Pedram Masoudi
email: pedram.masoudi@univ-rennes1.fr

Mohammad Ataei

Tahar Aïfa
email: tahar.aifa@univ-rennes1.fr

Hossein Memarian
email: memarian@ut.ac.ir

Methodology-selection by Fuzzy Analytic Hierarchy Process for Studying Net Pays

Keywords: net pay methodologies, fuzzy AHP for net pay, purposes

Net pay intervals are conventionally determined, applying cut-off values on geological well-logs. Recently developed methodologies utilize more complicated algorithms: Bayesian classifier, artificial neural network and Dempster-Shafer theory. The outputs of these methodologies are not completely compatible. It rises this question that which method should be recommended in each situation: (i) industrial use, (ii) research goal and (iii) general situations. Fuzzy analytical hierarchy process is used here to compare effectiveness of the four net pay determination methods in each three situations. Six criteria were defined: precision, generalization ability, fuzziness, simplicity of methodological concepts, user-friendly and speed of the algorithm. For the precision and generalization ability, mean squared error of training and generalization data are used, respectively. Mean squared error and speed (inverse of time) of the algorithm are continues variables, and provide quantitative comparison. While qualitative comparison is done for the criteria of simplicity and userfriendly. For the criterion of the fuzziness, a ranking, i.e. categorical variable, is used based on the number of classes that the classifier could provide. The comparison is done based on the results of net pay determination (the four methods) on sandy Burgan and carbonated Mishrif reservoirs, Iranian offshore oil-fields. The results show that from viewpoint of general situation, Bayesian and Dempster-Shafer-based methods are the best. Artificial neural network and Dempster-Shafer are the most suitable methods for industrial mode, and Bayesian and artificial neural network are the best methods for research applications. Finally, cut-off methodology is never prioritized.

INTRODUCTION

In cases that there are multiple solutions for a problem, how to select the most optimum methodology is an important question. Recently, novel net pay determination methods have been proposed that let user decide the process of evaluating productive zones by his own. Conventionally, pay zone identification methodologies were based on cut-off approaches, which simply applies two to three cut-offs on petrophysical parameters of shale percent, porosity and water saturation [1, 2 and 3]. However, it is also possible to study productive zones from other perspectives: reservoir engineering [START_REF] Masoudi | Developing a Method for Identification of Net Zones Using Log Data and Diffusivity Equation[END_REF], probabilistic view of Bayesian classifier [START_REF] Masoudi | Application of Bayesian in determining productive zones by well log data in oil wells[END_REF], using fuzzy aggregation for merging outputs of other net pay identification procedures [START_REF] Masoudi | Identifying productive zones of the Sarvak formation by integrating outputs of different classification methods[END_REF] or by the means of other intelligent methods as Dempster's rule of combination [START_REF] Masoudi | Net Pay Determination by Dempster Rule of Combination, Case Study on Iranian Offshore Oil Fields[END_REF] or Artificial Neural Networks (ANN) [8, 9 and 10].

Resolving a question by multiple approaches might be considered as a repetitive job, e.g. in above literature, net pay detection is addressed using six different methodologies. However, we believe that multiple approaches can provide options in different conditions. Here, fuzzy Analytic Hierarchy Process (fuzzy AHP) is used as a decision making tool to prioritize introduced methodologies in three cases: general situation, industrial use and research divisions.

AHP is a powerful and well-known process for ranking options/choices due to predefined criteria and goals; meanwhile, considering hierarchy of goals-criteriaoptions. From managerial viewpoint, AHP provides a link between "human's thoughts" and "decision making", through the tunnel of pairwise comparison process. If we use fuzzy numbers instead of crisp numbers for comparing options or criteria, this methodology is called fuzzy AHP. Using fuzzy numbers means that the process of pairwise comparison contains uncertainty. In another word, there is a vagueness or gloominess in our comparison procedure. Here, the aim of using AHP is to prioritize different net pay methodologies due to multiple goals (research, industry and general usage of methods).

WHAT IS NET PAY?

In general, net pay zone is a vague concept. It could be simply defined by words as "hydrocarbon producing horizon"; however, a mathematical and technical definition requires more considerations. In order to call a drilled interval a producing zone, it should firstly have a proper lithology, capable in storing and conducting hydrocarbon. Secondly, this reservoir rock should be hydrocarbonbearing, i.e. containing some minimum amount of hydrocarbon. Conventionally, it has been done by applying some cut-offs on petrophysical properties: shale percentage, porosity and water saturation.

A general cut-off-based approach in determining net pays in conventional reservoirs consists of three stages. At first, clean reservoir (net sand) have to be distinguished from the gross rock. It could be simply done by setting a constraint on log of shale volume. Range of cut-off of shale volume varies from 30% to 50% for all conventional reservoirs. Then, porous media -capacity for storage-have to be specified within net sand. For sandy reservoirs, this range is between 6% and 8%, and for carbonates it varies from 4% to 5%. Finally, within net sand, selecting those intervals that contain water saturation below cut-off value (usually 50% to 60%) is the most appropriate place for oil production [1 and 2]. The graphical definition is presented in Figure 1. Besides conventional cut-off-based method, recent works have defined net pays by well-test results, which provide less analytical and more data-driven-based method, i.e. Bayesian, ANN and fuzzy logic [9 and 11]. In a more precise language, net pays should be able in producing oil or gas during reservoir's life span, with the highest possible recovery rate.

METHODOLOGY

A. AHP in Simple Language

Analytic Hierarchy Process (AHP) is a decision making technique that organizes human's thoughts and psyches by comparing alternatives due to criteria, and comparing criteria due to goals. In this methodology, goals are described at first, then, few criteria should be defined due to predefined goals. In some cases, sub-criteria should be defined as well. Finally, alternatives are listed at the bottom of graph (Figure 2) [12 and 13]. Description of AHP method is well presented in [13 and 14]. Here, we are going to compare four net pay determination methodologies, i.e. Cut-off, Bayesian, ANN and Dempster-Shafer Theory (DST), due to six criteria under each specific goal. After constructing the graph, comparison matrices should be constructed by the user, based on his personal judgment or decision of a team, i.e. pairwise comparison should be carried out at each level (alternatives and criteria), separately. Then, comparison matrices of each level should be constructed. For each node of criterion, we will have one squared matrix with size of four. Four is the number of alternatives that will be compared to each other due to each criterion. For comparing criteria due to goals, three matrices (with size of six by six) should be constructed too. Six is the number of criteria, and three is the number of goals. Each matrix is for pairwise comparison between alternatives due to a specific criterion or between criteria due to a goal. When a matrix is for comparing alternatives due to a criterion, the element aij is the degree of relative preference of i th alternative (i.e. Ai) to j th alternative (i.e. Aj). The same concept is for comparison matrix of criteria due to goals.

