Rodney Michael Schmaltz

Mathieu Hainselin
email: mathieu.hainselin@u-picardie.fr

Emilie Lacot

Geoffrey Blondelle

From Bill Shankly to the Huffington Post: How to Increase Critical Thinking in Experimental Psychology Course?

Keywords: critical thinking, pedagogy, belief, reviewing process, authority

Although critical thinking and source checking are basic prerequisites to become a psychologist, or a scientist, it is usually difficult to have students interested in experimental methods courses. Most first year students are tempted not to attend these courses. Such behaviors are reinforced by arguments that "everybody is different" and "people are not numbers." Consequently, students have difficulties to develop source and evidence checking skills, and may be more prone to believe in any supposed expert. This paper presents two ways to involve students during lectures and seminars. The first method consists in presenting, during the initial lecture of the year, a fake scientific concept which students will believe as true. This phenomenon is called the "Bill Shankly syndrome" and it only exists if someone believes that the information is given by a serious lecturer, presenting oneself as a world-class researcher. The second method consists in training students to become reviewers using evidence checking of a mainstream media article which promises scientifically proven ways to be happy. The use of these methods may stimulate students' interest in research methods and its practical applications from week one.

INTRODUCTION

Psychology students, especially first year university students in France, attend their first lessons with many beliefs about psychology. Most of them think that it is not a scientific field, and are thus surprised with the number of courses on neuroscience, statistics and methodology. At the very beginning, students are confronted with the different steps of the scientific method and discover the mean of a sample, which is frequently and wrongly associated with the mean of a population. These concepts are usually in contradiction with their own conceptions such as "everybody is different" and "people are not numbers" [START_REF] Malekoff | A flexible organizing framework for group work with adolescents[END_REF]. Due to these misconceptions, students may encounter difficulties in understanding or seeing the usefulness of such courses [START_REF] Castro Sotos | Students' misconceptions of statistical inference: a review of the empirical evidence from research on statistics education[END_REF][START_REF] Gigerenzer | Helping doctors and patients make sense of health statistics[END_REF].

Moreover, for many reasons, it is hard for them to develop critical thinking or any experimental methodology skills. In high school, and more specifically in France, psychology is a tiny part of the Philosophy course, only studied through Freud's theories of psychoanalysis [START_REF] Lieury | Introduction à la Psychologie Cognitive[END_REF]. Although, APA have a resource manual for psychology teaching including critical thinking [START_REF] Apa | Resource Manual for New Teachers of High School Psychology[END_REF], students have barely been trained for critical thinking, as most part of their schooling is based on a passive listening model (e.g., [START_REF] Paul | Critical thinking: what, why, and how[END_REF]. Wegwert (2014, 141) highlighted that "fear is a powerful presence in schools, in teacher education, and in teacher identity" and consequently in students daily life. Because of the perceived power of lecturers, which can intimidate students, they feel compelled to integrate knowledge without checking its truthfulness. Consequently, students may be more prone to believe in any authority figure and are less inclined to show a critical mind [START_REF] Wegwert | Looking the tiger in the eye: overcoming fear-based teacher identities[END_REF]. In the same vein, [START_REF] Reeve | Why teachers adopt a controlling motivating style toward students and how they can become more autonomy supportive[END_REF], [START_REF] Skinner | Motivation in the classroom: reciprocal effects of teacher behavior and student engagement across the school year[END_REF], highlight that the same phenomenon occurred if lecturers do not adopt an autonomysupportive style. In our sense, these facts inhibit the development of great psychologists or scientists. However, a growing literature underlines that psychologists, scientists and thus lecturers can, due to a lack of evidence-based practice, make inappropriate inferences. For example, in their recent paper, [START_REF] Lilienfeld | Why ineffective psychotherapies appear to work: a taxonomy of causes of spurious therapeutic effectiveness[END_REF] highlight four barriers to scientific thinking: naïve realism, confirmation bias, illusory causation, and illusion of control. Thus, using concrete examples of inappropriate inferences in patients' treatment could help students identifying those biases. For example confirmation bias: I think I am a good therapist so patients have improved with me, when in fact it may come from an external event. The purpose is to show the many steps needed for any scientific reasoning, as well as for psychotherapies evaluation, and for possible clinical thinking.

We propose two different methods to engage first year psychology students and promote critical thinking. The first method, which must be used during the initial experimental methods lecture, consists of presenting a false scientific concept: the "Bill Shankly syndrome, " presented in the first part of this article. This method is supposed to sharpen the critical mind of the student and to lead them to make their own scientific research. However, first year students of psychology tend to search for information on the Internet without verifying the scientific quality of the sources. To change this behavior, we have used, during seminars, a participative method based on the criticism of research supposedly defined as scientific. This second method, described in the second part of this article, allows the students to distinguish between a scientific source and a non-scientific source and also to criticize the methodology used.

THE BILL SHANKLY SYNDROME: A SERIOUS JOKE FOR A LESSON

We propose a method that was used with first year students of psychology at Jules Verne University of Picardy. This method consists in presenting a scientific concept: the "Bill Shankly syndrome" during the initial lecture of the year. However, the success of this method benefits from a specific context (i.e., the lecturer must have some authority). Before the presentation of the concept, the university lecturer introduces oneself to the students. The lecturer boasts very seriously about his career, asking students to call him "Doctor" (which is very unusual in France if you are not a Medical Doctor) due to his PhD, explain that he is an Associate Professor at the Department of psychology, a trained neuropsychologist, works with highlevel research teams in different countries, publishes articles and is asked for his expertise (i.e., as a reviewer) for international scientific journals. Although, this is a standard resume for an Associate Professor, students are unaware of it. Then, this peremptory introductory speech takes place in an unusually silent amphitheater. This assertive presentation is determinant in the Bill Shankly syndrome. Immediately, the PowerPoint lecture starts, with the classical pavlovian-writing behavior (students blindly copying after the presentation of each slide) and moving on to the next slide. Note that this type of presentation is important too. In fact, the use of PowerPoint animations enable a chronological presentation (i.e., scrolling the sentences one by one), better note taking and, as all lecturers hope, a better understanding of courses [START_REF] Schmaltz | Death to weak PowerPoint: strategies to create effective visual presentations[END_REF]. The concept that we arbitrarily called the Bill Shankly syndrome, because the last author (MH) is a Liverpool F.C. fan, is presented as a main concept in psychology. The lecturer expresses it as naturally and seriously as possible. It is written on slides and read out to the students that the syndrome consists in believing that any truth is the truth because this truth is named, expressed, illustrated as a scientific truth. This definition is followed by a reference to a fake scientific reference "Shankly et al., 1959" (see Supplementary Material for the slide), with a fake concept. The only real thing in this part is the Bill Shankly black and white picture. The definition remains deliberately vague to reinforce social influence (see below). Above all, choosing a 1960's Scottish football manager allow us to emphasize that anything, including old sports references, can be seen as scientific if students do not improve their critical thinking.

The concept that was introduced was clearly flawed, so that someone with critical thinking skills should question the validity of such a claim. To believe that everything is true because someone says so should be an aberration for psychologists, scientists or even for students. In principle, critical thinking involves questioning concepts and existing theories. However, our example highlights that the majority of the first year students agree with this concept. In the past couple of years, about 1000 students attended this course. They took notes without one single objection, and none of them asked any questions. This silence can be interpreted as the students' idea of university lecturers as having a great deal of knowledge. Note that these effects (i.e., silence and note taking without questioning) can be increased by informational or normative social influences and by conformity behaviors (students might assume the actions of others in an attempt to reflect correct behavior for a given situation). Indeed, if the majority of students write in silence without questioning courses, the others are more likely to do the same. It highlights the strong influence of peer group, the compliance and the conformity, particularly in a new situation with possible anxiety [START_REF] Guimond | Attitude Change During College: Normative or Informational Social Influence?[END_REF][START_REF] Cialdini | Social Influence: compliance and conformity[END_REF]. Altogether, this is a great place to discuss informational social influence with students.

Following the presentation of the Bill Shankly syndrome, the university lecturer explained that this concept is false. His speech was supported by a new sentence appearing on the slide: "the was a Scottish footballer and manager of Liverpool [START_REF] Peace | Red or Dead[END_REF]. During the time of these explanations, it was interesting to note that the majority of students were still writing down the PowerPoint's sentences. Nevertheless, some of them understood the joke lesson and then they initiated discussion about the lecturer's speech. Indeed, despite his position as a lecturer/expert, all content should be supported by expert scientific support. At this point, the fake concept was disclosed and the lecture started again from the beginning with an as-normal-as-possible presentation of the lecturer and the course. The pedagogical aim, besides the academic message, is to generate in students the feeling that they can be victims of several cognitive biases and more largely they can be victims of social phenomena. Indeed, due to the social status of the university lecturer, students believe his speech without questioning the situation or the contents. Thus the message given here is: "enhance your critical thinking: don't believe everything you listen or read whoever the speaker/author is." Our aim is that students keep an open and critical mind. This requires searching for scientific information outside the courses.

However, in the same way that students believe in the lecturer, they can also think that all Internet retrieved information is true and scientific, particularly if there is an expert cited (i.e., some students have cited blog posts as a scientific reference because the blogger claimed, incorrectly, to be an expert). The lecturer must warn them against false sources, and to check for original scientific source rather than just blindly believing indirect sources (i.e., self-proclaimed expert) instead of fact checking. During this first lecture, the experimental method, scientific journals standards, including peer-review process, are presented. As a follow up to this exercise, students are presented with an opportunity to seek out valid scientific sources.

FIRST REVIEWING EXPERIENCE: FIRST DISAPPOINTMENT

Like the first method presented, the second one is also used with the first year students of psychology in Jules Verne University of Picardy during seminars, always occurring after the first lecture with the Shankly effect. Thus, students are already aware of the importance of reconsidering the lecturer's discourse content. They also know the need to verify supposedly scientific knowledge. Another important point is that this method must be used during seminars. As there are fewer students in seminars (taking place in classroom, 30-40) than in lectures (taking place in a lecture hall, 250-300), students are more likely to speak in public, which is usually a difficult exercise for them. This second method consists in training students to become reviewers using evidence checking. More specifically, three points should be broached: (i) distinguish a scientific source (i.e., peerreviewed journal) vs. a non-scientific source (i.e., without peerreview), (ii) criticize the proposed hypotheses, and (iii) propose another experiment to check, replicate or go further. In this way and after a brief presentation, the lecturer starts the PowerPoint course based on an article from the Le HuffPost (2013). It proposes 13 tricks scientifically proved to make people feel happy; each trick is displayed in Table 1. Tricks are seen one by one with the students.

First, only the trick and the original picture in the article were presented. Students were asked to say what they think about the trick and decide how they would validate the claim using source checking and critical thinking. For every trick, the first step is to find the source (scientific journal or not) by clicking on a link. The Huffington Post was chosen because it is a digital media allowing links, and because, except for this particular article, many good scientific popularization articles are available [START_REF] Eustache | A Partir de Quand Faut-il S'inquiéter Quand la Mémoire Nous Fait Défaut ? The Huffington Post[END_REF]. The aim is to make students check sources as often as possible, not to destroy any journal reputation.

