Juan Antonio Cordero

Aôut

Adjacency Persistency in OSPF MANET

Keywords: MANET, Mobile, Ad hoc, Network, Routing, OSPF, MPR, Persistency, Adjacency, Flooding, Synchronization, Link-State MANET, mobilité, ad hoc, réseu, routage, OSPF, MPR, persistance, adjacence, flooding, synchronization, état-lieu

Link-state routing remains as one of the most challenging issues in ad hoc networking, due to the special conditions and requirements that hold in such networks, which cannot be handled by classical routing protocols. In the last decade, several efforts have been deployed either to design new routing solutions adapted to ad hoc conditions, either to extend existing solutions for wired networks to the domain of wireless mobile scenarios. This paper elaborates on the latter alternative, focusing on the standard OSPF MANET extension RFC 5449. It analyzes the impact and interest of the persistency principle to the main OSPF MANET operations, in particular the adjacency synchronization and the other operations that relate to it (flooding and route construction). The presented results show that such persistent approach is appropriate for managing adjacencies in the context of RFC 5449, and significant improvements might be achieved by extending the persistent principle into the topology selection mechanism.

1 Introduction

Due to the challenging conditions in which communication is performed in ad hoc networks (short-lived links, radio channel unreliability, scarce bandwidth, etc.), link-state routing remains as one of the outstanding issues in this realm. Therefore, there have been many efforts in the last decade to provide efficient routing mechanisms in such networks, either by designing new link-state approaches specific for Mobile Ad hoc Networks (such as OLSR [START_REF] Clausen | RFC 3626, Optimized Link State Routing Protocol (OLSR)[END_REF]), either by extending existing protocols in order to facilitate operation in MANET conditions. This latter approach becomes particularly attractive for networks in which mobile ad hoc components may coexist with fixed infrastructure for which efficient routing solutions are already known. The most significant efforts in such approach have produced several extensions of the Open Shortest Path First protocol (OSPF) [START_REF] Moy | RFC 2328, OSPF version 2[END_REF] [START_REF] Coltun | RFC 5340, OSPF for IPv6[END_REF], which have been standardized by the IETF in [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF], [START_REF] Ogier | RFC 5614, Mobile Ad hoc Network (MANET) extension of OSPF using Connected Dominating Set (CDS) flooding[END_REF] and [START_REF] Roy | RFC 5820, Extensions to OSPF to support mobile ad hoc networking[END_REF]. This paper focus on the standard extension RFC 5449 [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF] and explores some aspects of the link-state database (LSDB) synchronization in the context of ad hoc networks. While the analysis is restricted to the particular configuration of MPR-OSPF, results might be interesting for more general conditions. Link synchronization is a basic operation in OSPF (and, more in general, in every link-state routing approach) which plays an essential role in assuring a consistent knowledge of the network topology shared by all routers, such that any of them can thus select optimal (shortest) paths to every possible destination. Due to the cost of such operation (in terms of overhead required for database exchange and utility in ad hoc networks), the study of synchronization properties and the analysis of different optimization possibilities constitute a relevant aspect to focus on.

In all the mentioned OSPF MANET extensions, synchronized links (so-called adjacencies in OSPF terminology) are selected among the set of network links according to a specific rule to form a reduced overlay in which reliable link-state synchronization is performed in a point-to-point fashion. In the case of RFC 5449, adjacent links are persistent, meaning that adjacencies may be conserved, once they have been formed, even when they would be no longer selected as adjacent.

This paper addresses the impact of such persistency policy in the adjacency management of OSPF MANET extensions, by studying the case of RFC 5449. More precisely, it discusses the advantages of using a persistent or non persistent adjacency rule and evaluates the effect of both approaches, in the context of RFC 5449, in terms of routing quality, performance of flooding operations and amount of control traffic dedicated to LSDB synchronization processes.

Paper Outline

The remaining of this paper is organized as follows. Section 2 briefly summarizes the architecture of OSPF and its standardized MANET extensions, with special attention to RFC 5449. Section 3 describes the role of adjacencies in such routing extensions and discusses the impact of forming persistent or non-persistent ones. Section 4 presents the results of simulating persistent and non-persistent approaches in a wide range of mobile scenarios. Finally, section 5 concludes the paper.

Background

This section provides a brief summary of OSPF architecture (section 2.1) and presents the main techniques used in the MANET extension standardized in RFC 5449 (section 2.2).

OSPF Architecture

OSPF [START_REF] Moy | RFC 2328, OSPF version 2[END_REF] [8] is one of the most prominent protocols for link-state IP routing within an Autonomous System (AS) [START_REF] Halabi | Internet Routing Architectures[END_REF]. As a link-state approach, it relies on the dissemination of the topology information across the network. Such dissemination allows all routers to keep a copy of the same distributed link-state database (LSDB). Maintaining a consistent LSDB enables every router to compute autonomously optimal routes (shortest paths) to all possible destinations. Such routes are computed through the Dijkstras algorithm.

Such maintenance requires the flooding of the topology information in an efficient way. This flooding is mostly handled by adjacencies. A link between two routers is called adjacent if both routers have synchronized their link-state databases. This operation requires a database exchange that is performed in a point-to-point fashion. When there are modifications in the network topology, they are advertised through the adjacent links, which thus need to form an overlay connecting all routers within the OSPF domain to each other. Topology updates are carried by Link State Advertisements (LSA).

Rules for determining which links must become adjacent depend on the type of the interfaces (point-to-point, broadcast, NBMA, point-to-multipoint or virtual link [START_REF] Moy | RFC 2328, OSPF version 2[END_REF]) involved in the links. For instance, a point-to-point link is always declared to be adjacent. For Non-Broadcast Multiple Access networks (NBMA), INRIA interfaces elect a Designated Router (DR) to be responsible for topology information flooding. Then, every interface in the network becomes adjacent to this elected DR and every topology update is first sent to the DR and then flooded by the DR to the whole network, through its adjacent links. An Autonomous System ruled by OSPF can be split in routing areas. An area is a group of routers that share the same link-state database and thus share the same view of the network topology. If there are more than a single area, all OSPF areas are connected to a backbone, so-called Area Zero (see Figure 1), which handles inter-area routing. Such a partition requires a 2-level hierarchy of routers (internal routers belonging to an area, IR, and area border routers connecting two or more areas, ABR), and enables the network to contain most of the impact of a topology change (in terms of control traffic overhead to update LSDBs) in the area in which that change occurred, keeping the rest of the network relatively unaffected.

The Multi-Point Relays Extension of OSPF RFC 5449

RFC 5449 [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF] specifies the standard MANET extension of OSPF based on the multi-point relays (MPR) technique [START_REF] Qayyum | Multipoint Relaying for Flooding Broadcast Messages in Mobile Wireless Networks[END_REF]. The Multi-Point Relaying technique provides a mechanism for performing efficient broadcast in wireless networks. It is known that nave broadcast operation leads to undesirable saturation effects (the broadcast storm problem) [START_REF] Ni | The Broadcast Storm Problem in a Mobile Ad Hoc Network[END_REF], so MPR aims to reduce the redundancy of such nave scheme by only allowing a limited subset of neighbors of the source to retransmit broadcast packets. Figure 2 shows the difference between a broadcast packet retransmission in a pure flooding (nave broadcast) fashion and under the MPR technique, only allowing selected relays (solid balls in the figure) to retransmit the received packet. Such multi-point relays are selected by the source node and the election can be done through different heuristics, but the subset of MPRs has to be able to reach every 2-hop neighbor of the computing source (MPR coverage criterion). The selection of the most convenient relays requires naturally that the computing source is aware of its own 2-hop neighbors, since any MPR selection is performed to provide coverage to a particular set of 2-hop neighbors, using a particular set of 1-hop neighbors.

Persistency in OSPF MANET Adjacencies

This section explores the notion of adjacency in the context of OSPF MANET and discusses the applicability and interest of persistent approaches in such domain. Section 3.1 elaborates on the impact of adjacent links in the main routing operations (database synchronization, control traffic flooding and route selection) for the standardized OSPF MANET extensions. Section 3.2 presents the persistency concept and the implications of its implementation in the different OSPF MANET operations. The case of RFC 5449 is described specifically for both subsections.

The Role of Adjacencies in OSPF MANET

It has been already mentioned in section 2.1 that adjacencies play an essential role in regular OSPF, in particular in NBMA interfaces, those from classic OSPF which best correspond to the semibroadcast situation in MANETs. In such cases, adjacencies populate Router-LSAs and the area-wide shortest paths are thus mainly computed among them.

In the standardized OSPF MANET extensions [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF] [10] [START_REF] Roy | RFC 5820, Extensions to OSPF to support mobile ad hoc networking[END_REF], this role is mostly kept, and hence adjacencies take part in the following aspects: ❼ Database synchronization. The adjacency rule enables every node to perform link-state database synchronization with a subset of its neighbors. Such rule is based in RFC 5449 in the Multi-Point Relaying criterion, as already mentioned in section 2.2. Flooding. Control traffic (Router-LSAs) is flooded across a subset of the adjacent links in [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF], [START_REF] Ogier | RFC 5614, Mobile Ad hoc Network (MANET) extension of OSPF using Connected Dominating Set (CDS) flooding[END_REF] and [START_REF] Roy | RFC 5820, Extensions to OSPF to support mobile ad hoc networking[END_REF]. In RFC 5449, nodes flood their Router-LSAs to their neighbors selected as MPR.

INRIA ❼ Topology selection. If the adjacency rule is able to select network-wide shortest paths, as in [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF], Router-LSAs advertise a subset of the adjacent links of the originating nodes (MPR selectors, for RFC 5449). Otherwise, Router-LSAs may need to list adjacent links and some additional bidirectional links, as in [START_REF] Ogier | RFC 5614, Mobile Ad hoc Network (MANET) extension of OSPF using Connected Dominating Set (CDS) flooding[END_REF] and [START_REF] Roy | RFC 5820, Extensions to OSPF to support mobile ad hoc networking[END_REF].

In all cases, changes in the adjacency rule have a direct impact in the three above-described main operations of OSPF.

Persistent Adjacencies

In the context of adjacency-related decisions, persistency is defined as an asymmetry between the condition for adding a link to the adjacent set and the condition for removing it, in which the latter is more restrictive than the former. Figure 3 illustrates an example of persistency in adjacency selection and maintenance, in which an adjacent link is only degraded in case it is no longer bidirectional. RFC 5449 applies such approach: a neighbor is selected by a node as adjacent if it is either an MPR or an MPR selector (neighbor that has selected the node as MPR) of the computing node. Once the adjacency has been formed (LSDB databases have been exchanged), the link is conserved as much as possible that is, as long as it is a two-way (bidirectional) link. When applied to LSDB synchronization, the main effect of this hysteresis is to reduce the number of database exchanges (in particular, for those neighbors which are alternatively added and removed from the MPR set) while increasing the overlay of reliable links (that is, those for which LSA exchange is secured by acknowledgements). Such approach is reasonable, given the short lifetime of multi-point relays (see Figure 4). As it was mentioned in section 2.2, MPR selection depends on the set of 1-hop and 2-hop neighbors of the computing node. Any change in this set of nodes may imply a recalculation of the whole MPR set, thus making the relay stability significantly smaller than the bidirectional neighbors.

Since adjacencies play a relevant role not only in terms of LSDB synchronization, but also for flooding and route selection operation (see section 3.1), adjacency hysteresis might be extended naturally to such aspects as well. Evaluation of these alternatives (flooding through persistent adjacencies and advertising persistent adjacent links in LSAs) is addressed in the following section.

Evaluation

This section analyzes the impact of adjacency persistency by evaluating the performance of four configurations based on RFC 5449, each of them exploring the implementation of the persistency principle into a different operation. Such configurations have been simulated for a wide range of density scenarios and mobility patterns. The results for moderately mobile scenarios (5 m/s), which are presented in the following subsections, are representative of the observations performed. For details of the simulations parameters, see the Appendix.

Considered Configurations

The following configurations are considered (see also ❼ Configuration MMM (non-persistent approach). Links are no longer adjacent when none of the involved nodes is MPR of the other.

Configurations P**2 are RFC5449-compliant [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF], configuration PMM following literally the specification. Configuration MMM breaks with the behavior prescribed in section 5.3.4 of [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF], in what concerns the treatment of adjacencies when the corresponding neighbor ceases to be part of the MPR set.

Persistent Adjacencies and Data Routing Quality

Figure 5 shows the data delivery ratio for each of the considered configurations.

In general terms, it can be observed that persistent approaches (P**) present a significantly better behavior that configuration MMM, the only one which does not apply the persistency principle in any main operation of the routing extension. Moreover, the figures shows that the configuration literally following the specification of RFC 5449 [START_REF] Baccelli | RFC 5449, OSPF Multipoint Relay (MPR) extension for dd hoc networks[END_REF] can be still improved in terms of delivery ratio by implementing the persistency principle in OSPF operations other than LSDB synchronization, as configurations PPM and PMP do. The simulations shows as well that such increase of the delivery ratio was achieved with no significant damages on the routing quality, measured in terms of data packets delay. This improvement of the routing quality is at the expense, first of all, of a relevant growth of the adjacent overlay, as it can be observed in Figure 6.a, which compares the size of the average adjacent set in persistent configurations (1, 2 and 3) with the non-persistent configuration (4). Figure 6.b, in turn, shows the gap between persistent adjacencies, with significantly longer lifetime, and non-persistent ones.

