_REF] Doukhan | A central limit theorem under non causal weak dependence conditions and sharp moment assumptions[END_REF]) Let (Y t ) t∈Z be a strictly stationary process with a finite moment of order m > 2 and let H satisfy the condition (23) for l = 0 and some nonnegative sequence

(b s ) s∈Z such that j b j < ∞. Then, • the process X n = H(Y n-j , j ∈ Z) := lim I→∞ H(Y n-j 1 {|j|≤I} , j ∈ Z) is a
strongly stationary process with finite moments of order m.

• if the input process (Y t ) t∈Z is λ-weakly dependent (the weak dependence coefficients are denoted λ Y,r ), then (X t ) t∈Z is λ-weakly dependent with

λ k ≤ inf 2r≤k   2 |i|≥r b i Y 0 1 + (2r + 1) 2 L 2 λ Y,k-2r   .
• if the input process (Y t ) t∈Z is η-weakly dependent (the weak dependence coefficients are denoted η Y,r ) then (X t ) t∈Z is η-weakly dependent and

η k ≤ inf 2r≤k   2 |i|≥r b i Y 0 1 + (2r + 1)Lη Y,k-2r   .
Beyond linear functions one may think to non causal ARCH(∞) inputs (with bounded inputs) and [START_REF] Doukhan | A central limit theorem under non causal weak dependence conditions and sharp moment assumptions[END_REF]

Proofs of the main theorems

First, note that it is not possible to adapt the classic method of proving the Bernstein inequality in the independent case since it makes heavy use of the independence; see Bennett (1962, pp. 33-45). One possible approach to proving an exponential inequality involves replacing blocks of weakly dependent random variables by independent ones and then applying an available inequality from the independent case. This was recently done by [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] who derived a Bennett inequality which then possibly implies a Bernstein-type inequality, however, with constants in the exponent different to ours. In particular, the leading term (asymptotically, as n → ∞) in the denominator of the exponent will then differ from var(S n ) which is possible in our case. We also note here that an abstract presentation of cumulant techniques involving Umbral Calculus is presented in [START_REF] Rota | On the combinatorics of cumulants[END_REF].

Our proof of the Theorem is based on a result of [START_REF] Bentkus | On exponential estimates of the distribution of random variables[END_REF] which we quote here for reader's convenience. Let ξ be an arbitrary realvalued random variable with Eξ = 0 and finite moments of all orders. The k-th cumulant of ξ is defined as

Γ k (ξ) = 1 i k d k dt k log Ee itξ t=0 .
If there exist γ ≥ 1 , σ 2 > 0 and B ≥ 0 such that

|Γ k (ξ)| ≤ k! 2 γ σ 2 B k-2
for all k = 2, 3, . . . , then, for all t ≥ 0 ,

P (ξ ≥ t) ≤ exp - t 2 /2 σ 2 + B 1/γ t (2γ-1)/γ . ( 24
)
Note that the quotation of this result in Lemma 2.4 in the monograph by Saulis and Statulevicius (1991, p. 19) contains a typo; it was correctly stated and proved in the initial paper, Bentkus and Rudzkis (1980, Lemma 2.1).

Before we proceed with the calculations, we recall some notions needed in the course of the proof. It follows from the definition of the cumulants that

Γ k (S n ) = 1≤t 1 ,...,t k ≤n Γ(X t 1 , . . . , X t k ), (25)
where

Γ(X t 1 , . . . , X t k ) = 1 i k ∂ k ∂u t 1 . . . ∂u t k log Ee i(u 1 X 1 +•••+unXn) u 1 =•••=un=0
are mixed cumulants. For any random variable Y with finite expectation, we define ulevicius (1970, Lemma 3) has shown that, for 1 ≤ t 1 ≤ • • • ≤ t k ≤ n, the mixed cumulants can be expressed in terms of centred moments as

Y = Y -EY . For 1 ≤ t 1 ≤ • • • ≤ t k ≤ n, define so-called centred moments as E(X t 1 , . . . , X t k ) = E X t 1 X t 2 • • • X t k-1 X t k (E(X t 1 ) = EX t 1 ). Stat-
Γ(X t 1 , . . . , X t k ) = k ν=1 (-1) ν-1 ν p=1 Ip=I N ν (I 1 , . . . , I ν ) ν p=1 EX Ip , (26)
where ν p=1 Ip=I denotes the summation over all unordered partitions in disjoint subsets I 1 , . . . , I ν of the set I = {1, . . . , k}; see also equation (1.63) in [START_REF] Saulis | Limit Theorems for Large Deviations, Mathematics and Its Applications[END_REF], as a more easily available reference. Given such a partition, EX Ip stands for E(X t

i (p) 1 , . . . , X t i (p) kp ) if I p = {i (p) 1 , . . . , i (p) kp } with i (p) 1 < • • • < i (p)
kp . We arrange the subsets in the partitions such that i

(1) 1 < • • • < i (ν)
1 . N ν (I 1 , . . . , I ν ) are certain nonnegative integers defined as follows. Let, for i ∈ I, n i (I 1 , . . . , I ν ) = #{p : i see equations (4.36) and (4.37) in Saulis and Statulevicius (1991, p. 80). According to this, it follows that N ν (I 1 , . . . , I ν ) = 0 if and only if {I 1 , . . . , I ν } is connected, that is, n i (p) 1 (I 1 , . . . , I ν ) > 0 for all p = 2, . . . , ν. Furthermore, we have that

k ν=1 ν p=1 Ip=I N ν (I 1 , . . . , I ν ) = (k -1)! ; ( 27
)
see Saulis and Statulevicius (1991, equation (4.43)).

As a first step to deriving estimates for the cumulants of

S n = X 1 + • • • + X n ,
we derive estimates for the centred moments.

Lemma 13 Suppose that X 1 , . . . , X n are zero mean random variables satisfying condition (1) from Theorem 1, for u + v ≤ k. Furthermore, assume that

E|X i | k-2 ≤ ((k -2)!) ν M k-2 . Then, for i ∈ {1, . . . , k -1} , E(X t 1 , . . . , X t k ) ≤ 2 k ((k -2)!) ν K 2 M k-2 ρ(t i+1 -t i ).
PROOF. For the four cases (a) to (d), most parts of this proof are the same. Accordingly, we distinguish between them only when we apply condition (1), and at the end of this proof when certain upper estimates are summed up.

For

t 1 ≤ • • • ≤ t k , k ∈ N , we define the short-hand notation Y k = X t k and, for 1 ≤ j < k, Y j = X t j X t j+1 • • • X t k-1 X t k .
Elementary calculations show, for 1 ≤ j ≤ i < k , that

Y j = X t j Y j+1 -X t j E[Y j+1 ] = • • • = X t j • • • X t i Y i+1 - i l=j X t j • • • X t l E[Y l+1 ] = X t j • • • X t i Y i+1 - i-1 l=j X t j • • • X t l E[Y l+1 ]. (28)
Since EX t k = 0, in the special case of i = k -1 this becomes

Y j = X t j • • • X t k - k-1 l=j X t j • • • X t l E[Y l+1 ]. ( 29
)
Without making use of the weak dependence assumption, we conclude recursively, for 3 ≤ j < k , that

E|Y j | ≤ 2 k-j ((k -2)!) ν M k-j+1 . (30)
Hence, we obtain again in conjunction with (29) that

C j,i := cov X t j • • • X t i , Y i+1 ≤ cov X t j • • • X t i , X t i+1 • • • X t k + k-1 l=i+1 cov X t j • • • X t i , X t i+1 • • • X t l E[Y l+1 ] ≤ K 2 M k-j-1 Ψ(i -j + 1, k -i) ρ(t i+1 -t i ) + k-1 l=i+1 K 2 M l-j-1 Ψ(i -j + 1, l -i) ρ(t i+1 -t i ) 2 k-l-1 ((k -2)!) ν M k-l ≤ K 2 ((k -2)!) ν M k-j-1 Ψ(i -j + 1, k -i) + k-1 l=i+1 Ψ(i -j + 1, l -i)2 k-l-1 ρ(t i+1 -t i ). ( 31
)
Before we turn to estimating |E(X t 1 , . . . , X t k )|, we estimate the term in curly braces on the right-hand side of (31) in the four cases (a) to (d).

