Ambre Davat
email: ambre.davat@gipsa-lab.fr

Véronique Aubergé
email: veronique.auberge@univ-grenoble-alpes.fr

Gang Feng
email: gang.feng@gipsa-lab.fr

Can we hear physical and social space together through prosody?

Keywords: speech localization, acoustic proxemics, socialaffective prosody, social space, ecological experimentation

When human listeners try to guess the spatial position of a speech source, they are influenced by the speaker's production level, regardless of the intensity level reaching their ears. Because the perception of distance is a very difficult task, they rely on their own experience, which tells them that a whispering talker is close to them, and that a shouting talker is far away. This study aims to test if similar results could be obtained for prosodic variations produced by a human speaker in an everyday life environment. It consists in a localization task, during which blindfolded subjects had to estimate the incoming voice direction, speaker orientation and distance of a trained female speaker, who uttered single words, following instructions concerning intensity and social-affect to be performed. This protocol was implemented in two experiments. First, a complex pretext task was used in order to distract the subjects from the strange behavior of the speaker. On the contrary, during the second experiment, the subjects were fully aware of the prosodic variations, which allowed them to adapt their perception. Results show the importance of the pretext task, and suggest that the perception of the speaker's orientation can be influenced by voice intensity.

Introduction

In the last century, several technologies have been developed to make humans ubiquitous. First, the telephone allowed people to speak to each other in real time, no matter their physical distance. The videophone further enhanced ubiquity, by also transmitting the image of its user. Nowadays, telepresence robots represent a new step in remote and ubiquitous immersion. This time, the goal is to bring the body of a person from one place to another, by using a remote controlled robot which embodies its user.

These technologies are currently developed for business or medical applications, where high social immersion of the remote user is needed. It is therefore interesting to reconsider the fidelity of voice transmission. The vocal artefacts produced by these robots have yet to be integrated into the ways we use them. In particular, it was observed that people interacting with a telepresence robot are reluctant to tune its volume, while tuning the volume of a phone when it's too high or too low is a perfectly common behavior. Unlike phones, correcting the volume of the robot would require physically interacting with their interlocutor's substitute body.

To improve the illusion of the remote users' presence, and allow them to conform to social rules, they need to be able to adapt their speech to the local environment. Previous research on this subject mostly consist in ensuring that the loudness of the robot is convenient for the local interlocutors [START_REF] Takahashi | A case study of an automatic volume control interface for a telepresence system[END_REF]- [START_REF] Kimura | Visual Feedback: Its Effect on Teleconferencing[END_REF]. This implies an artificial increase or lowering of the speaker's voice intensity to keep the voice intelligible. However, the impact of these variations on the interaction is yet to be studied.

Voice intensity depends on multiple elements of context, including acoustic properties of the environment, distance of hearing, as well as the speaker's social role [START_REF] Cooke | The listening talker: A review of human and algorithmic context-induced modifications of speech[END_REF]. Audio technologies enables these elements to be dissociated, leading to what [START_REF] Schafer | The tuning of the world[END_REF] referred as "schizophony". In particular, intimate voices with a small earshot can be amplified in order to be audible by a large audience [START_REF] Maasø | The Proxemics of the Mediated Voice: An Analytical Framework for Understanding Sound Space in Mediated Talk[END_REF], [START_REF] Collins | Sonic Proxemics and the Art of Persuasion: An Analytical Framework[END_REF]. Some results in psychoacoustics suggest that these artefacts could affect distance perception. In [START_REF] Gardner | Distance Estimation of 0° or Apparent 0°-Oriented Speech Signals in Anechoic Space[END_REF]- [START_REF] Eriksson | Perception of vocal effort and distance from the speaker on the basis of vowel utterances[END_REF], subjects were asked to estimate the position of a sound source, and gave closer distances when hearing whispers, and further distances when hearing shouts.

Tuning the volume of a telepresence robot may therefore affect the users' acoustic proxemics. This effect should be evaluated in realistic conditions. In this paper, we present a first study aiming at assessing if the perception of spatial information can be affected by prosodic variations. It consists in two experiments. One has already been described in [START_REF] Davat | Integrating Socio-Affective Information in Physical Perception aimed to Telepresence Robots[END_REF], and used a complex scenario, so that the subjects were focused on a task totally different from localization. In the second experiment, our aim was clear for the subjects, who were also asked to recognize the prosodic patterns. In this article, we will briefly summarize the methodology used, and compare the results of both experiments.

Method

The localization test took place in a reverberant room (reverberation time around 0.8 s). The subjects (S) were blindfolded and sat in the middle a square space ( Figure 1). As a part of the pretext task, eight tables with plastic cups were placed around them. One experimenter (E1) moved to twelve predefined positions in the room. She uttered single words from a list of 40 scents with varying number of syllables (ex: rose, eucalyptus). A second experimenter (E2) sat next to the subject. Loudspeakers were set up behind the subject, and rhythmic music was played between each utterance, in order to mask the speaker's footsteps when she changed position. ; these distances cover both close and far phases of social space according to Hall's proxemics theory [START_REF] Hall | Proxemics [and Comments and Replies[END_REF], while being close enough to be difficult to distinguish [START_REF] Zahorik | Auditory Distance Perception in Humans: A of Past and Present Research[END_REF]; • the social-affect she expressed: polite doubt (intended to bring the listener socially closer) or authoritative confidence (intended to push back the listener) (see section 3 for prosodic analyses); • her voice intensity: low or loud. Further details concerning the choice of these parameters can be found in [START_REF] Davat | Integrating Socio-Affective Information in Physical Perception aimed to Telepresence Robots[END_REF]. In order to shorten the duration of the experiment, the orientation varied only for middle distance: By default, the speaker was always facing towards the subject. For each test, a list of the 64 combinations of these 5 variables was randomly selected. It is worth noticing that the speaker E1 is a French native language phonetician, who has been studying audio-visual prosodic attitudes for thirty years. She is able to produce consistent French vocal attitudes, which were perceptually validated in previous studies (see, for instance, [START_REF] Shochi | Intercultural Perception of English, French and Japanese Social Affective Prosody[END_REF]).