Difference in the size of adjacent overlays is more significant as the network density increases: in 50 nodes networks, with a density ν = 50nodes 400m×400m = 312 nodes km 2 , about 80% of the adjacencies are persistent. It has to be noted, however, that the cost of such adjacencies is extremely low in terms of control traffic exchange. They correspond indeed to links which have been already synchronized when they become persistent, and thus can be conserved with very little additional overhead. While forming an adjacency is in general a quite expensive and hazardous process in an ad hoc network, mainly due to the rigid conditions in which databases are exchanged, conservation of such adjacency has almost no cost this argument will be more developed in section 4.3.

Control Traffic Structure

The implementation of the persistent approach in any of the OSPF main operations link synchronization, LSA flooding, route construction necessarily implies an increment of the density (number of links) of the corresponding overlay. One of the main objectives of the persistency evaluation is to identify the cost of this overlay density increase. Control traffic structure and amount are also some of the main differences among the considered configurations. Configuration PPM presents the highest overhead utilization, at least in terms of kbps. It is the only persistent approach that suffers from such a high overhead, being the other persistent configurations (PMP and PMM) the ones handling lowest amounts of control traffic. While link synchronization persistency (common to configurations P**) appears to have a positive impact in the overall traffic with respect to the non-persistent configuration, the application of the persistent principle to LSA flooding overcomes that effect by making explode the amount of control traffic dedicated to flooding. This is confirmed by Fig. 8, which represents separately the number of packets dedicated to the flooding operation (mostly Link State Update packets carrying Router LSAs). This figure points out the increase of flooding traffic happening under configuration PPM, which mobilizes more than three times the number of packets in the other configurations.

The fact that PPM performs flooding across an overlay containing all persistent adjacencies has a positive effect in the quality of the flooding operation, as the evolution of the LSA retransmission ratio indicates (Fig. 9). This ratio measures the average number of times that a Router-LSA needs to be retransmitted in order to be successfully received through an adjacent link. Other than the slight improvement for configuration PPM, it can be observed that flooding over non-persistent adjacencies might be seducing from the point of view of the overlay minimization, but results in a less stable set of adjacencies (see Fig. 4) and, consequently, a more unreliable operation that requires more LSA retransmissions that any persistent configuration.

The slight improvement of the flooding quality achieved by PPM with respect to confs. PMP and PMM (Fig. 5) are at the expense of a significant increase in the control traffic. This increase is mostly caused by the explosion of the flooding overhead (see Fig. 7 and8).

Topology information in OSPF is disseminated through LSA flooding and through point-to-point synchronization. There is a trade-off between control traffic dedicated to synchronization (adjacency-forming processes) and dedicated to LSA flooding via multicast, for similar levels of routing quality (data delivery ratio). Figure 10 shows the amount of synchronization control traffic (packets taking part in the database exchange) under each of the considered configurations. Configurations with low levels of flooding traffic are as well those with more significant amounts of synchronization control traffic, and vice versa. It can be observed (see Fig. 7) however that the implementation of persistency in the topology selection operation (configuration 3) produces a less significant impact on the overall control overhead than than in flooding (configuration 1), while achieving equivalent levels or delivery ratio (Fig. 5) and keeping reasonable flooding quality values (in terms of LSA retransmission ratio, below 10%, see Fig. 9). Figure 10 shows as well the main inconvenient of the non-persistent approach, confirming what was mentioned in section 4.2. Once formed, an adjacency can be conserved with almost no additional overhead only the corresponding to LSA reliable flooding over that link. But this additional overhead is far less relevant than the overhead caused by tearing down adjacencies that might be required again in a short time, as Figure 9 indicates. Therefore, keeping a small set of adjacencies, when the adjacency rule is as unstable as MPR (see Figure 4), can only be done by tearing down cheap existing adjacencies, while increasing the number of expensive database exchanges to perform. That leads to a significant synchronization control traffic growth, as it can be appreciated in Figure 10.

Conclusion

This paper has addressed the effect and interest of the principle of persistency in the context of link-state routing synchronization. The analysis has been done by focusing on the standard extension of OSPF for MANET specified by the IETF in RFC 5449. In such extension, like in other OSPF-like approaches, link-state database synchronization is handled by the notion of adjacency.

RFC 5449 specifies already a persistent rule for adding and removing links from the adjacent set. This paper analyzes the impact of such approach by comparing it with a non-persistent configuration. The results of such evaluation, performed by means of simulating both configurations in a wide range of mobile scenarios, clearly confirm the positive impact of persistency in terms of overhead optimization and routing quality.

The paper explores as well some other applications of the persistency principle, mainly focusing on the other main OSPF operations: LSA flooding and route construction. In these configurations, adjacency persistency is thus complemented with flooding and topology persistency, respectively. The analysis and evaluation of such configurations indicate that, while in both cases there are non-negligible benefits in terms of routing quality, flooding persistency causes a significant rise of the control traffic. In contrast, advertising persistent adjacent links has a very limited impact in terms of overhead while achieving a remarkable improvement of the overall performance.

Figure 1 :

 1 Figure 1: Area structure in an OSPF NBMA Autonomous System.

  RFC 5449 relies on this MPR technique to perform the classic operations of a link-state routing protocol such as OSPF. Hence, routers synchronize their LSDBs mostly with their MPRs, topology information traffic (LSAs) are only flooded through the overlay formed by links in which any of the endpoints is MPR of the other, and only MPR links are advertised in LSAs.

Figure 2 :

 2 Figure 2: (a) Pure flooding vs. (b) flooding through Multi-Point Relays.

Figure 3 :

 3 Figure 3: (a) Non-persistent and (b) persistent approaches for adjacency handling.

Figure 4 :

 4 Figure 4: Relay and neighbor lifetime in RFC5449 (5 m/s).

Figure 5 :

 5 Figure 5: Delivery ratio (5 m/s).

  grid, 5 m/s) Avg. # 2-way neighbors/node Confs. P** (pers. adj.) Conf. MMM (non-persistent) grid, 5 m/s) Confs. P** (pers. adj.) Conf. MMM (non-persistent)

Figure 6 :

 6 Figure 6: Adjacency characterization: (a) Adjacencies per node, (b) Adjacency lifetime (5 m/s).

  Conf. PMM (min. RFC 5449) Conf. MMM (non-persistent)

Figure 7 :

 7 Figure 7: Total control overhead, in kbps (5 m/s).

  grid, 5 m/s) Conf. PPM Conf. PMP Conf. PMM (min. RFC 5449) Conf. MMM (non-persistent)

Figure 8 :

 8 Figure 8: Flooding control traffic (LSUpdate packets via multicast), in number of packets (5 m/s).

Figure 9 :

 9 Figure 9: LSA retransmission ratio (5 m/s).

Figure 10 :

 10 Figure 10: Flooding control traffic (LSUpdate packets via multicast), in number of packets (5 m/s).

  Paper Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 OSPF Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Multi-Point Relays Extension of OSPF RFC 5449 . . . . . 5 3 Persistency in OSPF MANET Adjacencies 6 3.1 The Role of Adjacencies in OSPF MANET . . . . . . . . . . . . 6 3.2 Persistent Adjacencies . . . . . . . . . . . . . . . . . . . . . . . . 7 Considered Configurations . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Persistent Adjacencies and Data Routing Quality . . . . . . . . . 9 4.3 Control Traffic Structure . . . . . . . . . . . . . . . . . . . . . . . 11

	1 Introduction	3
	1.1 2 Background	4
	2.1 4 Evaluation	8
	4.1 5 Conclusion	14
	6 References	14

table 1 )

 1 1 :

	❼ Configuration PPM (adjacency and flooding persistency). Adjacency rule
	is persistent, and Router LSAs are flooded through all adjacent links (in-
	cluding those persistent).

Table 1 :

 1 Considered configurations.

	❼ Configuration PMP (adjacency and topology persistency). Adjacency rule
	is persistent, and all adjacent links (including those persistent) are adver-
	tised in LSAs.

❼ Configuration PMM (only adjacency persistency). Adjacency rule is persistent, but only non-persistent adjacencies (i.e., links to Path MPRs or Path MPR selectors) are advertised in the topology selection. LSAs are only flooded through Flooding MPR selectors.

RR n➦ 7390

Acronyms for the considered configurations stand for [P]ersistent / non-persistent [M]PR for (i) adjacencies, (ii) flooding and (iii) topology selection. INRIA

That is, all configurations with persistent adjacencies, regardless of the approaches implemented for the other OSPF operations. This includes PMP, PPM and PMM.RR n➦ 7390


N Ben Slimen

V Deniau

J Rioult

S Dudoyer

S Baranowski
email: sylvie.baranowski@univ-lille1.fr

Statistical Characterisation of the EM Interferences Acting on GSM-R Antennas Fixed Above Moving Trains

In order to harmonise the communication systems between the train and the railway control centres over the European territory, a GSM-R (Global System for Mobile communications -Railways) communication network is progressively deployed along the European railway network. However, the GSM-R communications on board high speed trains can be disturbed by the transient electromagnetic (EM) disturbances induced by the sliding contact between the catenary and the pantograph. In order to study the immunity of the embedded GSM-R communication system against these transient electromagnetic disturbances, the transient interferences induced on the GSM-R antennas on board trains, were characterised in terms of time and amplitude parameters. Measurement campaigns were carried out in France to collect a large number of induced EM interferences on GSM-R antennas fixed on the train roof. With the ultimate goal of generating transient noise scenarios representative of those detected by the antennas, and performing immunity tests in laboratory, statistical distributions of the characteristics (rise time, time duration, repetition rate, amplitude) of the transients are presented. This paper presents the different steps of the analysis of the transient disturbances and the generation of the transient EM scenarios.

Introduction

To ensure the interoperability of trains in Europe, the GSM-R communication system will be progressively deployed over all the railway network of the European Union. It is important to notice that the GSM-R is not similar to the GSM system but the use of the same communication protocol. The GSM-R is employed to guarantee the transmission of voice and signalling data between the train and the control centres by means of a mobile transmitter-receiver (mobile with respect to base stations but fixed inside the locomotive) connected to a GSM-R antenna fixed on the roof of the locomotive and base stations installed in the neighbourhood of the rail track. A shielded cable ensures the connexion between the GSM-R antenna and the mobile transmitter-receiver. Traffic data and voice collected throughout the GSM-R system are sent to the train driver. The reliability of the GSM-R system became an important parameter for the safety and the good operation of railway traffic. The signals received by the GSM-R antennas fixed on board the trains can be affected by transient electromagnetic emissions finding their source in an imperfect sliding contact between the catenary and the pantograph. Under some particular conditions, transient events can be observable with the naked eye as sparks. These EM transient events are penalizing for the embedded telecommunication systems, since they are broadband signals and can generate high levels of noise in the GSM-R frequency bands. In addition, the GSM-R is a digital communication system and there is no EMC standard well suited to study the EM immunity of this system against transient EM events. In this context, the immunity of the GSM-R system against the transient electromagnetic noises observed on board moving trains has to be studied carefully. Generally, the immunity of electric or electronic systems is defined as the ability of a device to perform in nominal operating conditions without degradation in the presence of standardized EM disturbances. The EM immunity standards describe the test configuration and define the time, frequency and amplitude characteristics of permanent and transient EM noises applied on the equipment under test (EUT). The tests are performed in EMC test facilities. The immunity of the EUT is verified if no failure is observed during and after the test. The EMC standards dealing with the immunity tests are adapted to electronic systems but are applicable with difficulty for digital communication systems. Our main objectives in this paper are to present and test the feasibility of a new method to characterize the EM immunity of the GSM-R system against the transient EM noise observed on board moving trains. This method is based on the statistical characterisation of some parameters of the transient EM noise observed through a GSM-R antenna fixed above a moving train. The digital communication systems use various communication protocols which combines different corrector codes. The corrector codes are used to correct some disturbed bits over the transmitted bursts. The immunity test methods have to be adapted to permit testing of the robustness of such complex systems against EM disturbances and to analyze relevant immunity criteria. The GSM-R uses different frequency channels over a given time. Thus, we will present in detail the temporal values that characterise the GSM-R protocol and which can be confronted with the temporal characteristics of transient noises. Then, a precise analysis of the time characteristics of the transient disturbances collected during measurement campaigns will be carried out. Statistical distributions of the different parameters which define the transient EM disturbances will be extracted. We will then propose a method to generate transient noise scenarios in laboratory, using the statistical distributions. Finally the immunity of the GSM-R system to EM noise conditions identical to those being on the train roof could be evaluated in laboratory.

Time characteristics of the GSM-R

The GSM-R system [START_REF] Uhlirz | Concept of a GSM-based Communication System for High-speed Trains[END_REF] is based on the standard GSM Phase 2+ in addition to some advanced functions specifically developed for railway applications. Moreover, the GSM-R system employs two specific frequency bands [921-925] MHz for the downlink (from the control centres to the train) and [876-880] MHz for the uplink (from the train to the control centres). Each frequency band is divided into 20 frequency channels of 200 kHz. For security purposes, the two bound channels of each frequency band are not used. The GSM-R is a TDMA (Time Division Multiple Access) system. Thus, for each frequency channel, data are organized per periodic TDMA frame, with a period of 4.615 ms. Each TDMA frame is divided into 8 time intervals of 577 µs called "Time Slots". Therefore, each user occupies a frequency channel only one eighth of the time and by period of 577µs. Each "Time Slot" includes 156 bits. The transmission time of one bit is 3.7 µs. Figure 1 provides a schematic representation of the GSM-R signal which is emitted during a "time slot" [START_REF] Steele | GSM, cdmaOne and 3G Systems[END_REF].