(a) If Ψ(u, v) = 2v, then

   Ψ(i -j + 1, k -i) + k-1 l=i+1 Ψ(i -j + 1, l -i) 2 k-l-1    = 2(k -i) + 2 k-1 l=i+1 (l -i) 2 k-l-1 ≤ 2 k-i-1 ∞ l =1 l 2 1-l = 2 k-i-1 d dp 1 1 -p p=1/2 = 2 k-i+1 =: λ (a) j,i . ( 32
) (b) If Ψ(u, v) = u + v, then    Ψ(i -j + 1, k -i) + k-1 l=i+1 Ψ(i -j + 1, l -i) 2 k-l-1    = (k -j + 1) + k-1 l=i+1 (l -j + 1) 2 k-l-1 = (k -j + 1) ∞ l=1 2 -l + k-1 l=i+1 (l -j + 1)2 k-l-1 ≤ ∞ l =k-j+1 l 2 -l +(k-j) + k-j l =i-j+2 l 2 -l +(k-j) = 2 k-j-1 ∞ l =i-j+2 l 2 1-l = (i -j + 3) 2 k-i-1 =: λ (b) j,i . (33)
(Here the last equation follows from

∞ l =i-j+2 l 2 1-l = d dp   ∞ l =i-j+2 p l   p=1/2 = d dp p i-j+2 1 -p p= 1 2 = (i -j + 3)2 j-i .) (c) If Ψ(u, v) = uv, then we obtain by (32)    Ψ(i -j + 1, k -i) + k-1 l=i+1 Ψ(i -j + 1, l -i) 2 k-l-1    = (i -j + 1)   (k -i) + k-1 l=i+1 (l -i) 2 k-l-1   ≤ (i -j + 1) 2 k-i =: λ (c) j,i . ( 34
) (d) If Ψ(u, v) = α(u + v) + (1 -α)uv
, then we obtain immediately by ( 33) and (34) that

   Ψ(i -j + 1, k -i) + k-1 l=i+1 Ψ(i -j + 1, l -i) 2 k-l-1    ≤ αλ (b) j,i + (1 -α)λ (c) j,i =: λ (d) j,i . (35)
Now we obtain from (28) that

|E[Y j ]| ≤ C j,i + i-1 l=j |E[X t j • • • X t l ]| • |E[Y l+1 ]|.
Therefore, we obtain recursively that

E(X t 1 , . . . , X t k ) = |E[Y 1 ]| ≤ C 1,i + i-1 l=1 M l |E[Y l+1 ]| ≤ • • • ≤ C 1,i + 1≤l 1 ≤i-1 M l 1 C l 1 +1,i + 1≤l 1 <l 2 ≤i-1 M l 2 C l 2 +1,i + • • • + 1≤l 1 <•••<l i-1 ≤i-1 M i-1 C i,i . (36)
At this point we have to distinguish again between the four cases (a) to (d).

From ( 36), ( 31) and ( 32) to ( 35) we obtain the common upper estimate

E(X t 1 , . . . , X t k ) ≤ K 2 ((k -2)!) ν M k-2 ρ(t i+1 -t i ) ×    λ (δ) 1,i + 1≤l 1 ≤i-1 λ (δ) l 1 +1,i + 1≤l 1 <l 2 ≤i-1 λ (δ) l 2 +1,i + • • • + 1≤l 1 <•••<l i-1 ≤i-1 λ (δ) l i-1 +1,i    , (37)
where δ = a, b, c, d refers to the four different cases. Now it remains to estimate the term in curly braces on the right-hand side of (37).

(a) If Ψ(u, v) = 2v, then

{. . .} = 2 k-i+1 i-1 l=0 i -1 l = 2 k . ( 38
) (c) If Ψ(u, v) = uv, then we obtain {. . .} =    λ (c) 1,i + 1≤l 1 ≤i-1 λ (c) l 1 +1,i + 1≤l 1 <l 2 ≤i-1 λ (c) l 2 +1,i + • • • + 1≤l 1 <•••<l i-1 ≤i-1 λ (c) l i-1 +1,i    = 2 k-i   i + 1≤l 1 ≤i-1 (i -l 1 ) + 1≤l 1 <l 2 ≤i-1 (i -l 2 ) + • • • + 1≤l 1 <•••<l i-1 ≤i-1 (i -l i-1 )   . Since 1≤l 1 <•••<lm≤i-1 (i -l m ) = i i -1 m - i-1 l=m l l -1 m -1 = (m + 1) i m + 1 -m i m + 1 = i m + 1 we get {. . .} = 2 k-i i + i-1 m=1 i m + 1 = 2 k-i i m=1 i m < 2 k . ( 39
) (b) If Ψ(u, v) = u + v, then we obtain analogously to (39) that {. . .} = 2 k-i-1   (i + 2) + 1≤l 1 ≤i-1 (i -l 1 + 2) + 1≤l 1 <l 2 ≤i-1 (i -l 2 + 2) + • • • + 1≤l 1 <•••<l i-1 ≤i-1 (i -l i-1 + 2)   < 2 k-i-1 i m=1 i m + 2 i-1 m=0 i -1 m < 2 k . ( 40
) (d) If Ψ(u, v) = α(u + v) + (1 -α)uv, then we easily obtain from λ (d) j,i = αλ (b) j,i + (1 -α)λ (c) j,i , (40) and (39) that {. . .} < 2 k . ( 41
)
The assertion of the lemma follows now from ( 37) and ( 38) to (41). 2

Equations ( 25), ( 26) and the result of Lemma 13 can now be used to derive estimates for the cumulants of S n .

Lemma 14 Suppose that the assertions of Lemma 13 are fulfilled. Then, for k ≥ 2 ,

|Γ k (S n )| ≤ n (k!) 2+ν 2 k K 2 (K ∨ M ) k-2 n-1 s=0 (s + 1) k-2 ρ(s).
PROOF. We deviate from the proof of similar results in Saulis and Statulevicius (1991) since we are not able to follow all of their arguments. In particular, we cannot verify their equation (4.55) on page 94 which is crucial for their approach.

From (25) we obtain that

|Γ k (S n )| ≤ k! 1≤t 1 ≤•••≤t k ≤n |Γ(X t 1 , . . . , X t k )| . (42)
According to (26) and Lemma 13, we have, for 1

≤ t 1 ≤ • • • ≤ t k ≤ n , that |Γ(X t 1 , . . . , X t k )| ≤ k ν=1 ν p=1 Ip=I N ν (I 1 , . . . , I ν ) ν p=1 E(X Ip ) ≤ k ν=1 ν p=1 Ip=I N ν (I 1 , . . . , I ν ) ν p=1 2 kp ((k p -2)!) ν K 2 M kp-2 min 1<j≤kp ρ(t i (p) j -t i (p) j-1
).