When any limit or lack of scientific reference is underlined, they have to suggest a new experiment to assess the trick validity. This was a first step into scientific methods.

PERSPECTIVE

The Shankly effect and the media source checking are engaging exercises to teach experimental methods. After showing some surprise in the first place, students seem to like this new approach to enhance critical thinking (i.e., the Bill Shankly syndrome and its explanation), and use it beyond the specific course.

Although generalization is expected, there are cautions for instructors who are going to use this approach. The first concern is an over-generalization of the "don't believe everything you listen or read, whoever the speaker/author is" to every single point of every lesson. If too many students ask lots of justification questions during the lecture, it might prevent the lecturer from saying everything s/he would have liked to say. In our experience, it might also be passed on to other lessons and lecturers, for whom over-questioning and fact checking could be unusual. To avoid this, we ask for constructive criticism and always give our sources so students can check themselves. While these approaches have worked well for the authors, the evidence is anecdotal and as of yet, limited to students in France. Our methods are however consistent with recent recommendation to develop critical thinking [START_REF] Schwanz | Academic performance of introductory psychology students: the importance of critical thinking[END_REF]. We strongly encourage instructors to try these methods, as well as other engaging methods, to help promote critical thinking among students. Future research is needed to assess the actual efficiency, short and long term, of the Shankly effect.

TABLE 1 |

 1 Examples of tricks and justification to be happy from the studied article, with found references and possible criticizes.

	Tricks	Justifications	References	Criticizes
	1. Make a gift.	According a study, kind souls are	Dunn et al., 2008	Scientific article in Science. Possible
		happy.		limits pinpointed because of financial
				resources of the participants.
	2. Count his/her moments of happiness.	A professor at the University of	Seligman, 2012	Book. Non-scientific. What does
		Pennsylvania has demonstrated		mean average? Compared to whom?
		[...] were happier than the
		average.
	3. Live new experiences.	Nevertheless, researches that	MSN Lifestyle.	Non-scientific.
		get out of the daily routine made
		happy.
	4. Anticipate good times.	None	Blog.	Non-scientific.
	5. See the life in blue.	Sussex University scientists have	Interview of the concerned	Non-scientific. Possibility of conflicts
		shown that [...].	scientists retrieved from the	of interests.
			website DailyMail, Macrae,
			2009.
	6. Define targets.	[...] according to the psychologist		Scientific article but it does not match
		Richard Davidson.	Jackson et al., 2000	with the tricks (different topic).
	7. Stop always wanting to be right.	The writer and author Deepak	Interview of Deepak Chopra,	Non-scientific. Does not match with
		Chopra recommends to his	FoxNews, 2011	the tricks.
		readers to remain neutral [...].
	8. Go to church.	[...] according to the Melbourne	DailyMail, 2010b	Non-scientific. The study is not
		University.		published.
	9. Sleep at least 6 hours a night.	In a study [...] assessed their	DailyMail, 2010a	Non-scientific. What does mean a
		level of happiness on a scale		level of happiness? Have we the
		varying from 1 to 5 [...].		same definition?
	10. 20min in means of transport.	British scientists advise to limit	DailyMail, 2010a	Non-scientific. Some factors will be
		the path that leads to work [...].		beyond the person's control.
	11. Have at least 10 friends.	Authentic friend can be counted	Website DailyMail,Macrae,	Non-scientific. How to define a friend
		on fingers. This expression is	2008	as being a real friend?
		familiar but for scientists [...].
	12. Keep it positive even at wrong times.	According to a psychologist [...].	Website Howstuffworks,	Non-scientific. Use of words as "it
			Layton, 2009	seems."
	13. Don't forget be in love!	None.	None.	The entire trick!

Bill Shankly syndrome" obviously does not exist...unless you believe it...and thus you become a victim. Then the lecturer explained that William (Bill) Shankly

Frontiers in Psychology | www.frontiersin.org

April 2016 | Volume 7 | Article 538

ACKNOWLEDGMENTS

The authors would like to thank Shan Williams, Pierre Hainselin and Yannick Gounden for their manuscript corrections to improve the English writing, and the reviewers for their comments that help improve the manuscript. The publication fees were supported by the CRP-CPO lab.

AUTHOR CONTRIBUTIONS

All authors listed, have made substantial, direct, and intellectual contribution to the work, and approved it for publication.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: http://journal.frontiersin.org/article/10.3389/fpsyg.

2016.00538

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Franc ¸oise

P Ène

LIMIT THEOREMS FOR BIRKHOFF SUMS AND LOCAL TIMES OF THE PERIODIC LORENTZ GAS WITH INFINITE HORIZON

This work is a contribution to the study of the ergodic and stochastic properties of Z d -periodic dynamical systems preserving an infinite measure. We establish functional limit theorems for natural Birkhoff sums related to local times of the Z d -periodic Lorentz gas with infinite horizon, for both the collision map and the flow. In particular, our results apply to the difference between the numbers of collisions in two different cells. Because of the Z d -periodicity of the model we are interested in, these Birkhoff sums can be rewritten as additive functionals of a Birkhoff sum of the Sinai billiard. Our proofs rely on a general argument valid in a general framework. For completness and in view of future studies, we state a general result of convergence of additive functionals of Birkhoff sums of chaotic probability preserving dynamical systems under general assumptions.

Introduction

Let d ∈ {1, 2}. The Z d -periodic Lorentz gas coming from [START_REF] Lorentz | The motion of electrons in metallic bodies[END_REF] models the displacement of a point particle moving at unit speed in R d × T 2-d (i.e. in the plane R 2 if d = 2 and on the tube R × T if d = 1) between a (non-empty) Z d -periodic configuration of obstacles, with elastic collisions off these obstacles. The obstacles are assumed to be open, convex, with boundary C 3 , non null curvature and with pairwise disjoint closures. The number of obstacles is assumed to be locally finite. We study the Z d -periodic Lorentz gas flow (Y t ) t∈R (continuous time model) via the Z d -periodic Lorentz gas map which describes the dynamics at collision times (discrete time model). Both for the continuous time and for the discrete time model, a state is a couple (q, v) made of a position and a unit velocity vector. For the Lorentz gas flow (Y t ) t∈R , the positions are taken in the domain (outside obstacles), and to avoid any confusion, we identify, when the position is on a obstacle, pre-collisional and post-collisional vectors. The flow Y t maps a state (q, v) to the state (q t , v t ) at time t of a point particle that was in state (q, v) at time 0. This map preserves the Lebesgue measure m. For the Lorentz gas map T , the states are the couples of a position on the boundary of an obstacle and of a unit post-collisional vector. We write M for the set of such states and we consider the collision map T : M → M which maps a state at a collision time to the state at the next reflection time. This maps T preserves a measure µ, invariant by translation, equivalent to the Lebesgue measure and normalized so that µ (M ∩ ([0; 1[ 2 ×S 1 )) = 1. This measure µ is infinite. The horizon is said to be finite if there exist no line touching no obstacle, then the horizon is actually bounded, meaning that the distance of a trajectory between two collisions is uniformly bounded. It is said to be infinite otherwise. Some results are now knwon to be true both in finite and infinite horizon. For d ∈ {1, 2}, the Z d -periodic Lorentz gas is known to be recurrent (both in finite horizon [START_REF] Conze | Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications (French)[END_REF][START_REF] Szász | Local limit theorem for the Lorentz process and its recurrence in the plane[END_REF] and in infinite horizon [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF]) and ergodic ( [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF][START_REF] Pène | Applications des propriétés stochastiques de billards dispersifs[END_REF]). A crucial point in the study of this model is that it is a Z d -extension of the Sinai billiard map ( M, T , μ), which enjoys a chaotic behaviour. This probability measure preserving system corresponding to the Lorentz gas modulo Z d for the position is mixing [START_REF] Ya | Dynamical systems with elastic reflections[END_REF], enjoys exponential decorrelation for Hölder observables (both in finite horizon [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF] and in infinite horizon [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF] and enjoys a standard Central Limit Theorem [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF] for Hölder observables (both in finite horizon [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF][START_REF] Bunimovich | Statistical properties of two-dimensional hyperbolic billiards, (Russian)[END_REF] and in infinite horizon [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF]).

When the horizon is finite, the position of a particle after n collisions satisfies a standard Central Limit Theorem [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF][START_REF] Bunimovich | Statistical properties of two-dimensional hyperbolic billiards, (Russian)[END_REF] (due to a general argument [START_REF] Zweimüller | Mixing limit theorems for ergodic transformations[END_REF], this result holds true with respect to any probability measure absolutely continuous with respect to the Lebesgue measure). Among the results proved for the Lorentz gas in finite horizon, let us mention mixing rate in infinite measure [START_REF] Szász | Local limit theorem for the Lorentz process and its recurrence in the plane[END_REF][START_REF] Pène | Mixing in infinite measure for Zd-extensions, application to the periodic Sinai billiard[END_REF][START_REF] Pène | Mixing and decorrelation in infinite measure: the case of the periodic Sinai Billiard[END_REF][START_REF] Dolgopyat | Asymptotic expansion of correlation functions for Z d -covers of hyperbolic flows[END_REF] including expansions of any order (both for the map [START_REF] Pène | Mixing and decorrelation in infinite measure: the case of the periodic Sinai Billiard[END_REF] and for the flow [START_REF] Dolgopyat | Asymptotic expansion of correlation functions for Z d -covers of hyperbolic flows[END_REF]), limit theorems for Birkhoff sums (both for integrable observables [START_REF] Dolgopyat | Recurrence properties of planar Lorentz process[END_REF], and for smooth integrable observables with null integral [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF][START_REF] Pène | Central limit theorems for the Z 2 -periodic Lorentz gas[END_REF]), quantitative recurrence estimates (estimate for the tail probability of the first return time in the initial cell [START_REF] Dolgopyat | Recurrence properties of planar Lorentz process[END_REF], limit theorem for the return time in a neighbourhood of the initial state or of its initial position [START_REF] Pène | Back to balls in billiards[END_REF]), limit theorem for the self-intersections number [START_REF] Pène | Self-intersections of trajectories of Lorentz process[END_REF][START_REF] Phalempin | Limit theorems for self-intersecting trajectories in Z-extensions[END_REF], study of differential equations perturbed by the Lorentz gas [START_REF] Phalempin | Théorèmes Limites en mesure infinie : auto-intersections et flots perturbés moyennés[END_REF], etc.