Definition of the time characteristics of the transient EM interferences

Generally, the immunity of electronic and electrical systems which the access ports (power, signals or control) are confronted to fast transient burst coming mainly from the inductive load disconnection, bouncing power relays…etc. is studied according to the recommendations described in the EN 61000-4-4 standard [START_REF]European Standard EN 61000-4-4 : Electromagnetic Compatibility Part 4: Testing and measurement techniques-Electrical fast transient/burst immunity test[END_REF]. The standard concerns the application of transient test disturbances at each different ports of Equipment Under Test (EUT), using a capacitive coupling device over the different lines connected to the EUT. Dealing with telecommunication equipments, a part of the test described in this standard focus on the effect of the disturbances affecting the input and output telecommunication signal ports. This standard is also the reference in the ETSI standards [START_REF]European Standard ETSI EN 300386, Electromagnetic compatibility and Radio spectrum Matters (ERM); telecommunication network equipment, ElectroMagnetic Compatibility (EMC), requirements[END_REF] for testing electromagnetic immunity of telecommunication network equipment. In the EN 61000-4-4 standard, the transient EM test signals which are intentionally generated to control the immunity of a device are notably defined in terms of maximal amplitude (A) and corresponding time interval (TI), rise time (RT) and time duration (TD). The maximal amplitude A is the peak amplitude of the transient. The TD presented Figure 2a, corresponds to the time duration when the EM noise level of the transient event is higher than 50 % of its peak amplitude A. The RT is defined as the time between 10 % and 90 % of A Figure 2b. The time interval TI is the time between two successive transients Figure 2c. These different characteristics have an important influence on the frequency bandwidth covered by the transient. However, the values of the time parameters indicated in the 61000-4-4 standard are not necessarily representative of the transient disturbances received by the GSM-R antenna on board the train. Moreover, a transient with a 5 ns rise time will not necessarily cover the GSM-R frequency bands. We have then decided to focus on the characterisation of the real time parameters of the on-board EM transients in order to compare them with the time characteristics of the GSM-R protocol [START_REF] Adriano | Prediction of the BER on the GSM-R communications provided by the EM transient disturbances in the railway environment[END_REF]. The characterisation of the transient events observed on board moving trains was performed according to the definitions of the rise time, time duration, amplitude, and time interval of the transient events given in the 61000-4-4 standard.

Measurement campaign configuration

The results presented in this paper are extracted from the transient events measured on a train including one locomotive and eight cars, as presented in Figure 3. Measurements were carried out in France between Saint Pierre des Corps (SPC) near the city of Tours and Nantes. The train path distance was about 200 km with a 25,000 V AC (50 Hz) supply voltage. The cruising speed of the train was about 160 km/h and the maximal speed was 200 km/h. In order to record the EM noise observed on the roof of the train, the measurements were performed with a digital oscilloscope connected to a GSM-R antenna fixed on the top of the fourth car, at approximately 80 m from the pantograph. The statistical results presented in this paper are obtained from the measurements performed during a round trip between Nantes and SPC.

Statistical study on the time characteristics and amplitude of the transient events

Statistical studies presented in this paper were performed with the distribution fitting tool of MATLAB and the analytical probability density functions used were extracted from [START_REF]NIST/SEMATECH 'e-Handbook of Statistical Methods[END_REF].

The study presented below does not aim to establish a link between the different characteristics of the transient events and the physical mechanisms that could control the generation of the transients.

In fact, it is obviously difficult to perform measurement campaigns where we can control all the operating conditions, such as the age of the catenary, the supply voltage, the quality of the pantograph, weather conditions…etc. However, we think that somehow the bandwidth of the GSM-R antenna could affect the measurement characteristics of the different transients, since the GSM-R antenna used along all the measurement campaigns laid the same for all the collected transient disturbances.

Moreover, a note of caution is in order here since the different statistical laws and parameters presented below do not narrowly illustrating the real statistical distributions of the studied parameters. Representative laws of the different characteristics of the transient events could be obtained with a flock of measurement campaigns performed with different configurations and operating conditions. However, our main objective in this part remains the presentation and the feasibility verification of a new method to characterize the immunity of the GSM-R system. The empirical distribution of RT follows a lognormal law, where the PDF is given by [START_REF] Uhlirz | Concept of a GSM-based Communication System for High-speed Trains[END_REF].

Rise Time and duration

        - - =       2 2 2 ) ) (ln( exp 2 1 , σ π σ σ µ RT RT µ RT f ( 1
)
where µ = -21.55 and σ = 0.36, µ and σ are the distribution parameters and are estimated by means of the maximum likelihood estimation method with 95 % confidence intervals.

The results show that the rise times are inferior to 1 ns and the more common rise time is 0.4 ns. The empirical distribution of TD follows an exponential law, where the PDF is given by (2).

( )

      - = λ λ λ TD TD f exp 1 (2)
where λ = 5.93.10-9 is estimated according to the maximum likelihood estimation method with 95% confidence interval.

The results show that the transient durations are systematically inferior to 20 ns and mean time duration is about 5 ns. We notice that, compared to the duration of the transmission of one GSM-R bit, the mean time duration of the transients is extremely short (5 ns << 3.7 µs). Supplementary measurement campaigns (not presented in this paper) carried out under other operating conditions (speed, locomotive model, train path, weather conditions, train length…), showed that the distribution of the rise time and time duration are constant, whatever the operating conditions. However, we have to keep in mind that the distributions of RT and TD are partially related to the frequency bandwidth of the GSM-R antenna which can limit the transient EM events' spectrums.

Consequently, by employing a reception antenna with different bandwidths, different distribution would be probably obtained.

We have to point out the fact that a Goodness of Fit (GOF) tests based on the Kolmogorov-Smirnov method [START_REF] Massey | _The Kolmogorov-Smirnov test for goodness of _t[END_REF] were performed to identify the validity of the presented laws. The results showed that the experimental distributions of TD and RT could be well fitted with the use of other distribution laws that we avoided the use in this paper.

Time interval between two successive transients

The time interval (TI) is a significant parameter for the study of the immunity of the GSM-R system since the interpretation of a GSM-R received bit could be spoiled when combined with an EM transient event.

On the one hand, if the time intervals between successive transients are short, that means that several bits within a GSM-R burst can be affected,.

On the other hand, if the time intervals are long, very few bits can be affected during a single burst. In that case, the transients cannot affect the information transmission. Knowing that each user employs a GSM-R frequency channel one eighth of the time and over periods of 577 µs, we aimed to measure the EM noise on board moving trains along comparable time durations, with a digital oscilloscope connected to the GSM-R antenna. Our objective being to detect the number of transient events covering the GSM-R frequency bands, and knowing that the bandwidth of the GSM-R antenna is limited to 1 GHz, the sampling frequency of the oscilloscope was fixed at 5 GHz. However, with such a sampling frequency, the memory depth (2.106 mega points) of the oscilloscope used limited the window to 400 µs. The results presented below are then based on 400 µs time windows.

During a period of 25 minutes (1,500 seconds), we recorded 1,568 files (or 1,568 time windows of 400 µs) and the time required to record the data of a 400 µs window on the oscilloscope hard disk is rather less than 1 second. Thus, the oscilloscope was continually activated by the presence of transient signals over the entire measurement period. Hence, we can suppose that the distribution of the time intervals obtained from the 1,568 time windows is representative of the entire duration measurement (25 minutes).

To obtain the time interval between the transients, the first step is to recognize the transient events in the 400 µs time window. The second step consists in calculating the time difference between the consecutive transient events in each window. In order to detect the transient events, we developed an algorithm allowing 3 µs resolution time, which approximately corresponds to the duration of one GSM-R bit. Thus, the number of transient events detected will correspond to the potential number of the disturbed GSM-R bits. This value of 3 µs then limits to the minimum time interval that we can obtain. Figure 5 gives examples of a collected time windows. The analysis of the recorded files shows that 1,030 files contain only one transient event in 400 µs. So, they cannot allow us to extract time intervals. 40 supplementary files were not employed due to their containing a high level of permanent noise probably coming from the GSM public transmitter nearby. Indeed, a relatively high permanent noise (illustrated in Figure 5) can hide the transient EM events and does not permit us to extract the correct number of the transient disturbances which occurred during the analysis time window. Finally, only 498 files containing a minimum of two transient events were employed. From these files, we extracted 1,121 time intervals less than 400 µs. The bars diagram in Figure 6 shows the experimental PDF of the 1,121 collected time intervals. According to the experimental PDF of the time intervals, empirical PDF approaching the experimental one is presented by the dashed line in Figure 6.

Experimental PDF Emperical PDF Experimental PDF Emperical PDF

f       - =       - - exp . . . , 1 for TI > (3)
where a = 9.22 10-5 is the scale parameter and b = 0.77 is the shape parameter. The parameters (a, and b) of the distribution are estimated by means of the maximum likelihood method with 95 % confidence intervals. The experimental PDF of the time intervals less than 400 µs corresponds perfectly to the Weibull's PDF presented above.

In Figure 6, we noticed that the time intervals are mainly inferior to 17 µs. Assuming that each transient interference can affect the interpretation of one transmitted bit, transient disturbances which occur with 17 µs time intervals could cause a 20 % bit error rate. Besides, we have to keep in mind that this distribution does not take into account the time intervals higher than 400 µs. Indeed, 48 % of the time intervals collected are higher than 400 µs. We will have to improve the method in order to include these time intervals at the time of the generation of the transient scenarios.

Peak Amplitude of the transients

With the same methodology employed for the extraction of the PDF of RT, TD, and TI we studied the PDF of the peak amplitudes A of the transient events. The analysis showed that A follows the Generalized Extreme Value (GEV) distribution law where the PDF is expressed by [START_REF]European Standard ETSI EN 300386, Electromagnetic compatibility and Radio spectrum Matters (ERM); telecommunication network equipment, ElectroMagnetic Compatibility (EMC), requirements[END_REF].

                      - + -             - + = - - ζ ζ σ ζ σ ζ σ ζ σ 1 1 1 1 exp 1 1 ) , , ; ( µ A µ A µ A f (4)
where A is the peak amplitude, µ = 0.06 is the location parameter, σ = 0.02 is the scale parameter, and ξ = 0.37 is the shape parameter. µ, σ, and ξ are estimated by means of the maximum likelihood method with 95 % confidence interval. However, under other measurement configuration, mainly the position of the GSM-R antenna, the distribution law parameters of the amplitude can vary regarding to the position gap between the GSM-R antenna and the pantograph. Thus, according to the position of the antenna, the amplitude distribution law parameters must be adjusted to fit the corresponding experimental distribution of the transients' peak amplitudes.

Typical transient disturbances collected on the train and comparison with the transient test signal definition applied in the Standard EN 61000-4-4

Nowadays, the immunity tests of the electronic systems against transient EM disturbances are performed respecting the recommendations given in the European Standard EN 61000-4-4. A comparison between the time characteristics of the transient events defined in this standard and the typical time characteristics obtained with GSM-R antenna on board moving train is presented in Table . 1. The time characteristics of the transients described in the EN 61000-4-4 standard are widely higher than the ones received by the GSM-R antenna on board the train. Previous studies [START_REF] Deniau | Railcom consortium, Characterisation of the electromagnetic disturbances received by GSM-R antennas in usual railway operating conditions[END_REF] showed that the EM noise amplitude generated by a transient event in the frequency band of the GSM-R system is related to time characteristics (RT and TD) of this transient event. In particular, the test signals defined in the EN 61000-4-4 do not cover the frequency bands of the GSM-R while the transient disturbances observed on board train systematically cover the GSM-R channels.

EN 61000-4-4

( Besides, intuitively we can think that the more short the time interval between two successive transient events, the more the system's immunity is affected. Regarding to the time interval between two successive transients in the standard, we notice that with a time interval included between 0.2 ms or 0.4 ms, a maximum of one bit can be affected during a 577 µs period (corresponding to the GSM-R time slot). Nevertheless, in practise, we observed that under certain operating conditions, a typical time interval can be 25 µs. In that case, 23 bits are potentially affected by the transient disturbances. Hence, the immunity tests based on the EN 61000-4-4 are not adapted for the immunity's study of the GSM-R system and are not representative of real EM risks induced by the transient disturbances on board trains. So, a relevant immunity test needs the generation of a transient events scenario similar to that observed on board a moving train. In the next part we propose to generate a transient events scenario statistically similar to that observed on board a moving train. ( ) ( )

Generation of the transient disturbance scenarios

t RT t FT t A t V γ               - -       - = exp exp (5)
V(t) is the transient signal voltage. A is the peak amplitude, RT is the rise time, FT is the fall time, and the causality of the signal is given by the echelon function γ. According to the TD and the RT of the signal, the FT is computed automatically [START_REF] Camp | Parameter estimation of double exponential pulses (EMP, UWB) with Least Squares and Nelder Mead algorithm[END_REF]. Once all the N transient events generated, the concatenation of the transients is performed with MATLAB to generate the transient noise sequence. 6: Experimental and empirical distributions of the time intervals between two consecutive transient events Figure 7: Transient events scenario statistically similar to the ones observed on board moving train with a GSM-R antenna Table 1: Time characteristics of the transients observed on board the train and the ones given in the EN 61000-4-4

Conclusions

Figures and Table Captions

Figure 1 :

 1 Figure 1: Time characteristics of one GSM-R Burst

Figure 2 :

 2 Figure 2: Definitions of the time characteristics of transient signals in EN 61000-4-4 The standardized values for each transient parameter described above given in the 61000-4-4 standard are as follows: RT: 5 ns ±30 %, TD: 50 ns ±30%, TI: (0.2 ms or 0.4 ms) according to the peak Amplitude A of the transients, where A remains constant for each test signal.