Note that we have, for any connected partition,

max 1≤p≤ν max 1<j≤kp {t i (p) j -t i (p) j-1 } ≥ max 1<i≤k {t i -t i-1 }.
Since N ν (I 1 , . . . , I ν ) = 0 if {I 1 , . . . , I ν } is not connected we therefore obtain, in conjunction with (27), that

|Γ(X t 1 , . . . , X t k )| ≤ k ν=1 ν p=1 Ip=I N ν (I 1 , . . . , I ν ) 2 k ((k -2)!) ν K 2 (K ∨ M ) k-2 min 1<i≤k ρ(t i -t i-1 ) ≤ (k -1)! 2 k ((k -2)!) ν K 2 (K ∨ M ) k-2 min 1<i≤k ρ(t i -t i-1 ).
This implies that (s + 1) k-2 ρ(s), this with ( 42) and ( 43), yields the assertion of the lemma. 2

1≤t 1 ≤•••≤t k ≤n |Γ(X t 1 , . . . , X t k )| ≤ n (k -1)! 2 k ((k -2)!) ν K 2 (K ∨ M ) k-
Proof of Theorem 1 From Lemma 14 we obtain, for k ≥ 3 , that

|Γ k (S n )| ≤ n (k!) 2+µ+ν 2 k K 2 L 1 ((K ∨ M )L 2 ) k-2 ≤ k! 2 2+µ+ν 2 4+µ+ν n K 2 L 1 (2 (K ∨ M ) L 2 ) k-2 ,
which implies that

|Γ k (S n )| ≤ k! 2 2+µ+ν A n B k-2
n holds for all k ≥ 2. The assertion of the Theorem follows now from (24). 2

Proof of Theorem 3 Recall that we have Γ 1 (S n ) = 0 and Γ 2 (S n ) = σ 2 n . Therefore, Leonov and Shiryaev's formula (see [START_REF] Saulis | Limit Theorems for Large Deviations, Mathematics and Its Applications[END_REF], formula (1.53) on page 11) writes as

ES p n = [p/2] u=1 1 u! k 1 +•••+ku=p p! k 1 ! • • • k u ! Γ k 1 (S n ) • • • Γ ku (S n ). ( 44
)
Note that Γ 1 (S n ) = 0 implies that the inner sums can be reduced to indices such that k i ≥ 2 for all i. If p is an even number, then the summand with u = p/2 on the right-hand side of ( 44) is equal to p! 2 p/2 (p/2)! (Γ 2 (S n )) p/2 = EZ p σ p n .

According to Lemma 14, we have, for 2 ≤ k ≤ p, that

|Γ k (S n )| ≤ n (k!) 2 2 k ρ k,n K 2 L 1 ((M ∨ K)L 2 ) k-2 . Γ k + 1 b ≤ √ 2π exp 1 12( k+1 b -1) k + 1 b -1 k+1 b -1 2 exp - k + 1 b -1 ≤ exp 1 - 1 12( k+1 b -1) 1 b k+1 b -1 2 √ 2π (k + 1) k+3/2 e -(k+1) 1/b ≤ C k ((k + 1)!) 1/b ≤ C k (k!) 1/b ,
for some appropriate C, C < ∞. Putting these bounds together we obtain the assertion. 2

Proof of Lemma 10 Set g T (x) = x ∨ T ∧ (-T ) and X i = g T (X i ) -Eg T (X i ), for some T > 0. Then

cov X s 1 • • • X su , X s u+1 • • • X sp ≤ p j=0 |A j |,
where A 0 = cov X s 1 • • • X su , X s u+1 • • • X sp , and for 1 ≤ j ≤ u,

A j = cov X s 1 • • • X s j-1 (X s j -X s j )X s j+1 • • • X su , X s u+1 • • • X sp , and if u < j ≤ p, A j = cov X s 1 • • • X su , X s u+1 • • • X s j-1 (X s j -X s j )X s j+1 • • • X sp .
First we bound |A j |, for 1 ≤ j ≤ p. We obtain by Hölder's inequality, with p-1 m + 1 m = 1, that

|A j | ≤ 2 i<j X s i m X s j -X s j m i>j X s i m .
From Jensen's inequality we get X s i m ≤ 2 g T (X s i ) m ≤ 2 X s i m ≤ 2M 1/m m . Furthermore, we have E|X s j -g T (X s j )| m ≤ E[|X s j | m I(|X s j | > T )] ≤ T m -m M m and |Eg T (X s j )| = |E(X s j -g T (X s j ))| ≤ X s j -g T (X s j ) m . This implies X s j -X s j m ≤ X s j -g T (X s j ) m +|Eg T (X s j )| ≤ 2T 1-m/m M 1/m m . Therefore, we obtain that

|A j | = 2 p-j+1 M p-1 m m X s i -X s i m ≤ 2 p-j+2 M p-1 m + 1 m m T 1-m m = 2 p-j+2 M m T p-m , which implies p j=1 |A j | ≤ 2 p+2 M m T p-m . ( 46
)
Now we bound |A 0 |. Coming back to the definition of weak dependence, we write it as cov(h(X s 1 , . . . X su ), k(X s u+1 , . . . X sp )) where h(x 1 , . . . , x u ) = u i=1 (g T (x i ) -Eg T (X s i )), and k(x u+1 , . . . , x p ) = p i=u+1 (g T (x i ) -Eg T (X s i )). Hence h ∞ ≤ 2 u T u , k ∞ ≤ 2 p-u T p-u , Lip h ≤ u2 u-1 T u-1 and Lip k ≤ (p -u)2 p-u-1 T p-u-1 and, according to the the weak dependence in use, we get

| cov(X s 1 • • • X su , X s u+1 • • • X sp )| ≤ (u 2 (2T ) u-1 (2T ) p-u + (p -u) 2 (2T ) p-u-1 (2T ) u )η r ≤ p 2 2 p-1 T p-1 η r , | cov(X s 1 • • • X su , X s u+1 • • • X sp )| ≤ (u(p -u)) 2 (2T ) p-2 κ r ≤ p 4 4 2 p-2 T p-2 κ r = p 4 2 p-4 T p-2 κ r or | cov(X s 1 • • • X su , X s u+1 • • • X sp )| ≤ p 2 p 2 8 ∨ 1 2 p-1 T p-2 (T ∨ 1)λ r .
To conclude, we equilibrate the bounds for the term |A 0 | with that of the other terms |A j |.

• Under η-dependence, set T = Mm

  for suitable Lipschitz functions A, B. The corresponding iterative model satisfies the previous relation with a = Lip (A) + E|ξ 0 | Lip (B) < 1.

  , . . . , I ν );

••

  Under κ-dependence, set T = Mm κr 1 m-2 . Then we obtain C p (r) ≤ 2 p+3 p 4 M Under λ-dependence, set T = Mm ηr 1 m-1 . Then we obtain C p (r) ≤ 2 p+3 p 4 M

  consider more general examples for which H does not satisfy eqn. (23) as polynomial Volterra models We now precise some examples of such Bernoulli shifts with dependent innovations:

	6 Proofs

  Since #{(s 2 , . . . , s k ) : 0 ≤ s i ≤ s, max{s 2 , . . . , s k } = s} ≤ (k -1)(s + 1) k-2 we obtain that

				∞
			2	s 2 ,...,s k =0	min 2≤i≤k	ρ(s i ).	(43)
	∞
	s 2 ,...,s k =0	min 2≤i≤k	ρ(s i ) ≤ (k -1)

∞ s=0

Applying Hölder's inequality to the Gamma function Γ we see that (k!) p/k ≤ p!. Hence, we obtain that

The assertion follows now from ( 44) and ( 45). 2

Proofs of some auxiliary results

Proof of Proposition 8 Inequality ( 7) is obvious.

The last term on the right-hand side can be further estimated by

Furthermore, it follows from Stirling's formula that


Paul Doukhan
email: doukhan@ensae.fr

Michael H Neumann
email: mi.neumann@tu-bs.de

The notion of weak dependence and its applications to bootstrapping time series

Keywords: 2000 Mathematics Subject Classification. Primary 60E15; secondary 62E99. Keywords and Phrases. Autoregressive processes, autoregressive bootstrap, mixing, weak dependence. Short title. Weak dependence

We give an introduction to the notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.