During the first experiment, 10 subjects (all native French speaker) were convinced that they were participating in a study about the interferences between olfaction and taste during social interactions. E1 pretended to be a professional Nose, and the words she pronounced were supposed to be the flavor of the pills she had to identify during the experiment. The subject S wore a blindfolding mask, supposedly for preventing both of them from reading emotion on the face of the other. S was asked to localize E1's position in the room, so we could monitor if s/he was still focused on the interaction task. Then, E1 gave to S a smelling jar, and s/he was able to have a short discussion with E1 to express their views on the flavor. After the task was completed, the subject was informed of the real aim of the experiment, and had the choice to ask to delete the data. If s/he agreed with the use of these data, s/he signed a new consent form canceling the one they signed initially.

During the second experiment, we didn't use the pretext task. This time, 8 new subjects (7 native French speakers) were asked to guess the speaker's distance, orientation and direction. Moreover, they had to label the words they heard as "low doubt", "loud doubt", "low confidence" or "loud confidence". They signed only one consent form, at the beginning of the experiment.

Before analyzing the results of both experiments, we need to validate the speaker's productions.

Validation

The speaker's performances were evaluated a posteriori, using recordings obtained with a Sennheiser HSP4 wireless headworn microphone. Every key-word was extracted by hand and labeled with the instructions given to the speaker. For brevity, the four classes of stimuli are labeled as "doubt", "DOUBT", "confidence" and "CONFIDENCE", uppercase letters corresponding to loud intensity.

First, the variations in intensity were checked (Table 1). Low stimuli clearly differ from loud stimuli, as they are 8.8 dB lower on average. This means that the speaker managed to respect the instructions she was given. Standard deviations are quite high, because the intensity also varies depending on the said word. Furthermore, the social-affect appears to have an impact on the intensity. Low doubt is indeed 2.4dB lower than low confidence, while loud confidence is 2.3dB louder than loud doubt.

Table 1: Intensity measures for each class.

Procedure: An A-weighting was applied to the numerical recordings.

Then the intensity was measured on 20ms-frames with the algorithm used in Praat [START_REF] Boersma | Praat: doing phonetics by computer[END_REF] in order to obtain readable values. The productions of social-affects are also interesting to analyze. The average intensity and pitch curves for each class of stimuli are shown in Figure 2. Intensity curves are consistent with the previous measures, being lower for the low stimuli, and higher for the loud stimuli. Pitch curves are also very specific: ascending for doubtful stimuli, versus descending for confident stimuli. Moreover, the word duration varies significantly, as shown in Table 2. On average, doubtful stimuli are 320ms longer than confident stimuli. The duration also depends on intensity, as loud doubt is 87ms longer than low doubt, and low confidence 98ms shorter than loud confidence. Voice quality was also considered, as shown in Table 3. Breathiness and laxness are strongly correlated with doubt, while tenseness is correlated with confidence. In particular, 95% of doubt stimuli are labeled as breathy or lax, and 95% of confidence stimuli are labeled as tense. On the contrary, only 8% of doubt stimuli are labeled as tense, and 5% of loud confidence stimuli are labeled as breathy or lax.

Class

It is worth noticing that the speaker found loud doubt and low confidence harder to produce. This initial feeling is confirmed by the analyses. As shown above, low confidence is not as low as low doubt, while loud doubt is not as loud as loud confidence. In order to produce these counter-intuitive stimuli, she exaggerated the word duration: loud doubt is therefore longer, and low confidence shorter. Moreover, the percentages in voice quality labels are more extreme when the intensity is coherent with the social-affect.

Results

The answers of the subjects were compared with the real productions of the speaker. Recognition scores were computed for each variable in different conditions. They are shown on Figure 3. Each plot is followed by a measure of p-value obtained with an ANOVA implemented with the R software.

Direction

The speaker's direction was well perceived by the subjects, who obtained very high recognition scores (> 90 %) in both experiments. Most of the errors were due to front/back confusion, which are common during localization tests [START_REF] Middlebrooks | Sound Localization by Human Listeners[END_REF]. It is worth noticing that these confusions occur generally in one direction: When the speaker was behind the subject, in 17 % of cases, they answered that she was in front of them, while the opposite occurred only in 2 % of cases. Neither the socialaffect nor the intensity seem to have any influence on the perception of direction, as the recognition scores are approximately constant in each condition.

Orientation

The orientation was more difficult to perceive, with recognition rates of 79 % and 85 % for experiments 1 and 2, respectively. Moreover, the variations in each condition vary significantly between the experiments. When the subjects were not aware of the aim of the experiment, their results were particularly poor in the low-intensity condition, because they tended to perceive that the speaker was back to them. In the experiment 1, the number of "back" stimuli perceived as "front" is three times higher in the low-intensity condition than in the loud-intensity condition, while in the experiment 2, both counts are approximately equal. There is also a visible effect of the social-affect in the first experiment, but considering the high inter-subject variability, it is not strong enough to be statistically significant.

Distance

The most difficult variable to estimate was the distance: In both experiments, the subjects were right only for 58 % of the stimuli. The close distance was generally well recognized (only 9% of the errors). Most of the errors (62 %) occurred for the middle distance, which was perceived as far in 70 % of cases. Generally, when the subjects were wrong, they chose the adjacent distance. Therefore, close was perceived as far only in 10 % of the wrong recognitions, and far was perceived as close only in 15 % of the wrong recognitions. Surprisingly, there is no improvement between the two experiments. Again, there is no clear effect of the social-affect or the intensity in the second experiment. However, the social-affect seems to have a little effect in the first experiment, the recognition scores being lower when the speaker was confident. During the second experiment, the subjects were asked to identify the labels of each stimulus. Their results are 4. The scores were very high when the social-affect was coherent with the intensity, i.e. for low doubt and loud confidence. In particular, loud confidence was never mistaken as doubt, and low doubt was never mistaken as loud confidence. Ambiguous stimuli were more difficult to classify and the subjects were right only half of the time. On average, social-affect was more easily perceived by the subject than intensity, as the recognition rates on both variables are respectively 89 % and 76 %.