Figure 3 :

 3 Figure 3: Measurement configuration on board the train The GSM-R antenna is a bi-band antenna working in the frequency band of the French railway radio system [420-520] MHz and of the GSM-R system [820-1000] MHz. Our objective being to extract the time characteristics of the EM transient events, experiments were carried out using a digital oscilloscope with a 20 GHz sampling frequency. The detection of transient signals was dependent on the triggering level of the oscilloscope. Generally, the trigger level is fixed arbitrarily by the operator at a value barely higher than the permanent noise level observed at low speed (less than 60 km/h). Each transient event detected during the train path was recorded in a 200 ns time window. During about 100 minutes of measurement, the oscilloscope collected about 25,700 transient signals. The randomness of the characteristics of the transient signals led us to perform a statistical analysis. Statistical studies on the different time characteristics and amplitude of the collected transients are presented in the next part.

ForFigure 4 :

 4 Figure 4: Experimental and empirical distributions of the Rise Time (a) and duration (b) of the transient events in ns

Figure 5 :

 5 Figure 5: Examples of measured files over 400 µs time windows

Figure 6 :

 6 Figure 6: Experimental and empirical probability density functions (PDF) of the time intervals between two consecutive transient events

Figure 7

 7 Figure 7 presents a transient events scenario where the distributions of the time characteristics of the transient events follow the probability density functions given above.Figure 7.a gives one scenario of 4 ms time long where 30 transient events are observed. Figure 7b is a Zoom-In of Figure 7a showing 10 transient events occurring along a 577 µs time duration corresponding to one GSM-R time slot.

Figure 7 .

 7 Figure 7 presents a transient events scenario where the distributions of the time characteristics of the transient events follow the probability density functions given above.Figure 7.a gives one scenario of 4 ms time long where 30 transient events are observed. Figure 7b is a Zoom-In of Figure 7a showing 10 transient events occurring along a 577 µs time duration corresponding to one GSM-R time slot.

Figure 7 :

 7 Figure 7: Transient events scenario statistically similar to the ones observed on board moving train with a GSM-R antenna It is, therefore, possible to identify sequences of EM transient interferences closer to reality than the electromagnetic noise conditions described in the 61000-4-4 standard. These results allow us to consider two major perspectives. Firstly, we plan to work on the generation of electromagnetic noise sequences (using arbitrary signal generators) within electromagnetic test facilities in order to test the immunity of GSM-R transmission given these conditions of transient EM noise. Secondly, the simulation chains of communication systems can generally simulate the behaviour of the system in the presence of Gaussian distribution EM noise. We might, therefore, consider undertaking these simulations with statistical distributions of EM noise more representative of certain areas.

A

  statistical study of the time characteristics of the transient events observed through a GSM-R antenna fixed on the roof of a moving train has been presented in this paper. Experimental and empirical distributions of each time parameter of the transient events have been presented and compared. These results show that the definitions of the transient disturbances currently employed in the standard 61000-4-4 are significantly different from the transient disturbances observed on the train roof. According to the empirical distributions obtained, an EM transient noise scenario statistically similar to the experimental one was generated. Thus, we envisage to carry out relevant immunity tests of the GSM-R system in a laboratory context. Currently, we are working on an adapted methodology allowing the generation of the transient EM noise scenario taking into account the possible dependency between the time characteristics of the transient events, in order to perform relevant GSM-R immunity tests. A new on-board methodology allowing the measurement of time interval higher than 400 µs has also to be developed. If this study would lead to standardized recommendations for the study of the EM immunity of such a digital communication system, an exhaustive study of the transient EM events characteristics has to be planned under different operating conditions and configurations.

Figure 1 :

 1 Figure 1: Time characteristics of one GSM-R Burst Figure 2: Time characteristics of the transient events Figure 3: Measurement configuration Figure 4: Experimental and empirical distributions of the Rise Time (a) and duration (b) of the transient events in ns Figure 5: Example of measured file with two transient events in a 400 µs time window Figure6: Experimental and empirical distributions of the time intervals between two consecutive transient events Figure7: Transient events scenario statistically similar to the ones observed on board moving train with a GSM-R antenna Table1: Time characteristics of the transients observed on board the train and the ones given in the EN 61000-4-4

Table 1 :

 1 Comparison between the time characteristics of the transient disturbances observed on board the train and the ones applied in the standard EN 61000-4-4

		1)	On train (2)	Comparison
	Rise Time	5 ns ± 30 %	0.4 ns	RT(1)>> RT(2)
	Duration 50 ns ± 30 %	5 ns	TD(1)>>TD(2)
	Time Interval	0.2 ms and 0.4 ms	25 µs	TI(1)>>TI(2)

Acknowledgements

This work was carried out within the framework of the RAILCOM project and benefited of the financial support of the PCRD 6. The measurement campaigns would not been able to be carried out without the contribution of the SNCF.


Françoise Pène

Transient random walk in Z 2 with stationary orientations

.

Introduction

Random walks in random environment in Z d have been studied by many authors. For a general reference on this subject, we refer to chapter 6 of the book of Hughes [START_REF] Hughes | Random walks and random environments[END_REF] . Random walks with random orientations have been less studied. However these two subjects are not far from each other. Indeed, random walks with random orientations can be viewed as a degenerate case of random walks in random environment in the sense that transition probabilities are allowed to be null. But this difference is significant. Moreover random walks in Z 2 with random orientations can also be viewed as a question of oriented percolation (see section 12.8 of the Book of Grimmett [START_REF] Grimmett | Percolation[END_REF]).

The present paper contains an extension of the model introduced by Campanino and Pétritis in [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF] in another direction than the one chosen by Guillotin-Plantard and Le Ny in [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF]. But our result will also apply to random walks of the form studied in [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF]. Now, let us present the different models introduced in [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], in [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF] and in the present paper with their common ideas and their differences. Let us construct a random walk (M n = ( Xn , Ỹn )) n≥0 in Z 2 with random orientations as follows. Let (ξ k ) k∈Z be a stationary sequence of centered random variables with values in {-1; 1}. The orientations of the k th horizontal floor of Z 2 is given by ξ k . Once the environment fixed, the random walk (M n = ( Xn , Ỹn )) n will be such that M 0 = (0, 0) and such that the distribution of M n+1 -M n conditioned to σ(M k ; k = 0, ..., n) is uniform on {(0, 1); (0, -1); (ξ Ỹn , 0)}.

In [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], Campanino and Pétritis prove the transience of the random walk (M n ) n when (ξ k ) k∈Z is sequence of independent identically distributed random variables. Moreover, they point out the fact that the random walk (M n ) n≥0 is recurrent in the 'alternate' case where ξ k only depends on the parity of k. Hence the behaviour of this random walk depends on the randomness of the orientations (ξ k ) k∈Z .

In [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF], Guillotin-Plantard and Le Ny give a first generalization of the work of Campanino and Pétritis. They envisage the case when the orientations of the floors are taken independently with stationary probabilities. More precisely, they consider the following situation : Let (f k ) k∈Z be a stationary sequence of random variables with values in [0; 1] and with expectation equal to 1 2 defined on some probability space (M, F, ν). Let us consider the probability space given by (Ω 1 := M ×[0; 1] Z , F 1 := F ⊗(B([0; 1])) ⊗Z 2. There exist some C > 0, some (ϕ p,s ) p,s∈N and some integer r ≥ 1 such that for all positive integers p and s, we have ϕ p+1,s ≤ ϕ p,s , such that we have lim s→+∞ s 6 ϕ rs,s = 0 and such that, for all integers n 1 , n 2 , n 3 , n 4 with 0 ≤ n 1 ≤ n 2 ≤ n 3 ≤ n 4 , for all real numbers α n1 , ..., α n2 and β n3 , ..., β n4 , we have :

Cov e i P n 2 k=n 1 α k ξ k , e i P n 4 k=n 3 β k ξ k ≤ C 1 + n2 k=n1 |α k | + n4 k=n3 |β k | ϕ n3-n2,n4-n3 .
Then the random walk (M n ) n is transient.

This result is proved in section 3. We will see in its proof that this question is linked with n-1 k=0 ξ S k where (S m ) m≥0 is a simple symmetric random walk on Z independent of (ξ k ) k∈Z . Let us give some examples of stationary sequences (ξ k ) k∈Z to which this result applies.

Theorem 2. [(α-mixing condition)] Let (g k ) k∈Z be a stationary sequence of bounded real-valued random variables defined on some probability space (Ω, A, P) satisfying the following α-mixing condition : Then :

(a) If g k takes its values in {-1; 1}, if M g k dν = 0 and if (ξ k := g k ) k∈Z , then (M n ) n is transient. (b) If g k takes its values in [0; 1], if M g k dν = 1 2 and if (ξ k := ξk,g k ) k∈Z , then (M n ) n is transient.
We will prove that the hypotheses of theorem 1 are satisfied in the general context of strongly mixing dynamical systems. We say that (M, F, ν, T ) is an invertible dynamical system if (M, F, ν) is a probability space endowed with an invertible bi-measurable transformation T : M → M . Definition 3. We say that an invertible dynamical system (M, F, ν, T ) is strongly mixing if there exists c 0 > 0, there exist two real sequences (ϕ n ) n≥0 and (κ m ) m≥0 and, for any function g : M → C, there exist

K (1) g ∈ [0; +∞] and K (2)
g ∈ [0; +∞] such that, for all bounded functions g, h : M → C :

1. for all integer n ≥ 0, we have :

|Cov ν (g, h • T n )| ≤ c 0 g ∞ h ∞ + h ∞ K (1) g + g ∞ K (2) h ϕ n ;
2. for all integer m ≥ 0, we have :

K (1) g•T -m ≤ c 0 K (1) g and : K (2) h•T m ≤ c 0 K (2) h (1 + κ m ); 3. we have : K (1) g×h ≤ g ∞ K (1) h + h ∞ K (1) g and : K (2) g×h ≤ g ∞ K (2) h + h ∞ K (2) g ;
4. the sequence (ϕ n ) n≥0 is decreasing, the sequence (κ m ) m≥0 is increasing and there exists an integer r ≥ 1 such that :

sup n≥1 n 6 (1 + κ n )ϕ rn < +∞.
Proposition 4. Let (M, F, ν, T ) be a strongly mixing dynamical system. Let the sequence (ξ k ) be of one the two following kinds :

(a) ξ k = f • T k with f : M → {-1; 1} a ν-centered function such that K (1) f + K (2) f
< +∞. We suppose that there exists some real number c 1 > 0 such that, for any real number α, we have :

K (1) exp(iαf ) + K (2) exp(iαf ) ≤ c 1 |α|. (b) ξ k = ξk,f•T k with f : M → [0; 1] such that M f dν = 1
2 and such that there exists some c 1 > 0 such that, for any a, b ∈ C, we have K (1)

af +b + K (2) af +b ≤ c 1 |a|.
Then (ξ k ) k satisfies the hypothesis of theorem 1.

Proposition 4 is proved in appendix A. Theorem 2 will appear as a direct consequence (see appendix B). Our strong mixing property is satisfied by a large class of dynamical systems (endowed with some metric) with K

(1)

f and K (2)
f dominated by the Hölder constant of f of order η. Interesting examples are given by hyperbolic or quasi-hyperbolic dynamical systems. We quickly give some examples of such dynamical systems. In the case of the billiard transformation, because of the discontinuity of the transformation, our class of allowed functions will contain discontinuous functions.

Examples 2.1.

1. Let (M, F, ν, T ) where T is an ergodic algebraic automorphism of the torus or a diagonal transformation on a compact quotient of Sl d0 (R) by a discrete group. Let η > 0. According to [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF], the strong mixing property holds with K (1) g some η-Hölder constant of g along the unstable manifolds and with K

(2) h some η-Hölder constant of h along the stable-central manifolds and with ϕ n = α n for some α ∈ (0, 1) and κ m = m β for some β ≥ 0. Moreover K (1) g and K

(2) g are dominated by the Hölder constant of order η of g.

2.