Mixing vs. weak dependence

For a long time mixing conditions have been the dominating type of conditions for imposing a restriction on the dependence between time series data. They are considered to be useful since they are fulfilled for many classes of processes and since they allow to derive tools similar to those in the independent case. On the other hand, it turns out that certain classes of processes which are of interest in statistics are not mixing although a successive decline of the influence of past states takes place. The simplest example of such a process is an AR(1)-process, X t = θX t-1 + ε t , where the innovations are independent and identically distributed with P (ε t = 1) = P (ε t = -1) = 1/2 and 0 < θ ≤ 1/2; see also [START_REF] Rosenblatt | Linear processes and bispectra[END_REF]. It is clear that this process has a stationary distribution supported on [-2, 2], and for a process in the stationary regime, it can be seen from the equality X t = ε t + θε t-1 + • • • + θ t-s-1 ε s+1 + θ t-s X s that a past state X s can always be recovered from X t . (Actually, since

|ε t | > θ|ε t-1 | + • • • + θ t-s-1 |ε s+1 | + θ t-s |X s | X t
has always the same sign as ε t which means that we can recover ε t and therefore X t-1 from X t . Continuing in this way we can finally compute X s .) This, however, excludes any of the commonly used mixing properties to hold. On the other hand, X s loses its impact on X t as t → ∞. Besides this somehow artificial example, there are many other processes of this type which are of great interest in statistics. For example, for bootstrapping a linear autoregressive process of finite order, it is most natural to estimate first the distribution of the innovations by the empirical distribution of the (possibly re-centered) residuals and to generate then a bootstrap process iteratively by drawing independent bootstrap innovations from this distribution. Now it turns out that commonly used techniques to prove mixing for autoregressive processes fail; because of the discreteness of the bootstrap innovations it is in general impossible to construct a coupling of two processes with different initial values.

Inspired by such problems, Doukhan and Louhichi (1999) and [START_REF] Bickel | A new mixing notion and functional central limit theorems for a sieve bootstrap in time series[END_REF] introduced the alternative notions of weak dependence and ν-mixing, respectively, which focus on covariances rather than the total variation norm between the joint distribution and the product of marginal distributions of random variables. A slightly simplified version of Doukhan and Louhichi's (1999) definition is given here: Definition 1.1. A process (X t ) t∈Z is called weakly dependent if there exists a universal null sequence ( r ) r∈N such that, for any k-tuple (s 1 , . . . , s k ) and any l-tuple (t 1 , . . . , t l ) with s 1 ≤ . . . ≤ s k < s k + r = t 1 ≤ . . . ≤ t l and arbitrary measurable functions g : R k → R, h : R l → R with g ∞ ≤ 1 and h ∞ ≤ 1, the following inequality is fulfilled:

|cov (g(X s 1 , . . . , X s k ), h(X t 1 , . . . , X t l ))| ≤ ψ(k, l, Lip g, Lip h) r .
Here Lip h denotes the Lipschitz modulus of continuity of h, that is,

Lip h = sup x =y |h(x) -h(y)| x -y l 1 , where z l 1 = i |z i |, and ψ : N 2 × R 2 + → [0, ∞) is an appropriate function.
Remark 1. In [START_REF] Bickel | A new mixing notion and functional central limit theorems for a sieve bootstrap in time series[END_REF], another type of weak dependence, called ν-mixing, was introduced. Similarly to Definition Remark 2. (Some classes of weak dependence) Specific functions ψ yield notions of weak dependence appropriate to describe various examples of models:

• κ-weak dependence for which ψ(u, v, a, b) = uvab; in this case we simply denote r as κ r . • κ (causal) weak dependence for which ψ(u, v, a, b) = vab; in this case we simply denote r as κ r . This is the causal counterpart of κ coefficients which we recall only for completeness. • η-weak dependence, ψ(u, v, a, b) = ua + vb; in this case we write r = η r for short.

• θ-weak dependence is a causal dependence which refers to ψ(u, v, a, b) = vb; we simply denote r = θ r [START_REF] Dedecker | A new covariance inequality and applications[END_REF]) for this causal counterpart of η coefficients. • λ-weak dependence ψ(u, v, a, b) = uvab + ua + vb; in this case we write r = λ r [START_REF] Doukhan | An invariance principle for new weakly dependent stationary models using sharp moment assumptions[END_REF]). Besides the fact that it includes η-and κ-weak dependence, this new notion of λ-weak dependence is convenient, for example, for Bernoulli shifts with associated inputs.

It turns out that the notion of weak dependence is more general than mixing and allows to treat, for example, also Markovian processes driven by discrete innovations as they appear with time series bootstrap. In the next section we consider as an instructive example linear autoregressive processes of finite order and a corresponding bootstrap version thereof. We will demonstrate that the desired property of weak dependence readily follows from a contraction property which is typical for such models under standard conditions on the parameters. The approach described there is also applicable to proving weak dependence for many other classes of processes. Section

Autoregressive processes and their bootstrap analogues

In this section we intend to give a gentle introduction to the basic ideas commonly used for verifying weak dependence. Most parts in this section are specialized to autoregressive processes of finite order and their bootstrap analogues.

We consider first a general real-valued stationary process (X t ) t∈Z . A simple and in many cases the most promising way of proving a property of weak dependence is via contraction arguments. For probability distributions P and Q on (R d , B d ) with finite mean, we define the metric

d(P, Q) = inf (X,Y ): X∼P,Y ∼Q E X -Y l 1 .
For d = 1 and the L 2 instead of the L 1 distance, we obtain Mallows distance; see [START_REF] Mallows | A note on asymptotic joint normality[END_REF]. It is well known that such distances are suitable for metrizing weak convergence, that is, d(P n , P ) -→ n→∞ 0 implies P n =⇒ P ; see e.g. [START_REF] Bickel | Some asymptotic theory for the bootstrap[END_REF]. Similar distances have also been used in the context of Markov processes to derive convergence of stationary distributions from convergence of the conditional distributions; see e.g. [START_REF] Dobrushin | Describing a system of random variables by conditional distributions[END_REF] and [START_REF] Neumann | Goodness-of-fit tests for Markovian time series models[END_REF]. The following lemma shows that closeness of the conditional distributions in the above metric gives rise to estimates for covariances.

Lemma 2.1. Suppose that (X t ) t∈Z is a real-valued stationary process. Furthermore, let

s 1 ≤ • • • ≤ s k ≤ t 1 ≤ • • • ≤ t l be
arbitrary and let g : R k → R and h : R l → R be measurable functions. Let (X t ) t∈Z be another version of the process, where

X s k = (X s k , X s k -1 , . . . ) is independent of X s k = (X s k , X s k -1 , . . . ). If g is bounded and E|h(X t 1 , . . . , X t l )| < ∞, then |cov (g(X s 1 , . . . , X s k ), h(X t 1 , . . . , X t l ))| ≤ g ∞ Lip h Ed P Xt 1 ,... ,Xt l |Xs k , P X t 1 ,... ,X t l |X s k .
This lemma shows that a property of weak dependence follows from a convergence of the conditional distributions as the time gap to the lagged variables tends to infinity. The latter property can often be shown by appropriate coupling arguments. Note that there is a close connection to the notion of τ -dependence introduced by [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF]. According to their Lemma 5, Ed(P Xt|Xs , P X t |X s ) is actually equal to their coefficient τ (t -s). Dedecker and Prieur used such coupling arguments to derive exponential inequalities and other interesting results, with applications to density estimation.

