Ahmed Soua
email: ahmed.soua@vedecom.fr

Oyunchimeg Shagdar
email: oyunchimeg.shagdar@vedecom.fr

Jean-Marc Lasgouttes
email: jean-marc.lasgouttes@inria.fr

Toward Efficient

Keywords: Platoon communication, simulation platform, large scale scenario, C-ITS

I. INTRODUCTION Cooperative vehicular communications, based on wireless data exchange between moving vehicles as well as with

Road Side Units, have attracted the interest of both industrial and research communities, thanks to their potential capacity to increase road safety and improve traffic efficiency. In fact, these emerging Cooperative Intelligent Transportation Systems (C-ITS) are promising safer roads by enabling several safety applications like lane-changing alert, accident-ahead notification and bad weather conditions alarm. On the other hand, different traffic management-related scenarios have been investigated by these cooperative networks, like lane-merging assistance, parking availability, and platoon communication. This latter use case is regarded as one of the most efficient remedies to congested roads issue in big cities. Vehicle platooning, where vehicles drive in a convoy with small relative distance, is expected to provide higher throughput, improved traffic safety [START_REF] Hall | Vehicle sorting for platoon formation: Impacts on highway entry and throughput[END_REF][2] and lower energy consumption due to slipstream [START_REF] Shladover | Cooperative (rather than autonomous) vehicle-highway automation systems[END_REF].

Platooning in general is realized through two functioning modes. The first one is based on Adaptive Cruise Control (ACC) [START_REF] Hosseinnia | Experimental application of hybrid fractional-order adaptive cruise control at low speed[END_REF] mechanism, which uses sensors such as radar, camera and laser to make an estimation of the relative distance and Fig. 1. Cooperative platoon in large scale environment velocity of two consecutive vehicles. Since ACC mode can not provide drivers with a very precise estimation of the distance gap or the velocity of vehicles, Cooperative Adaptive Cruise Control (CACC) [START_REF] Orosz | Connected cruise control: modeling, delay effects, and nonlinear behavior[END_REF] was proposed to tackle these limitations. In this cooperative mode, vehicles in the platoon exchange their dynamic state information like speed, position and heading using Vehicle to Vehicle (V2V) communications as well as Visible Light Communication (VLC) [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF]. This information, together with data gathered from on-board sensors, is used to adapt their own acceleration and speed to maintain the inter-vehicle distance at a desired value as shown in Fig. 1.

Most of the existing efforts on platooning have been concentrated on the control/behavior of the individual platoon members [3][4], or on the interaction/communication between the platoon members [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF][7] for a stable chain [START_REF] Hosseinnia | Experimental application of hybrid fractional-order adaptive cruise control at low speed[END_REF][5] [START_REF] Shagdar | Performance study of CAM over IEEE 802.11p for cooperative adaptive cruise control[END_REF]. Nevertheless, there is a lack of studies that have investigated the platoon in its environment, sharing the road with other vehicles under the strong traffic regulations (traffic lights, speed limits, and so on).

For the latter case, in order to validate its efficiency in alleviating the burden of traffic congestion, platoon communi-978-1-5386-3779-1/18/$31.00 c 2018 IEEE cation needs to be tested in a real scenario where the convoy of vehicles makes its journey through a large urban area with all its components such as moving vehicles, infrastructures, buildings, roads, radars, etc. Relying on experimental testbeds and real deployment is usually impractical since it requires an access to a large number of fleets and thousands of vehicles equipped with different technologies and communication abilities. For these reasons, simulation tools are regarded as an efficient technique to emulate large scale environments and accommodate their complexities with less technical and financial challenges.

While platooning control and interactions between the platoon members can be simulated using Matlab/Simulink [START_REF] Shagdar | Performance study of CAM over IEEE 802.11p for cooperative adaptive cruise control[END_REF] or ProSiVIC [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF], these tools cannot be used to simulate a platoon in its environment, where traffic behavior and V2X communication of a large number of vehicles need to be investigated.

In this paper, we consider the idea of enhancing the performance of iTETRIS [START_REF] Kumar | iTETRIS: adaptation of ITS technologies for large scale integrated simulation[END_REF], a dedicated simulation framework for vehicular communication, in order to tackle very large simulation scenarios. In particular, we present a new technique to adapt iTETRIS to a CACC-based platoon scenario in a city-wide context. iTETRIS consists of two main components: the network simulator ns-3 [START_REF]The ns-3 nework simulator[END_REF] and the mobility simulator SUMO [START_REF] Krajzewicz | Sumo (simulation of urban mobility)[END_REF]. The bottleneck is the execution time of ns-3, which is much more processing computation-intensive than SUMO. While ns-3 is able to model wireless networks with a high level of details, it has strong scalability limitations and struggles to simulate networks with only few thousands of nodes. As shown in [START_REF] Fernandes | Scalable VANET simulations with ns-3[END_REF], to process a simulation step of 1 second with 300 nodes sending 1 beacon per second, ns-3 requires approximately 75 seconds of real time. Consequently, a one minute simulation would take several hours to be carried out.

Our solution is based on constructing a specific zone called Restricted Zone (RZone) around the leader of the platoon and letting only the vehicles inside this zone exchange messages. The rest of vehicles are deactivated, since we are only interested on the platoon communication. The key idea behind this restriction is that vehicles situated beyond the interference zone of the platoon leader, are not concerned by the exchanged information between platoon members and not able also to interfere with these communications. It is thus possible to focus only on vehicles located inside the interference zone of the platoon leader.

The rest of this paper is organized as follows: Section II describes some of the state of the art in relation to ITS simulation platforms. In Section III, we present the simulation platform iTETRIS and we highlight its limitations to perform large scale simulations. Then, in Section III-D, we describe the key components of our proposed technique aiming at simulating cooperative platoon communications in large scale environments with low processing time requirements. Section IV investigates the potential of the proposed platform and its efficiency compared to iTETRIS. Conclusion and future work are outlined in Section V.

II. STATE OF THE ART

Before going in depth with our proposed enhancement to the iTETRIS platform for large scale simulation scenarios, it is worth taking an overview at the available literature on simulation platforms dedicated to cooperative transportation systems.

Usually, simulation tools dedicated to vehicular ad hoc networks (VANETs) are formed of two categories of simulators: networking and traffic simulators. These two components are dependent because, as vehicles move in the traffic platform, the network topology (as wireless nodes) in the networking simulator changes dynamically. While the traffic simulator is used to reproduce realistic vehicular mobility traces in accordance to the surrounding environments, like road topology, traffic lights, human interactions, etc., the networking simulator aims to perform detailed packet-level simulation of source, destinations, data traffic transmission, reception, background load, route, links, and channels [START_REF] Grzybek | Aspects and trends in realistic VANET simulations[END_REF].

Recent years have witnessed a growing interest in developing more realistic environments to validate ITS applications. Shawn [START_REF] Pfisterer | Comprehensive development support for wireless sensor networks[END_REF], Veins [START_REF] Sommer | Bidirectionally coupled network and road traffic simulation for improved ivc analysis[END_REF] and iTETRIS [START_REF] Kumar | iTETRIS: adaptation of ITS technologies for large scale integrated simulation[END_REF] are examples of simulation platforms that couple a mobility simulator with a network one: while SUMO is the common traffic simulator chosen by all the mentioned platforms, the network simulator is different for each tool. In fact, Shawn integrates its own specific network simulator, while Veins and iTETRIS are using OMNET [START_REF] Varga | The OMNET++ discrete event simulation system[END_REF] and ns-3 respectively. This coupling between the two simulators in such platforms, called federated architecture, allows to realistically analyze the influence of VANET-based applications on the road traffic and vice versa.

Another strategy to build VANET simulators is to conceive a unique tool that integrates both a network simulator and a traffic generator. NCTUns [START_REF] Wang | Nctuns 5.0: A network simulator for ieee 802[END_REF] stands as a relevant example of this technique. This simulator actually was at first developed as a network simulator, but then integrated some traffic simulation capabilities. NCTUns offers the possibility to simulate various protocols used in both wireless and wired networks and it uses Linux TCP/IP protocol stack to generate high-fidelity simulation results. In this simulator, since the code for the vehicles movement is integrated with the network simulation code, making some extension to the simulator is a very difficult task.

Since this current work is funded by SINETIC project [17], which aims at developing a complete framework to simulate C-ITS based on iTETRIS platform, we focus in this paper on enhancing iTETRIS to perform large scale simulations for platoon application.

III. SYSTEM MODEL

In this section, the system model of the enhanced platform is presented. We first describe in details the iTETRIS platform and its components, followed by presenting the cooperative platoon application. Then, we highlight our proposed technique to efficiently simulate platoons in large cities aiming at minimizing the processing time of simulations.

A. The iTETRIS Platform

We now describe the main components of the iTETRIS simulation framework. A general overview of the the platform is shown in Fig. 2.

The main objective of the platform is to efficiently simulate cooperative vehicular communication with real mobility patterns. The iCS interface is the component responsible for federating the traffic simulator SUMO, the network simulator ns-3 and the implemented ITS applications. In the following, we detail the different elements of this platform.