Social-affect and intensity

Discussion

The aim of this study was to test if the perception of a speaker's spatial position could be affected by prosodic variations in a non-anechoic environment. First, a complex scenario was designed in order to evaluate the subjects' localization skills when their attention is diverted from the real aim of the experiment by a pretext task. In these conditions, the subjects were fully focused on the localization task, but were not able to guess that the speaker's prosodic variations were part of the experiment. Then, a simplified version of this experiment without pretext task was implemented, for comparison purposes.

In the first experiment, we observed some effects of the prosody on the localization skills of the listener, despite a strong inter-subject variability. In particular, subjects tended to perceive that the speaker was facing away from them, when she pronounced words with a low intensity. Distance was also harder to perceive when the speaker expressed confidence. None of these effects could be obtained in the second experiment. This seems to validate our first choice of designing a complex protocol, in order to divert the subject's attention from the aim of the experiment. When they are aware of the prosodic variations, they probably adapt better to them and it is no longer possible to observe an influence of socialaffect and intensity.

However, the first experiment was difficult to set up, due to the availability of the speaker, as well as long and stressful for the experimenters. Therefore, only a small number of subjects passed the experiment. We tried to contact these first subjects for the second experiment, but most of them being students in their last year of university, they were no longer in the city at the time of the tests.

Another issue of this experimental protocol is the question of reproducibility. It was not possible to use prerecorded sounds, as it would have shattered the pretext of the experiment. The subjects needed to believe that the speaker was in the same room with them. The use of a portable loudspeaker was considered, but the subjects could have heard a difference between the pre-recorded words and the spontaneous talking generated by the experiment. Instead, the speaker's productions were analyzed a posteriori. The analyses showed four different prosodic patterns, i.e. one for each combination of social-affect and intensity. The speaker was better at following the instructions in intensity and voice quality, when the social-affect was coherent with the intensity. Coherent stimuli were better classified by the subjects in the second experiment. This is another proof that voice intensity is strongly linked to social-affect.

The results are therefore positive, but need to be confirmed by new reproductions. Our next step consists in an online listening test. This time, the stimuli have been recorded with our telepresence robot. It will be presented as a test to evaluate the quality of acoustic immersion for telepresence.

Figure 1 :

 1 Figure 1: Photo and top-view of the experimental setting. The crosses correspond to the speaker's spatial positions.Five parameters varied during the experiment:• the speaker's direction: left, right, behind or in front of the subject; • her orientation: she was either facing towards the subject (face), or turning her back to them (back): those first parameters represent basic spatial information, that subjects should be able to guess while driving a telepresence robot; they were included in anticipation of future experiments; • her distance: 1.7 m (close distance), 2.5 m (middle distance) or 3.3 m away from the subject (far distance); these distances cover both close and far phases of social space according to Hall's proxemics theory[START_REF] Hall | Proxemics [and Comments and Replies[END_REF], while being close enough to be difficult to distinguish[START_REF] Zahorik | Auditory Distance Perception in Humans: A of Past and Present Research[END_REF]; • the social-affect she expressed: polite doubt (intended to bring the listener socially closer) or authoritative confidence (intended to push back the listener) (see section 3 for prosodic analyses); • her voice intensity: low or loud. Further details concerning the choice of these parameters can be found in[START_REF] Davat | Integrating Socio-Affective Information in Physical Perception aimed to Telepresence Robots[END_REF]. In order to shorten the duration of the experiment, the orientation varied only for middle distance: By default, the speaker was always facing towards the subject. For each test, a list of the 64 combinations of these 5 variables was randomly selected. It is worth noticing that the speaker E1 is a French native language phonetician, who has been studying audio-visual prosodic attitudes for thirty years. She is able to produce consistent French vocal attitudes, which were perceptually validated in previous studies (see, for instance,[START_REF] Shochi | Intercultural Perception of English, French and Japanese Social Affective Prosody[END_REF]).During the first experiment, 10 subjects (all native French speaker) were convinced that they were participating in a study about the interferences between olfaction and taste during social interactions. E1 pretended to be a professional Nose, and the words she pronounced were supposed to be the flavor of the pills she had to identify during the experiment. The subject S wore a blindfolding mask, supposedly for preventing both of them from reading emotion on the face of the other. S was asked to localize E1's position in the room, so we could monitor if s/he was still focused on the interaction task. Then, E1 gave to S a smelling jar, and s/he was able to have a short discussion with E1 to express their views on the flavor. After the task was completed, the subject was informed of the real aim of the experiment, and had the choice to ask to delete the data. If s/he agreed with the use of these data, s/he signed a new consent form canceling the one they signed initially.

Figure 2 :

 2 Figure 2: Average pitch and intensity curves for each class of stimuli in both experiments

Figure 3 :

 3 Figure 3: Localization recognition rates. Comparison between two conditions: with or without a pretext task.

Table 2 :

 2 Duration measures for each class.

	Class	Mean (ms)	Standard deviation (ms)	Number of stimuli
	doubt	752	177	291
	DOUBT	839	207	281
	confidence	423	120	281
	CONFIDENCE	521	134	298

Table 3 :

 3 Voice quality for each class.

	Class	Lax	Breathy	Modal	Tense
		(%)	(%)	(%)	(%)
	doubt	95.9	96.5	4.5	0.7
	DOUBT	81.0	89.6	66.0	16.1
	confidence	6.8	8.5	41.6	91.5
	CONFIDENCE	0.7	0.7	11.3	98.3

Table 4 :

 4 Confusion matrix for the perception of intensity and social-affect in experiment 2.

				Production
			doubt DOUBT	conf.	CONF.
	Perception	doubt DOUBT conf. CONF	0.67 0.23 0.10 0	0.19 0.48 0.10 0.22	0.11 0.01 0.50 0.38	0 0 0.07 0.93


Sinda Ammous

Jérôme Dedecker

Céline Duval

Adaptive directional estimator of the density in R d for independent and mixing sequences

Keywords: Adaptive procedure, Anisotropy, Density estimation, Dependence, Stationary sequences, Fourier transform AMS Classification. Primary: 62G07, 62G20. Secondary: 62H12

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, α-mixing and τ -mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. In particular, if A is an invertible matrix, if the observations are drawn from X ∈ R d , d ≥ 1, it achieves the rate implied by the regularity of AX, which may be more regular than X. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.