In the present article, we focus on the case when the horizon is fully-dimensional infinite, i.e. when there exist d non parallel infinite lines touching no obstacle. In this case, the time between two consecutive collisions is not bounded anymore, and even worth it is not square integrable with respect to the invariant probability measure μ of the Sinai billiard map. It is still possible to apply operator techniques as in the finite horizon case, but with a loss of important nice properties, and the study requires much more delicate study. Roughly speaking, whereas the Fourier-perturbed operators family t → P t is C ∞ from R to L(B) for some nice Banach space B when the horizon is finite, when the horizon is infinite t → P t is only smooth (and less than C 2 ) from R to L(B → L 1 ), this complicates seriously the work with iterates or expansion this operator. Nevertheless some results have been established in this infinite horizon context, overcoming these difficulties by creative ideas combined with additional technicality. A first specific result is the non-standard Central Limit Theorem satisfied by the position at the n-th collision time [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF]. This result was established together with a non-standard local limit theorem for the cell label [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF], leading to a mixing rate in (n log n) -d 2 for the Lorentz gas map. Whereas a mixing expansion has been established in the finite horizon case, this does not seem reachable in the infinite horizon case because of the weak smoothness properties of t → P t . Nevertheless, further mixing estimates, including an error term and also different mixing rates for some specific null integral smooth observables have been established in [START_REF] Pène | Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon[END_REF]. These shows the variety of different possible mixing rates in the infinite horizon case. Among the recent results in infinite horizon, let us mention an estimate on the tail probability of the first return time of the map T to the initial cell [START_REF] Pène | Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon[END_REF], a Local Large Deviation (LLD) estimate [START_REF] Melbourne | Local large deviations for periodic infinite horizon Lorentz gases[END_REF], a mixing rate in (t log t) -d 2 for the flow [START_REF] Pène | Strong mixing for the periodic Lorentz gas flow with infinite horizon[END_REF] for natural observables (such as indicator functions of balls). The proof of this last result required a coupled version of the above mentioned LLD, combined with several new tricks such as a large deviation estimate on the time of the n-th collision, a joint mixing local limit theorem, a new tightness-type criteria, etc.

The goal of the present article is to investigate the behaviour of ergodic sums of the Lorentz gas map T , that is of partial sums of the form n-1 k=0 f • T k for integrable observables, including the case of null integral observables. We restrict our study to the case where the observables only depend on the cell label. This restriction allows us to treat natural interesting cases such as the number of visits to the 0-cell, or the difference between the number of visits to two different cells. Unfortunately the general study of [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF] does not apply to this context because of the lack of smoothness of t → P t ∈ L(B). Nevertheless, we found a way to implement the moment method used in [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF] despite these difficulties and to establish a limit theorem for some Birkhoff sums of null integral observables of the Z d -periodic Lorentz gas (Theorem 1.1). The observables we consider are functions of the local time in the Lorentz-gas cells. In particular it applies to the difference between the number of collisions (among the n first collisions) in the cell labeled by a ∈ Z d and the number of collisions in another cell labeled by b ∈ Z d with b = a. Setting a k := max(1, √ k log k), when the observable is integrable, the Birkhoff sum will be normalized by A n := n k=1 a -d k , whereas it will be normalized by √ A n when the observable has null integral (and satisfies some integrability assumption). We reinforce Theorem 1.1 in a joint functional limit theorem (Theorem 1.3) valid for the Birkhoff sum of a couple (g, f ) with g integrable and f having a null integral, and obtain as a consequence an analogous result for the Lorentz gas flow (Theorem 1.4). Observe that

A n ∼ 2 n log n if d = 1 , and A n ∼ log log n if d = 2 .
The present article is organized as follows. In Section 1 we present our main results for the periodic Lorentz gas flow in infinite horizon. These results will appear as an application of general results stated in a general framework in Section 2 completed with Appendix A. In Section 3, we present a general strategy to prove our general assumptions of Section 2 via Fourier type operator perturbation techniques and we use this approach to prove our main results stated in Section 1.

1. Main results for the periodic Lorentz gas in infinite horizon 1.1. Limit theorem for Birkhoff sums for the map. Let us start by introducing some additional notations. The obstacles are given by O i + ℓ with ℓ ∈ Z d and with i = 1, ..., I for some I ∈ N * (up to identifying Z 1 with Z × {0} when d = 1). We write C ℓ (and call it ℓ-cell) for the set of states (q, v) ∈ M based on I i=1 O i + ℓ. We identify M with C 0 and the Sinai billiard map T to the map T : M → M corresponding to the quotient map of T modulo Z d for positions. Let Ψ : M → Z d be the cell change function, i.e., for all x ∈ M = C 0 , T (x) ∈ CΨ (x) . This can be rewritten as follows

∀x = (q, v) ∈ M = C 0 , T (x) = (q ′ , v ′ ) ⇒ T (x) = q ′ + Ψ(x), v ′ .

More generally, by

Z d -periodicity, ∀x = (q, v) ∈ M = C 0 , ∀ℓ ∈ Z d , T (x) = (q ′ , v ′ ) ⇒ T ((q + ℓ, v)) = q ′ + ℓ + Ψ(x), v ′ .
It then follows by a direct induction that

∀x = (q, v) ∈ M = C 0 , ∀ℓ ∈ Z d , T n (x) = (q n , v n ) ⇒ T n ((q + ℓ, v)) = q n + ℓ + Sn (x), v n ,
where Sn := n-1 k=0 ψ• T k . This means that the dynamics of the Lorentz gas is totally determined by the joint dynamics of the Sinai billiard and of the Birkhoff sum Sn . In other words, (M, T, µ) can be represented as the Z d -extension of ( M, T , μ) by Ψ. Recall that Szász and Varjú proved in [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF] that ( Sn /a n ) n converges in distribution to a Gaussian distribution. Let us write Φ for the density function of this Gaussian random variable.

Theorem 1.1. Let f be a µ-integrable function constant on the cells C ℓ . Then n-1 k=0 f • T k /A n n converges in distribution (with respect to any probability measure absolutely continuous with respect to the Lebesgue measure on M) to Φ(0) M f dµ |Z|, where

• Z is a standard gaussian distribution if d = 1, • Z is a random variable with standard exponential distribution if d = 2. If furthermore M f dµ = 0 and M (1 + d(0, •)) 2+ε-d 2 |f | dµ < ∞ for some ε ∈ (0, 1/2), then n-1 k=0 f • T k / √
A n n converges in distribution (with respect to any probability measure absolutely continuous with respect to the Lebesgue measure on M) to σ 2 f Φ(0)|Z|N where N is a random variable, independent of Z, with standard gaussian and where

σ 2 f = k∈Z M f.f •T k dµ.
The first part of Theorem 1.1 is a direct consequence of [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF] via moment estimates used e.g. in [START_REF] Dolgopyat | Recurrence properties of planar Lorentz process[END_REF] for the periodic Lorentz gas with finite horizon. Even in the case of the finite horizon, the second part of Theorem 1.1 (study of Birkhoff sums of null integral) is more delicate to establish since it requires a delicate care of cancellations in order to identify the main order terms in compositions of perturbed operators using their expansions (see [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF]). When the horizon is infinite, taking care of these cancellations become even more challenging since the perturbed operators do not admit expansion as a family of operators, but only expansions as a family of linear maps from B to L 1 , forbidding direct compositions of these expansions. This additional difficulty comes from the fact that the cell change function Ψ is not square integrable with respect to μ (whereas it is bounded and so finite-valued when the horizon of the Lorentz process is finite). Remark 1.2. It follows from the first part of Theorem 1.1 with the notations therein combined with the Hopf ergodic theorem 1 that for any µ-integrable h : M → R, n-1 k=0 h • T k /A n n converges in distribution to Φ(0) M h dµ |Z|. This result can be seen as a weak Law of Large Number for the infinite measure preserving dynamical system (M, T, µ). 1.2. Functional limit theorem for the map and for the flow. Theorem 1.3. Let f, g : M → R be two integrable functions, with f constant on each cell C ℓ . Assume furthermore that M (1 + d(0, •))

2+ε-d 2 |f | dµ < ∞ for some ε ∈ (0, 1
2 ) and M f dµ = 0, then the following family of couple of processes

(1)     ⌊nt⌋-1 k=0 g • T k /A n   t ,   ⌊nt⌋-1 k=0 f • T k / A n   t   n≥1
converges in distribution (with respect to any probability measure absolutely continuous with respect to the Lebesgue measure on M) to ((

M g dµL t ) t , (B σ 2 f Lt ) t ) (in (D([0; T ])) 2 for all T > 0 if d = 1 and in (D([T 0 ; T ])) 2 for all 0 < T 0 < T if d = 2)
, where σ 2 f is the quantity introduced in Theorem 1.1, where B is a standard brownian motion independent of the process L t where

• if d = 1, L t is the local time at 0 in the time interval [0; t] of the Brownian motion W limit in distribution of ( S⌊nt⌋ / √ n) t as n → +∞, 1 The Hopf ergodic theorem states that, since (M, µ, T ) is recurrent ergodic, the sequence of ergodic ratios

n-1 k=0 h • T k / n-1 k=0 1 C0 • T k n≥1 converges almost surely to M h dµ/µ(C 0 ) = M h dµ.
• if d = 2, for all t > 0 L t = L 1 is a random variable with exponential distribution with mean Φ(0).

. Our proofs of Theorems 1.1 and 1.3 are given in Section 3. They rely on the general results (in a general framework) stated in the Section 2.

As an immediate consequence of Theorem 1.3 combined with the classical random time change result (see e.g. [START_REF] Billingsley | Convergence of probability measures[END_REF]Chapter 14]), we obtain the following result valid for the periodic Lorentz gas flow. Let us write M for the set of states of the Lorentz gas flow, and m for the Lebesgue measure on M, and N t (ℓ) for the number of collisions of the flow in the cell C ℓ up to time t. |β ℓ | for some ε > 0 and ℓ∈Z d β ℓ = 0, then the family of processes

(2)   A -1 n N nt (0), A -1 2 n ℓ∈Z d β ℓ N nt (ℓ) t   n≥1
converges in distribution (with respect to any probability measure absolutely continuous with respect to the Lebesgue measure on M) to ((

L ′ t ) t , (B σ 2 L ′ t ) t ) (in (D([0; T ])) 2 for all T > 0 if d = 1 and in (D([T 0 ; T ])) 2 for all 0 < T 0 < T if d = 2), where σ 2 is quantity σ 2 f introduced in Theorem 1.1 for the function f such that f |C ℓ ≡ β ℓ for all ℓ ∈ Z d , or equivalently, with σ 2 := k∈Z a,b∈Z d β a β b μ Sk = b -a = k∈Z a,b∈Z d β a β b μ Sk = b -a -μ Sk = b -μ Sk = -a + μ Sk = 0 .
where B is a standard brownian motion independent of the process Lt where

• if d = 1, L ′
t is the local time at 0 in the time interval [0; t] of the Brownian motion W ′ limit in distribution, as n → +∞, of ((q nt / √ n) t ) n where q nt is the first coordinate of the position of Y nt (•), i.e. the first coordinate of the position at time nt of the particle,

• if d = 2, for all t > 0 L ′ t = L ′
1 is a random variable with exponential distribution with mean Φ(0). . Proof of Theorem 1.4. Recall that the Sinai billiard flow (at unit speed) endowed with the Lebesgue measure m can naturally be represented by the suspension flow over ( M, T , 2 I i=1 |∂O i |μ) with roof function τ , the time before the next collision. Indeed, this representation consists in identifying each y ∈ M with the unique couple (x, s) such that x ∈ M, s ∈ [0; τ (x)) and y = Y s (x) (x corresponds to the state at the previous collision time and s to the time spent since this previous colliqion time). Since the Sinai billiard system is ergodic, it follows from the Birkhoff ergodic theorem that the number of collisions n nt for the billiard flow (or equivalently for the Z d -periodic Lorentz gas flow) in the time interval [0; nt] is almost surely equivalent to nt/E μ[τ ] as n → +∞. We conclude by applying Theorem 1.3 to g = 1 C 0 and to f = ℓ∈Z d β ℓ 1 C ℓ , and using the random time n nt , that

  A -1 nn N nt (0), A -1 2 nn ℓ∈Z d β ℓ N nt (ℓ) t   n
has the same limit in distribution (X (1) , X (2) ) as (1) for the above choice of (g, f ). Now, using the fact that, almost surely, as t → +∞,

A nn ∼ A n /(E μ[τ ]) 2-d
2 , we conclude that the joint process (2) converges in distribution to the joint process (( 2) ). But, by the same random time change argument, the Brownian motion W ′ corresponds to the Brownian motion limit of (( Snnt / √ n) t ) n , i.e. to the Brownian motion limit of ( Snt/Eμ

E μ[τ ]) 2-d 2 X (1) , (E μ[τ ]) 2-d 4 X (
[τ] / √ n) n , i.e. to W/ E μ[τ ]. Thus the local time (L ′ t ) t of W ′ at 0, is equal to ( E μ[τ ]L t ) t
, recalling that L is the local time of W at 0. This ends the proof of the corollary.