In the rest of this section we restrict our attention to a real-valued autoregressive process (X t ) t∈Z , which obeys the equation

X t = θ 1 X t-1 + • • • + θ p X t-p + ε t , t ∈ Z.
(2.1)

The innovations (ε t ) t∈Z are assumed to be independent and identically distributed with Eε t = 0. Furthermore, we make the standard assumption that the characteristic polynomial θ(z) = 1 -θ 1 z -

X s k is independent of X s k .
Note that (X t ) t∈Z can also be written as a linear process,

X t = ∞ k=0 α k ε t-k . Independence of X s k and X s k is equivalent to the fact that ε s k , ε s k -1 , . . . and ε s k , ε s k -1 , . . . are independent.
On the other hand, we have some freedom to couple the innovations after time s k . Here we only have to take care that both sequences (ε t ) t∈Z and (ε t ) t∈Z consist of independent random variables. A reasonably good coupling is obtained by feeding both processes after time s k with one and the same sequence of innovations, that is,

ε s k +1 = ε s k +1 , ε s k +2 = ε s k +2 , . . . . This gives that X t -X t = ∞ l=0 α t-s k +l (ε s k -l -ε s k -l ). (2.2) Since d(P Xt 1 ,... ,Xt l |Xs k , P X t 1 ,... ,X t l |X s k ) ≤ E l j=1 |X t j -X t j | X s k , X s k we obtain in conjunction with ( Lemma 2.2. Let (X t )
t∈Z and (X t ) t∈Z be two versions of the autoregressive process as described above. If E|ε 0 | < ∞, then

Ed P Xt 1 ,... ,Xt l |Xs k , P X t 1 ,... ,X t l |X s k ≤ 2 l K 1 1 -ρ ρ t 1 -s k E|ε 0 |.

Lemma

Corollary 2.1. Suppose that (X t ) t∈Z is a stationary process satisfying the above conditions. Furthermore, let

s 1 ≤ • • • ≤ s k ≤ t 1 ≤ • • • ≤ t l be arbitrary and let g : R k -→ R and h : R l -→ R be measurable functions. If g is bounded and E|h(X t 1 , . . . , X t l )| < ∞, then |cov (g(X s 1 , . . . , X s k ), h(X t 1 , . . . , X t l ))| ≤ g ∞ Lip h K 1 l ρ t 1 -s k ,
where

K 1 = 2 1 1-ρ K E|ε 0 |.
Now we define the autoregressive bootstrap. We assume that observations X 1-p , . . . , X n are available. Let θ n = ( θ n,1 , . . . , θ n,p ) be any consistent estimator of θ = (θ 1 , . . . , θ p ) , that is, θ n P -→ θ, as n → ∞. (The least squares and the Yule-Walker estimator are even √ n-consistent.) Let X t = (X t-1 , . . . , X t-p ) be the vector of the p lagged observations at time t. We define residuals

ε t = X t -X t θ n
and re-center them as

ε t = ε t -ε • ,
where ε • = (1/n) n t=1 ε t . Now we draw independent bootstrap innovations ε * t from the empirical distribution P n given by the ε t . A bootstrap version of the autoregressive process is now obtained as

X * t = θ n,1 X * t-1 + • • • + θ n,p X * t-p + ε * t . (2.3)
For simplicity, we assume that (X * t ) t∈Z is in its stationary regime. (This will be justified by (i) of the next lemma.) Before we state weak dependence of the bootstrap process, we show that it inherits those properties from the initial process which were used for proving weak dependence.

Lemma 2.3. Suppose that Eε 2 0 < ∞ and θ n P -→ θ are fulfilled.

(i) With a probability tending to 1, (X * t ) t∈Z can be written as a stationary causal linear process,

X * t = ∞ k=0 α n,k ε * t-k ,
where, for all > 0, there exists a K < ∞ such that, with ρ = (1 + )/ρ,

P | α n,k | ≤ K ρ k ∀k ∈ N -→ n→∞ 1. (ii) E(ε * t 2 | X 1-p , . . . , X n ) P -→ Eε 2 t .
Armed with the basic properties stated in Lemma Lemma 2.4. Suppose that the initial process (X t ) t∈Z satisfies the above conditions and that the bootstrap process (X * t ) t∈Z is in its stationary regime. Let (X * t ) t∈Z be another version of the bootstrap process, where

X * s k = (X * s k , X * s k -1 , . . . ) is (conditionally on X 1-p , . . . , X n ) independent of X * s k = (X * s k , X * s k -1 , . . . ).
For any > 0, let ρ = (1 + )/ρ and K < ∞ be an appropriate constant. Then there exists a sequence of events Ω n such that

P(Ω n ) -→ n→∞ 1 and if Ω n occurs, then Ed P X * t 1 ,... ,X * t l |X * s k , P X * t 1 ,... ,X * t l |X * s k ≤ 2 l K ρ t 1 -s k Eε 2 0 . From Lemma Corollary 2.2. Suppose that the conditions of Lemma Let s 1 ≤ • • • ≤ s k ≤ t 1 ≤ • • • ≤ t l be arbitrary and let g : R k -→ R and h : R l -→ R be measurable functions. If g is bounded and E|h(X * t 1 , . . . , X * t l )| < ∞, then cov g(X * s 1 , . . . , X * s k ), h(X * t 1 , . . . , X * t l ) ≤ g ∞ Lip h K 2 l ρ t 1 -s k ,
where

K 2 = 2 K Eε 2 0 .
Besides the useful property of weak dependence of the bootstrap process, the asymptotic validity of a bootstrap approximation requires that the (multivariate) stationary distributions of the bootstrap process converge to those of the initial process. Often, and in the case of the autoregressive bootstrap in particular, one has no direct access to these stationary distributions. However, according to Lemma 4.2 in [START_REF] Neumann | Goodness-of-fit tests for Markovian time series models[END_REF], convergence of the stationary distributions can be derived from an appropriate convergence of conditional distributions. The latter, however, follows directly from θ n P -→ θ and ε * Remark 3. Motivated by the desire to have some sort of mixing for a smoothed sieve bootstrap for linear processes, [START_REF] Bickel | A new mixing notion and functional central limit theorems for a sieve bootstrap in time series[END_REF] considered a condition called ν-mixing which is similar to the notion of weak dependence in our Definition

In contrast, the approach described here is fundamentally different. We intend to prove weak dependence for processes driven by innovations with a possibly discrete distribution and achieve this goal by exploiting a contraction property of the initial and the bootstrap process.

Remark 4. Arguing in the same way as above we could also establish the property of weak dependence for nonlinear autoregressive processes,

X t = m(X t ) + ε t , t ∈ Z,
where (ε t ) t∈Z is a sequence of independent and identically distributed innovations. If Lip m < 1, then we have obviously a contraction property being fulfilled which immediately yields weak dependence.

It is interesting to note that such a contraction property can still be proved if Lip m < 1 is not fulfilled. To this end, define the local Lipschitz modulus of continuity

∆(x) = sup y =x |m(y) -m(x)| |y -x|
and assume that

ρ = sup x E [∆(m(x) + ε 0 )] < 1.
Then

d P X t+k |Xt=x , P X t+k |Xt=y ≤ ρ k-1 • ∆(x) • |x -y|, (2.4)
which implies weak dependence by Lemma

Some examples of weakly dependent sequences

Note first that sums of independent weakly dependent processes admit the common weak dependence property where dependence coefficients are the sums of the initial ones. We now provide a non-exhaustive list of weakly dependent sequences with their weak dependence properties. Further examples may be found in Doukhan and Louhichi (1999). Let X = (X t ) t∈Z be a stationary process.