For constructing comparison matrix, usually, an odd digit (1, 3, 5, 7 and 9) is assigned to each element, expressing the relative degree of importance, e.g. 1 means similarity (there is no priority); 3 means one of them is a little more important than the other alternative; 5 means more important; 7 means much higher preference; and 9 has the meaning of extremely more important or absolute preference. It should be mentioned that comparison is not limited to odd numbers at all. Even digits could be used as well.

Pairwise comparison or assigning digits to the matrix elements could be done by an expert, a group of experts, measurement or a mixture of both (like here). But four principles should be considered in constructing comparison matrices [12 and 13]:

• Principle 1-Reciprocal Condition: if Ai (i.e. i th alternative) is n times more important than Aj; then, Aj is 1/n times less important than Ai. Because of this property, comparison matrix is a "reciprocal matrix".

• Principle 2-Homogeneity: relative importance of Ai to Aj should be limited. In mathematical language, aij could not be zero or infinite.

• Principle 3-Dependency: each element is linearly dependent on higher-level elements, and this dependency could be continuous upward towards the goal.

• Principle 4-Expectations: when a modification occurs in the structure of the graph, evaluation process should be redone.

In addition to above mentioned principles, another logical assumption has to be considered in constructing a consistent matrix. This principle is based on hypothetical syllogism, i.e. if the statement Q could be inferred from the statement P (i.e. P => Q); also if the statement T could be inferred from the statement Q (i.e. Q => T); therefore the statement T could be inferred from the statement P itself (i.e. P => T). This is an important reasoning in classical logic. In the literature, this property is called "cardinal" consistency property [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF]: when Ai is aij times more important than Aj; and if Aj is ajk times more important than Ak; then, Ak is aij × ajk times more important than Ai.

In the next step, weights of each criterion or alternative should be calculated by one of (logarithmic) least square method, eigenvectors or approximate methods. The final stage is calculating the final score by summing multiplication of corresponding weights in the hierarchy, i.e. multiplying weight of each alternative due to each criterion by the weight of corresponding criterion, and summing all multiplications to come up to the final score. Score the higher, the more optimum alternative to choose [12 and 16].

B. Why Fuzzy AHP?

How much are we sure about assigned numbers in pairwise comparison? In another word, is it precise enough fixing each element to a specific and fixed digit? In fact, because we are not sure about a single number (vagueness and inaccuracy in weights and rating), it is better to assign an interval or a fuzzy number (i.e. using fuzzy AHP or fuzzy TOPSIS) in an advanced mode [START_REF] Torfi | Fuzzy AHP to determine the relative weights of evaluation criteria and Fuzzy TOPSIS to rank the alternatives[END_REF]. [START_REF] Saaty | Decision-making with the AHP: Why is the principal eigenvector necessary[END_REF] has not approved using fuzzy numbers in the AHP process. His reasoning is due to fuzzy nature of the AHP, i.e. priorities have fuzzy meaning themselves [START_REF] Saaty | Decision-making with the AHP: Why is the principal eigenvector necessary[END_REF]. In spite of critics of Saaty about fuzzy-AHP, its application is growing (see below abstracted literature review) in different domains (place prioritizing, project selection, e-commerce, etc.). In fact, till there is no prove for using fuzzy AHP instead of AHP or vice-versa, both methods are valid. Authors support using fuzzy AHP in cases of vagueness in rating due to [START_REF] Torfi | Fuzzy AHP to determine the relative weights of evaluation criteria and Fuzzy TOPSIS to rank the alternatives[END_REF].

Fuzzy AHP has been developed in economic and business researches: customer requirements [START_REF] Kwong | A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment[END_REF], place prioritizing [START_REF] Kuo | A decision support system for selecting convenience store location through integration of fuzzy AHP and artificial neural network[END_REF], project selection [START_REF] Enea | Project selection by constrained fuzzy AHP[END_REF], e-commerce [START_REF] Kong | Applying fuzzy analytic hierarchy process to evaluate success factors of e-commerce[END_REF], comparing simple AHP and fuzzy AHP [17 and 22]. In newer publications, the use of fuzzy AHP is reported in textile industry [START_REF] Cebeci | Fuzzy AHP-based decision support system for selecting ERP systems in textile industry by using balanced scorecard[END_REF], and also in the project of Turkish national identity card [START_REF] Catak | Fuzzy Analytic Hierarchy Based DBMS Selection In Turkish National Identity Card Management Project[END_REF].

Fuzzy number is a fuzzy set on real numbers that should follow three properties: 1-must be a normal fuzzy set, 2-Its alpha cut must be a closed interval, 3-The support of the fuzzy set must be bounded [START_REF] Klir | Fuzzy Sets and Fuzzy Logic, Theory and Applications[END_REF]. One of the mostly used fuzzy numbers in engineering applications is triangular fuzzy number that corresponding non-fuzzy number is at its height, and two specific lower and upper bounds have also to be fixed. Triangular fuzzy number of [1/2,2,3] is an example of asymmetric fuzzy number, corresponding to none-fuzzy number of 2 (Figure 3). Defining fuzzy numbers is mostly based on engineering judgment. In extreme case that there is absolute certainty, fuzzy numbers would have no range, therefore would be exactly the same as simple crisp numbers. But in cases that there are large uncertainties, range of fuzzy numbers could be defined larger, subsequently large overlaps with neighbouring numbers, indicating uncertain space. In this work, defined fuzzy numbers are shown in Table I and Figure 4. They have both the following properties: 1inverse of fuzzy number one is equal to itself, like in real numbers, 2-all integer fuzzy numbers are symmetric, except fuzzy number one. All other parts of the algorithm of Fuzzy AHP are the same as conventional AHP [START_REF] Ataei | Fuzzy Multi-Criteria Decision Making[END_REF].

TABLE I. NUMBERS AND THEIR REVERSE IN THE FORM OF FUZZY NUMBERS (TRIANGULAR). [L,M,U] SHOWS TRIANGULAR FUZZY NUMBER

OF "M" WHEN "L" IS LOWER LIMIT AND "U" IS UPPER LIMIT.

Integer nonfuzzy Number

Fuzzy Number

Inverse of Integer Non-fuzzy Number

Fuzzy Number

1 [1/2,1,2] 2 [1,2,3] 1/2 [1/3,1/2,1] 3 [2,3,4] 1/3 [1/4,1/3,

C. Inconsistency Rate (IR)

There is an algorithm for checking consistency of comparison matrices. In this method, Inconsistency Index (II) can easily be calculated, after finding an eigenvector, corresponding to the largest eigenvalue, using relation 1.

Then dividing it by Random Inconsistency Index (RII) (relation 2), we will reach the Inconsistency Rate (IR) through relation 3. It is suggested in the literature that 0.1 is an appropriate cut-off for IR [START_REF] Ataei | Multi-Criteria Decision Making[END_REF]. There are some other approximate methods for calculating Max λ without using eigenvector that readers are referred to the literature [12 and 13].