Remark 1.5. Let G : M → R be an integrable function with respect to the Lebesgue measure m on M (velocity vectors v ∈ S 1 being identified with an angle in R/Z). The proof of Theorem 1.3 can be adapted, by taking defined on M by g(x) := τ (x) 0 G(Y s (x)) ds and by noticing that

M g dµ = M G dm 2 I i=1 |∂O i | to prove that the family of processes   A -1 n nt 0 G • Y s ds, A -1 2 n ℓ∈Z d β ℓ N nt (ℓ) t   n
converges in distribution (with respect to any probability measure absolutely continuous with respect to the Lebesgue measure on Q × S 1 ) to

1 2 I i=1 |∂O i | M G dm L ′ t ) t , ( B σ 2 L ′ t t
in the same sense as in Theorem 1.4, with σ 2 the quantity appearing in Theorem 1.4.

General results

Observe that the Birkhoff sums considered in Theorem 1.1 can be rewritten as additive functionals ( n-1 k=0 β Sk ) n of the Birkhoff sums ( Sn ) n with respect to the Sinai billiard system ( M , T , μ). We keep this formulation in the present section and state limit theorems for additive functionals of Birkhoff sums of a probability preserving dynamical systems under general assumptions expressed in terms of operators. We will see in Section 3 how these assumptions can be proved using Fourier-perturbations of the transfer operator and how this result can be used to prove Theorem 1.1. There exist a (1/α)-regularly varying sequence (a n ) n≥0 such that A n := n k=0 a -d n → +∞ as n → +∞ and a Banach space (B, • B ) preserved by the operators Q n,a such that

(3) 1 ∆ ∈ B ֒→ L 1 (ν) ,
where the notation ֒→ means a continuous inclusion, we assume furthermore that

(4) Q n,0 B = O(a -d n ) and that there exists Φ(0) > 0 such that 2 (5) Q n,0 = Φ(0)a -d n E ν [•] + o(a -d n ) in L(B → L 1 (ν)) .
In [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF], to study Birkhoff sums of the periodic Lorentz gas with finite horizon, we used the following condition

Q n,a = Φ(a/a n )a -d n E ν [•] + o(a -d n ) in L(B)
. A crucial difference between this condition and the assumptions of the present article is that ( 5) is much weaker since it holds in L(B → L 1 (ν)) instead of L(B). In practice, this weaker condition comes from the fact that the family of perturbed operators t → P t ∈ L(B) behind (see Section 3) is not continuous, but that t → P t ∈ L(B → L 1 (ν)) is continuous.

To study additive functionals n-1 k=0 β S k with a∈Z d β a = 0, we will reinforce the previous assumption as follows.

Hypothesis 2.2. Assume Hypothesis 2.1 and that [START_REF] Chernov | Chaotic billiards[END_REF] sup

a∈Z d Q k,a B = O(a -d n ) ,
and that, for all η ∈ [0; 1],

(7) Q ′ k,a,b := Q k,b -Q k,a B = O |b -a| η a -d-η k , and (8)
Q ′′ k,a,b := Q k,b-a -Q k,b -Q k,-a + Q k,0 B = O (|a||b|) η a -d-2η k , uniformly in a, b ∈ Z d .

2.2.

Limit theorem for additive functionals of Birkhoff sums.

Theorem 2.3. Assume Hypothesis 2.1. Then

n-1 k=0 1 {S k =0} /A n n≥1 converges in distribu- tion (and in the sense of moments), with respect to ν, to Φ(0)Y, where Y is a Mittag-Leffler distribution of index α-d α , i.e. E[Y N ] := N! Γ(1 + α-d α ) N Γ(1 + N α-d α )
.

If furthermore ℓ∈Z d |1 + |ℓ|| η |β ℓ | < ∞ with η := α+ε-d 2
for some ε ∈ (0, 1/2) and ℓ∈Z d β ℓ = 0, and if Hypothesis 2.2 holds true, then

n-1 k=0 β S k / √
A n n≥1 converges in distribution (and in 2 Φ(0) will appear to be the value at 0 of the density function Φ of the limit in distribution of (S n /a n ) n (see Section 3).

the sense of moments), with respect to ν, to σ 2 β Φ(0)YN where N is a random variable, independent of Z, with standard gaussian and where

σ 2 β := k∈Z a,b∈Z d β a β b ν S |k| = b -a (9) = k∈Z a,b∈Z d β a β b ν S |k| = b -a -ν S |k| = b -ν S |k| = -a + ν S |k| = 0 . ( 10
)
Remark 2.4. The summability assumption of β ℓ appearing in Theorem 2.3 is to our knowledge the optimal one even in the case of additive observables of random walks with i.i.d. increments.

Remark 2.5. It follows from our assumptions that, if β is not identically null, only the second sum [START_REF] Guivarc | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF]

defining σ 2 β is absolutely convergent in k, a, b. Indeed ν(S k = b -a) -ν(S k = b) -ν(S k = -a) + ν(S k = 0) = E ν [Q ′′ k,a,b (1)] is summable in (k, a, b) ∈ N × Z d × Z d , whereas ν(S k = 0) = E ν [Q k,0 (1)] ∼ Φ(0)a -d k is not
summable. The summability of (10) combined with the fact that

∀k ≥ 0, a,b∈Z d β a β b ν(S k = b -a) = a,b∈Z d β a β b E ν [Q ′′ k,a,b (1)]
implies the absolute convergence in k of the sum appearing in the right hand side of (9).

Proof of Theorem 2.3. Let (β ℓ ) ℓ∈Z d be as in the assumption of Theorem 2.3. We start by writing

E ν   n-1 k=0 β S k N   = n-1 k 1 ,...,k N =0 E ν N j=1 β S k j = 0≤k 1 ≤...≤k N ≤n-1 c (k 1 ,...,k N ) a 1 ,...,a N ∈Z d E ν N j=1 β a j 1 {S k j =a j } , (11)
where we denote by c (k 1 ,...,k N ) the number of N-uples (k ′ 1 , ..., k ′ N ) ∈ {0, ..., n -1} N such that there exists a permutation σ ∈ S N such that k ′ i = k σ(i) for all i = 1, ..., N. We observe that

E ν N j=1 β a j 1 {S k j =a j } = E ν N j=1 β a j 1 {S k j -k j-1 =a j -a j-1 } • F k j-1 = E ν P k N N j=1 β a j 1 {S k j -k j-1 =a j -a j-1 } • F k j-1 , ( 12
)
setting a 0 := 0 and P for the transfer operator of F with respect to ν, using the fact that

E ν [•] = E ν [P k N (•)]. Since P k (f.g • F k ) = gP k (f ),
we observe that, for any j = 1, ..., N, for any k 1 ≤ ... ≤ k j and any b 1 , ..., b j ∈ Z d ,

P k j j i=1 1 {S k i -k i-1 =b i } • F k i-1 = P k j -k j-1 1 {S k j -k j-1 =b j } P k j-1 j-1 i=1 1 {S k i -k i-1 =b i } • F k i-1 . (13)
It follows from ( 12) and ( 13) that

E ν N j=1 β a j 1 {S k j =a j } = E ν β a N Q k N -k N-1 ,a N -a N-1 (• • • (β a 2 Q k 2 -k 1 ,a 2 -a 1 (β a 1 Q k 1 ,a 1 (1))) • • • ) . (14)
For the first part of Theorem 2.3, we apply [START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF] with β ℓ = 1 {ℓ=0} with [START_REF] Lorentz | The motion of electrons in metallic bodies[END_REF]. In that case, applying repeatedly (5), combined with ( 4) and ( 3), the right hand side of ( 14) becomes

E ν Q k N -k N-1 ,0 (...(Q k 1 ,0 (1))) = (Φ(0)) N N j=1 a -d k j -k j-1 + N j=1 o(a -d k j -k j-1 )) i =j O(a -d k i -k i-1 )
and so that

E ν   n-1 k=0 1 {S k =0} N   = o(A N n ) + N! Φ(0) M f dµ N 1<k 1 <...<k N ≤n-1 N j=1 a -d k j -k j-1 = o(A N n ) + N!A N n Φ(0) M f dµ N Γ(1 + 2-d 2 ) N Γ(1 + N 2-d 2 )
+ o(1) , using [26, Lemma 2.7] for the last estimate. This ends the proof of the first part of Theorem 2.3. Now let us prove the second part. We assume from now on that a∈Z d β a = 0 and that

a∈Z d (1 + |a| η )|β a | < ∞ with η := α-d+ε 2
for some ε ∈ (0; 1 2 ). Recall that it follows from (11) combined with ( 14) that

E ν   n-1 k=0 β S k N   = 0≤k 1 ≤...≤k N ≤n-1 c (k 1 ,...,k N ) a 1 ,...,a N ∈Z d E ν β a N Q k N -k N-1 ,a N -a N-1 (• • • (β a 2 Q k 2 -k 1 ,a 2 -a 1 (β a 1 Q k 1 ,a 1 (1))) • • • ) . ( 15
)
In Formula (15), we decompose each

Q k,a in Q (0) k,a + Q (1) k,a , with Q (0) k,a := Q k,0 and Q (1) k,a := Q ′ k,a := Q ′ k,0,a = Q k,a -Q k,0 . Thus E ν   n-1 k=0 β S k N   = ε 1 ,...,ε N ∈{0,1} H (n,N ) ε 1 ,...,ε N , (16)
with

H (n,N ) ε 1 ,...,ε N = 0≤k 1 ≤...≤k N ≤n-1 c (k 1 ,...,k N ) H n,N k,ε setting k = (k 1 , ..., k N ), ε = (ε 1 , ..., ε N ) and H n,N k,ε := a 1 ,...,a N ∈Z d E ν β a N Q (ε N ) k N -k N-1 ,a N -a N-1 (• • • (β a 2 Q (ε 1 ) k 2 -k 1 ,a 2 -a 1 (β a 1 Q k 1 ,a 1 (1))) • • • ) = a N ∈Z d E ν   β a N   a N-1 ∈Z d β a N-1 Q (ε N ) k N -k N-1 ,a N -a N-1   • • •   a 1 ∈Z d β a 1 Q (ε 2 ) k 2 -k 1 ,a 2 -a 1 Q (ε 1 ) k 1 ,a 1 (1)   • • •       .
We observe that H