(1) If this process is either a Gaussian process or an associated process and lim t→∞ | cov(X 0 , X t )| = 0, then it is a κ-weakly dependent

process such that κ r = O sup t≥r | cov(X 0 , X t )| . It is also κ -weakly dependent with κ r = O   t≥r | cov(X 0 , X t )|   .
(2) ARM A(p, q) processes and more generally causal or non-causal linear processes: X = (X t ) t∈Z is a such that

X t = ∞ k=-∞ a k ξ t-k for t ∈ Z,
where (a k ) k∈Z ∈ R Z and (ξ t ) t∈Z is a sequence of independent and identically distributed random variables with Eξ t = 0. If a k = O(|k| -µ ) with µ > 1/2, then X is a η-weakly dependent process with η r = O 1 r µ-1/2 . In the general case of dependent innovations, properties of weak dependence are proved in [START_REF] Doukhan | An invariance principle for new weakly dependent stationary models using sharp moment assumptions[END_REF].

(3) GARCH(p, q) processes and more generally ARCH(∞) processes: X = (X t ) t∈Z is a such that

X t = ρ t • ξ t with ρ 2 t = b 0 + ∞ k=1 b k X 2 t-k for k ∈ Z,
with a sequence (b k ) k depending on the initial parameters in the case of a GARCH(p, q) process and a sequence (ξ t ) t∈Z of independent and • there exists C > 0 and µ ∈]0, 1[ such that ∀j ∈ N, 0 ≤ b j ≤ C • µ -j , then X is a θ-weakly dependent process with θ r = O(e -c √ r ) and c > 0 (this is the case of GARCH(p, q) processes).

identically distributed innovations. Then, if E(|ξ 0 | m ) < ∞,
• there exists C > 0 and Doukhan, Teyssière and Winant (2006) introduce vector valued LARCH(∞) models including the previous ones). ( 4) Causal bilinear processes introduced by [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] are such that

ν > 1 such that ∀k ∈ N, 0 ≤ b k ≤ C •k -ν , then X is a η-weakly dependent process with θ r = O r -ν+1 (
X t = ξ t a 0 + ∞ k=1 a k X t-k + c 0 + ∞ k=1 c k X t-k , for k ∈ Z.
Assume that there exists m ≥ 1 with

ξ 0 m < ∞ and ξ 0 m • ∞ k=1 |a k | + ∞ k=1 |c k | < 1. Then, if: • ∃K ∈ N such that ∀k > K, a k = c k = 0, or, ∃µ ∈]0, 1[ such that k |c k |µ -k ≤ 1 and ∀k ∈ N, 0 ≤ a k ≤ µ k , then X is a θ-weakly dependent process with θ r = O(e -c √ r ),
for some c > 0;

• ∀k ∈ N, c k ≥ 0, and ∃ν 1 > 2 and ∃ν 2 > 0 such that Doukhan, Teyssière and Winant (2006)). ( 5) Non-causal LARCH(∞) processes X = (X t ) t∈Z satisfying

a k = O(k -ν 1 ) and k c k k 1+ν 2 < ∞, then X is a θ-weakly dependent process with η r = O r log r d , d = max -(ν 1 -1); -ν 2 •δ δ+ν 2 •log 2 (see
X t = ξ t • a 0 + k∈Z\{0} a k X t-k , t ∈ Z,
where ξ 0 ∞ < ∞ (bounded random variables) and (a k ) k∈Z is a sequence of real numbers such that λ = ξ 0 ∞ • j =0 |a j | < 1 (stationarity condition). Assume that the sequence (a k ) k∈Z satisfies a k = O(|k| -µ ) with µ > 1, then X is a η-weakly dependent process with η r = O 1 r µ-1 (see Doukhan, Teyssière and Winant (2006)). ( 6) Causal and non-causal Volterra processes write as

X t = ∞ p=1 Y (p) t with Y (p) t = j 1 < j 2 < • • • < jp j 1 , . . . , jp ∈ Z a j 1 ,... ,jp ξ t-j 1 • • • ξ t-jp , for t ∈ Z. Assume ∞ p=0 j 1 < j 2 < • • • < jp j 1 , . . . , jp ∈ Z a j 1 ,... ,jp m ξ 0 p m < ∞
, with m > 0, and that there exists p 0 ∈ N \ {0} such that a j 1 ,... ,jp = 0 for p > p 0 . If [START_REF] Doukhan | A new weak dependence condition and application to moment inequalities[END_REF]). Finite order Volterra processes with dependent inputs are also considered in [START_REF] Doukhan | An invariance principle for new weakly dependent stationary models using sharp moment assumptions[END_REF]: again, η-weakly dependent innovations yields η-weak dependence and λ-weakly dependent innovations yields λ-weak dependence of the process. ( 7) Very general models are the causal or non-causal infinite memory processes X = (X t ) t∈Z such that X t = F (X t-1 , X t-2 , . . . ; ξ t ), and X t = F (X s , s = t; ξ t ),

a j 1 ,... ,jp = O max 1≤i≤p {|j i | -µ } with µ > 0, then X is a η-weakly de- pendent process with η r = O 1 r µ+1 (see
where the functions F defined either on R N\{0} × R or R Z\{0} × R satisfy

F (0; ξ 0 ) m < ∞, F ((x j ) j ; ξ 0 ) -F ((y j ) j ; ξ 0 ) m ≤ j =0 a j |x j -y j |,
with a = j =0 a j < 1. Then, works in progress by Doukhan and Wintenberger as well as Doukhan and Truquet, respectively, prove that a solution of the previous equations is stationary in L m and either θ-weakly dependent or η-weakly dependent with the following decay rate for the coefficients:

inf p≥1    a r/p + |j|>p a j    .
This provides the same rates as those already mentioned for the cases of ARCH(∞) or LARCH(∞) models.

Some probabilistic results

In this section, we present results derived under weak dependence which are of interest in probability and statistics. This collection clearly shows that the notion of weak dependence, although being more general than mixing, allows one to prove results very similar to those in the mixing case.

4.1. Donsker invariance principle. We consider a stationary, zero mean, and real valued sequence (X t ) t∈Z such that

µ = E|X 0 | m < ∞,
for a real number m > 2.

(4.1)

We also set

σ 2 = t∈Z cov(X 0 , X t ) = t∈Z EX 0 X t , (4.2)
W denotes the standard Brownian motion and

W n (t) = 1 √ n [nt] i=1 X i , for t ∈ [0, 1], n ≥ 1. (4.3)
We now present versions of the Donsker weak invariance principle under weak dependence assumptions.

Theorem 4.1 (Donsker type results). Assume that the zero mean stationary process (X t ) t∈Z satisfies ( λ-dependence. The process is λ-weakly dependent and satisfies λ r = O(r -λ ) (as r ↑ ∞) for λ > 4 + 2/(m -2). θ-dependence. The process is θ-weakly dependent and satisfies θ r = O(r -θ ) (as r ↑ ∞) for θ > 1 + 1/(m -2).

Remark 5. The result for κ -weak dependence is obtained in [START_REF] Bulinski | Strong invariance principle for dependent random fields[END_REF]. Results under κ-and λ-weak dependences are proved in [START_REF] Doukhan | An invariance principle for new weakly dependent stationary models using sharp moment assumptions[END_REF]; note that η-weak dependence implies λ-weak dependence and the Donsker principle then holds under the same decay rate for the coefficients. The result for θ-weak dependence is due to [START_REF] Dedecker | A new covariance inequality and applications[END_REF]. A few comments on these results are now in order:

• The difference of the above conditions under κ and κ assumptions is natural.

The observed loss under κ-dependence is explained by the fact that κ -weakly dependent sequences satisfy κ r ≥ s≥r κ s . This simple bound directly follows from the definitions.