1 - - n n λ = II Max (1) 1 - - n n λ = RII Max (2) RII II = IR ( 3
)
Where n is dimension of matrix; Max λ is maximum eigenvalue; Max λ is average of expected value of Max λ [27].

PAIRWISE COMPARISON

D. Comparing Alternatives

Based on six decision criteria, four net pay detection methods are going to be compared in carbonate reservoir rock of Mishrif and sandy Burgan reservoir (both in Iran). The six decision criteria are: precision, generalization ability, fuzziness, simplicity of methodology (science concepts), user-friendly and speed of the algorithm (Table II). Comparison, resulted in six reciprocal matrices (size of four by four), using the previously introduced network structure of AHP. II, the four methods were compared from perspective of each criterion, and the IR was calculated for each matrix. When IR is lower than 0.10, the matrix is assumed to be consistent [12 and 28]; otherwise, the matrix is reconstructed. Then, corresponding weights are calculated, and presented at the bottom of each matrix (Tables III). Comparison matrix of precision with: IR=0.0076<0.10

Based on Table

Cut-off ANN Bayes DST Cut-off [1/2,1,2] [1/5,1/4,1/3] [1/4,1/3,1/2] [1/2,1,2] ANN [3,4,5] [1/2,1,2] [1,2,3] [3,4,5] Bayes [2,3,4] [1/3,1/2,1] [1/2,1,2] [2,3,4] DST [1/2,1,2] [1/5,1/4,1/3] [1/4,1/3,1/2] [1/2,

E. Comparing Criteria

For building comparison matrix of criteria, priority of criteria to each other should be discussed, considering the goal. The authors found it difficult to prioritize precision, generalization ability and fuzziness to each other. We categorized these three criteria as Precision Group (PG). Also, ranking three criteria of simplicity of method, userfriendly and speed was impossible in general. We call these three criteria as Application Group (AG).

Criteria of PG address those metrics showing accuracy of net pay determination methods, i.e. precision shows precision of output in training dataset, generalization reveals precision of output in generalization dataset; here, a well of which the methodology is not trained within. Due to inherit of fuzziness of geosciences data: the fuzzier output, the closer to the reality. Therefore, it is believed that the degree of fuzziness of output is a measure of accuracy too.

Criteria of AG consist of attributes related to application of methods in pay zone determination. Understanding the science background of the method, applying it simply (user-friendly) and speed of the method. Here, no priority is set in comparing these attributes, because there is no specific goal in general mode. Prioritizing should be done case-based, e.g. in research applications, the easier and simpler the method, the more understandable methodology and path-way. Hence, it is easier to implement it in complex situations. The speed of running the algorithm becomes important in two situations: 1-in real-time industrial logs; 2-in high-dimensional or giant data.

1) General Mode

Based on equity of criteria of PG and its relative importance to AG, equity of criteria of AG too, the below matrix is constructed (Table IV). In the comparison matrix, it is assumed that PG is five times more important than AG. The IR of the matrix is calculated to be zero, i.e. absolutely consistent.

Then, final scores of each alternative are calculated in Table V. Bayes and DST were successful in getting the maximum score of 0.26, then ANN method with 0.25. Cutoff method got the lowest score: 0.23.

2) Industrial Mode

For the case of industrial applications, another comparison is accomplished, and Table IV is constructed. In this table, again PG is considered to be more important than AG due to importance of accurate and precise output. However, fuzziness is not as important as other criteria since most industrial decisions are made in categorical and non-fuzzy situations. In AG, being user-friendly is set prior to speed and simplicity of the methodology because a nonexperienced user in the industry should be able in running the algorithm too. Also, speed is prioritized to simplicity of the method for real-time applications.

Final scores in industrial mode show that both DST and ANN have the same scores (Table V). Bayes and cut-off methods were ranked in third and fourth stages, respectively.

3) Research Mode

For research issues, another comparison is fulfilled, and Table IV is constructed. In this table, again PG is considered to be more important than AG due to importance of having accurate and precise outputs. However, fuzziness is considered to be more important than other criteria because non-fuzzy output is completely incompatible with inherit of geosciences datasets. Within AG, simplicity of methodology is set prior to speed and user-friendly, because understanding the methodology fully is really critical for researchers to be able in evaluating complex models analytically. Also, speed is prioritized to user-friendly for coping with big data.

It is due to high importance of fuzziness in research mode (Table IV) that ANN and Bayes have been scored 0.26. DST scored just below ANN and Bayes: 0.25. Finally, cut-off was ranked fourth due to the lowest score: 0.22 (Table VI).

CONCLUSION

In this work, four previously developed methodologies for net pay determination were compared to each other within a fuzzy AHP. Six criteria for comparison were: precision, generalization ability, fuzziness, simplicity of methodology (science concepts), user-friendly and speed of the algorithm. First, pairwise comparison was done between alternatives due to each criterion. Then, these six criteria were compared to each other three times from three perspectives: from the viewpoint of the industry, standpoint of researchers and in general mode. In the last step, net pay identification methods were ranked due to their final scores (Table VI). None of ANN, Bayes and DST methods could be prioritized to each other, ANN is not the best method in general mode, Bayes is not the best in industrial mode and DST is not the best in research mode. However, cut-off method ranked 3 rd in all three modes. It shows that having different is not useless at all.

Figure 1 .

 1 Figure 1. Classical definition of nets; modified following [2].

Figure 2 .

 2 Figure 2. AHP structure, goals at top, alternatives at bottom, and criteria as mid-layer between goals and alternatives.

Figure 3 .

 3 Figure 3. Triangular fuzzy number of [1/2,2,3]

TABLE II .

 II COMPARISON OF METHODS, APPLIED ON TWO RESERVOIRS. ONE OF THEM IS CARBONATE RESERVOIR OF MISHRIF, THE OTHER ISSANDY BURGAN RESERVOIR[START_REF] Masoudi | Developing a New Data Fusion Approach for Fuzzy Net Pay Determination[END_REF].

			Precision			Generalization		Fuzziness	Simplicity of	Method	User-friendly	Speed
	Cut-off	acceptable in	carbonate,	poor in		acceptable		discrete	very simple	too simple	time	consuming
	ANN	86% in	carbonate;	94% in		not checked		discrete	complex	difficult	rather time	consuming
	Bayes	74% in	carbonate;	81% in	fuzzy: too	weak	crisp: very	discrete	simple	difficult	quick
	DST		acceptable			very good		continuous	very complex	simple	quick

TABLE III .

 III RECONSTRUCTED COMPARISON MATRICES DUE TO CRITERIA.

TABLE IV .