B. Platoon Communication Model

In this section, we specify the characteristics of the platoon system we want to simulate for the enhanced simulation platform. We suppose that the platoon system uses the CACC mode [START_REF] Orosz | Connected cruise control: modeling, delay effects, and nonlinear behavior[END_REF] to maintain its stability, which means that each vehicle in the platoon is equipped with a wireless interface and is able to exchange vehicular status (e.g., speed, orientation, acceleration) with other nodes as depicted by Fig. 1. The ETSI standardized Cooperative Awareness Message (CAM) [START_REF]Intelligent Transport Systems (ITS); Vehicular Communications; Basic Set of Applications; Part 2: Specification of Cooperative Awareness Basic Service[END_REF] can be used for intra-platoon communication to maintain a stable platoon, as well as for communication between the platoon and other non-platoon vehicles for collision avoidance [START_REF] Shagdar | Performance study of CAM over IEEE 802.11p for cooperative adaptive cruise control[END_REF]. Specifically, each member of the platoon announces its motion to other cars through CAMs. CAMs contain information about the sender like vehicle type, position, heading, velocity. The receiver of such messages interprets them and then adapts its velocity to keep the inter-distance from the preceding vehicle stable. As mentioned earlier, non-platoon vehicles on the road also send CAMs for the purpose of cooperative awareness.

A preliminary implementation of the platoon model inside iTETRIS platform was carried out within SUMO simulator. The main goal behind this implementation is to investigate how we can make iTETRIS faster at simulating platoon communication in the case of city-wide scenario.

C. Problem Statement

In this paper, we focus on CACC-enabled platoons where vehicles exchange periodic information to keep the platoon stable (and thus avoid the accordion effect) using iTETRIS platform. In CACC mode, CAMs are broadcast periodically, e.g. every 100ms, using dedicated short range communication devices (DSRC) [START_REF] Caveney | Cooperative vehicular safety applications[END_REF]. The networking load resulting from several platoons and non-platoon vehicles present in the simulation can lead to packet loss that can degrade the performance of CACC mode. This is the kind of effect that can be studied with a simulator like iTETRIS.

In addition, as stated above, we focus in this work on large scale simulation scenarios, where the total number of vehicles can reach easily thousands of nodes and the area to simulate hundred of km 2 . The complexity of such scenarios, coupled with the huge amount of CAM transmission, can only make the burden more overwhelming. The simulation of platoon communications in parallel with thousands of normal nodes becomes then a very challenging task.

Furthermore, in order to determine which nodes will receive a transmitted packet for a simulation of n nodes sharing the same wireless channel, ns-3 stores all the n nodes in a list (in the following, this list will be referred as ns3 list). This entire list has to be checked to locate the nodes within the transmission range of a given sender which leads to O(n) iterations. As vehicular transmissions are usually performed repeatedly and the number of sending vehicles can reach thousands of nodes, this list-checking process can dominate the computation time during the simulation and thus severely limit the scalability especially when the total number of vehicles n becomes very large.

As part of the project COLOMBO [START_REF]The COLOMBO Project[END_REF], a simple optimization, where communication with too distant nodes is not simulated, has been proposed. However, it's not efficient enough in our case. We will compare it to our approach in Section IV-B.

In order to mitigate the problem of scalability of iTETRIS and to simulate a platoon in its environment, we propose to enhance the performance of ns-3 by limiting the number of sending and receiving nodes to the ones located in a geographic zone centered around the leader of the platoon. This zone corresponds to the interference zone of the platoon leader. In fact, in the case of platooning communications, only the platoon members and nodes that are in the interference range from the platoon, are concerned by the messages transmitted either by the leader or the followers vehicles to maintain the stability of the platoon. Thus, limiting the number of active nodes in the simulation to the vehicles located in this restricted zone will significantly improve the execution time and memory usage, making it feasible to carry out simulations with thousands of nodes buy focusing only on platoon communications.

D. Proposed Approach

Our approach aims to make platoon simulations in large scale systems more efficient in term of execution time. Contrary to conventional vehicular simulation, where researchers are interested on evaluating their communication protocol in a large number of nodes from a bird's eye perspective, we focus in this work on a group of vehicles going in the same direction forming a platoon. The leader of this platoon is referred in the rest of paper as the ego-vehicle.

Platoon applications have specific characteristics that need to be taken into account when designing the new solution to make iTETRIS simulation faster in the case of platoon communications: they focus on the communication inside the platoon members to maintain the stability of the vehicles convoy as well as the possible messages, sent by non-platoon cars, that can interfere with the platoon members. Thus, only platoon members and the vehicles that interfere with the platoon's communication need to be taken into consideration for the wireless transmissions since platoon applications packets are only relevant for these nodes. However, although the contents of the packets sent by vehicles not belonging to the platoon but close enough to interfere with it is not relevant to the platoon application, these messages have to be simulated inside the wireless simulator in order to realistically evaluate wireless resource demands and bandwidth consumption.

Starting from these crucial distinction, we propose to define an area of interest centered around the ego-vehicle and in which the simulation of transmission packets is fulfilled. This area of interest, called Restricted Zone (RZone), moves during the simulation, since it is following the ego-vehicle as it moves through the city. Fig. 3 shows a schematic view of the dynamic restricted zone. This latter is defined as a disk centered around the leader of the platoon (red color). Dark Fig. 3. Restricted Zone of transmission for platoon communications blue vehicles (members of the platoon) are within that disk and are thus simulated in the wireless simulator ns-3. Other normal vehicles that are inside the zone (sky blue color) can also receive and transmit messages. These nodes, as it is the case in conventional VANETs, are periodically sending CAMs to advertise their positions to the neighboring cars. This is useful to depict the effects of the near surrounding transmissions on the platoon communication. The radius of the Restricted zone, namely R z , allows us to prevent vehicles outside the relevant zone (mauve color) to be simulated in ns-3. The idea is that these vehicles are too far from the ego-vehicle to interfere with it. Thus, by limiting the number of transmitting/receiving vehicles to only the nodes located inside the restricted zone, the size of ns3 list is much lower compared to "all nodes transmitting" technique and consequently the complexity of calculation dramatically decreases. We have to mention here that our proposal treats the interaction of a specific group of vehicles with its near environment, and then it can not be applied to accurately simulate the communication of a complete city-wide scenario.

While the above description of our proposal to reduce simulation temporal complexity defines a single ego-vehicle and then a single RZone, the concept can be generalized by defining multiple ego-vehicles to simulate a specific number of platoons driving through the city. In this case, a different RZone will be associated to each platoon leader, making our technique able to scale to very large vehicular systems and at the same time reducing simulation time complexity.

IV. EVALUATION

A. Scenario and Simulation Setup

In this section, we show simulation results and investigate the performances of our proposal in terms of simulation execution time. We compare our scheme to the original version of iTETRIS as well as its improved version COLOMBO. To carry out platoon simulations in a large scale system, the first step is to choose an appropriate traffic scenario that can reproduce realistic mobility patterns. To this end, we have selected the Luxembourg SUMO Traffic (LuST) scenario [START_REF] Codeca | Luxembourg SUMO Traffic (LuST) Scenario: 24 Hours of Mobility for Vehicular Networking Research[END_REF] as a synthetic example, since it provides us with realistic urban traffic traces of a mid-size European city, exhibiting different mobility patterns during the rush hour (high density), the day (moderate density) and the night (low density). The scenario, as described by Table I, consists in simulating a platoon inside Luxembourg, in an area of 156 km 2 , with a total length of roads equal to 931 km. The number of cars can reach 2500 during rush hours.

We performed the simulations on a PC equipped with an Intel Core CPU i7-5500U at 2.4 GHZ with 8 GB of RAM. The operating system is Ubuntu Linux 16.04 with 64 bit kernel. We used the latest version of COLOMBO to develop our restricted zone technique. The ego-vehicle starts in the center of Luxembourg city and starts moving as the simulation is running. The platoon we construct is composed of 4 members and is following the platoon leader during its journey. The radius of the restricted zone is fixed to 2000 m and is large enough to consider all the vehicles whose transmission can interfere with the communications of the platoon, since transmission range of V2V using the IEEE 802.11p is around 500 m. We simulate 10 min of wireless traffic at different moments of the day. In fact, we are interested on some specific 10 min time slices where the density of vehicles in the simulation is reflecting different traffic situations (low, medium and high density of traffic). We investigate in the following how long these 10 min of simulated time will require in term of real processing time.

B. Performance Evaluation

We conducted extensive simulation experiments to measure the duration that the simulation takes to perform 10 min of wireless packets exchange for our proposal and we compared it to iTETRIS and COLOMBO versions.

Fig. 4 shows the performance of the performed simulations in terms of processing time during different times of the day (with different vehicle densities). We can clearly see that our technique outperforms largely iTETRIS for all the different densities. For example, at 8am (which corresponds to a high density of vehicles), iTETRIS performs the 10 min simulation The same efficiency of the Restricted Zone approach can be noticed also at 10pm, where the densities of nodes is medium. This remarkable gain in term of simulation time is explained by the fact that with iTETRIS, all the nodes present in the simulations are able to transmit and receive packets (like beacons, event driven messages, etc.) in addition to the platoon communications which makes the wireless transmissions load very huge specially with high vehicles densities. On the other hand, using our technique, only cars present in the Restricted Zone are given the right to exchange messages which sharply limits the size of the ns3 list, reducing accordingly the time processing inside ns-3. This observation is confirmed by Fig. 5, where we compute the gain in simulation time as follows:

1 - SimulationTime(RZone) SimulationTime(iTETRIS) × 100
We can see clearly that our RZone approach maintains a minimum gain of 65% in term of simulation processing time and for different densities compared to iTETRIS. This gain can reach 82% in the case of low densities (2am). Fig. 5 confirms also the observations stated before regarding the number of implicated nodes and how it severely affects the simulation processing. In fact, it demonstrates that RZone can ensure a minimum gain in term of simulated vehicles equal to 56% and it can be over 70% at lower densities (2am). This gain Fig. 6. iTETRIS vs COLOMBO vs RZone: Gain in implicated nodes is explained by the fact that we limit the wireless transmissions to only the ones inside the circle of radius R z .