Let (M, F, ν, T ) where T is the Sinai billiard transformation (in T 2 ) with C 3 -convex scatterers and with finite horizon and where ν is the T invariant measure absolutely continuous with respect to the Lebesgue measure [START_REF] Sinai Ya | Dynamical systems with elastic reflections[END_REF]. Let m 0 ∈ Z + and η > 0. According to [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF] (theorem 4.3), the strong mixing property holds with ϕ n = α n for some α ∈ (0, 1) and κ m = m β for some β ≥ 0, K

being some Hölder constant of g along the T -m0 (γ u )'s (where the γ u 's are the unstable curves) and K

(2) h being some Hölder constant of h along the T m0 (γ s )'s (where the γ s 's are the stable curves). The quantities K The first example is a direct consequence of [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF]. The second example is a consequence of [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF]. In appendix C, we give a precise definition of K for these examples (and a definition of C m for the Sinai billiard). For these systems, we can say a little more : Theorem 5. Let η ∈ (0, 1) and let (M, F, ν, T ) be a strongly mixing dynamical system (endowed with some metric) such that there exists α ∈ (0, 1) and β ≥ 0 such that ϕ n = α n and κ m = m β and such that K

M g 0 dν = 1 2 , then (M n ) n is transient. (B) If (ξ k = 21 A • T k -1
) k∈Z with ν(A) = 1/2 and with A such that there exist c A > 0 and ζ > 0 such that, for every ε ∈]0; 1[, we have :

ν ({x ∈ M : d(x, A) < ε}) ≤ c A ε ζ , then (M n ) n is transient.
Conclusion (A) of theorem 5 follows directly from proposition 4. Conclusion (B) of theorem 5 is proved in appendix D.

Proof of theorem 1

Let us define T 0 := 0 and, for all n ≥ 1 : T n+1 := inf{k > T n : Ỹk = Ỹk-1 }. According to lemma 2.5 of [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], we have the following result :

Lemma 6. If (M Tn ) n≥0 is transient, then (M n ) n≥0 is transient
Now, still following [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], we construct a realization of (M Tn ) n . Let us consider a symmetric random walk (S n ) n on Z independent of (ξ k ) k∈Z . For any integer m ≥ 1 and any integer k, we define :

N m (k) := Card{j = 0, ..., m : S j = k}.
Let us also consider a sequence of independent random variables (ζ (y) i ) i≥1,y∈Z with geometric distribution with parameter 13 , and independent of ((ξ y ) y∈Z , (S p ) p≥1 ). In this lemma, ζ (y) i corresponds to the duration of the stay at the y th horizontal floor during the i th visit to this floor. According to the Borel-Cantelli lemma, it suffices to prove that : n≥1 P({(X n , S n ) = (0, 0)}) < +∞. We follow the scheme of the proof of [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF]. The difference will be in our way of estimating

I (1)
n and in the introduction of the sets U n . We will consider δ 1 , δ 2 , δ 3 , and γ such that :

0 < δ 1 < 2δ 2 , δ 1 + ( 27 2 + 16)δ 2 < 1 8 , δ 3 > 0, 1 4 -3δ 2 < δ 3 < 1 4 -5 2 δ 2 -δ 1 , δ3 2 -2δ 2 < β < δ3 2 -δ 2 , max(δ 1 , δ 2 ) < γ < 1 2 -22 max(δ 1 , δ 2 ).
The idea is that δ 1 , δ 2 , 1 4 -δ 3 and 1 8 -β are positive numbers very close to zero. As in [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF][START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF], let us define :

A n := {ω ∈ Ω : max ∈Z N n-1 ( ) ≤ n 1 2 +δ2 and max k=0,...,n |S k | < n 1 2 +δ1 }. Moreover, we define : U n := {ω ∈ A n : ∀x, y ∈ Z, |N n-1 (x) -N n-1 (y)| ≤ |x -y|n 1 2 +γ }.
The sketch of the proof is the following :

1. As in proposition 4.1 of [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], we have :

n≥1 P ({X n = 0 and S n = 0} \ A n ) < +∞. Actually we have : n≥1 P ({S n = 0} \ A n ) < +∞.
2. We will see in lemma 8 of the present paper that we have :

n≥1 P (A n \ U n ) < +∞. Therefore, we have : n≥0 P ({X n = 0 and S n = 0} \ U n ) < +∞; 3. Let us define B n := {ω ∈ U n : y∈Z ξ y N n-1 (y) > n (a)
As in lemma 4.5 of [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], there exists a real number C > 0 such that :

sup ω∈Un\Bn P ({X n = 0}|(S p ) p≥1 , (ξ k ) k∈Z ) ≤ C ln(n) n .
(b) We will prove that there exists some δ > 0 and some C > 0 such that :

∀ω ∈ U n , P (U n \ B n |(S p ) p ) (ω) ≤ C n -δ .
i. This probability is bounded by c n

1 2 +δ3 I n (ω) with I n (ω) = I (1)
n (ω) + I (2)
n (ω) and

I (1) n (ω) := {|t|≤n -1 2 -δ 3 +δ 2 } E e it P y∈Z ξyNn-1(y)(ω) (S p ) p e -t 2 n 1+2δ 3 2 dt and I (2) n (ω) := {|t|>n -1 2 -δ 3 +δ 2 } E e it P y∈Z Nn-1(y)(ω) (S p ) p e -t 2 n 1+2δ 3 2 dt.
ii. We will prove that n

1 2 +δ3 sup Un I (1)
n = O(n -δ ) for some δ > 0 (see our lemma 9); iii. On the other hand, following [START_REF] Campanino | Random walks on randomly oriented lattices, Markov Process[END_REF], we have :

n 1 2 +δ3 I (2) n ≤ {|s|>n δ 2 } e -s 2 2 ds ≤ 2n -δ2 e -n 2δ 2 2 (c) We have P(S n = 0) ≤ C n -1 2 .
(d) Hence we have :

P (U n ∩ {X n = 0 and S n = 0} \ B n ) ≤ C n -1-δ ln(n).
We have to prove that points 2 and 3(b)(ii) are true with our choices of parameters. Indeed, all the other points are true for any positive δ 1 , δ 2 , δ 3 and for any sequence of random variables (ξ k ) k∈Z independent of (S p ) p . We notice that, for any integer n ≥ 1, we have :

n-1 j=0 ξ Sj = k∈Z ξ k N n-1 (k).
In our proof, we need some real numbers δ 1 , δ 2 , δ 3 , δ 4 , β, γ and ε > 0. We will suppose that :

δ 1 > 0, δ 2 > 0, δ 1 + ( 27 2 + 16)δ 2 < 1 8 , δ 3 > 0, δ 1 < δ 4 < 1 4 -δ 3 -5 2 δ 2 , 1 4 -3δ 2 < δ 3 < 1 4 -5 2 δ 2 , 5 3 δ 2 < 1 2 δ 3 , δ3 2 -2δ 2 < β < δ3 2 -δ 2 , 5 2 δ 3 > 1 2 + 6δ 2 + δ 1 , max(δ 1 , δ 2 ) < γ < 1 2 -22 max(δ 1 , δ 2
) and :

n δ1+11δ2 m≥ (r+1)n β 2 |E[ξ 0 ξ m ]| = O(n -ε ). (we have : m≥N |E[ξ 0 ξ m ]| ≤ N -1 2 m≥N √ m|E[ξ 0 ξ m ]|).
All these inequalities are true with the following choices of parameters :

δ 1 = 1 3000 , δ 2 = 1 500 , δ 3 = 1 4 - 11 4 δ 2 = 489/2000, δ 4 = 1/2500, β = δ 3 2 - 3 2 δ 2 = 477/4000, γ = 1 4 .
Lemma 8. We have :

n≥1 P (A n \ U n ) < +∞.
Proof. Let us consider any x, y ∈ Z with x = y and |x -y| ≤ 3n 1 2 +δ1 . For any integer j ≥ 1, we define the time τ j (x) of the j th visit of (S p ) p to x and the number N j (x, y) of visits of (S p ) p to y between the times τ j (x) and τ j+1 (x). According to [START_REF] Kesten | A limit theorem related to a new class of self similar processes[END_REF][START_REF] Spitzer | Principles of random walk[END_REF] (see [START_REF] Kesten | A limit theorem related to a new class of self similar processes[END_REF] lemma 2), for any integer p ≥ 1, there exists K p > 0 such that, for any x = y, we have :

E[(N j (x , y )) p ] ≤ K p |x -y | p-1 .
According to [START_REF] Kesten | A limit theorem related to a new class of self similar processes[END_REF], on the set {τ 1 (x) ≤ τ 1 (y)}, we have :

(N n-1 (x) -N n-1 (y)) = Nn-1(x) j=1 (1 -N j (x, y)) + τ N n-1 (x)+1 (x) k=n 1 {S k =y} .
Let p be any positive integer. We have :

(N n-1 (x) -N n-1 (y)) 2p 1 {τ1(x)≤τ1(y)} ≤ 2 2p      Nn-1(x) j=1 (1 -N j (x, y))   2p +   τ N n-1 (x)+1 (x) k=n 1 {S k =y}   2p    . But, on A n , since we have N n-1 (x) ≤ n 1 2 +δ2 , we get :   τ N n-1 (x)+1 (x) k=n 1 {S k =y}   2p ≤ N Nn-1(x) (x, y) 2p ≤ j n 1 2 +δ 2 k j=1 (N j (x, y)) 2p .
Hence we have:

E      τ N n-1 (x)+1 (x) k=n 1 {S k =y}   2p 1 An    ≤ n 1 2 +δ2 K 2p |x -y| 2p-1 ≤ K 2p 3 p-1 |x -y| p n 1 2 +max(δ1,δ2) p .
Moreover, on A n , we have :

  Nn-1(x) j=1 (1 -N j (x, y))   2p ≤ max k=1,..., j n 1 2 +δ 2 k   k j=1 (1 -N j (x, y))   2p . Since k j=1 (1 -N j (x, y))
k≥1 is a martingale (see [START_REF] Kesten | A limit theorem related to a new class of self similar processes[END_REF] lemma 2), according to a maximal inequality, we have :

max k=1,..., j n 1 2 +δ 2 k   k j=1 (1 -N j (x, y))   2 L p ≤ p p -1 max k=1,..., j n 1 2 +δ 2 k   k j=1 (1 -N j (x, y))   2 L p .
Hence we have :

E      Nn-1(x) j=1 (1 -N j (x, y))   2p 1 An    ≤ p p -1 p max k=1,..., j n 1 2 +δ 2 k E      k j=1 (1 -N j (x, y))   2p    .
Let us write M 2p ν1,...,ν l = (2p)! Q l i=1 νi! . For any k = 1, ..., n 1 2 +δ2 , since the N jm 's are independent and since E [1 -N j (x, y)] = 0, we have :

E      k j=1 (1 -N j (x, y))   2p    = 2p l=1 ν1+...+ν l =2p; mini νi≥1 M 2p ν1,...,ν l j1<...<j l l m=1 E [(1 -N jm (x, y)) νm ] , ≤ 2p l=1 ν1+...+ν l =2p; mini νi≥2 M 2p ν1,...,ν l 1≤j1<...<j l ≤k l m=1 (2 νm E [1 + (N jm (x, y)) νm ]) ≤ 2p l=1 ν1+...+ν l =2p; mini νi≥2 M 2p ν1,...,ν l 1≤j1<...<j l ≤k l m=1 2 νm (1 + K νm |x -y| νm-1 ≤ Cp 2p l=1 |x -y| 2p-l (n 1 2 +δ2 ) l ≤ 2p3 p Cp |x -y| p (n 1 2 +max(δ1,δ2) ) p .
Hence we get :

E (N n-1 (x) -N n-1 (y)) 2p 1 An ≤ C p |x -y| p (n 1 2 +max(δ1,δ2) ) p .
Therefore, according to the Markov inequality, for any integer p ≥ 1, we have :

P (A n \ U n ) ≤ l n 1 2 +δ 1 m x,y=- l n 1 2 +δ 1 m P A n ∩ |N n-1 (x) -N n-1 (y)| > |x -y|n 1 2 +γ ≤ l n 1 2 +δ 1 m x,y=- l n 1 2 +δ 1 m E[(N n-1 (x) -N n-1 (y)) 2p 1 An ] |x -y| p (n 1 2 +γ ) p ≤ c p 5n 1 2 +δ1
2 n max(δ1,δ2)-γ p .

By taking p large enough, we get : n≥1 P(A n \ U n ) < +∞.

Estimates on U n

In this section, we suppose that we are in U n . We will estimate :

I (1) n (ω) := {|t|≤n -1 2 -δ 3 +δ 2 } E e it P y∈Z ξyNn-1(y) (S p ) p (ω) e -t 2 n 1+2δ 3 2 dt.
Lemma 9. There exists a real number δ > 0 such that :

sup n≥1 n δ sup ω∈Un n 1 2 +δ3 I (1)
n (ω) < +∞.

To prove this lemma, we will use the following formula :

n 1 2 +δ3 I (1) n (ω) = n δ2 {|u|≤1} E e iun -1 2 -δ 3 +δ 2 P y∈Z ξyNn-1(y) (S p ) p (ω) e -u 2 n 2δ 2 2 du.
The main idea is to prove that, in this formula, we can replace the term :

B n (u)(ω) := E e iun -1 2 -δ 3 +δ 2 P y∈Z ξyNn-1(y) (S p ) p (ω)
by the term :

A n (u)(ω) := e - u 2 2n 1+2δ 3 -2δ 2 P y,z E[ξyξz](Nn-1(y)(ω)) 2
. More precisely let us prove that we have :

Lemma 10. There exists a real number δ 0 > 0 such that we have :

sup n≥1 n δ0 sup ω∈Un n δ2 |u|≤1 |B n (u)(ω) -A n (u)(ω)| e -u 2 n 2δ 2 2 du < +∞. (1)
After proving 10, we will prove that lemma 9 is a consequence of it. We will use the following notation :

σ 2 ξ := m∈Z E[ξ 0 ξ m ].