 IV COMPARISON MATRIX FOR CRITERIA IN DIFFERENT MODES. WEIGHTS ARE CALCULATED AND BROUGHT AT THE BOTTOM OF EACH MATRIX.Comparison matrix for general mode with Inconsistency Rate: IR=0.0000

		Precision Group			Application Group
	Precision	Generalization	Fuzziness	Simplicity of Method	User-friendly	Speed

ACKNOWLEDGMENT

This work has been supported by the Center for International Scientific Studies & Collaboration (CISSC) and French Embassy in Iran through PHC Gundishapur Program. We want to thank Mahta Gholizadeh Ansari for sharing her experience in applying and verifying fuzzy AHP methodology.


Paul Doukhan
email: doukhan@ensae.fr

Hélène Madre
email: madre@ensae.fr

Mathieu Rosenbaum
email: rosenbaum@ensae.fr

Weak dependence for infinite ARCH-type bilinear models

Keywords: Time series, ARCH models, GARCH models, weak dependence, Markov chain. AMS codes : 60G10, 60F17, 62M10, 91B84

introduced ARCH-type bilinear models for their specific long range dependence properties. We rather consider weak dependence properties of these models. The computation of mixing coefficients for such models does not look as an accessible objective. So, we resort to the notion of weak dependence introduced by Doukhan and Louhichi (1999), whose use seems more relevant here. The decay rate of the weak dependence coefficients sequence is established under different specifications of the model coefficients. This implies various limit theorems and asymptotics for statistical procedures. We also derive bounds for the joint densities of this model in the case of regular inputs.

Introduction and motivations 1.Infinite ARCH-type bilinear models

A vast literature is devoted to the study of conditionally heteroskedastic models. One of the best-known model is the GARCH model (Generalized Autoregressive Conditionally Heteroskedastic) introduced by Engle [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF] and Bollerslev [START_REF] Bollerslev | Generalized autoregressive conditional heteroscedasticity[END_REF]. A usual GARCH(p, q) model can be written

r t = σ t ξ t , σ 2 t = α 0 + p i=1 β i σ 2 t-i + q j=1 α j r 2 t-j
where α 0 ≥ 0, β i ≥ 0, α j ≥ 0, p ≥ 0, q ≥ 0 are the model parameters and the (ξ t ) t are independent and identically distributed (iid). If the β i are null, we have an ARCH(q) model which can be extended in ARCH(∞) model, see Robinson [START_REF] Robinson | Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression[END_REF], Giraitis and Robinson [START_REF] Giraitis | Whittle estimation of ARCH Models[END_REF], Robinson and Zaffaroni [START_REF] Robinson | Pseudo-maximum likelihood estimation in ARCH(∞) models[END_REF], Kokozska and Leipus [START_REF] Kokozska | Change point estimation in ARCH models[END_REF], Kazakevicius and Leipus [START_REF] Kazakevicius | On stationarity in the ARCH(∞) model[END_REF]. These models are often used in finance because their properties are close to the properties observed on empirical financial data such as heavy tails, volatility clustering, white noise behaviour or autocorrelation of the squared series. To reproduce other properties of the empirical data such as leverage effect, a lot of extensions of the GARCH model have been introduced as EGARCH or T GARCH, see Zakoïan [START_REF] Zakoian | Threshold Heteroskedastic Models[END_REF], El babsiri and Zakoïan [START_REF] El Babsiri | Contemporaneous Asymetry in GARCH Processes[END_REF].

In this paper, we study weak dependence properties of ARCH-type bilinear models introduced by Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]. An ARCH-type bilinear model can be written

X t = ε t a + ∞ j=1 a j X t-j + b 0 + ∞ j=1 b j X t-j , (1)
where a, (a j ) and (b j ) are real coefficients and the (ε t ) t∈Z are iid centered. We usually consider b 0 = 0. ARCH(∞), GARCH(p, q) and LARCH models are particular cases of the bilinear models, see Giraitis, Leipus and Surgailis [START_REF] Giraitis | Recent advances in ARCH modelling[END_REF], Giraitis, Kokoszka, Leipus and Teyssière [START_REF] Giraitis | Rescaled variance an related tests for long memory in volatility and levels[END_REF]. Quote that Doukhan, Teyssière and Winant [START_REF] Doukhan | An ARCH(∞) vector valued model[END_REF] introduced a very general vector valued version of this model. Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] prove that under restrictions, there is a unique stationary solution for these models. This solution has a chaotic expansion.

The following assumption is necessary to define this solution.

Assumption H. The (ε t ) t∈Z are iid centered, IE|ε 1 | < ∞ and the power series

A(z) = ∞ j=1 a j z j and B(z) = ∞ j=1 b j z j exist for |z| ≤ 1.
We define ξ L p = (IE|ξ| p ) 1/p and h p p = ( ∞ j=0 |h j | p ) 1/p , for p ∈ [1, ∞), with usual extension to the supremum norm if p = +∞. We set

G(z) = {1 -B(z)} -1 = ∞ j=0 g j z j and H(z) = A(z){1 -B(z)} -1 = ∞ j=1 h j z j .
Let (a b) j = j i=0 a i b j-i denote the convolution and a

(n) j = a j 1(1 ≤ j ≤ n)
, where 1 is the indicator function. Giraitis and Surgailis established in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] the following proposition Proposition 1.1 (Giraitis, Surgailis) Assume that the (ε t ) are iid, centered at expectation and such that ε t 2 = 1. If (a (n) -a) g 2 and (b (n) -b) g 2 tend to zero as n goes to infinity and if h 2 < 1, g 2 < ∞, then there exists a solution of equation (1) which is unique, strictly stationary and given by

X t = a ∞ k=1 sk<•••<s1≤t g t-s1 h s1-s2 • • • h sk-1-sk ε s1 • • • ε sk .
(2)

Weak dependence

To our knowledge, there is no study of the weak dependence properties of ARCH or GARCH type models with infinite number of coefficients. In fact, the use of mixing coefficients is very technical and necessitates additional regularity assumptions, see Doukhan [START_REF] Doukhan | Mixing : Properties and Examples[END_REF]. In the case of finite memory ARCH models, Mokkadem derives in [START_REF] Mokkadem | Propriétés de mélange des processus autorégressifs polynomiaux[END_REF] the absolute regularity properties of such models. An extension to infinite memory case seems quite doubtful because of Andrew's example of a non mixing first order autoregressive process, see [START_REF] Doukhan | Mixing : Properties and Examples[END_REF]. We add that mixing conditions also necessitate some regularity properties of the innovation process. In order to derive limit theorems for functionals of such models, we prove in this paper that a causal version of the weak dependence property introduced by Doukhan and Louhichi in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] holds. Indeed, such weak dependence conditions is a simple alternative to mixing. It also yields all kinds of limit theorems. We recall the definition of θ-weak dependence introduced by Doukhan and Louhichi [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] and Dedecker and Doukhan [START_REF] Dedecker | A new covariance inegality and applications[END_REF] :