To highlight the efficiency of our proposal compared to other enhancements made to iTETRIS, we compare in Fig. 6 iTETRIS and COLOMBO (seen as an improvement to iTETRIS) in term of simulation duration time. We observe clearly that even if COLOMBO improves the overall simulation duration time when simulating the platoon, this reduction still smaller than the one introduced by our RZone technique. In fact, COLOMBO ensures a minimum gain of 20% compared to iTETRIS but can not exceed the value of 40% in best cases. This can be explained by the fact that COLOMBO removes only the nodes that are very distant (located in the interference zone) from the ns-3 list to check (ns3 list) but maintains all the nodes present in the simulation able to communicate and thus the reduction in the total of nodes to simulate is modest compared to iTETRIS.

The above performance evaluations corroborate that the restricted zone approach for platoon communications can spectacularly alleviate the burden of high simulation temporal complexity when dealing with large scale systems and consequently open widely the doors to more efficient simulation platforms for platooning.

V. CONCLUSION

In this paper, we have introduced a concept for making large scale simulations for cooperative platoons more efficient in term of simulation processing time. The presented approach is based on the definition of a dynamic zone of interest called Restricted Zone (RZone), centered around the leader of the platoon and which defines the vehicles that can transmit and receive packets. The performance evaluation of our proposal shows a dramatic reduction of the computation time in comparison to iTETRIS and COLOMBO platforms. This is consistent with the huge reduction of active wireless nodes in ns-3 achieved by limiting the transmission and reception zone to the circle area defined by RZone. As next steps, we are investigating the set up of multiple restricted zones to be able to simulate several vehicular platoons inside the same scenario and then move further towards simulating city-wide scenarios with efficient time processing performances.

Fig. 2 .

 2 Fig. 2. iTETRIS: Architecture overview

Fig. 4 .Fig. 5 .

 45 Fig. 4. Simulation execution time comparison: iTETRIS vs Restricted Zone

ACKNOWLEDGMENTS

This work benefited from the support of the project SINETIC of the French "Fonds unique interministriel" (FUI).


Dalibor Volný
email: dalibor.volny@univ-rouen.fr

Yizao Wang
email: yizao.wang@uc.edu

ERRATUM TO 'AN INVARIANCE PRINCIPLE FOR STATIONARY RANDOM FIELDS UNDER HANNAN'S CONDITION'

DALIBOR VOLN Ý AND YIZAO WANG

In this erratum, we point out a mistake in a statement in Volný and Wang [4, Section 6], published in Stochastic Processes and their Applications, 124(12):4012-4029, on the relation between Wu's physical dependence measure [START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF][START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF] and Hannan's condition [START_REF] Hannan | Central limit theorems for time series regression[END_REF][START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] on stationary random fields. These conditions have been introduced in the past as easy-to-verify conditions for stationary random fields that lead to the central limit theorem and invariance principle. It had been known before that Wu's condition is strictly stronger than Hannan's condition in dimension one, as shown in Wu [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF]Theorem 1]. However, it was stated in [4, p. 4026] that the argument for dimension one 'can be easily adapted to high dimension and the details are omitted'. This statement is not true.

We are grateful to Davide Giraudo for having read carefully [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] and pointed out the mistake to us. It turned out that Wu and Hannan's condition are no longer comparable in dimension d ≥ 2. We have provided an example of stationary random field in [4, Proposition 6.1] that satisfies Hannan's condition, but not Wu's. Here, in Proposition 1 below, we provide another example in dimension d ≥ 2 that satisfies Wu's condition, but not Hannan's.

Similar statements as the aforementioned one on the relationship between Wu and Hannan's conditions in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] can also be found in [START_REF] Klicnarová | Limit theorems for weighted Bernoulli random fields under Hannan's condition[END_REF]. The mistake does not affect other results in [START_REF] Klicnarová | Limit theorems for weighted Bernoulli random fields under Hannan's condition[END_REF][START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF], which are on limit theorems under Hannan's condition for stationary random fields.

Recall that we consider stationary random fields in the form of (1)

X i = f • T i (ǫ), i ∈ Z d where ǫ = {ǫ i } i∈Z d is a collection of i.i.d. random variables, f is a measurable function from R Z d to R, and
T i given by [T i (ǫ)] j = ǫ i+j , i, j ∈ Z d , are the canonical shift operators on R Z d .
For the sake of simplicity, we only recall the two conditions for d = 2 in (2) below, and refer to [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] for the case when d ≥ 3.

Proposition 1. For every d ≥ 2, there exists a random field in the form of (1), such that Wu's condition holds, but Hannan's condition does not.

Proof. We first recall the two conditions of interest for d = 2. The Hannan's condition is based on the so-called projection operators P i , i ∈ Z 2 defined as, for F i = σ(ǫ j : j ≤ i, j ∈ Z 2 ) with j ≤ i in the coordinate-wise sense,

P i g(ǫ) = E(g(ǫ) | F i ) -E(g(ǫ) | F i-(1,0) ) -E(g(ǫ) | F i-(0,1) ) + E(g(ǫ) | F i-(1,1)
),

1
for any integrable function g. On the other hand, to introduce Wu's condition we set ǫ * = {ǫ * i } i∈Z 2 with

ǫ * i =    ǫ i if i = (0, 0) ǫ * (0,0) otherwise,
where ǫ * (0,0) is an independent copy of ǫ (0,0) also independent of {ǫ i } i∈Z 2 , and set

f * i (ǫ) = f • T i (ǫ *
). Hannan's condition and Wu's condition for the random field {f • T i (ǫ)} i∈Z 2 correspond to the following inequalities, (2)

i∈Z 2 P i f 2 < ∞ and i∈Z 2 f -f * i 2 < ∞,
respectively, where here and below we write h 2 = (Eh 2 (ǫ)) 1/2 . Now, we provide a concrete example of f . Let ǫ = {ǫ i } i∈Z 2 be a family of i.i.d. random variables with P(ǫ i = ±1) = 1/2. Introduce, for a sequence of strictly positive integers {n k } k∈N to be determined later,

g (k) (ǫ) = i∈{1,...,n k } 2 ǫ (i1,0) ǫ (0,i2) , k ∈ N,
and here and in the sequel we use the notation i = (i 1 , i 2 ) ∈ Z 2 . Next, for a sequence of non-negative real numbers {a k } k∈N to be determined later, consider

f (ǫ) = ∞ k=1 a k g (k) • T (m k ,0) (ǫ), with m 1 = 0, m k = k-1 i=1 n i , k ≥ 2.
For the so-constructed f , we can compute explicitly the two series in [START_REF] Hannan | Central limit theorems for time series regression[END_REF]. For the series corresponding to Hannan's condition, observe that by independence, (3)

P i g (k) (ǫ) =    ǫ (i1,0) ǫ (0,i2) if i ∈ {1, . . . , n k } 2 0 otherwise.
So we have

i∈Z 2 P i g (k) 2 = n 2 k . Furthermore for each i ∈ Z 2 , P i [g (k) • T (m k ,0) ](ǫ) = ǫ (i1,0) ǫ (0,i2) if i -(m k , 0) ∈ {1, .
. . , n k } 2 and zero otherwise. By the choice of n k and m k , it also follows that P i [g (k) • T (m k ,0) ](ǫ) = 0 for at most one k. So we can write

P i f 2 = ∞ k=1 a k P i g (k) • T (m k ,0) 2 = ∞ k=1 a k P i-(m k ,0) g (k) 2 .
Hence,

i∈Z 2 P i f 2 = ∞ k=1 a k i∈Z 2 P i-(m k ,0) g (k) 2 = ∞ k=1 a k i∈Z 2 P i g (k) 2 = ∞ k=1 a k n 2 k . (4)
For the series corresponding to Wu's condition, observe first that

f * i (ǫ) = ∞ k=1 a k g (k), * i (ǫ) with g (k), * i (ǫ) = g (k) • T i (ǫ * ),
and

g (k) -g (k), * i 2 =    √ 2n 1/2 k if i ∈ ({1, . . . , n k } × {0}) ∪ ({0} × {1, . . . , n k }) 0 otherwise. So i∈Z 2 g (k) -g (k), * i 2 = 2 √ 2n 3/2 k .
Since for every i ∈ Z 2 there exists at most one k ∈ N such that g (k) (ǫ)-g (k), * i (ǫ) = 0, we arrive at (5)

i∈Z 2 f -f * i 2 = i∈Z 2 ∞ k=1 a k g (k) -g (k), * i 2 = 2 √ 2 ∞ k=1 a k n 3/2 k .
Thus, in order to have an example as desired, in view of ( 4) and ( 5) it suffices to choose {a k } k∈N and {n

k } n∈N such that ∞ k=1 a k n 2 k = ∞ but ∞ k=1 a k n 3/2 k
< ∞. This can be achieved by, for example, taking n k = 2 k and a k = n -2 k . The idea can be adapted to d ≥ 3, and we only sketch the key steps. For each i = (i 1 , . . . , i d ) ∈ Z d , let i (q) be the vector in Z d such that the q-th coordinate equals i q , and all other ones equal zero (e.g. i (1)

= (i 1 , 0, . . . , 0) ∈ Z d ). Consider g (k) (ǫ) = i∈{1,...,n k } d d q=1 ǫ i (q) .
An extension of the projection operator P i is introduced for i ∈ Z d in [4, (2.3) and (2.4)]. The key step is to verify that P i g (k) (ǫ) = d q=1 ǫ i (q) for i ∈ {1, . . . , n k } d and 0 otherwise, a generalization of (3). This is a consequence of [4, (2.3)] and the independence assumption on ǫ. The rest of the calculation is similar to d = 2, and in particular one can obtain, with similar notation for g The details are omitted.