• Actually, it is enough to assume the θ-weak dependence inequality for any positive integer u and only for v = 1. Hence, for any 1-bounded function g from R u to R and any 1-bounded Lipschitz function

h from R to R with Lips- chitz coefficient Lip(h), it is enough to assume cov g X i 1 , . . . , X iu , h X iu+i ≤ θ i Lip(h), for any u-tuple i 1 ≤ i 2 ≤ • • • ≤ i u .
4.2. Empirical process. Let (X t ) t∈Z a real-valued stationary process. We use a quantile transform to obtain that the marginal distribution of this sequence is the uniform law on [0, 1]. The empirical process of the sequence (X t ) t∈Z at time n is defined as

1 √ n E n (x)
where

E n (x) = n k=1 I 1 (X k ≤x) -P(X k ≤ x) .
Note that E n = n (F n -F ) if F n and F denote the empirical distribution function and the marginal distribution function, respectively. We consider the following convergence result in the Skohorod space D([0, 1]) when the sample size n tends to infinity:

1 √ n E n (x) d -→ B(x).
Here ( B(x)) x∈[0,1] is the dependent analogue of a Brownian bridge, that is B denotes a centered Gaussian process with covariance given by

E B(x) B(y) = ∞ k=-∞ (P(X 0 ≤ x, X k ≤ y) -P(X 0 ≤ x)P(X k ≤ y)) . (4.4)
Note that for independent sequences with a marginal distribution function F , this turns into B(x) = B(x) for some standard Brownian bridge B; this justifies the name of generalized Brownian bridge. We have:

Theorem 4.2. Suppose that the stationary sequence (X t ) t∈Z has uniform marginal distribution and is either η-weakly dependent with η r = O(r -15/2-ν ), or κ-weakly dependent κ r = O(r -5-ν ) , for some ν > 0. Then the following empirical functional convergence holds true in the Skohorod space of realvalued càdlàg functions on the unit interval, D([0, 1]):

1 √ n E n (x) d -→ B(x).
Remark 6. Under strong mixing, the condition ∞ n=0 α n < ∞ implies fidi convergence. The empirical functional convergence holds if, in addition, for some a > 1, α n = O(n -a ) (see [START_REF] Rio | Théorie asymptotique pour des processus aléatoires faiblement dépendants[END_REF]). In an absolutely regular framework, [START_REF] Doukhan | Invariance principle for the empirical measure of a weakly dependent process[END_REF] obtain the empirical functional convergence when, for some a > 2, β n = O(n -1 (log n) -a ). [START_REF] Shao | Weak convergence for weighted empirical processes of dependent sequences[END_REF] and [START_REF] Shao | Maximal inequalities for partial sums of ρ-mixing sequences[END_REF] obtain the empirical functional convergence theorem when the maximal correlation coefficients satisfy the condition

∞ n=0 ρ (2 n ) < ∞.
To prove the result, we introduce the following dependence condition for a stationary sequence (X t ) t∈Z :

sup f ∈F |cov (f (X t 1 )f (X t 2 ), f (X t 3 )f (X t 4 ))| ≤ r , (4.5)
where

F = {x → I 1 s<x≤t , for s, t ∈ [0, 1]}, 0 ≤ t 1 ≤ t 2 ≤ t 3 ≤ t 4 and r = t 3 -t 2 (
in this case a weak dependence condition holds for a class of functions R u → R working only with the values u = 1 or 2).

Proposition 4.1. Let (X n ) be a stationary sequence such that ( 4.3. Central limit theorems. First central limit theorems for weakly dependent sequences were given by Corollary A in Doukhan and Louhichi (1999) and Theorem 1 in Coulon-Prieur and [START_REF] Coulon-Prieur | A triangular central limit theorem under a new weak dependence condition[END_REF]. While the former result is for sequences of stationary random variables, the latter one is tailor-made for triangular arrays of asymptotically sparse random variables as they appear with kernel density estimators. Using their notion of ν-mixing [START_REF] Bickel | A new mixing notion and functional central limit theorems for a sieve bootstrap in time series[END_REF] proved a CLT for linear processes of infinite order and their (smoothed) bootstrap counterparts. Below we state a central limit theorem for general triangular schemes of weakly dependent random variables. Note that the applicability of a central limit theorem to bootstrap processes requires some robustness in the parameters of the underlying process since these parameters have to be estimated when it comes to the bootstrap. A result for a triangular scheme is therefore appropriate since the involved random variables have changing properties as n varies, but also for a fixed n. An interesting aspect of the following result is that no moment condition beyond Lindeberg's is required. Its proof uses the variant of Rio of the classical Lindeberg method. Suppose that (X n,k ) k∈Z , n ∈ N, is a triangular scheme of (row-wise) stationary random variables with EX n,k = 0 and EX 2 n,k ≤ C < ∞. Furthermore, we assume that

1 n n k=1 EX 2 n,k I(|X n,k |/ √ n > ) -→ n→∞ 0 (4.6)
holds for all > 0 and that

σ 2 n = k∈Z EX n,0 X n,k -→ n→∞ σ 2 ∈ [0, ∞). (4.7)
For n ≥ n 0 , there exists a monotonously nonincreasing and summable sequence (θ r ) r∈N such that, for all indices s 1 < s 2 < • • • < s u < s u + r = t 1 ≤ t 2 , the following upper bounds for covariances hold true: for all measurable and quadratic integrable functions g : R u → R,

|cov (g(X n,s 1 , . . . , X n,su ), X n,t 1 )| ≤ Eg 2 (X n,s 1 , . . . , X n,su ) θ r , (4.8)
for all measurable and bounded functions g : R u -→ R,

|cov (g(X n,s 1 , . . . , X n,su ), X n,t 1 X n,t 2 )| ≤ g ∞ θ r , (4.9)
where g ∞ = sup x∈R u |f (x)|. Then

1 √ n (X n,1 + • • • + X n,n ) d -→ N (0, σ 2 ).
Remark 7. Conditions ( The following multivariate central limit theorem, simply applicable to triangular schemes of weakly dependent random vectors, was derived in [START_REF] Bardet | A Lindeberg central limit theorem for dependent processes and its statistical applications[END_REF]. It fits better results where dependence does not play any role in the limit.

Theorem 4.4. (Theorem 1 in [START_REF] Bardet | A Lindeberg central limit theorem for dependent processes and its statistical applications[END_REF])

Suppose that (X n,k ) k∈N , n ∈ N, is a triangular scheme of zero mean random vectors with values in R d . Assume that there exists a positive definite matrix Σ such that

n k=1 Cov(X n,k ) -→ n→∞ Σ and that for each > 0 n k=1 E( X n,k 2 I 1 { X n,k > } ) -→ n→∞ 0,
where • denotes the Euclidean norm. Furthermore, we assume the following condition is satisfied:

n k=2 cov(e it (X n,1 +•••+X n,k-1 , e it X n,k ) -→ n→∞ 0.
Then, as n → ∞,

n k=1 X n,k d -→ N d (0 d , Σ).
Remark. One common point of those two results is the use of the classical Lindeberg assumption. Note that this assumption is more often checked by using a higher order moment condition. A main distinction between those two results is that while the first one yields direct application to partial sums, the second one is more adapted to triangular arrays where limit does not write as a sum. In this setting [START_REF] Doukhan | An invariance principle for new weakly dependent stationary models using sharp moment assumptions[END_REF] use Bernstein blocks to prove a CLT for partial sums.