Introduction

The problem of adaptive density estimation was mainly studied in a context of i.i.d. (independent and identically distributed) random variables, but also for dependent data since the late 90's. Given the considerable amount of articles on the subject, the following review does not pretend to be an exhaustive rendition of all the works on the topic. On minimax adaptive multivariate density estimation from direct independent random vectors, we mention projection methods, with the preliminary work of [START_REF] Hasminskii | On Density Estimation in the View of Kolmogorov's Ideas in Approximation Theory[END_REF] [START_REF] Hasminskii | On Density Estimation in the View of Kolmogorov's Ideas in Approximation Theory[END_REF] or more recently [START_REF] Birgé | Model selection for density estimation with L 2 -loss[END_REF] [START_REF] Birgé | Model selection for density estimation with L 2 -loss[END_REF], wavelet techniques with [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF] [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF] and kernel estimators investigated e.g. in [START_REF] Chacón | Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices[END_REF] [START_REF] Chacón | Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices[END_REF], the works of Goldenshluger and Lepski [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF][START_REF] Goldenshluger | On adaptive minimax density estimation on R d[END_REF] or [START_REF] Rebelles | L p adaptive estimation of an anisotropic density under independence hypothesis[END_REF] [START_REF] Rebelles | L p adaptive estimation of an anisotropic density under independence hypothesis[END_REF].

Let us look in more detail at two recent papers. In [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF], Comte and Lacour consider the problem of deconvolution in R d , but their results also apply to the case where the variables are observed directly. The authors propose adaptive density estimators for both the pointwise L 2risk and the integrated L 2 -risk, by adapting the method of Goldenshluger and Lepski. For the integrated L 2 -risk, they obtain adaptive estimators on general anisotropic Sobolev classes, using tensor products of the sinus-cardinal (sinc) kernel. In [START_REF] Lacour | Estimator selection: a new method with applications to kernel density estimation[END_REF], Lacour et al. propose a new method for selecting the bandwidth of kernel density estimators. For a class of kernels described in [START_REF] Goldenshluger | On adaptive minimax density estimation on R d[END_REF] they obtain adaptive estimators for anisotropic Nikolskii classes and the integrated L 2 -risk.

Regarding dependent sequences, The only article that we know of that deals with adaptive multivariate density estimation in a mixing context, is that of Bertin et al. [START_REF] Bertin | Adaptive density estimation on bounded domains under mixing conditions[END_REF] . These authors deal with density estimation on a bounded domain of R d , for geometrically β-mixing sequences (see Volkonski and Rozanov(1959) [START_REF] Volkonskiȋ | Some limit theorems for random functions[END_REF] for the definition of β-mixing coefficients).

The other articles we briefly describe below only concern the estimation of the density in the univariate case. [START_REF] Tribouley | Lp adaptive density estimation in a β mixing framework[END_REF] [START_REF] Tribouley | Lp adaptive density estimation in a β mixing framework[END_REF] propose L p adaptive estimators for the common density f of a stationary β-mixing sequence using wavelets, that are minimax optimal (in the sense that the procedure leads to the same minimax rate as in the i.i.d. case). Considering discrete or continuous time stationary processes, [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] study the adaptive density estimation of the common density f of a α-mixing or β-mixing process (see Definition 3.1 below for the definition of α-mixing coefficients in the sense of [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF] [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]). They use a penalized least square method to compute the adaptive estimators. [START_REF] Comte | Adaptive density deconvolution with dependent inputs[END_REF] [START_REF] Comte | Adaptive density deconvolution with dependent inputs[END_REF] propose a model selection procedure for projection estimators on the Shannon basis to estimate f on the whole real line for stationary β-mixing sequences. Lerasle [START_REF] Lerasle | Adaptive density estimation of stationary β-mixing and τ -mixing processes[END_REF][START_REF] Lerasle | Optimal model selection for density estimation of stationary data under various mixing conditions[END_REF] proposes an adaptive estimator based on model selection for the density of a stationary process which is either β-mixing or τ -mixing (see Definition 3.2 below for the definition of τ -mixing coefficients). [START_REF] Gannaz | Adaptive density estimation under weak dependence[END_REF] [START_REF] Gannaz | Adaptive density estimation under weak dependence[END_REF] consider other type of dependence; they estimate the common marginal density f by a wavelet type estimator, under some conditions on the λ or ϕ coefficients (see [START_REF] Dedecker | Weak dependence: with examples and applications[END_REF] for their definition). Asin and Johannes (2017) [START_REF] Asin | Adaptive nonparametric estimation in the presence of dependence[END_REF] give a data-driven non-parametric estimation procedure for a density and a regression function. They use an orthogonal series estimator for β-mixing sequences. [START_REF] Bertin | Pointwise adaptive estimation of the marginal density of a weakly dependent process[END_REF] [START_REF] Bertin | Pointwise adaptive estimation of the marginal density of a weakly dependent process[END_REF] estimate the common density f of a weakly dependent process, giving upper bounds for the pointwise risk, and propose a data-driven procedure based on Goldenshluger and Lepskii [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] for α-mixing and λ-dependent sequences.