Jana Klicnarov´a

Dalibor Volny
email: dalibor.volny@univ-rouen.fr

Yizao Wang
email: yizao.wang@uc.edu

DALIBOR VOLN Ý Jana Klicnarov Á

Jana Klicnarová

Dalibor Volný

LIMIT THEOREMS FOR WEIGHTED BERNOULLI RANDOM FIELDS UNDER HANNAN'S CONDITION

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Introduction

We are interested in limit theorems for partial sums of weighted stationary random fields {X j } j∈Z d , in form of (1.1)

S n = j∈Z d b n,j X j , n ∈ N,
where {b n,j } j∈Z d are coefficients such that j b 2 n,j < ∞. We will impose further conditions on the dependence of {X j } j∈Z d so that S n is well defined in the L 2 sense.

Limit theorems for partial sums of dependent random variables have a long history. In the case d = 1, limit theorems for stationary sequences have been extensively developed. The most considered case is the unweighted case with b n,j = 1 {j∈{1,...,n}} , which yields

S n = X 1 + • • • + X n .
In this case, various conditions on the weak dependence of stationary sequence {X j } j∈Z have been developed. Heuristically, when the sequence of {X j } j∈Z is said to have weak dependence, S n should behave asymptotically as partial sums of i.i.d. random variables; in particular one expects

(1.2) S nt √ n t∈[0,1] ⇒ σ{B t } t∈[0,1]
for a standard Brownian motion B and some constant σ > 0. Results of this type are referred to as (weak) invariance principles or functional central limit theorems. See for examples [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF][START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF][START_REF] Merlevède | Recent advances in invariance principles for stationary sequences[END_REF][START_REF] Peligrad | Central limit theorem for linear processes[END_REF][START_REF] Peligrad | Central limit theorem for stationary linear processes[END_REF][START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF] and references therein for recent developments. In case that b n,j is a more general 2 sequence, the same rational applies: S n in (1.1) with weakly dependent {X j } j∈Z should behave as if {X j } j∈Z are independent and identically distributed (i.i.d.) random variables. Different types of invariance principles may arise, see for example [START_REF] Dedecker | Invariance principles for linear processes with application to isotonic regression[END_REF] where the limiting processes are fractional Brownian motions. A few conditions have been known that lead to the above invariance principle (1.2). See for example [START_REF] Durieu | Independence of four projective criteria for the weak invariance principle[END_REF][START_REF] Durieu | Comparison between criteria leading to the weak invariance principle[END_REF] on comparisons between different conditions. In this paper we focus on Hannan's condition [START_REF] Hannan | Central limit theorems for time series regression[END_REF] and Wu's condition [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF], and particular their extensions to random fields. In order to compare with our results for random fields, assume in addition that the stationary random variables {X j } j∈Z have the form

X j = f • T j ({ k } k∈Z )
for some measurable function f : R Z → R and the shift operator T on R Z defined by [T j (w)] k = w j+k for w = {w k } k∈Z ∈ R Z , and a sequence of stationary random variables { k } k∈Z . We assume EX j = 0 and EX 2 j < ∞. Furthermore, introduce F j := σ( k : k ≤ j), F -∞ := j F j and F ∞ := j F j , and assume f to be regular in the sense that E(f | F -∞ ) = 0, E(f | F ∞ ) = f . With these notations, Hannan's condition reads as

(1.3) j∈Z E(f • T j | F 0 ) -E(f • T j | F -1 ) 2 < ∞.
To introduce Wu's condition, furthermore set first * j = j for j = 0 and * 0 to be an independent copy of 0 and independent of all the other random variables, and then set X * j = f • T j ({ * k } k∈Z ). Then Wu's condition reads as (1.4) j∈Z X j -X * j 2 < ∞.

Both conditions have been extensively investigated (e.g. [START_REF] Cuny | A quenched invariance principle for stationary processes[END_REF][START_REF] Dedecker | On the weak invariance principle for non-adapted sequences under projective criteria[END_REF][START_REF] Wu | Asymptotic theory for stationary processes[END_REF]). It has been shown in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] that Wu's condition is strictly stronger than Hannan's condition, in the sense that one can find an example such that the invariance principle as in (1.2) holds for Hannan's condition, but Wu's condition is violated. On the other hand, Wu's condition is very practical in proving limit theorems [START_REF] Wu | Asymptotic theory for stationary processes[END_REF], and conditions of similar types have lead to strong invariance principles [START_REF] Berkes | Komlós-Major-Tusnády approximation under dependence[END_REF][START_REF] Wu | Strong invariance principles for dependent random variables[END_REF].

Limit theorems for stationary random fields (d ≥ 2) have also been investigated since long time ago. There is a vast literature on limit theorems for general stationary random fields, and we refer to [START_REF] Bolthausen | On the central limit theorem for stationary mixing random fields[END_REF][START_REF] Bradley | Introduction to strong mixing conditions[END_REF][START_REF] Dedecker | A central limit theorem for stationary random fields[END_REF][START_REF] Dedecker | Exponential inequalities and functional central limit theorems for a random fields[END_REF] and the references therein. A main motivation of the recent developments was to extend the well-investigated dependence conditions for stationary sequences to random fields. However, the success so far has been mostly limited to Bernoulli random fields {X j } j∈Z d :

(1.5)

X j = f • T j ({ k } k∈Z d )
where

f : R Z d → R is a measurable function, {T j } j∈Z d are the shift operators on R Z d such that for w = {w k } k∈Z d ∈ R Z d , [T j (w)] k = w j+k for all k, j ∈ Z d ,
and { k } k∈Z d are i.i.d. random variables. We assume EX j = 0 and EX 2 j < ∞.

It is important to point out that in the case d = 1, { j } j∈Z is only assumed to be stationary. At the technical level, the reason for restricting { j } j∈Z d to be i.i.d. when d ≥ 2 is the following. Conditional expectations of random variables, in the form of E(Y | F j ) with F j = σ( k : k ≤ j, k ∈ Z d ), are involved in the proof. Such conditional expectations are extensively used to express random variables of interest in terms of sums of orthogonal random variables. For example in one dimension, one can write

(1.6) Y = j∈Z [E(Y | F j ) -E(Y | F j-1 )] =: j P j Y
for any regular random variable Y with finite moment, and for all j = j , P j Y and P j Y are uncorrelated. To extend this key decomposition to the random-field setup (Lemma 2.1 below) and maintain the orthogonality of the terms, one needs the following commuting property of the filtration:

(1.7) E[E(Y | F j ) | F k ] = E[E(Y | F k ) | F j ] = E(Y | F j∧k )
with (j ∧ k) q = min(j q , k q ), q = 1, . . . , d. This identity, unfortunately, is not true for filtrations generated by general stationary { j } j∈Z d except when they are i.i.d. Therefore, many arguments based on the telescoping (1.6) fail to be generalized to high dimensions for arbitrary stationary { j } j∈Z d .

A deeper reason for working under the assumption of { j } j∈Z d being i.i.d. is that the so-constructed random fields as in (1.5) can be approximated by m-dependent random fields, but it is not clear how to construct an m-dependent approximation for general stationary random fields. For general stationary sequences in one dimension, instead of m-dependent approximation one can proceed alternatively by establishing martingale approximation, using the projections based on conditional expectations, and then apply the martingale central limit theorem. However, the martingale central limit theorem has been well known (e.g. [START_REF] Bolthausen | On the central limit theorem for stationary mixing random fields[END_REF]) to be much less practical in high dimensions than in one dimension, and limit theorems for stationary random fields established via martingale approximations have very stringent conditions [START_REF] Basu | On functional central limit theorem for stationary martingale random fields[END_REF][START_REF] Morkvėnas | The invariance principle for martingales in the plane[END_REF][START_REF] Nahapetian | Billingsley-Ibragimov theorem for martingaledifference random fields and its applications to some models of classical statistical physics[END_REF][START_REF] Poghosyan | Invariance principle for martingaledifference random fields[END_REF]. So for general stationary random fields in form of (1.5) with stationary { j } j∈Z d , it is still an open question that what should be the general approach to establish central limit theorems. This is the limitation of most of the results so far on weighted Bernoulli random fields. It might be true that the situation with stationary { j } j∈Z d is actually much more complicated than in one dimension. An exceptional result for general stationary random fields is due to Dedecker [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for a random fields[END_REF] via a very involved Lindeberg method, and the condition on weak dependence is much more complicated than the one-dimensional case. Another very recent result that goes beyond the i.i.d. regime is in [START_REF] Volný | A central limit theorem for fields of martingale differences[END_REF], although it is still not yet as satisfactory as in one dimension.