Definition 1.2 (X n ) n∈Z is θ-weakly dependent if there is a sequence (θ i ) i such that lim i θ i = 0 and (X n ) n satisfies Cov f X i1 , . . . , X iu , g X j1 , . . . , X jv ≤ θ i v f ∞ Lip (g) for all u, v, i 1 ≤ i 2 ≤ • • • ≤ i u ≤ i u +i ≤ j 1 ≤ j 2 ≤ • • • ≤ j v and any measurable functions f : R u → R and g : R v → R such that f ∞ , Lip (g) < ∞.
This condition implies limit theorems such as -Donsker invariance principle, see Dedecker and Doukhan [START_REF] Dedecker | A new covariance inegality and applications[END_REF] :

1 √ n [nt] k=1 X k D[0,1] ----→ n→∞ σW t ,
where W t is a standard Brownian motion and σ 2 = ∞ k=-∞ Cov(X 0 , X k ) ≥ 0 is well defined, if for some positive δ,

IE|X 0 | 2+δ < ∞ and i>0 i 1/δ θ i < ∞.
-Empirical central limit theorem, see Prieur [START_REF] Prieur | An empirical functional central limit theorem for weakly dependent sequences[END_REF] :

1 √ n n k=1 {1(X k ≤ x) -F (x)} D[ R] ---→ n→∞ Z(x),
where {Z(x)} x∈ R is the centered Gaussian process with covariance function

Γ(x, y) = ∞ k=-∞ Cov[1(X 0 ≤ x), 1(X k ≤ y)], if θ i = O(i -a ) for a > 2 + 2 √
2 and the marginal distribution of X 0 is atomless.

-Other statistical asymptotic features are considered in [START_REF] Ango Nze | Weak dependence, models and applications to econometrics[END_REF], [START_REF] Ango Nze | Weak dependence beyond mixing and asymptotics for non parametric regression[END_REF], [START_REF] Dedecker | A new covariance inegality and applications[END_REF], [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF], [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] and [START_REF] Doukhan | Functional estimation for weakly dependent stationary time series[END_REF]. See also section 3.

Under restrictions on the model coefficients, we derive explicit bounds for these weak dependence coefficients and thus directly obtain asymptotic results for this process.

We first establish sufficient conditions for the existence of the chaotic solution (2) in L p . Indeed in proposition 1.1, the existence condition in L 2 is based on the coefficients (h j ). We prefer conditions on the coefficients (a j ) and (b j ). Thus we extend the result of existence of the chaotic expansion to L p . After this, we establish bounds for the decay of the weak dependence coefficients sequence in different cases. We define under H the following assumption :

Assumption H . Let h = h 1 ε 1 | L 1 , h < 1 and g 1 < ∞.
Remark. Assumption H is granted as soon as the coefficients (b j ) are non negative and

a 1 ε 1 L 1 + b 1 < 1, see section 4.1.
We now define our different specifications :

(a) Markovian case : H, η = J j=1 IE|a j ε 1 + b j | 1/J < 1 with J such that ∀j > J, a j = b j = 0. (b) Geometric decay : H, H and ∃α ∈]0, 1[, β > 1 : j β j |b j | ≤ 1 and |a j | ≤ α j .
Remark. Our definition of the geometric case gets for particular case a more classical definition where we suppose there exists 0 < ζ < 1, and

0 < λ < 1-ζ ζ such that, for all j ≥ 1 we have, 0 ≤ b j ≤ λζ j . (c) Riemannian decay : H, H , b 1 < 1 and ∃β > 1, α > 1 : j j β |b j | = B < ∞, and j j α |a j | = A < ∞.
In the following, we shall systematically refer to the previous conditions (a), (b) and (c).

Applications of our results are given in section 3. We also prove in section 3 that if all the coefficients are non negative, then each vector (X 1 , . . . , X n ) admits a density conditional on the past of the process if this is the case for the innovations. We also show that we can uniformly control the density of any couple (X 1 , X i ). Such results are very useful for functional estimation as stressed in [START_REF] Ango Nze | Weak dependence beyond mixing and asymptotics for non parametric regression[END_REF], [START_REF] Doukhan | Functional estimation for weakly dependent stationary time series[END_REF] and [START_REF] Bardet | A Lindeberg central limit theorem for dependent processes and its statistical applications[END_REF]. Section 4 contains the proofs.

2 Properties of bilinear models

2.1 Existence of the solution in L p (Ω, A, IP) Theorem 2.1 If the (ε t ) are iid and belong to L p (Ω, A, IP), (i) a sufficient condition of existence in L p (Ω, A, IP) of expansion (2) is g 1 < ∞ and h 1 ε 1 L p < 1.
(ii) The preceding condition is implied by the non negativity of the coefficients (b j ) and the inequality a

1 ε L p + b 1 < 1.
Remark. The result extends to dependent innovations (ε t ). Existence in

L 1 (Ω, A, IP) of the chaotic expansion (2) holds if sup t ε t L ∞ ≤ M , g 1 < ∞ and M h 1 < 1.
A condition on the coefficients (a j ) and (b j ) is the following : if the (b j ) are non negative, sup t ε t L ∞ ≤ M and a 1 M + b 1 < 1 then the expression (2) exists in L 1 (Ω, A, IP).

Weak dependence coefficients

From now on, we assume that the solution exists in L 1 (Ω, A, IP), is stationary and is given by equation ( 2). .

3 Applications Theorem 2.2 leads to various applications in the bilinear context. We first precise below conditions to get Donsker invariance principle and empirical central limit theorem. A last subsection devoted to conditional densities is more specific to our models. The results of this section are particularly relevant for functional estimation. Notice also that theorem 2.2 enables to obtain exponential inequalities, see [START_REF] Doukhan | A Bernstein type inequality for times series[END_REF]. Results for stochastic algorithms, Whittle estimator, and copula can be respectively found in [START_REF] Brandière | Dependent noise for stochastic algorithms[END_REF], [START_REF] Bardet | A uniform central limit theorem for the periodogram and its applications to Whittle parametric estimation for weakly dependent time series[END_REF] and [START_REF] Doukhan | Copula of a stationary vector valued weakly dependent process[END_REF]. For a general review of these properties, see [START_REF] Dedecker | Weak dependence : models, theory and applications[END_REF].

Donsker invariance principle

Let P δ be the following condition : for some δ > 0, IE|X 0 | 2+δ < ∞. Using the results of Dedecker and Doukhan [START_REF] Dedecker | A new covariance inegality and applications[END_REF] together with theorem 2.2, we easily obtained the Donsker invariance principle (see section 1.2) under the following assumptions (a) in the Markovian case : P δ and η < 1, (b) under geometric decay : P δ , (c) under Riemannian decay : P δ and d > 1 + 1/δ with d defined in theorem 2.2.

Note in particular that in the important case of LARCH models (b j = 0), in the Markovian case, the second condition is reduced to IE|ε 1 | J j=1 |a j | < 1 and under Riemannian decay to α > 1 + 1/δ.