We observe that many papers on adaptive density estimation in a dependent context deal with β-mixing sequences. The reason is a technical one: for β-mixing sequences, one can use Berbee's coupling lemma [START_REF] Berbee | Random walks with stationary increments and renewal theory[END_REF], which enables to go back to the i.i.d. case and to apply the powerful concentration inequalities of Talagrand [START_REF] Talagrand | New concentration inequalities in product spaces[END_REF]. This approach is much trickier in the α-mixing case because the coupling tools for α-mixing sequences are much less efficient (see for instance [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF] Chapter 5). τ -mixing sequences have better coupling properties (see [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF]), but they are quite complicated to handle because of their hereditary properties (individual functions of τ -mixing sequences are not necessarily τ -mixing).

Let us now briefly underline why the α-mixing case is so attractive. It is well known that the notion of α-mixing is the weakest type of mixing among the usual mixing conditions (see e.g. Bradley [START_REF] Bradley | Basic properties of strong mixing conditions[END_REF]). It contains two large classes of examples: irreducible, aperiodic and positively recurrent Markov chains (for which β-mixing is equivalent to α-mixing, see [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF], Chapter 9), as well as functions of Gaussian processes which are naturally ρ-mixing (see [START_REF] Bradley | Basic properties of strong mixing conditions[END_REF] for the definition of ρ-mixing).

However, α-mixing has some limitations: it does not apply in general to non-irreducible Markov chains. For instance the Markov chains associated with most dynamical systems are not α-mixing (see for instance [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF]). By contrast, the τ -mixing coefficients can be computed for a large class of dynamical systems, as well as for many other examples (such as functions of linear processes with discrete innovations). Of course, other notions of dependence can also be defined: we refer to [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF] and [START_REF] Dedecker | Weak dependence: with examples and applications[END_REF] for an overview.

The purpose of this paper is to introduce a unified adaptive density estimator for multivariate stationary random variables that are independent, α-mixing or τ -mixing (see Equation (2.1) below for the definition of the estimator). It is inspired by the recent adaptive procedure introduced by Duval and Kappus (2019) [START_REF] Duval | Adaptive procedure for Fourier estimators: application to deconvolution and decompounding[END_REF] to select the optimal cutoff parameter for univariate density estimators based on a Fourier method, in the i.i.d. case. This procedure is numerically easy to implement and the concentration tool used to establish the oracle inequality is a basic Hoeffding inequality for partial sum of bounded random variables, which makes it eligible for an extension to the case of dependent variables. More precisely, we shall use Fuk-Nagev type inequalities, that can be derived from [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF] in the α-mixing case (see Lemma 4.4), and from [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] in the τ -mixing case (see Lemma 4.8). Equation (2.1) below proposes a complete rewriting of the estimator of [START_REF] Duval | Adaptive procedure for Fourier estimators: application to deconvolution and decompounding[END_REF] for multivariate random variables.

The contributions of the present work are the following. We propose a unique thresholded estimation procedure which is parameter free. Indeed, if the procedure depends on a thresholding constant (unrelated to the smoothness of the unknown density), this constant can be easily calibrated through an innovative tool using the Euler characteristic of thresholded area. Oracle inequalities are established for independent and dependent sequences of multivariate random variables (which seems quite new in a dependent context, in particular regarding the adaptivity to anisotropic regularity classes). The proofs of the oracle inequalities rely on fine cuttings of the quadratic risk, which makes it elegant and easy to adapt in other contexts such as indirect observations. The bias term appearing in these inequalities has such a general shape that it allows anisotropic regularity classes with a possible change of base (not necessarily orthonormal) that may lead to faster convergence rates. A property that, to the knowledge of the authors, has not been studied in the literature and is not shared with existing estimators. We call this property "Directional Adaptivity".

The paper is organized as follows. Section 2 is devoted to the multivariate independent case; we first define our adaptive estimator and state Theorem 2.1 which is an oracle inequality. Some extensions are discussed as well as a comparison with wavelet thresholded estimators. Rates of convergence are derived on general anisotropic Sobolev classes, where a linear bijective transformation of the data is allowed (see Equation (2.4)). As emphasized by Example 2.1 this transformation can improve the rates of convergence. The remaining of the Section is devoted to its numerical study. In Section 2.3 we propose a numerical procedure for the calibration of the only parameter appearing in the definition of the estimator, using the Euler characteristic of the thresholded areas. Section 2.4 provides numerical illustrations in dimension 2. Section 3 generalizes Theorem 2.1 for stationary sequences that are either α-mixing (see Definition 3.1 and Theorem 3.1) or τ -mixing (see Definition 3.2 and Theorem 3.2). In each case the numerical performances of the estimator, combined with the selection procedure of Section 2.3, are investigated in dimension 1. All the proofs are gathered in Section 4.

Notations Given a random variable Z taking values in R d , ϕ Z (u) = E[e i u,Z ] denotes the characteristic function of Z. For f ∈ L 1 (R d ), Ff (u) = R d e i u,x f (x)dx, u ∈ R d ,
is understood to be the Fourier transform of f . Moreover, we denote by

• the L 2 -norm of functions, f 2 := R d |f (x)| 2 dx. Let A be the class of invertible d × d matrices such that A([-1, 1] d ) ⊆ [-1, 1] d . Note that, if A ∈ A, then |det(A)| ≤ 1.
A necessary and sufficient condition for an invertible matrix A to belong to A is that the 1 norm of each row is less than 1

. Given f ∈ L 1 (R d )∩L 2 (R d ) and A ∈ A, we denote by f A,m , m ∈ R d + , the uniquely defined function with Fourier transform Ff A,m = (Ff )1 A([-m,m]) , where [-m, m] = [-m 1 , m 1 ] × • • • × [-m d , m d ].
The interest of introducing the matrix A is to capture directional adaptivity and consider general anisotropy classes as defined in (2.4) (see also Example 2.1).