4.4. Probability and moment inequalities. In this section we state inequalities of Bernstein and Rosenthal type. In the case of mixing, such inequalities can be easily derived by the well-known technique of replacing dependent blocks of random variables (separated an appropriate gap) by independent ones and then using the classical inequalities from the independent case; see for example [START_REF] Doukhan | Mixing: Properties and Examples[END_REF] and [START_REF] Rio | Théorie asymptotique pour des processus aléatoires faiblement dépendants[END_REF]. The concept of weak dependence is particularly suitable for deriving upper estimates for the cumulants of sums of random variables which give rise to rather sharp inequalities of Bernstein and Rosenthal type which are analogous to those in the independent case. Based on a Rosenthal-type inequality, a first inequality of Bernstein-type was obtained by Doukhan and Louhichi (1999), however, with √ t instead of t 2 in the exponent. [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] proved a Bennett inequality which can possibly be used to derive also a Bernstein inequality. A first Bernstein inequality with var(X 1 + • • • + X n ) in the asymptotically leading term of the denominator of the exponent has been derived in [START_REF] Kallabis | An exponential inequality under weak dependence[END_REF], under a weak dependence condition tailor-made for causal processes with an exponential decay of the coefficients of weak dependence. The following result is a generalization which is also applicable to possibly non-causal processes with a not necessarily exponential decay of the coefficients of weak dependence.

Theorem 4.5. (Theorem 1 in [START_REF] Doukhan | Probability and moment inequalities for sums of weakly dependent random variables, with applications[END_REF])

Suppose that X 1 , . . . , X n are real-valued random variables defined on a probability space (Ω, A, P) with EX i = 0 and P(|X i | ≤ M ) = 1, for all i = 1, . . . , n and some M < ∞. Let Ψ : N 2 → N be one of the following functions:

(a) Ψ(u, v) = 2v, (b) Ψ(u, v) = u + v, (c) Ψ(u, v) = uv, (d) Ψ(u, v) = α(u + v) + (1 -α)uv,
for some α ∈ (0, 1).

We assume that there exist constants K, L 1 , L 2 < ∞, µ ≥ 0, and a nonincreasing sequence of real coefficients (ρ(n)) n≥0 such that, for all u-tuples (s 1 , . . . , s u ) and all v-tuples (t 1 , . . . , t

v ) with 1 ≤ s 1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v ≤ n the following inequality is fulfilled: |cov (X s 1 • • • X su , X t 1 • • • X tv )| ≤ K 2 M u+v-2 Ψ(u, v) ρ(t 1 -s u ), (4.10) where ∞ s=0 (s + 1) k ρ(s) ≤ L 1 L k 2 (k!) µ ∀k ≥ 0. (4.11) Then P n k=1 X k ≥ t ≤ exp - t 2 /2 A n + B 1/(µ+2) n t (2µ+3)/(µ+2)
, (4.12)

where A n can be chosen as any number greater than or equal to σ 2 n and

B n = 2 (K ∨ M ) L 2 2 4+µ n K 2 L 1 A n ∨ 1 .
A first Rosenthal-type inequality for weakly dependent random variables was derived by Doukhan and Louhichi (1999) via direct expansions of the moments of even order. Unfortunately, the variance of the sum did not explicitly show up in their bound. Using cumulant bounds in conjunction with Leonov and Shiryaev's formula the following tighter moment inequality was obtained in [START_REF] Doukhan | Probability and moment inequalities for sums of weakly dependent random variables, with applications[END_REF].

Theorem 4.6. (Theorem 3 in [START_REF] Doukhan | Probability and moment inequalities for sums of weakly dependent random variables, with applications[END_REF])

Suppose that X 1 , . . . , X n are real-valued random variables on a probability space (Ω, A, P) with zero mean and let p be a positive integer. We assume that there exist constants K, M < ∞, and a non-increasing sequence of real coefficients (ρ(n)) n≥0 such that, for all u-tuples (s 1 , . . . , s u ) and all v-tuples (t 1 , . . . , t v ) with 1 ≤ s

1 ≤ • • • ≤ s u ≤ t 1 ≤ • • • ≤ t v ≤ n and u + v ≤ p, condition (

Proofs

Proof of Lemma Let, for simplicity of notation, s k = 0. Then cov (g(X s 1 , . . . , X s k ), h(X t 1 , . . . , X t l ))

= E [g(X s 1 , . . . , X s k ) (E(h(X t 1 , . . . , X t l ) | X s 1 , . . . , X s k ) -Eh(X t 1 , . . . , X t l ))] . Now we obtain by Jensen's inequality for conditional expectations that |cov (g(X s 1 , . . . , X s k ), h(X t 1 , . . . , X t

l ))| ≤ E [|g(X s 1 , . . . , X s k )| • |E(h(X t 1 , . . . , X t l ) | X 0 ) -Eh(X t 1 , . . . , X t l )|] ≤ E |g(X s 1 , . . . , X s k )| • E(h(X t 1 , . . . , X t l ) | X 0 ) -E(h(X t 1 , . . . , X t l ) | X 0 ) ≤ Lip h E   |g(X s 1 , . . . , X s k )| • E   l j=1 |X t j -X t j | | X 0 , X 0     .
The assertion follows now immediately. (5.1)

Proof of Lemma

Thus there exists a stationary solution to equation ( Proof of inequality ( We prove this result by a simple coupling argument. Let (X t ) t∈Z and (X t ) t∈Z be two versions of the autoregressive process with X t = x and X t = y. We contruct a coupling simply by feeding both processes after time t with the same sequence of innovations ε t+1 , ε t+2 , . . . , that is, we have X t+l+1 = m(X t+l ) + ε t+l+1 and X t+l+1 = m(X t+l ) + ε t+l+1 (l ≥ 0). We obtain by this construction that

|X t+k -X t+k | ≤ ∆(X t+k-1 )|X t+k-1 -X t+k-1 | ≤ • • • ≤ ∆(X t+k-1 ) • • • ∆(X t+1 )∆(x)|x -y|.
Therefore, we obtain

d P X t+k |Xt=x , P X t+k |Xt=y ≤ E|X t+k -X t+k | ≤ E ( ∆(X t+k-1 ) • • • ∆(X t+1 )| X t = x) • ∆(x) • |x -y|.
Using the Markov property we see that

E ( ∆(X t+k-1 ) • • • ∆(X t+1 )| X t = x) = E [E(∆(X t+k-1 ) | X t+k-2 ) • • • E(∆(X t+1 ) | X t = x)] ≤ ρ k-1 ,
which yields the assertion.

  t . Therefore, consistency of the autoregressive bootstrap can be shown by simple arguments which were already used for proving weak dependence of the bootstrap; for details see Section 4.2 in[START_REF] Neumann | Goodness-of-fit tests for Markovian time series models[END_REF].

  Theorem 4.3. (Theorem 6.1 in[START_REF] Neumann | Goodness-of-fit tests for Markovian time series models[END_REF]

  The assertion follows immediately from ( Proof of Lemma (i) Let ξ n,1 , . . . , ξ n,p be the roots of the characteristic poly-nomial θ(z) = 1 -θ n,1 z -• • • -θ n,p z p of the bootstrap process. Since θ P -→ θ we obtain by Theorem 1.4 in Marden (1949) that min{| ξ n,1 |, . . . , | ξ n,p |} P -→ ρ = min{|ξ 1 |, . . . , |ξ p |}. Therefore, we have, for any > 0, that P min{| ξ n,1 |, . . . , | ξ n,p |} ≥ ρ/(1 + /2) -→ n→∞ 1.

  Xt 1 ,... ,Xt l |Xs k , P

		• • • -θ p z p has no zero in the
	unit circle. It is well known that there exists then a stationary solution to
	the model equation (
	Convergence of Ed(P X t 1	,... ,X t l	|X s k ) as t 1 -s k → ∞ can
	now be shown by a simple coupling argument. For this purpose, we consider
	a second (stationary) version of the autoregressive process, (X t ) t∈Z , where