(n,N ) ε 1 ,...,ε N = 0 as soon as ε N = 0 or if there exists j 0 = 1, ..., N -1 such that ε j 0 = ε j 0 +1 = 0 (since a j 0 ∈Z d β a j 0 = 0). Thus, we restrict our study to the case of the ε ′ j s for which the j 0 's such that ε j 0 = 0 are isolated and do not include N. Let such an ε = (ε 1 , ..., ε N ). Observe that there are at most N/2 indices j's such that ε j = 0. We set, by convention, ε N +1 = ε 0 = 0. We will prove that (17)

H n,N ε 1 ,...,ε N = o A N 2 n unless if #{j : ε j = 0} = N 2 , i.e. H n,N ε 1 ,...,ε N = o A N 2
n unless if N is even and if (ε 1 , ..., ε N ) = (0, 1, ..., 0, 1). For reader's convenience, we first give a short proof of this estimate in a particular case. A proof of this estimate in the general case

a∈Z d |a| α-d+ε 2 |β a | < ∞ with η = α-d+ε 2 , with ε ∈ (0; 1 2 ) is given in Appendix A.
• A short proof of (17) in a particular case. Assume in this item only that α < d + 1 and

a∈Z d (1 + |a|) 2η |β a | < ∞ with α -d < η < 1. It follows from Hypothesis 2.2 that Q (ε j ) k j -k j-1 ,a j -a j-1 = O |a j -a j-1 | ε j ηa -d-ε j η k j -k j-1 = O ((1 + |a j |)(1 + |a j-1 |)) ε j ηa -d-ε j η k j -k j-1
, and so that

H n,N ε 1 ,...,ε N = O   a∈Z d (1 + |a|) 2η |β a | N N j=1 n-1 k j =0 a -d-ε j η k   = O A #{j:ε j =0} n ,
where we used the fact that a∈Z d (1+|a|) 2η |β a | < ∞ combined with k≥0 a -d-η k < +∞, since d + η > α and since (a n ) n is 1 α -regularly varying. This concludes the proof of [START_REF] Pène | Applications des propriétés stochastiques de billards dispersifs[END_REF]. In particular, Formula (17) ensures that

(18) E ν   n-1 k=0 β S k N   = o A N 2 n if N is odd ,
and that

(19) E ν   n-1 k=0 β S k N   = H (n,N ) 0,1,...,0,1 + o A N 2 n if N is even .
Assume from now on that N is even, then

H (n,N ) 0,1,...,0,1 = 0≤k 1 ≤...≤k N ≤n-1 c k 1 ,...,k N a 1 ,...,a N ∈Z d N j=1 β a j E ν [Q ′ k N -k N-1 ,a N -a N-1 (Q k N-1 -k N-2 ,0 (...(Q ′ k 2 -k 1 ,a 2 -a 1 (Q k 1 ,0 (1)))...))] Hence H (n,N ) 0,1,...,0,1 = 0≤k 1 ≤...≤k N ≤n-1 c k 1 ,...,k N E ν [ Qk N -k N-1 (Q k N-1 -k N-2 ,0 (...( Qk 2 -k 1 (Q k 1 ,0 (1)))...))] , (20)
with Qk := a,b∈Z d

β a β b Q ′ k,b-a = a,b∈Z d β a β b Q k,b-a = a,b∈Z d β a β b Q ′′ k,a,b , (21)
where we used the fact that a∈Z d β a = 0 and the notation

Q ′′ k,a,b = Q k,b-a -Q k,b -Q k,-a + Q k,
E ν [ Qk ′ (Q k,0 (h))].
Unfortunately we cannot compose directly ( 8) and ( 5) since this last estimate is in L(B → L 1 (ν)) and not in L(B). But, proceeding in two steps, we will prove that

(23) E ν [ Qk ′ (Q k,0 (h))] -Φ(0)a -d k E ν Qk ′ (1) E ν [h] = O a -α-ε ′ k ′ h B o(a -d k )
,

where ε ′ ∈ (0, 1 2 ) is small enough so that (α-d+2ε ′ )(α+ε ′ ) α+2ε ′ ≤ α -d + ε = 2η.
First, dominating separately both terms, it follows from (8) with η ′ = α+2ε ′ -d 2 ∈ (0, 1) and from ( 3) and ( 4) that

E ν [Q ′′ k ′ ,a,b (Q k,0 (h))] -Φ(0)a -d k E ν Q ′′ k ′ ,a,b (1) E ν [h] = O |a| η ′ |b| η ′ a -α-2ε ′ k ′ a -d k h B , (24) since
-d -2η ′ = -α -2ε ′ . Second, it follows from the definition of Q ′′ k ′ ,a,b that E ν [Q ′′ k ′ ,a,b (h 0 )] = E ν (1 {S k ′ =b-a} -1 {S k ′ =b} -1 {S k ′ =-a} + 1 {S k ′ =0} ).h 0 = O h 0 L 1 (ν)
. This combined with [START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF] ensures that

E ν [Q ′′ k ′ ,a,b (Q k,0 (h))] -Φ(0)a -d k E ν Q ′′ k ′ ,a,b (1) E ν [h] = E ν Q ′′ k ′ ,a,b (Q k,0 (h) -Φ(0)a -d k E ν [h]) = O( h B )o(a -d k ) . (25)
Thus, combining [START_REF] Pène | Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon[END_REF] and ( 25), we obtain that

E ν [Q ′′ k ′ ,a,b (Q k,0 (h))] -Φ(0)a -d k E ν Q ′′ k ′ ,a,b (1) E ν [h] = O(|a| η ′ |b| η ′ a -α-2ε ′ k ′ a -d k h B ) (α+ε ′ )/(α+2ε ′ ) O( h B )o(a -d k ) ε ′ /(α+2ε ′ ) = O(|a| η ′ (α+ε ′ ) α+2ε ′ |b| η ′ (α+ε) α+2ε ′ a -α-ε ′ k ′ h B )o(a -d k ) . After summation over a, b ∈ Z d , we obtain (23), since η ′ (α+ε ′ ) α+2ε ′ ≤ η and since a∈Z d (1+|a| η )|β a | < ∞.
Using [START_REF] Pène | Quantitative recurrence for T, T -1 -tranformations[END_REF] inductively in (20) (combined with the fact that Q k,0 B = O(a -d k ) and that Qk B is summable which follows from ( 22)), we conclude that, when N is even

H (n,N ) 0,1,...,0,1 = 0≤k 1 ≤...≤k N c k 1 ,...,k N N/2 j=1 Φ(0) a -d k 2j-1 -k 2j-2 E ν [ Qk 2j -k 2j-1 (1)] + O   n k=1 a -d k N/2-1 n k=1 o(a -d k )   . Recall that σ 2 β = k∈Z E ν Q|k| (1) = E ν Q0 (1) + 2 k≥1 E ν Qk (1) .
Therefore, proceeding exactly as in [26, p. 1918-1919], we obtain that

H (n,N ) 0,1,...,0,1 = Γ(1 + α-d α ) N/2 Γ(1 + N 2 α-d α ) N! 2 N/2 Φ(0)σ 2 β A n N/2 + o A N/2 n = A N/2 n E σ 2 β Φ(0)|Z|N N + o A N/2 n .
This, combined with ( 18) and ( 19), ends the proof of the convergence of every moments. We conclude the convergence in distribution by the Carleman criterion [START_REF] Schmüdgen | The moment problem[END_REF].

S k /A n   t ,   ⌊nt⌋-1 k=0 β (1) S k / A n   t   n≥1 converges in distribution, with respect to ν, to (( a∈Z d β (0) a L t ) t , (σ β (1) B Lt ) t ), (in (D([0; T ])) 2 for all T > 0 if d = 1 and in (D([T 0 ; T ])) 2 for all 0 < T 0 < T if d = 2), where σ 2 β (1)
is defined in Formula (9) of Theorem 2.3 taking β = β (1) , where B is a Brownian motion and where L t is the following process

• if α > d, L t is the local time at 0 in the time interval [0; t] of an centered α-stable process W , independent of B, such that W 1 has density probability Φ with Φ(0) satisfying (5),

• if α = d, L t = 1 {t>0} L 1 ,
where L 1 is a random variable with exponential distribution with mean Φ(0).

Proof of Theorem 2.6. We start by proving the convergence of the finite distributions and we will then prove the tightness. For the convergence of the finite distributions, we use again the convergence of moments. It is enough to study the asymptotic behaviour as n goes to infinity of every moments of the following form

E n := E ν    M j=1   ⌊nt j ⌋-1 k j =⌊nt j-1 ⌋ β (0) S k   N (0) j   ⌊nt j ⌋-1 k ′ j =⌊nt j-1 ⌋ β (1) S k   N (1) j    , for any M ∈ N * , any N (0) j , N (1) j
∈ N, any t 0 = 0 < t 1 < ... < t M . We set Γ k := M j=1 N (k) j for k ∈ {0, 1} and will prove that

lim n→+∞ A -Γ 0 - Γ 1 2 n E n = (Φ(0)) Γ 0 + Γ 1 2 a∈Z d β (0) a Γ 0 σ Γ 1 β (1) E M j=1 L t j -L t j-1 N (0) j + N (1) j 2 M j=1 E[N N (1) j ] .
We set m i := i r=1 (N

(0) r + N (1)
r ) and N := m M . Proceeding as in the proof of Theorem 1.1, we observe that E n can be rewritten as follows

γ 1 ,...,γ N (k 1 ,...,k N )∈Kn N i=1 c (γ m i-1 +1 ,...,γm i ) (k m i-1 +1 ,...,km i ) a 1 ,...,a N ∈Z d N j=1 β (γ j ) a j ν (∀i = 1, ..., N, S k i = a i ) , (27)
where K n is the set of increasing N-uples (k 1 , ..., k N ) such that ⌊nt j-1 ⌋ ≤ k m ≤ ⌊nt j ⌋ -1 for all m such that m j-1 < m ≤ m j , where the first sum is taken over γ j ∈ {0, 1} such that, for all γ ∈ {0, 1} and all i = 1, ..., M, #{j = m i-1 + 1, ..., m i :

γ j = γ} = N (γ) i and where c (γ m i-1 +1 ,...,γm i ) (k m i-1 +1 ,...,km i ) is the number of (γ ′ m i-1 +1 , , ..., γ ′ m i , k ′ m i-1 +1 , ..., k ′ m i ) ∈ {0, 1} N (1) i +N (2) i × {⌊nt i-1 ⌋, ..., ⌊nt i ⌋ -1} N (0) i +N (1) i
such that there exists a permutation σ of {m i-1 + 1, ..., m i } such that γ ′ σ(r) = γ r and k ′ σ(r) = k r for all r ∈ {m i-1 + 1, ..., m i }. Furthermore we use ( 14) to express ν(∀i = 1, ...j -1, N, k i = a i ) using a composition of operators Q k j -k j-1 ,a j -a j-1 and, as in Appendix A, for each j, we decompose each Q k j -k j-1 ,a j -a j-1 in a sum of Q (ε) . This leads to