2 Directional adaptive procedure: the independent case

Adaptive thresholded estimator

Consider n i.i.d. realisations X j , 1 ≤ j ≤ n, of a R d -valued random variable X, with Lebesguedensity f . We build an estimator ϕ X,n of ϕ X as follows

ϕ X,n (u) = 1 n n j=1 e i u,X j , u ∈ R d .
Using Fourier inversion, we define an estimator of f as follows

f n (x) = 1 (2π) d [-n,n] d e -i u,x ϕ X,n (u)du, x ∈ R d . (2.1)
where for κ n := (1 + κ √ log n), for some positive κ, we set

ϕ X,n (u) = ϕ X,n (u)1 {| ϕ X,n (u)|≥κnn -1/2 } , u ∈ R d . (2.2)
The quantity in (2.1) has no reason to be positive therefore in practice we take

f n = Re( f n ) + .
We underline that contrary to classical Fourier estimators the cutoff parameter is taken equal to n and that adaptivity will be obtained by thresholding the estimated characteristic function.

Therefore the only parameter in (2.1) that requires a choice is the constant κ appearing in the threshold. This quantity plays a role only in the order of remainder terms (see Theorem 2.1). This is why we choose not to make explicit the dependency in κ of f n as the choice of κ is not related to the smoothness of f . In Section 2.3, we propose an innovating adaptive procedure to select κ using the Euler characteristic of the set {u, | ϕ X,n (u)| ≥ κ n n -1/2 }, which performs quite well numerically.

Theorem 2.1. Let κ > 0. The following inequality holds

E[ f n -f 2 ] ≤ inf m∈[0,n] d ,A∈A 18 f -f A,m 2 + 10 + 2(1 + (κ + 2) √ log n) 2 |det(A)| π d n m 1 • • • m d + 64 π d n d-κ 2 /4 .
Note that a choice of κ > 2 √ d + 1 ensures that the last term n d-κ 2 /4 is negligible (see Remark 2.1 and 2.2 below for modifications of the estimator f n allowing a choice of κ independent of d). Proof of Theorem 2.1 is given in Section 4.1, it is self contained and relies on fine cuttings of the quadratic risk. The inequality involved is an Hoeffding inequality, which makes the proof robust to other contexts such as α or τ mixing sequences as shown in Section 3. A discussion on the resulting rates of convergence is given in the next Section. The generalization to indirect measurements such as a deconvolution framework with known noise distribution could be easily derived (see [START_REF] Duval | Adaptive procedure for Fourier estimators: application to deconvolution and decompounding[END_REF]) by replacing ϕ X by ϕ Y /ϕ ε if one observes i.i.d. realizations of Y = X + ε where X is independent of ε.

The spirit of estimator (2.1) is closely related to wavelet thresholded estimator where adaptivity is achieved by thresholding empirical wavelet coefficients that are too small (namely smaller than a constant times log n/n) to suppress noise artefacts (see e.g. [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF] [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF] Sections 5 and 6 (in particular Theorem 6.1) for the multivariate isotropic density estimation (see also [START_REF] Donoho | Wavelet shrinkage: asymptopia?[END_REF] [START_REF] Donoho | Wavelet shrinkage: asymptopia?[END_REF]) or [START_REF] Neumann | Multivariate wavelet thresholding in anisotropic function spaces[END_REF] [START_REF] Neumann | Multivariate wavelet thresholding in anisotropic function spaces[END_REF] (in particular Theorem 2.3) for anisotropic estimation in a Gaussian white noise model). As for estimator (2.1), thresholding enables to directly define an adaptive minimax estimator which requires no calibration of a parameter depending on the unknown smoothness of the function, at the cost of a logarithmic loss. If wavelets method have the advantage of facilitating the control of L p loss functions, the proof of Theorem 2.1 relies on direct arguments allowing the generalization to mixing sequences as shown in Section 3.

Remark 2.1. The remaining term 4n d-κ 2 /4 in Theorem 2.1 can be improved in C κ,d n -κ 2 /4 by making the threshold in (2.2) depend on u. For a function h :

R d → R + , let φX,n (u) = ϕ X,n (u)1{| ϕ X,n (u)| ≥ κ log(h(u)n)/n}.
It is then straightforward to replace the bound in Lemma 4.1 below by 4(nh(u)) -κ 2 /4 . Depending on the choice of h the remaining term of Theorem 2.1 changes (see Inequality (4.8)). For h ≡ 1 we recover the remaining term 4n d-κ 2 /4 ; choosing

h(u) = (1 + |u 1 |) . . . (1 + |u d |) enables to replace this term by C κ,d n -κ 2 /4 which is negligible for κ > 2.
Remark 2.2. If one does not look for the directional adaptivity given by the infimum over the class A, there is another way to get rid of the quantity n d appearing in the remaining term of Theorem 2.1. It suffices to modify the estimator f n as follows : let

D n = [-n, n] d ∩ {u ∈ R d : |u 1 • • • u d | ≤ n} and f (x) = 1 (2π) d Dn e -i u,x ϕ X,n (u)du .
Following exactly the proof of Theorem 2.1 and letting f m = f Id,m , one gets the oracle inequality:

E[ f -f 2 ] ≤ inf m∈[0,n] d ,m 1 •••m d ≤n 18 f -f m 2 + 10 + 2(1 + (κ + 2) √ log n) 2 π d n m 1 • • • m d + C d n 1-κ 2 /4 (log n) d-1 ,
where C d is a constant depending only on d. Here, the remaining term in the oracle inequality comes from the estimate

Dn du = [-n,n] d 1 |u 1 •••u d |≤n du 1 . . . du d ∼ n→∞ 2 d (d -1) d-1 (d -1)! log(n) d-1 n. (2.3)
Proof of Equation (2.3) is given in Section 4.