Proof of lemma 10

Our proof uses a method introduced by Jan (cf. [START_REF] Jan | Vitesse de convergence dans le TCL pour des processus associés à des systèmes dynamiques et aux produits de matrices aléatoires[END_REF], [START_REF] Jan | Vitesse de convergence dans le TCL pour des chaînes de Markov et certains processus associés à des systèmes dynamiques[END_REF]). This method also gives a result of convergence in distribution for n -3/4 n-1 k=0 ξ Sn n≥1

(see [START_REF] Borgne | Exemples de systèmes dynamiques quasi-hyperboliques  décorrélations lentes[END_REF][START_REF] Borgne | Exemples de systèmes dynamiques quasi-hyperboliques à décorrélations lentes[END_REF]). Let n be an integer such that n β ≥ 2. Let us fix

ω ∈ U n and u ∈ [-1; 1]. Let us recall that 0 < β < δ3 2 -δ 2 et let us define : L n := 2 j n 1 2 +δ 1 k +1 n β
(we have : ξyNn-1(y) and a k := e

L n ≤ 4n
- u 2 2n 1+2δ 3 -2δ 2 Pα (k+1) -1 y=α (k) σ 2 ξ (Nn-1(y)) 2
. We have to estimate :

n δ2 E Ln k=0 b k (S p ) p (ω) - Ln k=0 a k (ω) .
Hence it is enough to estimate :

n δ2 Ln k=0 E k-1 m=0 b m (b k -a k ) Ln m =k+1 a m (S p ) p (ω) .
• We explain how we can restrict our study to the sum over the k such that (r + 1) 4 ≤ k ≤ L n -1. Let k ∈ {0, ..., L n }. We have :

E   α+θ =α+1 ξ N n-1 ( ) 2 |(S p ) p   (ω) ≤ α+θ =α+1 α+θ m=α+1 |E[ξ ξ m ]| N n-1 ( )(ω)N n-1 (m)(ω) ≤ θ m∈Z |E[ξ 0 ξ m ]|n 1+2δ2 .
Hence we have :

E [|b k -1||(S p ) p ] (ω) ≤ n -1 2 -δ3+δ2 E α (k+1) -1 y=α (k) ξ y N n-1 (y) |(S p ) p (ω) ≤ n -1 2 -δ3+δ2 n β 2 m∈Z |E[ξ 0 ξ m ]|n 1 2 +δ2 ≤ n -3 4 δ3+ 3 2 δ2 m∈Z |E[ξ 0 ξ m ]|,
since we have β < δ3 2 -δ 2 . Moreover we have :

|a k (ω) -1| ≤ σ 2 ξ α (k+1) -1 y=α (k) (N n-1 (y)(ω)) 2 2n 1+2δ3-2δ2 ≤ σ 2 ξ n 1+2δ2 2n 1+2δ3-2δ2 n β σ 2 ξ n 1+2δ2 ≤ n -3 2 δ3+3δ2 σ 2 ξ 2 .
From which, we get :

n δ2 (r+1) 4 -1 k=0 E [|b k -a k ||(S p ) p ] (ω) + E [|b Ln -a Ln ||(S p ) p ] (ω) ≤ c 0 n -3 4 δ3+ 5 2 δ2 + n -3 2 δ3+4δ2 , (2)
with c 0 := ((r + 1)

4 + 1) m∈Z |E[ξ 0 ξ m ]| + 1 2 σ 2 ξ .
Let us recall that 5 3 δ 2 < 1 2 δ 3 . Hence, it remains to estimate :

n δ2 Ln-1 k=(r+1) 4 E k-1 m=0 b m (b k -a k ) Ln m =k+1 a m |(S p ) p . (3)
• Let us introduce some holes in the indices m in order to use our decorrelation hypothesis. Let us control the following quantity :

Bn := n δ2 Ln-1 k=(r+1) 4 E     k-(r+1) 4 m=0 b m   3 j=1     k-(r+1) j m=k-(r+1) j+1 +1 b m   -1   k-1 m =k-r b m (b k -a k ) Ln m =k+1 a m (S p ) p  
We have :

Bn (ω) ≤ n δ2 Ln-1 k=(r+1) 4 3 j=1   k-(r+1) j m=k-(r+1) j+1 +1 b m   -1 L ∞ (Un) b k -a k L ∞ (Un) .
On U n , we have :

|b k -1| ≤ n -1 2 -δ3+δ2 n β n 1 2 +δ2 ≤ n -δ3+2δ2+β .
Analogously, we get :

k-(r+1) j m=k-(r+1) j+1 +1 b m -1 ≤ r(r + 1) j n -δ3+2δ2+β
. On the other hand, we have :

|a k -1| ≤ 1 2 n -2δ3+4δ2+β σ 2 ξ .
Therefore, since we have β < δ3 2 -δ 2 , we get :

Bn ≤ 4n δ2 n 1 2 +δ1-β r 3 (r + 1) 6 1 + 1 2 σ 2 ξ n -δ3+2δ2+β 4 = O n 1 2 -5 2 δ3+6δ2+δ1 .
The control of the quantity Bn comes from the fact that 5 2 δ 3 > 1 2 + 6δ 2 + δ 1 .

It remains to estimate : n δ2 Ln-1 k=(r+1) 4 +1 1≤j0<j1≤j2≤4 C n,k,j0,j1,j2 , where C n,k,j0,j1,j2 is the following quantity :

E     k-(r+1) 4 m=0 b m     k-(r+1) j 1 m=k-(r+1) j 2 +1 b m     k-1 m=k-(r+1) j 0 +1 b m   (b k -a k ) Ln m =k+1 a m (S p ) p   ,
with the convention :

β m=α b m = 1 if β < α.
Let j 0 , j 1 , j 2 be fixed. We have : C n,k,j0,j1,j2 ≤ D n,k,j0,j1,j2 + E n,k,j0,j1,j2 , with :

D n,k,j0,j1,j2 := Cov |(Sp)p (∆ n,k,j1,j2 , Γ n,k,j0 ) Ln m =k+1 a m and E n,k,j0,j1,j2 := E [ ∆ n,k,j1,j2 | (S p ) p ] E [ Γ n,k,j0 | (S p ) p ] Ln m =k+1 a m . with ∆ n,k,j1,j2 := k-(r+1) 4 m=0 b m k-(r+1) j 1 m =k-(r+1) j 2 +1 b m and Γ n,k,j0 := k-1 m=k-(r+1) j 0 +1 b m (b k -a k ).
• Control of the covariance terms (thanks to our decorrelation hypothesis). Let j 0 , j 1 , j 2 be fixed.

Let k = (r + 1) 4 , ..., L n -1. We have :

D n,k,j0,j1,j2 ≤ Cov |(Sp)p   ∆ n,k,j1,j2 , k m=k-(r+1) j 0 +1 b m   Ln m =k+1 a m + + Cov |(Sp)p   ∆ n,k,j1,j2 , k-1 m=k-(r+1) j 0 +1 b m   Ln m =k a m .
But we have : ξ Nn-1( )

. Therefore, according to point 2 of the hypothesis of our theorem, we have :

D n,k,j0,j1,j2 ≤ 2C 1 + n -1 2 -δ3+δ2 ∈Z N n-1 ( ) ϕ p,s
with p := n β ((r + 1) j1 -(r + 1) j0 ) and s := n β (r + 1) j0 -1. Let us notice that we have : p ≥ rs.

Since

∈Z N n-1 ( ) = n, we have :

n δ2 Ln-1 k=(r+1) 4 D n,k,j0,j1,j2 ≤ 4C n 1-δ3+δ1-β+2δ2 n -6β sup s≥n β s 6 ϕ rs,s ≤ 4C n 1-9 8 +δ1+( 27 2 +16)δ2 sup s≥n β s 6 ϕ rs,s , since β > δ3 2 -2δ 2 and δ 3 > 1 4 -3δ 2 .
We end this point by noticing that δ 1 + ( 27 2 + 16)δ 2 < 1 8 . • Control of the term with the product of the expectations. Let j 0 , j 1 , j 2 be fixed. Let k = (r + 1) 4 , ..., L n -1. We can notice that E n,k,j0,j1,j2 is bounded by the following quantity :

F n,k,j0 := E   k m=k-(r+1) j 0 +1 b m -   k-1 m=k-(r+1) j 0 +1 b m   a k (S p ) p   .
We approximate the terms with exponential using Taylor expansions.

-First we explain that, in F n,k,j0 , we can replace

k m=k-(r+1) j 0 +1 b m = exp   iun -1 2 -δ3+δ2 α (k+1) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( )  
by the formula given by the second order Taylor expansion of the exponential function :

1 + iun -1 2 -δ3+δ2 α (k+1) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( ) - u 2 2n 1+2δ3-2δ2   α (k+1) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( )   2 . (4)
Indeed, the induced error is less than :

1 6 n -3 2 -3δ3+3δ2 E α (k+1) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( ) 3 |(S p ) p .
Moreover, we have :

E    α (k+1) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( ) 4 |(S p ) p    ≤ α (k+1) -1 y1,y2,y3,y4=α (k-(r+1) j 0 +1) |E[ξ y1 ξ y2 ξ y3 ξ y4 ]| n 1 2 +δ2 4 ≤ c 0 n 2+4δ2 (r + 1) 6 n 2β ,
according to the hypothesis of our theorem. Hence, taking the sum over k = (r + 1) 4 , ..., L n -1 and multiplying by n δ2 , this substitution induces a total error bounded by :

(c 0 ) 3/4 6 n δ2+ 1 2 +δ1-β n -3 2 -3δ3+3δ2 n 3 2 +3δ2 (r + 1) 9 2 n 3 2 β
and so by :

(c 0 ) 3/4 6 n 7δ2+ 1 2 +δ1-3δ3+ 1 2 β (r + 1) 9 2 . Since β < δ3 2 -δ 2 , δ 3 > 1 4 -3δ 2 and δ 1 + ( 27 2 + 16)δ 2 < 1 8 , we have : 7δ 2 + 1 2 + δ 1 -3δ 3 + 1 2 β ≤ -1 16 . -Let us introduce Y k := α (k) -1 =α (k-(r+1) j 0 +1) ξ N n-1 ( ) and Z k := α (k+1) -1 =α (k) σ 2 ξ N n-1 ( ) 2 .
We explain that, in F n,k,j0 , we can replace

k-1 m=k-(r+1) j 0 +1 b m a k = e iu n 1 2 +δ 3 -δ 2 Y k - u 2 2n 1+2δ 3 -2δ 2 Z k
by the formula given by the Taylor expansion of the exponential function at the second order :

1 + iu n 1 2 +δ3-δ2 Y k - u 2 2n 1+2δ3-2δ2 Z k + 1 2 iu n 1 2 +δ3-δ2 Y k - u 2 2n 1+2δ3-2δ2 Z k 2 .
(

) 5
Indeed the modulus of the error between these two quantities is less than :

1 6 E iu n 1 2 +δ3-δ2 Y k - u 2 2n 1+2δ3-2δ2 Z k 3 |(S p ) p ≤ 4 3 E 1 n 1 2 +δ3-δ2 Y k 3 + 1 2n 1+2δ3-2δ2 Z k 3 |(S p ) p .
We control the first term as in the preceding point. Moreover, we have :

n -1-2δ3+2δ2 Z k 3 ≤ n -3-6δ3+6δ2 σ 2 ξ 3 n 3β n 3+6δ2 ≤ n -6δ3+12δ2+3β σ 2 ξ 3 .
Hence, taking the sum over k = (r + 1) 4 , ..., L n -1 and multiplying by n δ2 , we get a quantity bounded by : 2n

1 2 +δ1-6δ3+13δ2+2β σ 2 ξ 3
and we have :

1 2 + δ 1 -6δ 3 + 13δ 2 + 2β < 0.
-Now, we show that in formula (5), we can omit the term with (Z k ) 2 . Indeed, we have :

n δ2 Ln-1 (r+1) 4 n -1-2δ3+2δ2 Z k 2 ≤ 2n δ2+ 1 2 +δ1-β-2-4δ3+4δ2 n 2β (σ 2 ξ ) 2 n 2+4δ2 ≤ 2n -1 5 -2 5 δ1-2 5 δ2 (σ 2 ξ ) 2 since β < δ3 2 -δ 2 and 5 2 δ 3 > 1 2 + 6δ 2 + δ 1 .
-Hence, it remains to estimate the following quantity called G n,k,j0 :

E iu n 1 2 +δ3-δ2 (Y k + W k ) - u 2 2n 1+2δ3-2δ2 (Y k + W k ) 2 - iu n 1 2 +δ3-δ2 Y k + + u 2 2n 1+2δ3-2δ2 Z k + u 2 2n 1+2δ3-2δ2 (Y k ) 2 + iu n 1 2 +δ3-δ2 Y k u 2 2n 1+2δ3-2δ2 Z k (S p ) p , with W k := α (k+1) -1 =α (k)
ξ N n-1 ( ). We get :