E n := γ=(γ 1 ,...,γ N ) ε=(ε 1 ,...,ε N ) H(n) ε (γ) , (28)
with

H(n) ε 1 ,...,ε N (γ 1 , ..., γ N ) = (k 1 ,...,k N )∈Kn M i=1 c (h m i-1 +1 ,...,hm i ) (k m i-1 +1 ,...,km i ) a 1 ,...,a N ∈Z d E ν β (γ N ) a N β (γ N-1 ) a N-1 Q (ε N ) k N -k N-1 ,a N-1 ,a N • • • β (γ 1 ) a 1 Q (ε 2 ) k 2 -k 1 ,a 1 ,a 2 Q (ε 1 ) k 1 ,0,a 1 (1) • • • ,
with the use of the operators

Q (ε)
(k,a,b) defined in Appendix A and where the sum over ε 1 , ..., ε N is taken over ε 2 , ..., ε N ∈ {0, 1} 2 and ε 1 ∈ {(0, 0), (0, 1)}. We write ε j = (ε j,1 , ε j,2 ). Since

a∈Z d β (1)
a = 0, H ε 1 ,...,ε N = 0 if there exists j = 1, ..., N such that γ j = 1 and ε j,2 + ε j+1,1 = 0 (with convention ε N +1,1 = 0). Therefore we assume from now on that (γ, ε) is such that ε j,2 + ε j+1,1 ≥ γ j , for all j (with the convention ε N +1,1 = 0, and with ε 1,1 = 0) and we call admissible such a pair (γ, ε). Then

H(n) ε (γ) = O   (k 1 ,...,k N ) a 1 ,...,a N ∈Z d N j=1 |β (γ j ) a j | Q (ε j ) k j -k j-1 ,a j-1 ,a j B   = O   (k 1 ,...,k N ) a 1 ,...,a N ∈Z d N j=1 |β (γ j ) a j | (1 + |a j |) η a -d-η j,1 ε j,1 -η j,2 ε j,2 k j -k j-1   ,
with (η j,i ) j,i a sequence in {0, η} such that η j,2 ε j,2 + η j+1,1 ε j+1,1 = ηγ j and such that

• (η j,1 , η j,2 ) = ηε j if ε j = (1, 1), • η j,1 = η(1 -ε j-1,2 ) if ε j = (1, 1), • η j,2 = η if ε j = (1, 1) and ε j+1 = (1, 0), • η j,2 = 0 if ε j = (1, 1) and ε j+1 = (1, 0).
As seen in Appendix A, we use the fact that there exists

u 0 ∈ (0, 1] such that n-1 k=0 a -d-η k = O A 1-u 0 2 n
and that k≥0 a -d-2η k < ∞. Using the summability assumption on β (γ) a , we infer that

H(n) ε (γ) = O   (k 1 ,...,k N ) j:ε j,1 +ε j,2 =0 a -d-η j,1 ε j,1 -η j,2 ε j,2 k j -k j-1   = O A E η 0 + 1-u 0 2 E η 1 n
, where E η k := #{j = 1, ..., N : η j,1 ε j,1 + η j,2 ε j,2 = kη} . We also set E k := #{j = 1, ..., N : ε j,1 + ε j,2 = k} and observe that

E 1 ≤ E η 1 . Recall Γ k = N j=0 N (k) j = #{j : γ j = k}. Then N = E η 0 + E η 1 + E η 2 = E 0 + E 1 + E 2 = Γ 0 + Γ 1 .
On the other side

ηΓ 1 = η N j=1 γ j = N j=1 (η j,2 ε j,2 + η j+1,1 ε j+1,1 ) = N j=1 (η j,1 ε j,1 + η j+1,1 ε j,2 ) -η 1,1 ε 1,1 = η (E η 1 + 2E η 2 ) ,
and so

E η 0 + 1 -u 0 2 E η 1 = N -E η 2 -E η 1 + 1 -u 0 2 E η 1 = (Γ 0 + Γ 1 ) - Γ 1 -E η 1 2 - 1 + u 0 2 E η 1 = Γ 0 + Γ 1 2 - u 0 2 E η 1 .
Hence we have proved that

H (n) ε (γ) = o A Γ 0 + Γ 1 2 n if E η 1 > 0 ,
Now we assume that E η 1 = 0, this implies that E 1 = 0 and so that the j's such that ε j = (1, 1) are isolated and

E 2 = E η 2 = Γ 1 2 . Observe that this implies that Γ 1 = M j=1 N
(1) j

is even and we have proved that (29)

E n = o A Γ 0 + Γ 1 2 n if n is odd .
The above conditions on γ and ε imply that ε is a sequence of (0, 0) and (1, 1) (which are isolated) and that ε j,2 + ε j+1,1 ≥ γ j , thus ε j,2 + ε j+1,1 = γ j and

• if γ j = 0, then both ε j = (ε j,1 , ε j,2 ) and ε j+1 = (ε j+1,1 , ε j+1,2 ) are (0, 0).

• if γ j = 1, then either ε j = (ε j,1 , ε j,2 ) or ε j = (ε j,1 , ε j,2 ) is [START_REF] Billingsley | Convergence of probability measures[END_REF][START_REF] Billingsley | Convergence of probability measures[END_REF], and the other one is (0, 0). This means that the j's such that γ j = 1 appear in pairwise disjoint pairs (j -1, j) such that (γ j-1 , γ j ) = (1, 1), and that ε j = (1, 1) if and only if (j -1, j) is such a couple. Fixing such a pair (γ, ε), let us write J for the set of such j such that ε j = (1, 1). Then, using repeatedly ( 5) and ( 24), we obtain

A -Γ 0 - Γ 1 2 n H(n) ε (γ) = (k 1 ,...,k N )∈Kn M i=1 c (γ m i-1 +1 ,...,γm i ) (k m i-1 +1 ,...,km i ) j ′ :γ j ′ =0 a -d k j ′ -k j ′ -1 A n Φ(0) a∈Z d β (0) a j∈J Φ(0)a -d k j-1 -k j-2 2 A n a,b∈Z d β (1) a β (1) b E ν Q ′′ k j -k j-1 ,a,b (1) + o(1)
and so

A -Γ 0 - Γ 1 2 n H(n) ε (γ) = o (1) + Φ(0) Γ 0 + Γ 1 2 a∈Z d β (0) a Γ 0 (k 1 ,...,k N )∈Kn:k j <k j+1 M i=1 (N (0) i )!(N (1)
i )!   j:γ j =0 a -d k j -k j-1 A n   j∈J a -d k j-1 -k j-2 2 A n a,b∈Z d β (1) a β (1) b E ν Q ′′ k j -k j-1 ,a,b (1)
.

It can be worthwhile to notice that we can restrict the above sum on the (k 1 , ..., k N ) ∈ K n such that k jk j-1 < log n if j ∈ J , and k jk j-1 > log n for the other values of j's. This implies that

A -Γ 0 - Γ 1 2 n H(n) ε (γ) = o(1)
as soon as there exist j ∈ J and i ∈ {1, ..., M} such that k j-1 < ⌊nt i ⌋ ≤ k j (indeed this combined with k jk j-1 < log n implies that 0 < k jnt i < log n and 0 < nt ik j-1 < log n.

In particular

E n = o A Γ 0 + Γ 1 2 n if ∃j ∈ {1, ..., M}, N (1) j ∈ 2Z + 1 .
We assume from now on that the N (1) j 's are even and that J is such that, for every j ∈ J , there exists i = 1, ..., M such that k j-1 , k j are in a same set {⌊nt i-1 ⌋, ..., ⌊nt i ⌋ -1}. Then

A -Γ 0 - Γ 1 2 n H(n) ε (γ) = o (1) + M j=1   N (0) j ! N (1) j !(Φ(0)) N ′′ j a∈Z d β (0) a N (0) j   1 2 k≥0 (1 + 1 {k =0} ) a,b∈Z d β (1) a β (1) b E ν Q′′ k j -k j-1 ,a,b (1)
  N (1) j 2 (k ′ 1 ,...,k ′ N ′′ )∈K ′ n N ′′ j=1 a -d k ′ j -k ′ j-1 A n   , with N ′′ j := N (0) j + N (1) j 2
and N ′′ := M j=1 N ′′ j , and where K ′ n is the set of strictly increasing sequences k ′ 1 ≤ ... ≤ k ′ N ′′ with exactly N ′′ j elements between ⌊nt j-1 ⌋ and ⌊nt j ⌋ -1 and with the convention k ′ 0 = 0. Thus

H(n) ε (γ) A M j=1 N ′′ j n = o (1) + E ′ n M j=1   N (0) j !N (1) j ! a∈Z d β (0) a N (0) j 2 - N (1) j 2 σ N (1) j β (1) 1 N ′′ j !   with E ′ n := (Φ(0)) N ′′ M j=1 N ′′ j ! (k ′ 1 ,...,k ′ N ′′ )∈K ′ n N ′′ j=1 a -d k ′ j -k ′ j-1 A n = o(1) + A -N ′′ n E ν    M j=1   ⌊nt j ⌋-1 k j =⌊nt j-1 ⌋ 1 {S k j =0}   N ′′ j    . (30)
We observe that there exist

N ′′ j ! N (0) j !(N (1) j /2)! sequences (γ m j +1 , ..., γ m j+1 ) ∈ {0, 1} N (0) j +N (1) j in which the 1's appear in N (1) j /2 pairwise distinct pairs (γ j-1 , γ j ). Therefore E n A M j=1 N ′′ j n = o (1) + E ′ n M j=1    N (1) j ! (N (1)
j /2)!2 N (1) j 2 a∈Z d β (0) a N (0) j σ N (1) j β (1)    = o (1) + E ′ n M j=1   a∈Z d β (0) a N (0) j E (σ β (1) N ) N (1) j 
 .

It remains to study the asymptotics of E ′ n .