Empirical central limit theorem

Let P be the following condition : the marginal distribtion of X 0 is atomless. The results of Prieur together with theorem 2.2 enable us to get the empirical central limit theorem (see section 1.2) under the following assumptions (i) in the Markovian case : P and η < 1, (ii) under geometric decay : P , (iii) under Riemannian decay :

d > 2 + 2 √ 2 with d defined in theorem 2.2.
Note also that for LARCH models, in the Markovian case the second condition is reduced to IE|ε 1 | J j=1 |a j | < 1 and under Riemannian decay to α > 2+2 √ 2.

Conditional densities

We give here a useful result for the density of (X 1 , . . . , X n ).

Theorem 3.1 (Density of n-th marginals) We define

A t = a + ∞ j=t a j X t-j , B t = ∞ j=t b j X t-j and C i = A i + i-1 k=1 a k x i-k , with C 1 = A 1 .
Assume all variables and coefficients are non negative. If the ε t are independent with marginal density f εt , then (X 1 , . . . , X n ) has a density L(x 1 , . . . , x n ) conditionally to the past of the process. Forgetting the defi-nition set, we have

L(x 1 , . . . , x n ) = n i=1 1 |C i | f εi x i -b 1 x i-1 -b 2 x i-2 -• • • -b i-1 x 1 -B i C i .
Corollary 3.2 (Control of the density) Under the same assumptions as in theorem 3.1, if a is different from zero and if the densities of the (ε t ) are all bounded by M , then, for all (x 1 , . . . , x n ),

L(x 1 , . . . , x n ) ≤ M a n .
Corollary 3.3 (Density of a couple) Under the same assumptions as in theorem 3.1, and if the (ε t ) are iid with density f , then the density

p i of the couple (X 1 , X i ), satisfies p i ∞ ≤ f 2 ∞ /A 1 for all i ∈ Z Z.
Those lemmas are useful respectively for subsampling and functional estimation, see [START_REF] Ango Nze | Weak dependence beyond mixing and asymptotics for non parametric regression[END_REF], [START_REF] Doukhan | Functional estimation for weakly dependent stationary time series[END_REF] and [START_REF] Bardet | A Lindeberg central limit theorem for dependent processes and its statistical applications[END_REF]. For example, a standard kernel estimate of the density is classically proved to have variance ∼ f (x) K 2 (s)ds/nh n (with kernel function K, density f and bandwidth h n ) with corollary 3.3 and an additional dependence assumption θ r = O(r -a ) for a > 3.

Proofs

We give in this section the proofs of theorem 2.1, theorem 2.2 and theorem 3.1. In the following, c denotes a constant that may vary from line to line.

Proof of theorem 2.1

We begin the proof by a useful lemma Lemma 4.1 Assume that the coefficients (b j ) are non negative and b 1 < 1, then the coefficients (g j ) are non negative and

g 1 = (1 -b 1 ) -1 .
Proof of lemma 4.1. Since the (b j ) are non negative and b 1 < 1, a classical result shows that the development in power series +∞ j=0 g j z j of the function G(z) = {1 -B(z)} -1 has a radius bigger than 1. Moreover, after direct computations, we get the non negativity of the (g j ). Hence, as G is increasing on [0, 1], we have for all positive integer n and for all z ∈ [0, 1]

n j=0 g j z j ≤ G(z) ≤ G(1).
Consequently, +∞ j=0 g j ≤ G(1). We conclude by a continuity argument.

We now prove theorem 2.1. We use the normal convergence in L p of the series defined by (2), indeed

∞ k=1 sk<•••<s1≤t g t-s1 h s1-s2 • • • h sk-1-sk ε s1 • • • ε sk L p ≤ ∞ k=1 sk<•••<s1≤t |g t-s1 h s1-s2 • • • h sk-1-sk | ε s1 • • • ε sk L p ≤ g 1 k h k-1 1 ε k L p . Hence h 1 ε 1 L p < 1 implies X t L p < ∞. Since H(z) = A(z)G(z), we derive h 1 ≤ a 1 g 1 . Non negativity of the (b j ) implies g i ≥ 0. Thus sup |z|<1 |G(z)| = sup |z|<1 ∞ i=0 g i z i = sup |z|<1,z>0 ∞ i=0 g i z i = ∞ i=0 g i = g 1 . Hence, h 1 ≤ a 1 sup |z|<1 |1 -B(z)| -1 ≤ a 1 inf |z|<1 |1 -B(z)| -1 . Now j b j < 1 implies h 1 ≤ a 1 (1 -j b j ) -1 ≤ a 1 1-b 1 . Finally, if a 1 ε 1 L p + b 1 < 1, then ε L p h 1 < 1. 4.2 Proof of theorem 2.2
Markovian case (a). We use the general Markov chain theory. Write

X n = M (X n-1 , . . . , X n-J , ε n ). Then Y n = (X n , . . . , X n-J+1 ) is a Mar- kov chain with Y n = F (Y n-1 , ε n ) where x = (x 1 , . . . , x J ) and F (x, ε) = (M (x, ε), x 1 , . . . , x J-1 ). Thus IE|M (x, ε 1 ) -M (y, ε 1 )| ≤ IE J j=1 (a j ε 1 + b j )(y j -x j ) ≤ max j |y j -x j | IE J j=1 |a j ε 1 + b j | .
We define x = max 1≤j≤J η j-1 |x j |. We have

F (x, ε 1 ) -F (y, ε 1 ) ≤ max |M (x, ε 1 ) -M (y, ε 1 )| , max 1≤i<J η i |x i -y i | IE F (x, ε 1 ) -F (y, ε 1 ) ≤ max η J max 1≤i≤J (|x i -y i |) , max 1≤i<J (η i |x i -y i |) . Finally, IE F (x, ε 1 ) -F (y, ε 1 ) ≤ η x -y . ( 3
)
We then use the following lemma which is a vectorial extension of a result of Doukhan and Louhichi [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF].