G n,k,j0 = E - u 2 2n 1+2δ3-2δ2 (Y k + W k ) 2 + u 2 2n 1+2δ3-2δ2 Z k + u 2 2n 1+2δ3-2δ2 (Y k ) 2 (S p ) p = u 2 2n 1+2δ3-2δ2 E (W k ) 2 + 2W k Y k -Z k (S p ) p .
Let us notice that we have :

Z k := α (k+1) -1 =α (k)   E[(ξ ) 2 ]N n-1 ( ) 2 + 2 m≤ -1 E[ξ ξ m ]N n-1 ( ) 2   .
-Let us show that, in the last expression of G n,k,j0 , we can replace Z k by :

Zk := α (k+1) -1 =α (k)   E[(ξ ) 2 ]N n-1 ( ) 2 + 2 m≤ -1 E[ξ ξ m ]N n-1 ( )N n-1 (m)   .
Indeed, by definition of U n , we have :

u 2 2n 1+2δ3-2δ2 E Z k -Zk (S p ) p ≤ ≤ 1 n 1+2δ3-2δ2 α (k+1) -1 =α (k) m≤ -1 |E[ξ ξ m ]|N n-1 ( )|N n-1 (m) -N n-1 ( )| ≤ n -1 4 -2δ3+3δ2+β+ γ 2 m≥1 √ m|E[ξ 0 ξ m ]|.
Hence, taking the sum over k = (r + 1) 4 , ..., L n -1 and multiplying by n δ2 , we get a quantity bounded by : 4n

1 4 +δ1-2δ3+4δ2+ γ 2 m≥1 √ m|E[ξ 0 ξ m ]|. But, since δ 3 > 1 4 -3δ 2 and γ < 1 2 - 22 max(δ 1 , δ 2 ), we have : 1 4 + δ 1 -2δ 3 + 4δ 2 + γ 2 < 0 -Hence we have to estimate : Gn,k,j0 = u 2 2n 1+2δ3-2δ2 E (W k ) 2 + 2W k Y k -Zk (S p ) p .
We have :

E (W k ) 2 (S p ) p = α (k+1) -1 =α (k)   E[(ξ ) 2 ](N n-1 ( )) 2 + 2 -1 m=α (k) E[ξ ξ m ]N n-1 ( )N n-1 (m)   .
Hence we have :

E (W k ) 2 + 2W k Y k (S p ) p = α (k+1) -1 =α (k)   E[(ξ ) 2 ](N n-1 ( )) 2 + 2 -1 m=α (k-(r+1) j 0 +1) E[ξ ξ m ]N n-1 ( )N n-1 (m)   .
We get :

Gn,k,j0 = u 2 n 1+2δ3-2δ2 α (k+1) -1 =α (k) m≤α (k-(r+1) j 0 +1) -1 E[ξ ξ m ]N n-1 ( )N n-1 (m) ≤ u 2 n 1+2δ3-2δ2 n β m≥ (r+1)n β 2 |E[ξ 0 ξ m ]|n 1+2δ2 ≤ n -2δ3+4δ2+β m≥ (r+1)n β 2 |E[ξ 0 ξ m ]|.
Hence, taking the sum over k = (r + 1) 4 , ..., L n -1 of these quantities and multiplying by n δ2 , we get a quantity bounded by :

4n 1 2 +δ1-2δ3+5δ2 m≥ (r+1)n β 2 |E[ξ 0 ξ m ]| ≤ 4n δ1+11δ2 m≥ (r+1)n β 2 |E[ξ 0 ξ m ]|, since δ 3 > 1 4 -3δ 2 .
To conclude it suffices to notice that :

n δ1+11δ2 m≥ (r+1)n β 2 |E[ξ 0 ξ m ]| = O(n -ε ).

Proof of lemma 9

Let us consider n ≥ 2. According to lemma 10, it suffices to prove that there exists a real number δ > 0 such that we have :

sup n≥1 n δ sup ω∈Un n δ2 |u|≤1 exp - u 2 2n 1+2δ3-2δ2 y,z E[ξ y ξ z ](N n-1 (y)(ω)) 2 e -u 2 n 2δ 2 2 du < +∞.
Let us take ω ∈ U n . We have :

exp - u 2 2n 1+2δ3-2δ2 y,z E[ξ y ξ z ](N n-1 (y)(ω)) 2 = exp - u 2 2n 1+2δ3-2δ2 σ 2 ξ y (N n-1 (y)(ω)) 2 .
Let us define :

p n := Card{y ∈ Z : N n-1 (y) ≥ n 1 2 -δ 4 3
}. We have :

n = j n 1 2 +δ 1 k y=- j n 1 2 +δ 1 k N n-1 (y) ≤ p n n 1 2 +δ2 + n 1 2 -δ4 3 3n 1 2 +δ1 -p n ≤ p n n 1 2 +δ2 1 - n -(δ2+δ4) 3 + n 1+δ1-δ4 .
Since δ 1 < δ 4 , we have :

p n ≥ n -1 2 -δ2 n -n 1-(δ4-δ1) ≥ n 1 2 -δ2 1 -n -(δ4-δ1) ≥ c 0 n 1 2 -δ2 , with c 0 := 1 -2 -(δ4-δ1
) . Hence we have :

y∈Z (N n-1 (y)(ω)) 2 ≥ p n n 1 2 -δ 4 3 2 ≥ c0n 3 2 -δ 2 -2δ 4 9 and e - u 2 P y σ 2 ξ (N n-1 (y)(ω)) 2 2n 1+2δ 3 -2δ 2 ≤ e - u 2 σ 2 ξ c 0 n 3 2 -δ 2 -2δ 4 18n 1+2δ 3 -2δ 2 ≤ e -u 2 18 σ 2 ξ c0n 1 2 +δ 2 -2δ 3 -2δ 4 .
Therefore, we have :

n δ2 |u|≤1 e - u 2 2n 1+2δ 3 -2δ 2 P y,z E[ξyξz](Nn-1(y)(ω)) 2 e -u 2 n 2δ 2 2 du ≤ ≤ n δ2 |u|≤1 e -u 2 18 σ 2 ξ c0n 1 2 +δ 2 -2δ 3 -2δ 4 du ≤ n -1 4 +δ4+ 1 2 δ2+δ3 R e -v 2 18 σ 2 ξ c0 dv.
This ends the proof since

δ 4 + δ 3 + 1 2 δ 2 < 1 4 .
4 About the model of Guillotin-Plantard and Le Ny

In this section, we prove that the hypothesis

M 1 √ f0(1-f0)
dν < +∞ of Guillotin-Plantard and Le Ny in [START_REF] Guillotin-Plantard | Transient random walks on 2d-oriented lattices[END_REF] can be replaced by the existence of p ≥ 1 such that M 1 (f0(1-f0)) p dν < +∞, for some p > 0. In this situation, there is no need to introduce the set U n ; we take U n = A n . If we take δ 1 > 0, δ 2 > 0 and δ 3 > 0, all the points (of the sketch of the proof of section 3) except the point 3(b)(ii) come in the same way without the need of the hypothesis

M 1 √ f0(1-f0)
dν < +∞. It remains to estimate :

sup ω∈An n 1 2 +δ3 I (1) n (ω) := n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E e it P y∈Z ξyNn-1(y) (S p ) p (ω)e -t 2 n 1+2δ 3 2 dt.
Let us take ω ∈ A n . We suppose

δ 3 > 2δ 2 and δ 1 < δ 4 < 1 4 -δ 3 -δ2 2 .
The idea of Guillotin-Plantard and Le Ny is to write :

n 1 2 +δ3 I (1) n ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E   y∈Z |cos(tN n-1 (y)) + i(2f y -1) sin(tN n-1 (y))| (S p ) p   e -t 2 n 1+2δ 3 2 dt ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E   y∈Z 1 -4f y (1 -f y ) sin 2 (tN n-1 (y)) (S p ) p   e -t 2 n 1+2δ 3 2 dt ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E   y∈Z 1 -f y (1 -f y ) 16 π 2 (tN n-1 (y)) 2 (S p ) p   e -t 2 n 1+2δ 3 2 dt ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E   y∈Z e -8 π 2 fy(1-fy)t 2 Nn-1(y) 2 (S p ) p   e -t 2 n 1+2δ 3 2 dt since |tN n-1 (y)| ≤ n -1 2 -δ3+δ2 n 1 2 +δ2 = n 2δ2-δ3
. Hence, if n is large enough, then |tN n-1 (y)| will be uniformly less than π 2 and |sin(tN n-1 (y))| ≥ 2 π |tN n-1 (y)|. We also use the fact that, for positive u, we have : 1 -u ≤ e -u . According to the Hölder inequality with y Nn-1(y) 2 P k Nn-1(k) 2 = 1, we have :

n 1 2 +δ3 I (1) n ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E e -8 π 2 f0(1-f0)t 2 P k Nn-1(k) 2 (S p ) p e -t 2 n 1+2δ 3 2 dt.
Now, we use the fact that, since δ 4 > δ 1 , there exists a constant c such that we have :

∀ω ∈ A n , y∈Z (N n-1 (y)) 2 (ω ) ≥ cn 3 2 -δ2-2δ4 .
This has been proved in the previous section entitled 'proof of lemma 9'. Hence, under the hypothesis

M 1 √ f0(1-f0)
dν < +∞ of Guillotin-Plantard and Le Ny, we have :

n 1 2 +δ3 I (1) n (ω) ≤ n 1 2 +δ3 {|t|≤n -1 2 -δ 3 +δ 2 } E e -8c π 2 f0(1-f0)t 2 n 3 2 -δ 2 -2δ 4 e -t 2 n 1+2δ 3 2 dt ≤ n -1 4 +δ3+ δ 2 2 +δ4 R E 1 f 0 (1 -f 0 ) e -8 π 2 v 2 dv with the change of variable v = t f 0 (1 -f 0 )n 3 2 -δ2-2δ4
. This gives the result of Guillotin-Plantard and Le Ny since -1 4 + δ 3 + δ2 2 + δ 4 < 0. We adapt this argument to our hypothesis. Now let us replace the hypothesis M 1 √ f0(1-f0) dν < +∞ by M 1 [f0(1-f0)] p dν < +∞ for some p > 0. Let us take δ 3 > 2δ 2 and δ 1 < δ 4 < 1 4 -δ 3 -δ2 2 -δ2 p . We have :

n 1 2 +δ3 {|t|≤n -3 4 + δ 2 2 +δ 4 + δ 2 p } E e -8 π 2 f0(1-f0)t 2 n 3 2 -δ 2 -2δ 4 e -t 2 n 1+2δ 3 2 dt ≤ 2n 1 2 +δ3 n -3 4 + δ 2 2 +δ4+ δ 2 p ≤ 2n -1 4 +δ3+ δ 2 2 +δ4+ δ 2 p .
On the other hand, let c p = sup u>0 u p e -u , we have :

n 1 2 +δ3 {n -3 4 + δ 2 2 +δ 4 + δ 2 p <|t|<n -1 2 -δ 3 +δ 2 } E e -8 π 2 f0(1-f0)t 2 n 3 2 -δ 2 -2δ 4 e -t 2 n 1+2δ 3 2 dt ≤ ≤ 2n 1 2 +δ3 n -1 2 -δ3+δ2 M e -8 π 2 f0(1-f0)n 2δ 2 p dν ≤ n -δ2 c p π 2 8 p M [f 0 (1 -f 0 )] -p dν.
A Proof of proposition 4

In cases (a) and (b), (ξ k ) k is a stationary sequence of bounded centered random variables

A.1 Proof of (a)

We have :

p≥0 √ 1 + p|E[ξ 0 ξ p ]| = p≥0 √ 1 + p|E ν [f.f •T p ]| which is less than : c 0 f ∞ f ∞ + K (1) f + K (2) f p≥0
√ and hence is finite. Let us consider an integer N ≥ 1. We have :

1 N 2
k1,k2,k3,k4=0,...,N -1

|E[ξ k1 ξ k2 ξ k3 ξ k4 ]| ≤ 24 N 2 0≤k1≤k2≤k3≤k4≤N -1 |E[ξ k1 ξ k2 ξ k3 ξ k4 ]| .
Let us consider the set E

(1)

N of (k 1 , k 2 , k 3 , k 4 ) such that 0 ≤ k 1 ≤ k 2 ≤ k 3 ≤ k 4 ≤ N -1 and k 4 -k 3 ≥ N 1 3
We have :

(k1,k2,k3,k4)∈E (1) N |E[ξ k1 ξ k2 ξ k3 ξ k4 ]| = (k1,k2,k3,k4)∈E (1) N Cov ν f • T k1-k3 f • T k2-k3 f, f • T k4-k3 ≤ c 0 N 4 f 4 ∞ + f 3 ∞ (K (2)
f + 3c 0 K (1) f ) ϕ N 1 3 ≤ c 0 N 2 f 4 ∞ + f 3 ∞ (K (2)
f + 3c 0 K (1) f ) sup n≥1 n 6 ϕ n .
Let us consider the set E

(2)

N of (k 1 , k 2 , k 3 , k 4 ) such that 0 ≤ k 1 ≤ k 2 ≤ k 3 ≤ k 4 ≤ N -1 and k 4 -k 3 < N 1 3
and k 3 -k 2 ≥ rN 1 3 . We have :