• If d = 1 < α, we consider a Z-valued non-arithmetic random walk ( S n ) n (with i.i.d. increments) such that ( S ⌊nt⌋ /a n ) n converges in distribution to the α-stable process W . The previous computations hold also true (more easily) for S n instead of S n and lead to

E ′ n ∼ A -N ′′ n E    M j=1   ⌊nt j ⌋-1 k j =⌊nt j-1 ⌋ 1 { S k j =0}   N ′′ j    as n → +∞ .
But the process

⌊nt⌋-1 k=0 1 {S k =0} t
of local time at 0 of S n converges in distribution to the process (L t ) t of local time at 0 of W . This combined with the dominations of the moments of any order ensures that (

E ′ n ) n converges in distribution to E M j=1 L t j -L t j-1 N ′′ j . We conclude that E n A M j=1 N ′′ j n = o (1) + E M j=1 L t j -L t j-1 N ′′ j M j=1   a∈Z d β (0) a N (0) j E (σ β (1) N ) N (1) j   = o (1) + E   M j=1 a∈Z d β (0) a L t j -L t j-1 N (0) j M j=1 σ β (1) (N Lt j -N Lt j-1 ) N (1) j   . • If d = α, then A -N ′′ n E ν      ⌊N t M ⌋-1 k=⌊N T 0 ⌋ 1 {S k =0}   N ′′    = O    A -N ′′ n   ⌊nt M ⌋-1 k=⌊N T 0 ⌋ a -d k   N ′′    = O (A ⌊nt M ⌋ -A ⌊nT 0 ⌋ )/A N ′′ n N ′′
, converges to 0 as n → +∞, since (A n ) n≥0 is slowly varying. Thus

E ′ n = o(1) if M ≥ 2 . Furthermore, it follows from the proof of Theorem 1.1 that if M = 1, E ′ n = o(1) + A -N ′′ n E ν      ⌊nt j ⌋-1 k j =⌊nt j-1 ⌋ 1 {S k j =0}   N ′′ 1    = (Φ(0)) N ′′ 1 N ′′ 1 ! = (Φ(0)) N ′′ 1 E[E N ′′ 1 ] ,
where E is a random variable with standard exponential distribution due to theorem 2.3. We infer that

E n A M j=1 N ′′ j n = o (1) + E (Φ(0)E) N ′′ 1 M j=2 (Φ(0)E -Φ(0)E) N ′′ j M j=1   a∈Z d β (0) a N (0) j E (σ β (1) N ) N (1) j   = o (1) + E   M j=1 a∈Z d β (0) a Φ(0)E(1 {t j >0} -1 {t j-1 >0} ) N (0) j M j=1 (σ β (1) (N Φ(0)E1 {t j >0} -N Φ(0)E1 {t j >0} )) N (1) j   .
This combined with [START_REF] Phalempin | Limit theorems for self-intersecting trajectories in Z-extensions[END_REF] and the Carleman's criteria [START_REF] Schmüdgen | The moment problem[END_REF] ends the proof of the convergence of the finite dimensional distributions. Let us write ((X

(1,n) t , X (2,n) t
) t ) n≥1 for the joint process [START_REF] Pène | Potential kernel, hitting probabilities and distributional asymptotics[END_REF] and let us prove its tightness. When d = 1 < α, we set T 0 = 0, otherwise we fix some T 0 ∈ (0; T ). We use the tightness criterion of [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 13.5,(13.4)]. We have proved the convergence of the finite dimensional distributions. It remains to prove that there exist α 1 > 1 and C > 0 such that, for every r, s, t such that T 0 ≤ r ≤ s ≤ t ≤ T , for all j ∈ {1, 2}, [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF]

∃p j ∈ N * , E ν |X (j,n) t -X (j,n) s | p j |X (j,n) s -X (j,n) r | p j ≤ C|t -r| α 1 .
Observe first that, if 0 ≤ r ≤ s ≤ t ≤ T and t-r < 1/n, then

X (n) t -X (n) s = 0 or X (n) s -X (n) r
= 0, thus the left hand side of ( 31) is null and so (31) holds true. Assume from now on that T 0 ≤ r ≤ s ≤ t ≤ T and that tr ≥ 1/n. We will use the following inequality

E ν |X (j,n) t -X (j,n) s | p j |X (n) s -X (n) r | p j ≤ X (j,n) t -X (j,n) s p j L 2p j (ν) X (j,n) s -X (j,n) r p j L 2p j (ν) .
Thus [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF] will follow from the fact that, for any T 0 ≤ r < t ≤ T , |t -r| > 1/n, and j ∈ {1, 2}, (32)

∃p j ∈ N * , sup a,b:r≤a<b≤t X (j,n) b -X (j,n) a 2p j L 2p j (ν) ≤ C|t -r| α 1 .
It follows from our previous moment computation that

E µ C 0 |X (j,n) b -X (j,n) a | 2p = O      A -1 n ⌊nt⌋-1 k=⌊nr⌋ a -d k   2p j    .
Thus it is enough to prove that (33) ∃α 0 > 0, sup r,t:T 0 ≤r<t<T, |t-r|≥1/n

A -1 n ⌊nt⌋-1 k=⌊nr⌋ a -d k = O ((t -r) α 0 ) .
Indeed, we will conclude by taking p j = j⌈(2α 0 ) -1 ⌉ so that [START_REF] Ya | Dynamical systems with elastic reflections[END_REF] and so (31) hold true with α 1 := 2p j α 0 j > 1. Since (a n ) n≥0 is (1/α)-regularly varying, it follows from Karamata's theorem [START_REF] Karamata | Sur un mode de croissance regulière. Théorèmes fondamentaux[END_REF][START_REF] Bingham | Regular variation[END_REF] that there exist three bounded convergent sequences (c(n)) n≥0 (positive, with positive limit), (b(n)) n≥0 (converging to 0) and (θ n ) n≥0 (positive, converging to

A -1

n ⌊nt⌋-1 k=0 a -d k = O A ⌊nt⌋ A n = O t 1-1 α -α 0 = O ((t -r) α 0 ) ,
implying [START_REF] Szász | Local limit theorem for the Lorentz process and its recurrence in the plane[END_REF] and so [START_REF] Ya | Dynamical systems with elastic reflections[END_REF] and [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF] in this case.

We assume from now on that T 0 ≤ r < t ≤ T and 2/n < r < t ≤ T (so that ⌊nr⌋ -1 > 0) and tr ≥ 1/n (so that ⌊nt⌋ -⌊nr⌋ ≤ 2(ntnr)). Then, using the uniform dominations on (c(n), θ n , b(n)) n combined with a series-integral comparison, we obtain that

A -1 n ⌊nt⌋-1 k=⌊nr⌋ a -d k = ⌊nt⌋-1 k=⌊nr⌋ α -1 αk c(k) c(n) k α-d α n α-d α θ k e k n b(u) u du = O   ⌊nt⌋-1 k=⌊nr⌋ 1 n (k/n) -d α -α 0   . • If d = 1 < α, it follows that A -1 n ⌊nt⌋-1 k=⌊nr⌋ a -d k = O ⌊nt⌋ -1 n α 0 - ⌊nr⌋ -1 n α 0 = O ⌊nt⌋ -⌊nr⌋ n α 0 = O ((t -r) α 0 ) , (35)
ending the proof of [START_REF] Szász | Local limit theorem for the Lorentz process and its recurrence in the plane[END_REF], from which we infer [START_REF] Ya | Dynamical systems with elastic reflections[END_REF] and [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF]. This ends the proof of the tightness when d < α.

• When d = α, we obtain

A -1 n ⌊nt⌋-1 k=⌊nr⌋ a -d k = O ⌊nr⌋ n -α 0 - ⌊nt⌋ n -α 0 = O ⌊nt⌋ -⌊nr⌋ n = O (t -r) , (36)
from which we infer [START_REF] Szász | Local limit theorem for the Lorentz process and its recurrence in the plane[END_REF], [START_REF] Ya | Dynamical systems with elastic reflections[END_REF], [START_REF] Simanyi | Towards a proof of recurrence for the Lorentz process[END_REF], and so the tightness in the case where d = α.

This ends the proof of Theorem 2.6.

Proof of Theorems 1.1 and 1.3 via Fourier perturbations

A strategy to prove Assumptions 2.1 and 2.2 consists in noticing that (37)

Q k,a = 1 (2π) d [-π;π] d
e -i t,a P k t (•) dt , with P t (h) := P (e i t,Ψ ) , and to establish nice properties for P t as the one listed in the next result. Recall that P n t (h) = P n (he itSn ). Proposition 3.1. Assume B is a Banach space satisfying (3) and that there exist two constants b ∈ (0, π) and α 0 > 0 such that :

(38) ∀t ∈ [-b, b] d , P k t = λ k t Π t + O(e -α 0 k ) and sup b<|u|∞<π P k u B = O(e -α 0 k ) ,
with (λ k t/a k ) k≥0 converging to the characteristic function ϕ of an α-stable distribution, with

Π t = E ν [•] + o(1) in L(B → L 1 (ν)) as t → 0, and with (39) sup t∈[-b;b] d Π t B < ∞ and R (1 + |t| 2 ) sup k≥1 |λ k t/a k |1 {|t|<ba k } dt < ∞ .
Then Hypotheses 2.2 (and so 2.1) hold true with Φ(0) the value at 0 of the density function Φ of the α-stable distribution with characteristic function ϕ.

Proof of Proposition 3.1. It follows from our assumptions that

Q k,0 = 1 (2π) d [-π;π] d P k t dt = a -d k (2π) d [-ba k ;ba k ] d λ k t/a k Π t/a k dt + O(e -α 0 k ) = Φ(0)a -d k + o a -d k , (40)
in L(B → L 1 (ν)), via the dominated convergence theorem since lim n→+∞ λ n t/an Π t/an = ϕ(t)E ν [•] in L(B → L 1 (ν)) and since Φ(0) = 1 (2π) d R ϕ(t) dt. Thus (5) holds true. Furthermore, for all η ∈ [0; 2] and a ∈ Z d , in L(B),

[-π;π] d |t| η P k t B dt = a -d-η k [-ba k ;ba k ] d |t| η λ k t/a k Π t/a k B dt + O e -α 0 k = O a -d-η k . (41)
Using the expression (37) of Q k,a combined with (41) with m = 0, we obtain (4) and [START_REF] Chernov | Chaotic billiards[END_REF]

. Fix η ∈ [0; 1]. Then Q ′ k,a,b = Q k,b -Q k,a = 1 (2π) d [-π;π] d (e i t,b -e i t,a )P k t (•) dt = 1 (2π) d [-π;π] d O( t, b -a ) η P k t (•) dt = O |b -a| η a -d-η k in L(B) ,
where we used again (41) combined with the bound |e ixe iy | ≤ min(2, |x -y|) ≤ 2 1-η |x -y| η , and so we have proved [START_REF] Conze | Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications (French)[END_REF]. For [START_REF] Dolgopyat | Asymptotic expansion of correlation functions for Z d -covers of hyperbolic flows[END_REF], in the same way, we obtain

Q ′′ k,a,b = Q k,b-a -Q k,b -Q k,-a + Q k,0 = 1 (2π) d [-π;π] d (e i t,b-a -e i t,b -e -i t,a + 1)P k t (•) dt = 1 (2π) d [-π;π] d (e i t,b -1)(e -i t,a -1)P k t (•) dt = 1 (2π) d [-π;π] d O( t, b t, a ) η P k t (•) dt = O (|a||b|) η a -d-2η k in L(B) , (42)
and so [START_REF] Dolgopyat | Asymptotic expansion of correlation functions for Z d -covers of hyperbolic flows[END_REF].