As discussed above, latest advances on limit theorems for stationary random fields have been mostly limited to Bernoulli random fields in form of (1.1) and (1.5). In particular, progress has been made on Wu's condition in the past few years. Observe that Wu's condition in one dimension (1.4) can be naturally extended to high dimensions, formally as

(1.8) j∈Z d X j -X * j 2 < ∞, with X * j = f •T j ({ * k } k∈Z d ) and { * k } k∈Z d similarly as in the case d = 1.
This extension is first considered by El Machkouri et al. [START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF], where invariance principles for unweighted partial sums are established. Limit theorems have been also established for fractional Brownian sheets [START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF] and set-indexed random fields [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF][START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF]. All these results can be formulated as limit theorems for weighted Bernoulli random fields as in (1.1), under Wu's condition and certain assumptions on b n,j . Recent results on Bernoulli random fields under other conditions include also [START_REF] Machkouri | Orthomartingalecoboundary decomposition for stationary random fields[END_REF].

In this paper, we consider limit theorems for Bernoulli random fields under Hannan's condition ((2.1) below), and continue the development in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF]. It is proved in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] that for stationary Bernoulli random field under Hannan's condition,

   1 n d/2 1≤i≤nt X i    t∈[0,1] d ⇒ σ{B t } t∈[0,1] d as n → ∞ in D([0, 1] d )
, where the limiting random field is the Brownian sheet B up to a multiplicative constant σ, as in the case for {X j } j∈Z d being i.i.d. Here, we complement results in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF] by considering more general weights b n,j : we extend a few of aforementioned results [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF][START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF][START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF] on limit theorems for weighted Bernoulli random fields under Wu's conditions to the strictly weaker Hannan's condition (except in one case where the results are not comparable; see Remark 4.4).

There are two key ingredients in the proofs here. One is the assumption on the i.i.d. random variables discussed above: in particular, this assumption allows the approximation of the stationary random fields by m-dependent ones. The other is a moment inequality for weighted partial sums, in form of

j∈Z d b n,j X j p ≤ C   j∈Z d b 2 n,j   1/2
for some p ≥ 2. We establish such an inequality in Lemma 2.2 under Hannan's condition. It plays the key role in bounding the error term in the m-dependent approximation of the random fields. Such an inequality has been known under Wu's condition (1.8) [17, Proposition 1]. (See also for a different extension of Wu's condition proposed by [START_REF] Truquet | A moment inequality of the Marcinkiewicz-Zygmund type for some weakly dependent random fields[END_REF], where a similar moment inequality was established for the unweighted partial sums.) Our proof of the main result, Theorem 2.5, makes essential use of the two keys, and the proof is inspired by Biermé and Durieu [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF] (see also Remark 2.9). Here we present a variation of the same idea, using m-dependent approximation instead of m n -dependent approximation.

The paper is organized as follows. The main result, Theorem 2.5, is established in Section 2. As consequences, we present two applications. First, central limit theorems for partial sums over arbitrary index sets are investigated in Section 3. Second, invariance principles in [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF][START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF][START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF] are established under the Hannan's condition in Section 4.

A central limit theorem

Consider i.i.d. random variables { i } i∈Z d defined in a probability space (Ω, B, P). Set F i = σ( j : j ∈ Z d , j ≤ i), i ∈ Z d and F (q) iq = σ( j : j ∈ Z d , j q ≤ i q ), q = 1, . . . , d, i q ∈ Z. Because of the independence of { j } j∈Z d , {F j } j∈Z d are
commuting in the sense of (1.7). Next, as in [START_REF] Volný | An invariance principle for stationary random fields under Hannan's condition[END_REF], introduce the projection operator

P i = d q=1 P (q) iq with P (q) iq (•) = E(• | F (q) iq ) -E(• | F (q) iq-1 ).
In this way, P (q) iq and P i are commuting operators from L 2 (Ω, B, P) to L 2 (Ω, B, P), due to the commuting property of the filtration. For more properties of these filtrations and operators, see [START_REF] Khoshnevisan | Multiparameter processes[END_REF][START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF]. The following decomposition based on these projection operators is useful.

Lemma 2.1. Let Y be a random variable measurable with respect to the σ-algebra

F ∞ = σ( j : j ∈ Z d ), with EY = 0, E|Y | p < ∞, for some p ≥ 2. Then, Y = j∈Z d P j Y := lim m→∞ j∈{-m,...,m} d P j Y in L p .
Proof. By definition of P j , j∈{-m+1,...,m} d

P j Y = E(Y | F m1 ) + δ∈{-1,1} d \{1} (-1) δ E(Y | F mδ ) with δ = d q=1 1 {δq=-1} , 1 = (1, . . . , 1) ∈ Z d and mδ, m1 ∈ Z d . By martingale convergence theorem, E(Y | F m1 ) → Y almost surely and in L p .
The other 2 d -1 terms all converge to zero in L p . Indeed, observe that for each q = 1, . . . , d,

lim m→∞ E(Y | F (q) -m ) = E Y m∈N F (q) -m
almost surely and in L p , by backwards martingale convergence theorem. By Kolmogorov's zero-one law, the limit is a constant and hence necessarily zero since EY = 0. To complete the proof, it suffices to observe for each

δ ∈ {-1, 1} d \ {1} with δ q = -1 for some q ∈ {1, . . . , d}, E(Y | F mδ ) p ≤ E(Y | F (q) m ) p for j ∈ Z d .
In view of Lemma 2.1, throughout, an infinite sum of random variables is understood as the limit of partial sums over sequence of finite sets in the L p sense.

With projection operators defined above, the Hannan's condition states as

(2.1) ∆ p (X) := i∈Z d P 0 X i p < ∞,
for some p ≥ 2. We first give two lemmas on Bernoulli random fields under Hannan's condition.

Lemma 2.2. Suppose ∆ p (X) < ∞ for some p ≥ 2. Then for all {a i } i∈Z d ∈ 2 (Z d ), (2.2)
i∈Z d a i X i p ≤ C p,d   i∈Z d a 2 i   1/2 ∆ p (X) with C p,d = (p -1) d/2 .
Proof. Observe that it suffices to show

i∈Λ a i X i p ≤ C p,d i∈Λ a 2 i 1/2 ∆ p (X)
for all finite Λ ⊂ Z d . Then, by Lemma 2.1,

i∈Λ a i X i = i∈Λ a i j∈Z d P j X i = j∈Z d P j i∈Λ a i X i .
Now let Y be any random variable with E|Y | p < ∞ in the same probability space, and recall that

P j = d q=1 P (q) jq . Then, j∈{-m,...,m} d P j Y 2 p = m j 1 =-m P j 1   j 2 ,...,j d ∈{-m,...,m} d q=2 P jq Y   2 p ≤ (p -1) m j 1 =-m P j 1   j 2 ,...,j d ∈{-m,...,m} d q=2 P jq Y   2 p (2.3) = (p -1) m j 1 =-m j 2 ,...,
a i X i 2 p ≤ C 2 p,d j∈Z d i∈Λ |a i | P 0 X i-j p 2 ≤ C 2 p,d j∈Z d i∈Λ a 2 i P 0 X i-j p ∈Λ P 0 X -j p = C 2 p,d ∆ 2 p (X) i∈Λ a 2 i ,
where we applied triangle inequality and Cauchy-Schwarz inequality in the second and third steps, respectively. Thus, we have shown (2.2).

Remark 2.3. It is not clear to us whether the constant C p,d = (p -1) d/2 is optimal for d ≥ 2. It is proved by Rio [31] that it is optimal when d = 1. The constant C p,d will play a role when establishing tightness with entropy conditions for invariance principles. See Remark 4.4 below.

Lemma 2.4. Suppose ∆ 2 (X) < ∞. Then j∈Z d |Cov(X 0 , X j )| ≤ ∆ 2 2 (X) < ∞.
Proof. Hannan's condition enables to write

X i = j P j X i . Since {P j } j∈Z d are orthogonal in the sense that E[(P j X)(P k Y )] = 0 for all j, k ∈ Z d , j = k and X, Y ∈ L 2 (Ω, B, P), it follows that k∈Z d |E(X 0 X k )| ≤ k∈Z d i∈Z d E|(P i X 0 )(P i X k )| ≤ k∈Z d i∈Z d P i X 0 2 P i X k 2 = ∆ 2 2 (X).
As a consequence, we introduce (2.5)

σ 2 := j∈Z d Cov(X 0 , X j )
which is finite under Hannan's condition.

To state the main result, introduce b

n = {b n,j } j∈Z d ∈ 2 (Z d ), b n := ( j∈Z d b 2 n,j ) 1/2 . For { b n } n∈N ⊂ 2 (Z d ), we are interested in S n = j∈Z d b n,j X j ,
which by Lemma 2.2 is defined in the L 2 sense under ∆ 2 (X) < ∞, and moreover for σ

2 n := Var(S n ), (2.6) σ 2 n ≤ Cb 2 n ∆ 2 2 (X) < ∞, n ∈ N for some constant C > 0.
T eq b n -b n 2 b n = 0, for all q = 1, . . . , d hold, then (2.11) lim n→∞ σ n b n = σ
with σ defined as in (2.5), and

S n b n ⇒ N (0, σ 2 ).
Proof of Corollary 2.6. We first show (2.11). Recall (2.6). Observe that -2

k∈Z d b n,k b n,k+j = T j b n -b n 2 2 -b n 2 2 -T j b n 2 2 ,
and Proof of Theorem 2.5. We proceed an m-dependent approximation argument. For each m ∈ N, set G

T j b n -b n 2 = o(b n ) for all fixed j ∈ Z d ,
(m) j = σ( i : i ∈ Z d , |j -i| ∞ ≤ m), X (m) j = E(X j | G (m) j ), j ∈ Z d . In this way, {X (m) j
} j∈Z d is a (2m + 1)dependent stationary random field. Write

S (m) n = j∈Z d b n,j X (m) j and σ 2 m,n = Var(S (m) n ).
Observe that

P 0 X (m) j = δ∈{0,1} d (-1) δ 1 +•••+δ d E E(X j | G (m) j ) F -δ = δ∈{0,1} d (-1) δ 1 +•••+δ d E E(X j | F -δ ) G (m) j = E P 0 X j G (m) j
, where in the second equality we used the fact that the σ-algebras G (m) j and F are conditionally independent and hence commuting, because they are generated by independent random variables { j } j∈Z d . Thus,

(2.14) ∆ p (X (m) ) ≤ ∆ p (X),
and S

(m) n is well defined in the L p sense if ∆ p (X) < ∞, p ≥ 2. We will approximate S n by S (m) n . To establish a central limit theorem for m-dependent random variables, we will apply a result due to Heinrich [START_REF] Heinrich | Asymptotic behaviour of an empirical nearestneighbour distance function for stationary Poisson cluster processes[END_REF], We first summarize a few estimates in the following lemma.