Lemma 4.2 Assume Y n = (X n , . . . , X n-J+1 ) is a Markov chain with Y n = F (Y n-1 , ε n ) and the (ε n ) are iid with IE|ε 1 | < +∞. Then, if equation (3) holds, θ r = O(η r ) as r → ∞. Proof of lemma 4.2. For f : IR u → IR with f ∞ ≤ 1, g : IR v → IR, Lip g < ∞, i 1 ≤ • • • ≤ i u ≤ i u + r ≤ n 1 ≤ • • • ≤ n v and i u -i 1 > J, we set X i = (X i1 , .
. . , X iu ) and X n = (X n1 , . . . , X nv ). We have

Cov[f (X i ), g(X n )] = f (x i1 , . . . , x iu ){IEg(X y u n ) -IEg(X n )}dP Xi 1 ,Xi 1 +1...,Xiu ,
where y u = (x iu , x iu-1 , . . . , x iu-J+1 ) and X yu n denotes the vector

X n knowing that Y iu = y u . Now, it is clear that |IEg(X y u n ) -IEg(X n )| ≤ µ(dỹ u )IE|g(X y u n ) -g(X ỹu n )| ≤ Lip (g) v z=1 µ(dỹ u )IE|X y u nz -X ỹu nz |.
From inequality (3) and using that η < 1 , we get

IE Y y u nz -Y ỹu nz ≤ η nz-iu y u -ỹu ≤ η r y u -ỹu and consequently IE|X y u nz -X ỹu nz | ≤ η r y u -ỹu .
Finally, we get

dP Xi 1 ,Xi 1 +1...,Xiu µ(dỹ u )IE|X y u nz -X ỹu nz | ≤ η r dP Xi 1 ,Xi 1 +1...,Xiu µ(dỹ u ) y u -ỹu ≤ η r dP Xi 1 ,Xi 1 +1...,Xiu µ(dỹ u ) i |y u i -ỹu i | ≤ cη r .
An explicit bound on θ r follows. In the cases (b) and (c), we shall need lemmas describing the behavior of the coefficients (g j ) and (h j ) involved by expansion [START_REF] Ango Nze | Weak dependence beyond mixing and asymptotics for non parametric regression[END_REF]. We derive this behavior from the decay rates of the initial parameters (a j ) and (b j ). The study of the coefficients (g j ) et (h j ) aims at controlling the tails of the coefficients series. We begin by some useful lemmas.

Lemma 4.3 Let γ J = max j>J |g j |. Then |g k | ≤ γ J b 1 + g 1 k j=k-J |b j |, ∀k ∈ N, ∀J ∈ {1, . . . , k -1}. Proof of lemma 4.3. By definition of G, G(z){1 -B(z)} = ∞ k=0 α k z k = 1. Put b 0 = -1, then 1 -B(z) = -∞ j=0 b j z j thus α k = k j=0 g j b k-j
. By identification : α k = δ 0k . Thus we get recursive equations on the coefficients

g j : α 0 = -g 0 b 0 ⇒ g 0 = 1. Then, for k ≥ 1, g k = k-1 j=0 g j b k-j . For all J ∈ {1, . . . , k -1}, g k = k-J-1 j=1 g k-j b j + k j=k-J g k-j b j ≤ γ J k-J-1 j=1 |b j | + k j=k-J |g k-j ||b j | ≤ γ J b 1 + g 1 k j=k-J |b j |.
We give now an explicit bound for the coefficients in the geometric case.

∀ 1 < β 1 < β, γ k = O(β -k 1 ) as k → ∞.
Proof of lemma 4.4. The series ∞ j=1 b j r j has a radius bigger than β. Thus, the development in power series of {1 -B(z)} -1 has a radius of at least β. Consequently, for all 0 ≤ r < β the sequence (g j r j ) j is bounded and so, there exists M r > 0 such that g j ≤ M r r -j for all j.

We now turn to the tails of the series generated by (g j ) and (h j ).

Lemma 4.5 (Control of q K thanks to r K ) Let r K = ∞ k=K+1 |g k | and q K = ∞ k=K+1 |h k |. For all J < K, q K ≤ g 1 l≥K+1-J |a l | + 2 r J a 1 .
Proof of lemma 4.5. We have

q K ≤ ∞ k=K+1 k j=0 |a k-j g j | ≤ ∞ j=0 |g j | ∞ k=max(K+1,j+1) |a k-j | ≤ K j=0 |g j | ∞ k=K+1 |a k-j | + ∞ j=K+1 |g j | ∞ k=j+1 |a k-j |.
We now control each term,

∞ j=K+1 |g j | ∞ k=j+1 |a k-j | ≤ r K a 1
and for all J < K,

K j=0 |g j | ∞ k=K+1 |a k-j | = J j=0 |g j | ∞ k=K+1 |a k-j | + K j=J+1 |g j | ∞ k=K+1 |a k-j | ≤ g 1 l≥K+1-J |a l | + r J a 1 .
Lemma 4.6 (Controls of r K and q K ) -Geometric case (b) :

r K = O(β -K 1 ) and q K = O α -K log β 1 log(α/β 1 )
as K → ∞.

-Riemannian case (c) :

r K = O K (1-β) log ρ (1-β) log 2+log ρ and q K = O K max( (1-β) log ρ (1-β) log 2+log ρ ,-α) as K → ∞, with ρ = 1 1+ 1-b 1 B < 1.
Proof of lemma 4.6, (b). As |g j | ≤ cβ -j 1 , we easily get

r K = O(β -K 1 )
and q K ≤ c(e (K-J) log α + e -J log β1 ).

We take J = K log α log(α/β1) and we obtain

q K = O α -K log β 1 log(α/β 1 )
as K → ∞.

Proof of lemma 4.6, (c). We have

r K ≤ ∞ k=K+1 k-1 j=0 |g j b k-j | ≤ ∞ j=0 |g j | ∞ k=max(K+1,j+1) |b k-j | ≤ K j=0 |g j | ∞ k=K+1 |b k-j | + ∞ j=K+1 |g j | ∞ k=j+1 |b k-j | ≤ K j=0 |g j | ∞ k=K+1-j |b k | + r K b 1 . Thus, r K (1 -b 1 ) ≤ K j=0 |g j | ∞ k=K+1-j |b k |. Moreover, ∞ k=K+1-j |b k | ≤ ∞ k=K+1-j |b k |k β (K + 1 -j) -β ≤ B(K -j) -β .
Then, for all J < K,

r K 1 -b 1 B ≤ K j=0 |g j |(K -j) -β .
Consequently,

r K 1 -b 1 B ≤ J j=0 |g K-j |j -β + K j=J+1 |g K-j |j -β ≤ r K-J -r K + g 1 K j=J+1 j -β ≤ r K-J -r K + g 1 K 1-β -J 1-β 1 -β .
In particular, we get

r 2K 1 + 1 -b 1 B ≤ r K + g 1 1 -β K 1-β (2 1-β -1).
Thus we derive for all j the inequality r 2 j+1 ≤ ρ r 2 j + γ 2 j(1-β) , with 0 < ρ < 1 and γ > 0. By induction,

0 ≤ r 2 j+1 ≤ j k=0 γρ k 2 (1-β)(j-k) + ρ j g 1 .
We now control the first term. We have

j k=0 γρ k 2 (j-k)β = j k=0 γρ j-k 2 k(1-β) = J k=0 γρ j-k 2 k(1-β) + j k=J+1 γρ j-k 2 k(1-β) ≤ γ 1 -2 1-β (ρ j-J + 2 J(1-β) ).
We balance both terms putting J = j log ρ (1-β) log 2+log ρ and we get

r 2 j = O 2 j (1-β) log ρ (1-β) log 2+log ρ as j → ∞.
Let K such that 2 j < K ≤ 2 j+1 , we have 0 ≤ r K ≤ r 2 j . Finally,

r K = O K (1-β) log ρ (1-β) log 2+log ρ , as K → ∞.
Using that for all J < K,

l≥K+1-J |a l | ≤ A(K -J) -α , taking J = K/2 , we get q K = O K (1-β) log ρ (1-β) log 2+log ρ + K -α as K → ∞, which concludes.
Lemma 4.7 (Bounding θ) For all r, L, J > 0 such that LJ < r, θ r ≤ c( hL + r J + q J ).