(k1,k2,k3,k4)∈E (2) N |Cov (ξ k1 ξ k2 , ξ k3 ξ k4 )| = (k1,k2,k3,k4)∈E (2) N Cov ν f • T k1-k2 f, (f.f • T k4-k3 ) • T k3-k2 ≤ 2 6 c 0 N 2 f 4 ∞ + 2c 0 f 3 ∞ (K (2) f + K (1) f ) sup n≥1 n 6 (1 + κ n )ϕ rn .
Moreover, we have :

(k1,k2,k3,k4)∈E (2) N |E[ξ k1 ξ k2 ]E[ξ k3 ξ k4 ]| ≤   0≤k1≤k2≤N -1 |E[ξ k1 ξ k2 ]|   2 ≤   N k≥0 E ν [f.f • T k ]   2 ≤ N 2   c 0 f 2 ∞ + f ∞ (K (1)
f + K (2) f ) k≥0 ϕ k   2 .
Let us consider the set E

(3)

N of (k 1 , k 2 , k 3 , k 4 ) such that 0 ≤ k 1 ≤ k 2 ≤ k 3 ≤ k 4 ≤ N -1 and k 4 -k 3 < N 1 3 and k 3 -k 2 < rN 1 3 and k 2 -k 1 ≥ r(1 + r)N 1 3
. By the same method, we get :

(k1,k2,k3,k4)∈E (3) N |E [ξ k1 ξ k2 ξ k3 ξ k4 ]| ≤ N 2 c 0 2 6 (1 + r) 6 f 4 ∞ + 3c 0 f 3 ∞ (K (2)
f + K (1) f ) sup n≥1 n 6 (1 + κ n )ϕ rn .
Since the number of (k

1 , k 2 , k 3 , k 4 ) such that 0 ≤ k 1 ≤ k 2 ≤ k 3 ≤ k 4 ≤ N -1 and that do not belong to E (1)
N ∪ E

N ∪ E

N is bounded by N 2 2(r + 1) 3 , we get :

sup N ≥1 1 N 2
β k ξ k = Cov ν e i P n 2 k=n 1 α k f •T -(n 2 -k) , e i P n 4 k=n 3 β k f •T k-n 3 • T n3-n2 ≤ c 0 1 + K (1)
exp

" i P n 2 k=n 1 α k f •T -(n 2 -k) " + K (2) exp " i P n 4 k=n 3 β k f •T k-n 3 " ϕ n3-n2 ≤ c 0 1 + n2 k=n1 c 0 c 1 |α k | + n4 k=n3 c 0 c 1 |β k |(1 + κ n4-n3 ) ϕ n3-n2 .
This gives the point 2 of the hypothesis of theorem 1 with ϕ p,s := (1 + κ s )ϕ p .

A.2 Proof of (b)

Let us define the function g = 2f -1. This function is ν-centered. More generally, for any integer m ≥ 1, let us define : g 2m = 1 and g 2m+1 = g. We observe that, conditionally to ω ∈ M , the expectation of (ξ k (ω, •)) m is equal to g m • T k (ω). Using the Fubini theorem and starting by integrating over [0; 1] Z , we observe that, for any integers p ≥ 1, we have : E[ξ 0 ξ p ] = E ν [g.g • T p ] and that, for any integers

k 1 , k 2 , k 3 , k 4 , we have : E [ξ k1 n1 ξ k2 n2 ξ k3 n3 ξ k4 n4 ] = E ν 4
j=1 g nj • T kj . Hence, we can prove the point 1 of theorem 1 as we did for (a). Now, let us prove the point 2 of the hypothesis of theorem 1. We observe that, conditionally to ω ∈ M , the ξ k (ω, •) are independent and that the expectation of exp

(iuξ k (ω, •)) is h u • T k (ω) with (h u := e -iu + 2i sin(u)f • T k .
The modulus of this function is bounded by 1 and we have : max K

(1) hu , K (2) hu ≤ 2c 1 |u|.
Let n 1 , n 2 , n 3 and n 4 be four integers such that 0 ≤ n 1 ≤ n 2 < n 3 ≤ n 4 . Let us consider any real numbers α n1 , .., α n2 and β n3 , ..., β n4 . We have :

Cov e i P n 2 k=n 1 α k ξ k , e i P n 4 k=n 3 β k ξ k = Cov ν n2 k=n1 h α k • T k , n4 k=n3 h β k • T k ≤ c 0 1 + 2c 0 c 1 n2 k=n1 |α k | + n4 k=n3 |β k | (1 + κ n4-n3 )ϕ n3-n2 .

B Proof of theorem 2 : α-mixing condition

Let us define (M, F) = (R Z , B(R) ⊗Z ). Let T : M → M be such that T ((ω k ) k∈Z ) = (ω k+1 ) k∈Z . Let ν be the image probability measure on (M, F) of Π : Ω → R Z with Π(ω) = (ξ k (ω)) k∈Z . The process (ξ k ) k∈Z (with respect to P) has the same distribution as (f • T k ) k∈Z (with respect to ν) with f : M → R given by f ((ω k ) k∈Z ) = ω 0 . According to [START_REF] Ibragimov | Some limit theorems for stationary processes[END_REF], lemma 1.2, (M, F, ν, T ) is strongly mixing (in the sense of our definition 3) with the following choice of K

• and of K

• . If g is σ(f • T k , k ≤ 0)-measurable, we have K

(1) g := 0; otherwise we have K Let η > 0. Let us denote by Γ (s,e) the set of stable-central manifolds and by Γ u the set of unstable manifolds. In [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF], each γ u ∈ Γ u is endowed with some metric d u and each γ (s,e) ∈ Γ (s,e) is endowed with some metric d (s,e) such that there exist c0 > 0, δ 0 ∈]0; 1[ and β > 0 such that, for any integer n ≥ 0, for any γ u ∈ Γ u and any γ (s,e) ∈ Γ (s,e) , we have :

• For any y, z ∈ γ u , d u (y, z) ≥ d(y, z) and for any y , z ∈ γ (s,e) , d (s,e) (y , z ) ≥ d(y , z ).

• For any y, z ∈ γ u , there exists γ u (n) ∈ Γ u such that T -n (y) and T -n (z) belong to γ u (n) and we have : d u (T -n (y), T -n (z)) ≤ c0 (δ 0 ) n d u (y, z).

• For any y, z ∈ γ (s,e) , there exists γ (s,e) (n) ∈ Γ (s,e) such that T n (y) and T n (z) belong to γ (s,e) (n) and we have : d (s,e) (T n (y), T n (z)) ≤ c0 (1 + n β )d (s,e) (y, z). For these examples, the result follows from [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF] (cf. lemme 1.3.1 in [START_REF] Borgne | Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques[END_REF]).

C.2 Case 2 : Sinai billiard

Since the early work of Sinai [START_REF] Sinai Ya | Dynamical systems with elastic reflections[END_REF], this billiard system has been studied by many authors ([1, 2, 3, 4, 8] and others). Let us recall that a point of M is a couple (q, v) corresponds to a reflected unit speed vector v at the position q on some obstacle O i and is parametrised by (i, rϕ) where i is the index of the obstacle O i , r the curvilinear of x on it and ϕ the measure of the angle (taken in [-π/2; π/2]) made by v with the unit normal vector n(q) to O i at q directed to the outside of the obstacle. We endow M with a metric d such that : d((i, r, ϕ), . For all k = -m, ..., m, T k is C 1 on each C belonging to C m . We will use the notations of Chernov in [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF]. Let us consider the set Γ s of homogeneous stable curves and the set Γ u of homogeneous unstable curves and the two separation times s + (•, •) (in the future) and s -(•, •) (in the past) considered in [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF]. We recall that there exist two constants c 1 > 0 and δ 1 ∈]0; 1[ such that, for any nonnegative integer n, for any y and z in M , we have : This gives the point 2 of the hypothesis of theorem 1 with ϕ p,s = p -7 + (1 + s β )p 7/ζ δ p .

• If

  , ν 1 := 2 Main result, examples, strong mixing property Theorem 1. Let (ξ k ) k∈Z be a stationary sequence of centered random variables with values in {-1; 1} such that : 1. we have : p≥0 √ 1 + p |E[ξ 0 ξ p ]| < +∞ and c 0 := sup N ≥1 N -2 k1,k2,k3,k4=0,...,N -1 |E[ξ k1 ξ k2 ξ k3 ξ k4 ]| < +∞.

sup n≥1 n 6

 6 α n < +∞, with α n := sup p≥0; m≥0 sup A∈σ(g-p,...,g0) sup B∈σ(gn,...,gn+m) |P(A ∩ B) -P(A)P(B)| .

h

  will be dominating by C (η,m0) h = sup C∈Cm sup x,y∈C, x =y |h(x)-h(y)| max(d(T k (x),T k (y)); k=-m,...,m) η , where C m is a set of open subsets of M on which T m and T -m are C 1 .

h

  are both dominated by the η-Hölder constant of h. Then : (A) If (ξ k := ξk,g0•T k ) k∈Z with g 0 : M → [0; 1] a Hölder continuous function (of order η) such that

Lemma 7 .

 7 The process (X n , S n ) n≥1 with X n := y∈Z ξ distribution as (M Tn ) n≥1 .

1 2 1 2 1 2 +δ1 +1; b k := e iun - 1 2-δ 3 +δ 2

 11112 +δ1-β ) and, for all integer k = 0, ..., L n : α (k) := -n +δ1 +k n β and α (Ln+1) := n Pα (k+1) -1 y=α (k)

θ1+θ2 m=θ1+1 b m = e iun - 1 2-δ 3 +δ 2

 12 Pα (θ 1 +θ 2 +1) -1 =α (θ 1 )

1 C. 1

 11 . If h is σ(f • T k , k ≥ 0)-measurable, we have K (2) h := 0; otherwise we have K (2)h := ∞. We conclude with proposition 4.C Proof of example 2.Case 1

  γ u ∈Γ u sup y,z∈γ u :y =z |f (y) -f (z)| (d u (y, z)) η and K (2) f := sup γ (s,e) ∈Γ (s,e) sup y,z∈γ (s,e) :y =z |f (y) -f (z)| (d (s,e) (y, z)) η .

  (i, r , ϕ )) = |r -r | + |ϕ -ϕ |.Let us denote by R 0 the set of points in M corresponding to a reflected vectors tangent to the obstacles, i.e. such that ϕ = ±π/2. The transformationT n defines a C 1 -diffeomorphism from M \ n k=0 T -k (R 0 ) onto M \ n k=0 T k (R 0 ). Let us consider the set C m of connected components of M \ m k=-m T k (R 0 )

1 n 2 ( 1 + (n 4 -≤ C 1 7 ζ

 121417 y and z belong to the same homogeneous unstable curve, then s + (x, y) ∈ Z + , moreover T -n (y) and T -n (z) belong to a same homogeneous unstable curve and we have :d(T -n (y), T -n (z)) ≤ c 1 δ and s + (T -n (x), T -n (y)) ≥ n + s + (x, y). n 3 ) β )α n3-n2 k | (1 + (n 4 -n 3 ) β )(n 3 -n 2 ) α n3-n2 .

  |E[ξ k1 ξ k2 ξ k3 ξ k4 ]| < +∞.Now, let us prove the point 2 of the hypothesis of theorem 1. Let n 1 , n 2 , n 3 and n 4 be four integers such that 0 ≤ n 1 ≤ n 2 ≤ n 3 ≤ n 4 . Let us consider any real numbers α n1 , .., α n2 and β n3 , ..., β n4 . We have :

				k1,k2,k3,k4=0,...,N -1
	Cov e i	P n 2 k=n 1	α k ξ k , e i	P n 4 k=n 3

+δ3 }. As in proposition 4.3 of [5], we have : n≥0 P(B n ∩ {X n = 0 and S n = 0}) < +∞. It remains to prove that : n≥0 P (U n ∩ {X n = 0 and S n = 0} \ B n ) < +∞.

• If y and z belong to the same homogeneous stable curve, then s -(x, y) ∈ Z + , moreover T n (y) and

T n (z) belong to a same homogeneous stable curve and we have : d(T n (y), T n (z)) ≤ c 1 δ 1 n and s -(T n (x), T n (y)) ≥ n + s -(x, y).

With these notations, according to [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF] (theorem 4.3 in [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF] and the remark after theorem 4.3 in [START_REF] Chernov | Advanced statistical properties of dispersing billiards[END_REF]), this system is strongly mixing with :

D Proof of conclusion (B) of theorem 5

We will use b and δ of proposition 2.1. First let us notice that there exists c A > 0 such that, for every ε ∈]0; 1[, there exists a Lipschitz continuous function f ε such that :

This quantity can be rewritten :

and is less than :

. Moreover, we have :

• Let us prove that : sup

k1,k2,k3,k4=0,...,N -1

We use the notations E

N , E

N and E

N and the calculations done in section A.1. -To estimate (k1,k2,k3,k4)∈E (1)

. This substitution makes a total error in O(N 2 ). Moreover, according to the calculations of section A.1, we have :

-With the same technique, we get :

). Moreover, as in section A.1, we have : (k1,k2,k3,k4)∈E (2)

and we have already proved that :

N ∪ E

N ∪ E

N is controlled as in section A.1.

• Let us prove point 2 of hypothesis of theorem 1. By replacing each ξ k by ξ(n3-n2)

ζ ] , we make a total error in 1 +