Lemma 2.8. With the construction described above,

(2.15) lim m→∞ lim sup n∈N Var(S n -S (m) n ) σ 2 n = 0, lim m→∞ lim sup n∈N |σ 2 m,n -σ 2 n | σ 2 n = 0,
and with the choice of V n described above, for every m large enough,

(2.16) lim n→∞ Var(S (m) n -S (m) Vn ) σ 2 m,n = 0, lim n→∞ σ 2 m,Vn σ 2 m,n = 1.
Proof of Lemma 2.8. In the sequel, we let C denote constant number independent from n and m, but may change from line to line. We first show the first part of (2.15). Indeed, by Lemma 2.2,

Var(S n -S (m) n ) ≤ Cb 2 n   j∈Z d P 0 (X j -X (m) j ) 2   2 .
Observe that for each j, P 0 (X j -X (m) j

) 2 ≤ X j -X (m) j 2 → 0 as m → ∞, and that

j∈Z d P 0 (X j -X (m) j ) 2 ≤ j∈Z d P 0 (X j ) 2 + P 0 (X (m) j ) 2 ≤ ∆ 2 (X (m) ) + ∆ 2 (X) ≤ 2∆ 2 (X),
which is finite under Hannan's condition. By the dominated convergence theorem, lim m→∞ sup n∈N Var(S (m) n -S n )/b 2 n = 0, and the first part of (2.15) follows from the assumption (2.8). To see the second part, it suffices to observe

σ 2 m,n -σ 2 n ≤ Var 1/2 (S (m) n -S n )Var 1/2 (S (m) n + S n ).
We have seen that σ 2 n ≤ Cb 2 n in (2.6). Again by Lemma 2.2 and (2.14),

σ 2 m,n ≤ Cb 2 n ∆ 2 2 (X (m) ) ≤ Cb 2 n ∆ 2 2 (X).
Therefore, Var(S

(m) n + S n ) ≤ 2(σ 2 m,n + σ 2 n ) ≤ Cb 2 n , for all m, n ∈ N. It then follows lim sup n→∞ |σ 2 m,n -σ 2 n | σ 2 n ≤ C lim sup n→∞ b n σ n Var 1/2 (S (m) n -S n ) σ n .
The second part of (2.15) now follows from the first part and (2.8).

For (2.16), to show the first part, using the same argument as above it suffices to observe Var(S

(m) n -S (m) Vn ) σ 2 m,n ≤ C j / ∈Vn b 2 n,j σ 2 m,n ∆ 2 2 (X (m) ) ≤ C j / ∈Vn b 2 n,j b 2 n b 2 n σ 2 n σ 2 n σ 2 m,n ∆ 2
σ 2 n σ 2 m,n ≤ 2, m ≥ m 0 .
Therefore the first part of (2.16) follows, for m ≥ m 0 . For the second part, observe that

|σ 2 m,n -σ 2 m,Vn | ≤ Var 1/2 (S (m) n -S (m) Vn )Var 1/2 (S (m) n + S (m)
Vn ), and by Lemma 2.2 and (2.14),

σ 2 m,Vn ≤ C   j∈Vn b 2 n,j   ∆ 2 2 (X (m) ) ≤ Cb 2 n ∆ 2 2 (X).
Thus,

(2.18) |σ 2 m,n -σ 2 m,Vn | σ 2 m,n ≤ C Var(S (m) n -S (m) Vn ) σ 2 m,n 1/2 b n σ n σ n σ m,n
.

By (2.8), (2.17) and the first part of (2.16), for m ≥ m 0 the second part of (2.16) follows.

Now we prove the desired central limit theorem (2.9) in three steps.

E b 2 n,j X (m)2 j ≤ b 2 n σ 2 m,Vn Var(X (m) 0 ) ≤ C < ∞
for some constant C and n large enough, and for all > 0, and

m 2d σ 2 m,Vn j∈Vn E b 2 n,j X (m)2 j 1 {|X (m) j |≥ m -2d σ m,Vn |b n,j | } ≤ m 2d b 2 n σ 2 m,Vn E X (m)2 0 1 {|X (m) 0 |≥ m -2d / sup j |b n,j | σ m,Vn } → 0 as n → ∞
where the last step is due to (2.20) and the assumption (2.7).

2) Observe that σ m,n ⇒ N (0, 1).

S (m) n σ m,n = S (m) n -S (
3) At last, to show (2.9), observe that

S n σ n - S (m) n σ m,n = 1 σ n (S n -S (m) n ) + σ m,n -σ n σ n σ m,n S (m) n .
By

) b 2 n = 0
holds (in the same way as in the proof of the first part of (2.15)) in place of [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF]Eq. (3.4)] for an appropriately chosen increasing sequence {m n } n∈N , and the rest of the proof therein can be carried out with minor changes. In order not to introduce too much duplication, we chose to present a different proof. Our result is more general also in the sense that we consider the normalization of σ n instead of b n .

Central limit theorems for set-indexed partial sums

In this section, we consider the case

S n ≡ S Γn = i∈Γn X i
for a sequence of subsets {Γ n } n∈N of Z d with the cardinality of subsets |Γ n | → ∞ as n → ∞. This corresponds to the case b n,j = 1 {j∈Γn} and b n = |Γ n | 1/2 . Then, in view of Corollary 2.6, it is easy to notice that (2.10) is equivalent to

(3.1) lim n→∞ |∂Γ n | |Γ n | = 0,
where

∂Γ n = {i ∈ Γ n : ∃j / ∈ Γ n , |i -j| ∞ = 1}
is the boundary set of Γ n . Indeed, if we identify Γ n with an element in 2 (Z d ) via b n,j = 1 {j∈Γn} , then for each q = 1, . . . , d we have

T eq Γ n -Γ n 2 2 ≤ 2|∂Γ n | ≤ d m=1 T em Γ n - Γ n 2 2 .
We have thus obtained the following. Corollary 3.1. For a Bernoulli random field with ∆ 2 (X) < ∞, and a sequence of subsets

{Γ n } n∈N of Z d satisfying |Γ n | → ∞ and (3.1), (3.2) S Γn |Γ n | 1/2 ⇒ N (0, σ 2 )
with σ 2 given in (2.5).

In the rest of this section, we discuss what happens if we are interested in the convergence of (3.3) S n σ n ⇒ N (0, 1). Example 3.2. Consider Γ n = {0, 1, . . . , n -1}. We construct an example such that S n /σ n converges to different limits along different subsequences.

Suppose that there exists a collection of mutually independent random variables {ζ

W (k) n = α k (ζ (k) n -ζ (k) n-n k ), k ∈ N and X n = ∞ k=1 W (k) n .
Observe that

P 0 X n = ∞ k=1 P 0 W (k) n , which equals -α ξ ( ) 0 if n = n for some ∈ N, and 0 otherwise. Thus, ∆ 2 (X) = ∞ =1 |α | < ∞. Write S n = S Γn = n-1 i=0 X i and S n (W (k) ) = n-1 i=0 W (k) i . So S n = ∞ k=1 S n (W (k) ).
By independence,

E(S n k (W ( ) )) 2 = 2n α 2 ≤ k 2n k α 2 > k , and
Var(S n k ) = ∞ =1 Var(S n k (W ( ) )) = k-1 =1 2n α 2 + ∞ =k+1 2n k α 2 + 2n k α 2 k .
One can choose α k and n k so that

(3.4) Var(S n k ) ∼ Var(S n k (W (k) )) = 2n k α 2 k as k → ∞. For example, taking α k = 2 -k 2 and n k = 2 3k 2 k ∈ N, it yields Var(S n k ) ∼ Var(S n k (W (k) )) = 2 k 2 +1 .
Now in view of (3.4), for our purpose it suffices to choose ζ k appropriately such that (3.5)

Z k := S n k (W (k) ) α k √ n k
converge to different limits along even and odd sequences.

To do so, we now give an explicit construction of {ζ

(k)
n } n∈Z,k∈N . For the sake of simplicity, consider (Ω,

B, P) = ([0, 1] Z , B([0, 1]) Z , Leb Z ), and n (ω) = ω n , ω ∈ Ω.
Here, T is the shift operator although we do not use it explicitly. In this way, for any sequence {d k } k∈N with d k ∈ [0, 1], we choose a family of sets

{A ± k } k∈N ⊂ B([0, 1]) such that A + k ∩ A - k = ∅, µ(A ± k ) = d k /2, and set ζ (k) n (ω) = 1 √ d k (1 A + k -1 A - k )(ω n ), n ∈ Z, k ∈ N. So for n = n , ζ (k)
n and ζ (k ) n are independent. In order that {ζ

(k)
n } n∈Z,k∈N satisfy the conditions that we assumed at the beginning, it remains to choose

{A + k , A - k } k∈N such that for each fixed n, {ζ (k)
n } k∈N are mutually independent.