Proof of Theorems 1.1 and 1.3. Let us write β ℓ for the constant to which f is equal on the ℓ-cell C ℓ . The integrability assumption means that ℓ∈Z |β ℓ | < ∞. Due to [START_REF] Zweimüller | Mixing limit theorems for ergodic transformations[END_REF], since µ is equivalent to the Lebesgue measure on M, it is enough to prove the results with respect to the measure µ C 0 (the restriction of µ to C 0 ). Thus, we consider this reference measure and establish the convergence of every moment with respect to this probability measure. We observe that, with the identification of M to C 0 , µ C 0 is identified with μ and f • T k is identified with β Sk . As in [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF][START_REF] Pène | Sharp error term in local limit theorems and mixing for Lorentz gases with infinite horizon[END_REF], we use the two Young towers [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF][START_REF] Chernov | Decay of correlations and dispersing billiards[END_REF]. We write ( ∆, F , ν) for the hyperbolic tower which is an extension of ( M, T , μ), and write (∆, F, ν) for the expanding tower obtained by quotienting ( ∆, F , ν) along stable curves. We write π : ∆ → M and π : ∆ → ∆ for the two measurable maps such that π *

ν = µ, π * ν = ν, T • π = π • F and F • π = π • F . Since Ψ is constant on stable curves, there exists a function Ψ : ∆ → Z d such that Ψ • π = Ψ • π. Setting S n := n-1 k=0 Ψ • F k , it follows that S n • π = Sn • π.
For the first part of Theorem 1.1, as noticed in Remark 1.2, it is enough to prove the first convergence result for β ℓ = 1 {ℓ=0} . We will conclude by Theorem 2.3. To prove the assumptions of Theorem 2.3, we show that the criterion given in Proposition 3.1 is satisfied here with our choice of a n , with α = 2 and with Φ the characteristic function of the Gaussian limit distribution of (S n /a n ) n . The assumptions of Proposition 3.1 have been proved in [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF] with the use of the Banach spaces introduced in [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF] combined with the use of the Nagaev-Guivarch perturbation method [START_REF] Nagaev | Some limit theorems for stationary Markov chains (Russian)[END_REF][START_REF] Guivarc | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF][START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF] via the Keller and Liverani theorem [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF] (see also [START_REF] Pène | Stochastic properties of dynamical systems[END_REF] for a general reference on this method). The fact that (λ k t/a k ) k≥1 converges pointwise to the characteristic function of a Gaussian random variable follows from the existence of a positive symmetric matrix A such that 1λ t ∼ At, t | log |t|| as t → 0 (this was proved in [START_REF] Szász | Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon[END_REF]). For the second part of (39), one can e.g. use the fact that

|λ k t/a k |1 {|t|<ba k } ≤ e -c 0 min(|t| 2-ε ,|t| 2+ε ))
. Thus Proposition 3.1 holds true and Theorems 2.3 and 2.6 apply. Finally, we identify the formulas of the asymptotic variances σ 2 f and σ 2 β by noticing that a,b∈Z 2

β a β b ν(S |k| = b -a) = M f.f • T |k| dµ = M f.f • T k dµ .
For Theorem 1.3, we deduce the result for general g using ergodicity of (M, T, µ) together with the Hopf ergodic ratio theorem.

Appendix A. Proof of (17) in the general case

We assume here that a∈Z d (1 + |a| η )|β a | < ∞ with η := α-d+ε 2 for some ε ∈ (0; 1/2).

In Formula (15), we decompose Q k,a using the operators Q ′′ k,a,b and

Q ′ k,c := Q ′ k,0,c = Q k,c -Q k,0 as follows Q k,a = Q ′′ k,a,b + Q k,b + Q k,-a -Q k,0 = Q ′′ k,a,b + Q ′ k,b + Q ′ k,-a + Q k,0 .
In [START_REF] Melbourne | Local large deviations for periodic infinite horizon Lorentz gases[END_REF], we replace each Q k j -k j-1 ,a j -a j-1 by this decomposition, we develop and obtain

E ν   n-1 k=0 β S k N   = ε 1 ,...,ε N D (n,N ) ε 1 ,...,ε N , (43)
summing a priori over (ε 1 , ..., ε N ) ∈ ({0, 1} 2 ) N such that ε 1 ∈ {(0, 0), (0, 1)} the following quantity D (n,N ) ε 1 ,...,ε N =

β a j E ν [ Q (ε N ) k N -k N-1 ,a N-1 ,a N ... Q (ε 2 ) k 2 -k 1 ,a 1 ,a 2 Q (ε 1 )
k 1 ,a 0 ,a 1 (1)] , with a 0 = 0 and where

Q (0,0) k,a,b = Q k j -k j-1 ,0 , Q (1,0) k,a,b = Q ′ k,-a , Q (0,1) k,a,b = Q ′ k,b , Q (1,1)
k,a,b = Q ′′ k,a,b . We first restrict the sum over ε 1 , ..., ε N . We observe that, since a j ∈Z d β a j = 0, D n,N ε 1 ,...,ε N = 0 if there exists j = 1, ..., N such that ε j,2 + ε j+1,1 = 0 (with convention ε N +1,1 = 0). Thus we restrict the sum in (43) to the sum over the ε 1 , ..., ε N such that for all j = 1, ..., N, such that ε j,2 + ε j+1,1 ≥ 1. We call admissible any such sequence ε := (ε 1 , ..., ε N ). Let ε := (ε 1 , ..., ε N ) be an admissible sequence.

• We observe that #{j : ε j = (0, 0)} ≤ N/2. unless if N 1 = 0, i.e. unless if N is even and if ε 1 , ..., ε N is the alternate sequence (0, 0), (1, 1), ..., (0, 0), (1, 1). • Assume now that there exists some j 0 ∈ {1, ..., N} such that ε j 0 ,2 + ε j 0 +1,1 = 2. Recall that it follows from Hypothesis 2.2 that, for all η ′ j,1 , η ′ j,2 ∈ {0, η}, We choose a family (η ′ j,i ) j=1,...,N ;i=1,2 of {0, 1} such that, for all j = 1, ..., N, η ′ j,2 ε j,2 + η ′ j+1,1 ε j+1,1 = η = α+ε-d where we set ε ′ j := (η ′ j,1 ε j,1 , η ′ j,2 ε j,2 )/η. We also consider the sequence ε ′′ 1 , ..., ε ′′ N obtained from ε ′ by permuting the values of ε ′ j 0 ,2 and ε ′ j 0 +1,1 . Both sequences ε ′ and ε ′′ are admissible and satisfy ε ′ j,2 + ε ′ j+1,1 = ε ′′ j,2 + ε ′′ j+1,1 = 1 for all j = 1, ..., N. Thus it follows from the previous item that D (n,N ) ε 1 ,...,ε N = O min d

Theorem 1 . 4 .

 14 For any real valued sequence (β ℓ ) ℓ∈Z d such that ℓ∈Z d (1 + |ℓ|) 2+ε-d 2

2. 1 .

 1 General assumptions. Hypothesis 2.1. Let d ∈ {1, 2} and α ∈ [d; 2]. Let (∆, F, ν) be a probability preserving dynamical system with transfer operator P . Let Ψ : ∆ → Z d . For any a ∈ Z d and any non-negative integer n, we set S n := n-1 k=0 Ψ • F k and we set Q n,a for the operator given by Q n,a := P n (1 {Sn=a} •) .

  0 introduced in (8). Combining (8) with η = α+ε-d 2 with (21), since a∈Z d (1 + |a|) η |β a | < ∞, we infer that (22) Qk = O a -α-ε k in L(B) , since d + 2η = α + ε, which ensures the summability of Qk B . The study of (20) leads us to the question of estimating

( 1 )

 1 a = 0. Then the following family of couples of processes

  α-1 α if d < α and to 0 if d = α, see [2, Proposition 1.5.9.b]) such that ∀n ≥ 0, na -d n = A n θ n and A n = c(n)n Now let us choose α 0 . If d = 1 < α, we set α 0 := α-1 2α . If d = α,we take α 0 := 1. Up to change, if necessary, the first terms of (b(n)) n≥0 and (c(n)) n≥0 , we assume without loss of generality that the sequence (b(n)) n≥0 is bounded by α 0 . If d = 1 < α, if r ≤ 2/n (observe that, when d = α, this case does not happen for large values of n since r ≥ T 0 > 0) and |t -r| ≥ 1/n, then

0≤k 1 ≤

 1 ...≤k N ≤n-1 c (k 1 ,...,k N ) a 1 ,...,a N ∈Z d N j=1

••.N 0 +N 1 1-u 0 2 n, 2 n= o A N/ 2 n

 1222 The contribution to (43) of an admissible sequence ε = (ε 1 , ..., ε N ) isD (n,N ) ε 1 ,...,ε N = O   a 1 ,...,a N ∈Z d We observe that there exists u 0 > 0 such that (44) n k=0 a -d k = A n , Indeed d+2η = α+ε > α.For the last estimate, we use the fact that (a k ) k is 1 α -regularly varying, and infer that (n k=0 a -d-η k ) 2 is either bounded or 2 -2d+2η α -regularly varying whereas (A n ) n is (1 -d α )-regularly varying and diverges to infinity (and 2-2d+2η α = 1 -d+ε α < 1 -d α ).• If, for all j = 1, ..., N, ε j,2 + ε j+1,1 = 1, then, it follows from Hypothesis 2.2 thatD (n,N ) ε 1 ,...,ε N = O d (n,N ) ε 1 ,...,ε N , with (45) d (n,N ) ε 1 ,...,ε N := a 1 ,...,a N ∈Z d N j=1 |β a j | |a j | η(ε j,2 +ε j+1,1) using (44), thatD (n,N ) ε 1 ,...,ε N = O d (n,N ) ε 1 ,...,ε N = O Awhere N k := #{j :ε j,1 + ε j,2 = k}, since a∈Z d (1 + |a|) η |β a | < ∞. Observe that N 0 + N 1 + N 2 = N and N = 2 k=1 N j=1 ε j,k = N 1 + 2N 2 and so N 2 = N 0 and N 0 = N -N 12 . Therefore, in this case,D (n,N ) ε 1 ,...,ε N = O d (n,N ) ε 1 ,...,ε N = O A N-u 0 N 1

.

  ) k,a j ,a j-1 B = O |a j-1 /a k | η ′ j,1 ε j,1 |a j /a k | η ′ j,2 ε j,2 a -d kIndeed this follows from (6), (7) and (8) combined with the two following facts∀η ∈ [0; 1], Q ′′ k,a,b B = Q ′ k,-a,b-a -Q ′ k,0,b B = O(|b| η )a -d-η k and ∀η ∈ [0; 1], Q ′′ k,a,b B = Q ′ k,b,b-a -Q ′ k,0,-a B = O(|a| η )a -d-η k .

2 and η ′ 1 , 1 

 11 = 0, with convention η ′ N +1,1 = 0. ThereforeD (n,N ) ε 1 ,...,ε N = O   a 1 ,...,a N ∈Z d N j=1 |β a j | |a j |

  (n,N ) ε ′ 1 ,...,ε ′ N , d (n,N ) ε ′′ 1 ,...,ε ′′ N = o A N/2 n ,since ε ′ and ε ′′ cannot both coincide with the alternate sequence (0, 0), (1, 1), ..., (0, 0), (1, 1).

  2.3. Joint Limit theorem for additive functional of Birkhoff sums. ) a∈Z d be two families of real numbers such that a∈Z d (1 + |a|) η |β

	Theorem 2.6. Assume Assumptions 2.1 and 2.2. Let η := α+ε-d 2 (β (0) a ) a∈Z d and (β (1) and a∈Z d β	for some ε ∈ (0, 1/2). Let (j) a | < ∞

a

Acknowledgement : The author conduced this work within the framework of the Henri Lebesgue Center ANR-11-LABX-0020-01 and is supported by the ANR project GALS (ANR-23-CE40-0001).