Proof of lemma 4.7. The chaotic expansion (2) writes

X t = a ∞ l=1 ∞ j1=0 ∞ j2=1 • • • ∞ jl=1 g j1 h j2 • • • h jl ε t-j1 • • • ε t-(j1+•••+jl) . (4)
Consider the vectors X i = (X i1 , . . . , X iu ), and X j = (X j1 , . . . , X jv ) where j 1 -i u ≥ r. Take f : R u → R, g : R v → R two functions such that Lip (g) < ∞ and f ∞ ≤ 1. We define X * j = (X * j1 , . . . , X * jv ), where X * t denotes the sums for which l stops at L and j 1 , . . . , j l stop at J in the expansion (4). Note that X * j and X i are independent if L and J satisfy LJ < r, thus

|Cov[f (X i ), g(X j )]| ≤ |Cov[f (X i ), g(X j ) -g(X * j )]| + |Cov[f (X i ), g(X * j )]|.
The second term vanishes by independence. For simplicity, we forget the constant value a. We have

|Cov[f (X i ), g(X j ) -g(X * j )]| ≤ 2 f ∞ E|g(X j ) -g(X * j )| ≤ 2 Lip (g) f ∞ v k=1 E|X jk -X * jk | ≤ 2vLip (g) f ∞ E|X 0 -X * 0 |,
by the stationarity of (X t , X * t ). Thus, we may set

θ r = E|X 0 -X * 0,r,J,L |,
where

X * 0,r,J,L = L l=1 J j1=0 • • • J jl=0 g j1 h j2 • • • h jl ε -j1 • • • ε -(j1+•••+jl) , if JL < r.
Thus, we finally obtain

θ r ≤ ∞ l=L+1 g 1 hl-1 ε L 1 + ∞ l=1 r J hl-1 ε L 1 + ∞ l=1 q J g 1 (l -1) hl-2 ε 2 L 1 ≤ c( hL + r J + q J ).
End of the proof of the theorem 2.2. The end of the proof is dedicated to explicit this bound in terms of (a j ) and (b j ) decay rates.

-Geometric case (b) :

θ r = O( hL + β -J 1 + m J ) = O( hL + m J ) such that JL < r, as J, L → ∞. Consequently, θ r = O e - √ L log( h)J log(m) e - q L log( h) J log(m) + e -q J log(m) L log( h) = O e - √ L log( h)J log(m) = O e - √ r log( h) log(m) as r → ∞.
-Riemannian case (c) : Take L = C log J . We get θ r = O(J m (α,β,ρ) ). Consider now the largest possible integer J = J r such that J C log J < r. Since J r ∼ r C log r , we finally obtain θ r = O r log r m (α,β,ρ) , as r → ∞. We work conditional on the past of {X s , s ≤ 0}. We set

M =       1 -a 1 ε n -b 1 -a 2 ε n -b 2 . . . -a n-1 ε n -b n-1 0 1 -a 1 ε n-1 -b 1 . . . -a n-2 ε n-1 -b n-2 . . . . . . . . . . . . . . . 0 0 0 1 -a 1 ε 2 -b 1 0 0 0 0 1       , then, M     X n X n-1 . . . X 1     =     A n ε n + B n A n-1 ε n-1 + B n-1 . . . A 1 ε 1 + B 1     ,
which can be written

             ε 1 = X1-B1 A1 ε 2 = X2-b1X1-B2 a1X1+A2 . . ε n = Xn-b1Xn-1-b2Xn-2-•••-bn-1X1-Bn a1Xn-1+a2Xn-2+•••+an-1X1+An
, and we may set (ε 1 , . . . , ε n ) = φ(X 1 , . . . , X n ). Now IEg(X 1 , X 2 , . . . , X n ) = g{φ -1 (u 1 , . . . , u n )}f ε1 (u 1 )

• • • f εn (u n )du 1 • • • du n .
We put (u 1 , . . . , u n ) = φ(x 1 , . . . , x n ). The Jacobian is diagonal, hence we compute

∂u 1 ∂x 1 = C -1 1 , ∂u 2 ∂x 2 = C -1 2 , . . . , ∂u n ∂x n = C -1 n .
Proof of corollary 3.3. We prove the result for the density of the couple (X 1 , X 4 ), we can prove the general result the same way. We have p(x 1 , x 4 ) = L(x 1 , . . . , x 4 )dx 2 dx 3

≤ f 2 |C 1 | 1 |C 2 C 3 | f x 2 -b 1 x 1 -B 2 C 2 f x 3 -b 1 x 2 -b 2 x 1 -B 3 C 3 dx 2 dx 3 .
Hence we put :

u = C -1 2 (x 2 -b 1 x 1 -B 2 ) and v = C -1 3 (x 3 -b 1 x 2 -b 2 x 1 -B 3
). Direct computations give that the Jacobian matrix is diagonal and that its absolute value writes

|C 2 C 3 |, thus p(x 1 , x 4 ) ≤ f 2 A1 f (u)f (v)dudv ≤ f 2 A1 .

  Let f : R d → R, we define Lip (f ) = sup (x1,...,xd) =(y1,...,yd) |f (x 1 , . . . , x d ) -f (y 1 , . . . , y d )| |x 1 -y 1 | + • • • + |x d -y d | .

Theorem 2 . 2 (

 22 Bounds on θ) (a) In the Markovian case, θ r = O(η r ) as r → ∞. (b) Under geometric decay, for any β 1 such that 1 < β 1 < β, θ r = O e -c √ r as r → ∞, where c = (log h log m) 1/2 with m = α log β 1 log(α/β 1 ) . (c) Under Riemannian decay, θ r = O r log r -d as r → ∞, where d = min (β-1) log ρ (1-β) log 2+log ρ , α with ρ = 1 1+ 1-b 1 B

Lemma 4 . 4 (

 44 Control of γ k under geometric decay) -Geometric case (b) :

θ

  r = O hL + J max (1-β) log ρ (1-β) log 2+log ρ ,-α ,where J, L, r → ∞ in such a way that JL < r. We definem (α,β,ρ) = max (1 -β) log ρ (1 -β) log 2 + log ρ , -α and C = m (α,β,ρ) log h .

4. 3

 3 Proof of theorem 3.1