This can be done via a variation of dyadic expansion as follows. First, pick

A + 1 := (0, d 1 /2], A - 1 := (d 1 /2, d 1 ]. Suppose A + k , A - k have been selected for k ∈ N. Then each of A + k , A - k and (0, 1] \ (A + k ∪ A - k
) can be expressed as a disjoint union of left-open-right-closed intervals, and together these intervals form a partition of (0, 1], say (0, 1] = j k j=1 (a j,k , b j,k ]. Now, set

A + k+1 := j k j=1 a j,k , a j,k + (b j,k -a j,k ) d k+1 2
A - k+1 := j k j=1 a j,k + (b j,k -a j,k ) d k+1 2 , a j,k + (b j,k -a j,k )d k+1 .
The so-constructed {ζ

(k)
n } n∈Z,k∈N are then mutually independent. Now set d k = 1 for k even and d k = 1/n k for k odd. For {A ± k } k∈N and {ζ

(k) n } n,k∈N described above, when k is even, Z k in (3.5) becomes 1 √ n k n k -1 i=0 (ζ (k) i -ζ (k) i-n k )
which is the normalized sum of 2n k Rademacher random variables, and thus Z 2k ⇒ N (0, 2) as k → ∞. At the same time, for k odd, Z k in (3.5) becomes

n k -1 i=0 (ζ (k) i -ζ (k) i-n k ),
which is the sum of 2n k i.i.d. random variables with P(ζ

(k) 1 = ±1) = 1/(2n k ) and P(ζ (k) 1 = 0) = 1 -1/n k . Clearly as k → ∞ Z 2k-1 has a non-degenerate limiting distribution which is not Gaussian. So S n /σ n does not converge. Example 3.3. Consider X i = i -i-1 .
Observe that X i and X j are uncorrelated if |i -j| ≥ 2. Therefore, this stochastic process satisfies ∆ 2 (X) < ∞. We now construct a sequence of subsets {Γ n } n∈N such that lim inf n→∞ σ n /b n > 0 but lim n→∞ σ n /b n does not exist.

We construct Γ n iteratively. Set Γ Then, the convergence of finite-dimensional distributions (4.4) holds.

We highlight that to apply Proposition 4.1, the essential work consists of verifying the assumptions on b n,j , and computing the covariance (4.5). Both of these two steps are independent from the choice of dependence assumption on {X j } j∈Z d . For invariance principles to be established below, these computations have been carried out in earlier proofs (under stronger assumptions on {X j } j∈Z d ) and can be borrowed here without any changes.

For the tightness, the moment inequality (2.2) in Lemma 2.2 plays an important role. Similar inequalities have been used to establish tightness in the aforementioned work, and the proofs can be adapted with little extra effort in most the cases. See, however, Remark 4.4 for an exception.

Below we present two improvements of earlier results. We only sketch the proofs in order not to introduce too much duplications. b n,j (A)X j with b n,j (A) := µ(nA ∩ R j ) 1/2 , A ∈ A where R j is the set of unit cube in R d with lower corner j ∈ Z d , and A is a class of Borel sets of R d , equipped with pseudo-metric ρ(A, B) = µ(A B) 1/2 . For µ being the Lebesgue measure, this framework has been considered for example in [START_REF] Alexander | A uniform central limit theorem for set-indexed partial-sum processes with finite variance[END_REF][START_REF] Dedecker | Exponential inequalities and functional central limit theorems for a random fields[END_REF][START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF]. The generalization to other measures, even for the i.i.d. {X j } j∈Z d , was first proposed by Biermé and Durieu [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF]. In particular, they assume the measure µ to satisfy the following. Assumption 4.2. µ is a σ-finite measure on (R d , B(R d )), absolutely continuous with respect to the Lebesgue measure, and such that (i) There exists β > 0 such that µ(nA

) = n β µ(A) for all n ∈ N, A ∈ B(R d ). (ii) lim sup π(j)→∞ µ(R j ) < ∞ and lim π(j)→∞ |µ(R j+eq ) -µ(R j )| µ(R j ) = 0, q = 1, . . . , d with π(j) = min q=1,...,d |j q |, j ∈ Z d .
Furthermore, they also worked with regular Borel sets A, that is, for the boundary set ∂A of A ⊂ R d , Leb(∂A) = 0. The size and complexity of such classes are normally described via covering numbers: the smallest number of ρ-balls with radius to cover A, denoted by N (A, ρ, ). The entropy numbers are given by H(A, ρ, ) = log N (A, ρ, ).

The following result is the counterpart of [4, Theorem 4.5], replacing Wu's condition by Hannan's condition. See Remark 4.4 below on the comparison with conditions under Wu's condition. For concrete examples on self-similar set-indexed random fields as applications, see [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF]. (i) There exists p ≥ 2 such that (4.6)

1 0 N (A, ρ, ) 1/p d < ∞ and ∆ p < ∞.
(ii) There exists γ ∈ (0, 2/d], such that

(4.7) 1 0 H(A, ρ, ) 1/γ d < ∞ and sup p>2 ∆ p (X) p 1/γ-d/2 < ∞. Then, S n (A) n β/2 A∈A ⇒ σ{G(A)} A∈A
in the space of continuous functions on A equipped with supremum norm, where σ is as in (2.5) and G is a zero-mean Gaussian process with covariance Cov(G(A), G(B)) = µ(A ∩ B).

Proof. First, by [4, Proposition 4.2], for each A ∈ A, {b n,j (A)} n,j satisfy the assumptions of Theorem 2.5, and b 2 n (A) = n β µ(A). This tells the order of normalization should be n β/2 .

To show the convergence of finite-dimensional distributions, we apply Proposition 4.1. The verifications of conditions and the computations of covariance, all based on definitions of b n,j (A) and properties of µ only, have been carried out in the proof of [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF]Theorem 4.3]. Now we show the tightness. First consider assumption (i). As in [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF], we apply [START_REF] Ledoux | Probability in Banach spaces[END_REF]Theorem 11.6], which states, if for some constant C > 0, p ≥ 2,

X ψγ ≤ C sup p>2 X p p 1/γ . Now Lemma 2.2 implies for all n ∈ N, A, B ∈ A, (4.9) 1 n β/2 S n (A) -S n (B) ψγ ≤ C n β/2 sup p>2 S n (A) -S n (B) p p 1/γ ≤ Cρ(A, B) sup p>2 C p,d p 1/γ ∆ p (X).
Recall that C p,d = (p -1) d/2 . Again by [START_REF] Ledoux | Probability in Banach spaces[END_REF]Theorem 11.6] the tightness now follows from assumption (4.7).

Remark 4.4. assumption (i), compared to [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF] we simply replace Wu's condition by Hannan's condition, and thus strictly improve the results. However, the results obtained here in part (ii) are not comparable with those under Wu's condition: when working with Wu's condition, for the second part of (4.7) one can actually assume the strictly weaker assumption

(4.10) sup p>2 j∈Z d X j -X * j p p 1/γ-1/2 < ∞.
To establish this condition, the only difference from the above argument is in the second inequality of (4.9), where a similar moment inequality as in (2. In other words, when replacing Wu's condition by Hannan's condition on the weak dependence of the stationary Bernoulli random fields, the condition on entropy numbers (4.7) is strictly strengthened here. This is due to different constants in the moment inequalities of weighted partial sums as in (2.2) in Lemma 2.2 under different conditions: this constant C p,d plays a key role in the second inequality in (4.9). For our approach here, the constant is essentially due to a Marcinkiewicz-Zygmund type inequality applied iteratively d times to orthomartingale differences in (2.4); so the power d reflects the dimension. For the moment inequality under Wu's condition in [START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF], a dimension-free argument is applied: essentially a one-dimensional martingale is embedded into the random field, and thus the constant inherits the one from one-dimensional martingale inequalities.

It is not clear to us whether the constant C p,d in Lemma 2.2 can be chosen to be independent from d. Such a choice would weaken the assumption (ii) in Theorem 4.3. The following theorem generalizes [START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF]Theorem 3]. In particular, [START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF] considered the case that {a j } j∈Z d is of the product form: there exist real numbers {a (q) jq } jq∈Z , q = 1, . . . , d such that

a j = d q=1 a (q)
jq .

Introduce also b

n,j = n i=1 a

i-jq and b n (q) = ( j∈Z b (q)2 n,j ) 1/2 . Examples on coefficients satisfying the assumption below can be found in [START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF]Example 2]. We also point out that the conditions in [START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF] were actually redundant: in [35, Definition 1], Eq.( 8) implies Eq.( 9) by Cauchy-Schwarz inequality. ∆ p (X), which could lead to the desired condition (4.11). This plan can be carried out as in [35, Proposition 2], with Eq. ( 23) therein replaced by (4.12) above and no other changes. We omit the details.

2 n

 2 which requires each partial sum to be of finite number of random variables. Therefore, we introduce a finite set V n ⊂ Z d for each n such that |V n | → ∞ and lim n→∞ b -

  (k) n } n∈Z,k∈N such that for each k ∈ N, {ζ (k) n } n∈Z are i.i.d., and for each n, ζ (k) n is σ( n )-measurable. We further assume that Eζ n ) = 1. A detailed construction is given at the end. For coefficients {α k } k∈N satisfying k |α k | < ∞ and a sequence of increasing positive integers {n k } k∈N , set

  (i) for all λ ∈ R m , t ∈ T m , m ∈ N, { b n,j } j∈Z d ,n∈N satisfy the assumptions in Theroem 2.5 and that b n /b n converges to a constant as n → ∞, and (ii) for a zero-mean Gaussian process G, n (t)S n (τ )) = E(G t G τ ), for all t, τ ∈ T.