Rates of convergence on general anisotropic Sobolev classes

To derive the resulting rate of convergence we require some regularity on the density f to control the order of the bias term. Theorem 2.1 implies that our adaptive estimator f n is rate optimal on Sobolev regularity classes, with a regularity direction given by any matrix A ∈ A, up to a logarithmic loss. Indeed, consider a Sobolev class S(A, s, L) for s = (s 1 , . . . , s d ), L > 0 and A ∈ A, defined as follows

S(A, s, L) = f ∈ L 1 (R d ), d k=1 R d |Ff (Au)| 2 (1 + |u k |) 2s k du ≤ L . (2.4)
For f ∈ S(A, s, L), the bias term can be controlled by

f -f A,m 2 ≤ L|det(A)|(m -2s 1 1 + • • • + m -2s d d ) .
Minimizing in (m 1 , . . . , m d ) the bound of Theorem 2.1 we find for

1 s = d k=1 1 s k (2.5) that the optimal cutoff is such that m * k n log n 2s 2s k (2s+1) leading to a rate n -2s 2s+1 (log n) 2s 2s+1
. In case A = Id, this rate is optimal, up to a power of log n (see [START_REF] Hasminskii | On Density Estimation in the View of Kolmogorov's Ideas in Approximation Theory[END_REF] [START_REF] Hasminskii | On Density Estimation in the View of Kolmogorov's Ideas in Approximation Theory[END_REF]). Other regularity classes such as super smooth classes of densities can also be considered to control the bias term and different minimax optimal rates of convergence emerge (see [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF] [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF]).

Note that Ff (Au) is the characteristic function of the random variable Y = t AX, if X has Lebesgue density f . Introducing A allows to define a regularity class dependent free of the specific frame in which the observations X j are displayed, since in Theorem 2.1 we take the infimum over the set A. Example 2.1 below shows that it is a relevant transformation that can improve the rates, since A and s are connected, namely s = s(A). Taking the infimum over the set A permits to attain the rate induced by the best orientation of the axes, that is such that

s = s(A) is maximal (see (2.5)).
The introduction of a linear transformation of the data is also evoked in Corollary 1 of Varet et al. (2019) [START_REF] Varet | Numerical performance of penalized comparison to overfitting for multivariate kernel density estimation[END_REF]: it is underlined that considering such transformations can improve the rates. However, the matrix transformations considered therein are orthonormal and their procedure numerically investigates all possible transformations matrix whereas thresholding appears to intrinsically adapt to the optimal transformation.

Example 2.1. Let d = 2, b > 1 and 0 < a < b(1 -b). Define X = (bX 1 , aX 1 + bX 2 ) where X 1 and X 2 are independent, X 1 ∼ Γ(α + 1 2 , 1) and X 2 ∼ Γ(β + 1 2 , 1), where 0 < β < α. Denote by f the density of X. It is straightforward to check that |Ff (u)| 2 = 1 (1 + |bu 1 + au 2 | 2 ) α+ 1 2 1 (1 + |bu 2 | 2 ) β+ 1 2 .
Denote the associated bias term, for m 1 ≥ 1 and

m 2 ≥ 1, by B f (m 1 , m 2 ) = [-m,m] c |Ff (u)| 2 du. Consider the matrix A a,b = 1 b 2 b -a 0 b ∈ A for which |Ff (A a,b u)| 2 = 1 (1 + |u 1 | 2 ) α+ 1 2 1 (1 + |u 2 | 2 ) β+ 1 2
and the associated bias term defined by

B f A a,b (m 1 , m 2 ) = [-m,m] c |Ff (A a,b u)| 2 du.
We prove in Section 4 the following result for 0 < β < α and

m 1 ≥ 1 ∨ C 1 α,β,b , m 2 ≥ 1 ∨ C 2 α,β,b B f (m 1 , m 2 ) ≥ c α 2bβ(1 + b) 2β 1 m 2β 2 + 1 2m 2β 1 ≥ c β 2α 1 m 2α 1 + c α 2β 1 m 2β 2 ≥ B f A a,b (m 1 , m 2 ), (2.6)
for some positive constants C j α,β,b and where

c γ = R (1 + z 2 ) -(γ+ 1 2 ) dz, γ = α, β. Hence, if 0 < β < α, the rate obtained by minimizing in (m 1 , m 2 ) the quantity 18 f -f A,m 2 + 10 + 2(1 + (κ + 2) √ log n) 2 |det(A)| π d n m 1 m 2
will be strictly better for A = A a,b than for A = Id : for A = A a,b , we obtain a rate of order n -2s 2s+1 (log n) 2s 2s+1 with s = αβ/(α + β), while for A = Id, we obtain a rate of order n

-β β+1 (log n) β β+1 .

Empirical procedure to select κ

Most adaptive methods require the calibration of a constant, here this constant is κ. This is usually done by preliminary simulation experiments. Calibration strategies (dimension jump and slope heuristics) exist for penalized methods, see [START_REF] Baudry | Slope heuristics: overview and implementation[END_REF] [START_REF] Baudry | Slope heuristics: overview and implementation[END_REF] and Lerasle (2012) [START_REF] Lerasle | Optimal model selection in density estimation[END_REF] for theoretical justifications.

Here, we propose a strategy to select κ adapted to our context and we implement it in the numerical Sections 2.4 and 3.3. This procedure is the result of two observations. Firstly, the set

{| ϕ X,n | ≥ κ n n -1/2 } (recall that κ n = (1 + κ √ log n)) appearing in (2.1)
is the excursion set of the process Z = | ϕ X,n | above the level κ n n -1/2 . For these sets there exists an important literature on their geometry which provide an information that we exploit here (see [START_REF] Adler | Random fields and geometry[END_REF] [START_REF] Adler | Random fields and geometry[END_REF]). One of these geometrical measure is the Euler characteristic; in dimension d = 1 it is a count of the number of connected components and if d = 2 it is a count of the number of connected components minus the number of holes. Secondly, the constant κ determines how much information from ϕ X,n is dropped. If κ is selected too large relevant information on f is lost and if κ is selected too small artefact noise may jeopardize its estimation.