4. 1 .

 1 Invariance principles for self-similar set-indexed Gaussian fields. Let µ be a σ-finite measure on R d . Consider S n (A) := j∈Z d

Theorem 4 . 3 .

 43 Let µ be a measure on R d satisfying Assumption 4.2, and let A be a class of regular Borel sets of R d such that µ(A) < ∞ for all A ∈ A. Assume further that one of the following conditions holds.

  2 S n (A) -S n (B) p ≤ Cρ(A, B) for all n ∈ N, A, B ∈ A, and the first part of (4.6) holds, then lim ρ(A,B)<η|S n (A) -S n (B)|n βyields the tightness. It remains to remark that (4.8) follows from (2.2). Now consider assumption (ii). Consider the Young function ψ γ (y) = exp(y γ ) -1 and the Orlicz norm X ψγ := inf{a > 0 : Eψ γ (|X|/a) ≤ 1}. It is well known (e.g.[START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF] Lemma 4]) that

4. 2 .

 2 An invariance principle for fractional Brownian sheet. Consider a linear random field {Y j } j∈Z d in form ofY j = k∈Z d a k X j-k , j ∈ Z d , with k a 2 k < ∞. Invariance principles for S n (t) = 1≤j≤nt Y j , t ∈ [0, 1] dwith nt = (nt 1 , . . . , nt d ) have been studied in the literature. Observe thatS n (t) = j∈Z db n,j (t)X j with b n,j (t) = 1≤i≤nt a i-j .

Theorem 4 . 5 .-

 45 Suppose there exists H ∈ (0, 1) d such thatlim n→∞ b 2 ns (q) b 2 n (q)= s 2Hq , for all s ∈ [0, 1], q = 1, . . . , d, and there exists p such thatp ≥ 2, p > max q=1,...,d 1 H q and ∆ p (X) < ∞. Then, {S n (t)/b n } t∈[0,1] d converges weakly in D([0, 1] d ) to a fractional Brownian sheet G H with Hurst index H, a zero-mean Gaussian process with covariance |t q -s q | 2Hq , s, t ∈ [0, 1] d .Proof. To show the convergence of finite-dimensional distributions, the conditions in Proposition 4.1 have been verified as in [35, proof of Proposition 1]; actually, there a different set of conditions in [35, Definition 1] on b n,j were verified. The equivalence between conditions there and ours were pointed out by Biermé and Durieu [4, Remark after Theorem 3.1] (see also Remark 2.7).

  where in the first and last equalities above we used the commuting property of the projection operators, and the inequality is a Marcinkiewicz-Zygmund type inequality for one-dimensional martingales due to Rio[START_REF] Rio | Moment inequalities for sums of dependent random variables under projective conditions[END_REF] Theorem 2.1]. Iterating the same argument we arrive at, for all m ∈ N, Taking Y = i∈Λ a i X i , we obtain for all m ∈ N,

			2
	(2.4)		P j Y	≤ C 2 p,d	P j Y 2 p .
	j∈{-m,...,m} d	p	j∈{-m,...,m} d
	P j	a i X i	≤ C 2 p,d	P j
	j∈{-m,...,m} d	i∈Λ	p	j∈{-m,...,m} d	i∈Λ

j d ∈{-m,...,m} d q=2 P jq (P j 1 Y ) 2 p ,

  Our main result is the following. Theorem 2.5. Let {X i } i∈Z d be a stationary Bernoulli random field as in (1.5) satisfying Hannan's condition (2.1). If ) is subtle as it involves both the coefficients and the dependence of underlying random fields (via σ n ). The following corollary is more convenient, as it imposes only conditions on coefficients. However, we see later in Example 3.2 that there are examples that satisfy the conditions in Theorem 2.5, but the conclusion of Corollary 2.6 does not hold. Recall that for k ∈ Z d , the shift operator yields T k b n = {b n,j+k } j∈Z d . Let e 1 , . . . , e d be the d canonical unit vector in R d . Corollary 2.6. Let {X i } i∈Z d be a stationary Bernoulli random field as in (1.5) satisfying Hannan's condition (2.1). Under the notations as in Theorem 2.5, if

	(2.7)	lim n→∞	sup j∈Z d	|b n,j | b n	= 0
	and
	(2.8)	lim inf n→∞	σ 2 n n b 2	> 0
	hold, then
	(2.9)	S n σ n	⇒ N (0, 1).
	The condition (2.8(2.10) lim n→∞

  n,k b n,k+j = 1 for all j ∈ Z d .

				a consequence of (2.10).
	Therefore,
	(2.12)		lim n→∞	1 b 2 n k∈Z d
	Thus, by the dominated convergence theorem, (2.6) and (2.12) imply (2.11).
	If σ = 0, then σ 2 n /b 2 n → 0, and the central limit theorem is degenerate and
	trivially holds. If σ > 0, then (2.8) holds. By Cauchy-Schwarz inequality,
	(2.10) implies
	(2.13)	lim n→∞	1 b 2 n j∈Z d

b b 2

n,j+eq -b 2 n,j = 0, for all q = 1, . . . , d.

It has been shown in

[START_REF] Biermé | Invariance principles for operator-scaling Gaussian random fields[END_REF] Lemma 8]

, using an idea from

[START_REF] Peligrad | Central limit theorem for linear processes[END_REF]

, that (2.13) implies (2.7). The desired result now follows from Theorem 2.5. Remark 2.7. Condition (2.10) was introduced in Biermé and Durieu [4, Theorem 3.1]. Condition (2.7) was also assumed there. It has been pointed out in [5, Remark 3] that (2.7) was redundant.

  Second, by Example 3.3, we show that the assumption in Corollary 2.6 is strictly stronger than (2.8), in the sense that there are examples satisfying (2.8), but the conclusion of Corollary 2.6 does not hold. Note also that Example 3.2 also shows that when S n /b n ⇒ N (0, σ 2 ) with σ 2 = 0, one should not expect S n /σ n to converge, without further assumptions.For the sake of simplicity, both examples are given in one dimension. Let { i } i∈Z be the i.i.d. random variables that generate the Bernoulli random field (1.5).

This follows from

(2.8)

, by Theorem 2.5. To see the role of the condition (2.8), we provide two examples. First, by Example 3.2, we show that condition (2.8) cannot be removed: otherwise (3.3) may no longer hold under Hannan's condition.

  1 = {0, 1}. For n ∈ N, set Γ n+1 = Γ n ∪ B n with B n = {a n + 2,a n + 3, . . . , a n + 2 n + 1} n even {a n + 2, a n + 4, . . . , a n + 2 • 2 n } n odd with a n = max{j : j ∈ Γ n }. By construction, Var(S Γ n+1 ) = Var(S Γn ) + Var(S Bn ), and Var(B n ) = 2E 2 0 for n even, and 2 n+1 E 2 0 for n odd. At the same time, |Γ n | = 2 n . It is clear that the desired result follows.

  To show the tightness, by[START_REF] Lavancier | Processus empirique de fonctionnelles de champs gaussiens à longue mémoire[END_REF] Corollary 3], it suffices to show, for some β > 1, p > 0,

			d
	(4.11)	S n (t) p ≤ Cb n	t β/p q , t ∈ [0, 1] d .
			q=1
	For this purpose, by (2.2),
					 1/2
	(4.12)	S n (t) p ≤ C		b 2 n,j (t)	
			j∈Z d

Acknowledgement. We would like to thank Jérôme Dedecker for pointing the reference [31] to us, and the AE and two anonymous reviewers for several suggestions that helped us improve the paper. Klicnarová's research was partially supported by Czech Science Foundation (project number P201/11/P164). Wang's research was partially supported by NSA grant H98230-14-1-0318.

Invariance principles for Gaussian random fields

In this section, we present two invariance principles for weighted Bernoulli random fields. Let T be an index set equipped with a pseudo-metric. Consider random fields in form of (4.1)

where G is a zero-mean Gaussian process. The space of weak convergence will be specified below. Most results improve earlier ones [START_REF] Biermé | Invariance principles for self-similar set-indexed random fields[END_REF][START_REF] El Machkouri | A central limit theorem for stationary random fields[END_REF][START_REF] Wang | An Invariance Principle for Fractional Brownian Sheets[END_REF], in the sense that Wu's condition is replaced by Hannan's condition.

We first provide an overview on how to establish (4.2), illustrating how previous proofs can be adapted without much changes. To establish such an invariance principle, we proceed as in the standard two-step proof: we first show convergence of finite-dimensional distributions and then tightness. To show the convergence of finite-dimensional distributions, we first remark that marginally, for b n (t) := ( j∈Z d b 2 n,j (t))

The linear combinations of finite-dimensional distributions can again be represented as a linear random field via

to which one can apply Theorem 2.5 again. This is the standard procedure to establish finite-dimensional convergence of linear random fields. In our setup we have thus proved the following as a consequence of Theorem 2.5. Write b n = ( j b 2 n,j ) 1/2 . Proposition 4.1. Consider random fields in form of (4.1) with {X j } j∈Z d satisfying Hannan's condition ∆ 2 (X) < ∞. Suppose there exists a sequence of real numbers {b n } n∈N such that