We expect that the function

κ → χ({| ϕ X,n | ≥ κ n n -1/2 })
, where χ denotes the Euler characteristic, will stabilize once all uninformative areas are thresholded. Then, the first κ where χ is constant will perform a good compromise between keeping enough information and removing most of the noise. This motivates the following adaptive procedure to select κ. Set δ > 0 and define

χ x := χ({| ϕ X,n | ≥ (1 + xδ √ log n)n -1/2 }) κ n := inf {κ ∈ {kδ, k ∈ N}, χ κ = χ κ-1 } . (2.7)
Note that this quantity does not depend on any additional calibration constant, except for the mesh-grid δ that should be taken small. In the case d = 2 we can visualize the set {| ϕ X,n | ≥ κ n n -1/2 } as a black and white image (see Figure 1). On this example X is a Gamma-Beta random variable (see [GB] example in the next Section), we observe that if κ is two small there are many uninformative areas that are taken into account in the estimator (left image) and that the Euler characteristic indeed gets constant for large enough κ (right image).

Numerical study in the independent case

Examples considered We illustrate the performances of the estimator (2.1) with the adaptive choice of κ described in (2.7) in dimension d = 2. We consider three examples:

[N] Gaussian: X ∼ N 0 0 , Σ 2 , with Σ 2 = 1 0.5 0.5 3 , [Mix-NN] Mixture of Gaussian: X ∼ 0.4 N -2 -2 , 1 0.2 0.2 3 +0.6 N 2 2 , 1 0.3 0.3 1 , [GB] Gamma-Beta : X ∼ W Y , where W = 1 0.1 0.2 1 and Y = Y 1 Y 2 where Y 1 ∼ Γ(5, 1) is independent of Y 2 ∼ Beta(2, 2).
To evaluate the performances of the procedure we compute normalized L 2 -risks to allow the numerical comparaison of the different examples for which f 2 may vary. Namely, we evaluate empirically

E[ f n -f 2 ]
f 2 and the associated deviations, from N = 100 independent datasets with different values of sample size n = 10 3 , 10 4 and 10 5 .

Results and comments They are displayed in Table 1 and confirm the theoretical results. As expected the estimated risks decrease as n increases. Since we compute the normalized L 2 -risks we can compare them for the different examples. As anticipated the GB case is the most difficult to estimate and has the largest risk, indeed the Beta distribution is compactly supported which makes it difficult for a Fourier estimator to recover. Interestingly, the choice of κ by (2.7) leads in each example to a constant close to 1, except for the Gaussian case where it is a little smaller. Its dependency in n can also be observed: it has a tendency to decrease with n. 3 Directional adaptive procedure: mixing sequences

In this section, we consider X 1 , ..., X n a stationary sequence of R d -valued dependent random variables with density f with respect to the Lebesgue measure. As in the independent case, we estimate f on R d with the estimator (2.1) and evaluate its performances using the quadratic loss on R d . We consider the cases of α-mixing and τ -mixing sequences, for which we give analogues of Theorem 2.1.

α-mixing sequences

To measure the dependence between the variables X i , we consider the usual α-mixing coefficients introduced by Rosenblatt (1956) [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. Let us recall the definition of these coefficients.

Definition 3.1. Let (Ω, A, P) be a probability space, and

(X i ) i∈Z be a sequence of R d -valued random variables. Let F k = σ(X i : i ≤ k) and G k = σ(X i : i ≥ k).
The strong mixing coefficients (α(n)) n≥0 of (X i ) i∈Z (Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]), are defined by

α(0) = 1 2 and α(n) = sup k∈Z α(F k , G k+n ) for any n > 0,
where, for two σ-algebra F, G,

α(F, G) = 2 sup A∈F ,B∈G |P(A ∩ B) -P(A)P(B)| .
Note that we use here the convention of the book by Rio [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF], so that the definition of α(F, G) differs from that of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF] from a factor 2.

For α-mixing sequences, we prove the following oracle inequality :

Theorem 3.1. Let κ > 0. Assume that ∞ k=1 α(k) < ∞, and let C α = 1 + 4 ∞ k=1 α(k). There exist constants K i , i ∈ {1, ..., 7}, depending on C α such that E[ f n -f 2 ] ≤ inf m∈[0,n] d ,A∈A K 1 f -f A,m 2 + K 2 + 2(1 + (κ + √ K 3 ) √ log n) 2 |det(A)| π d n m 1 • • • m d + 2 7 π d n d-κ 2 K 3 + f α (n, d, κ) ,
where the residual term f α (n, d, κ) is given by

f α (n, d, κ) = K 4 n (2d+1)/2 √ log n α √ nK 5 √ log n + K 6 n (2d+1)/2 κ √ log n α √ nK 7 κ √ log n .
Explicit upper bounds for the constants K i , i ∈ {1, ..., 7}, can be computed from the proof of Theorem 3.1.

Remark 3.1. The term n (2d+1)/2 in the expression of f α (n, d, κ) implies that the polynomial rate of mixing must depend on d if this residual term is to be of order O(1/n). More precisely, we need a rate of order α(n) = O(n -a ), for a > 3 + 2d. As in Remark 2.2, if one does not look for the directional adaptivity given by the infimum over the class A, then the term n (2d+1)/2 can be considerably weakened by considering the estimator f of Remark 2.2. Doing so, the term n (2d+1)/2 in the expression of f α (n, d, κ) is replaced by C d n(log n) d-1 , and the constraint on the rate of mixing becomes a > 5 (not depending on d).

According to Remark 3.1, if α(n) = O(n -a ) for a > 3 + 2d and for κ large enough (κ > K 3 (d + 1)), the upper bound of Theorem 3.1 is the same, except for the constants, as in the independent case (Theorem 2.1). Then, the resulting rates of convergence on the Sobolev class S(A, s, L) defined by (2.4) are minimax optimal up to a logarithmic loss. Note that the latter choice for κ depends on the unknown constant C α that involves the mixing coefficients. In practice, the choice of κ selected by the adaptive procedure described in Section 2.3 leads to very good numerical results and does not rely on any knowledge on C α .

Note that the residual term f α (n, d, κ) tends to zero as soon as α(n) = O(n -a ) for a > 2d+1, and is of order

f α (n, d, κ) = O (log n) (a-1)/2 n (a-(2d+1))/2 .
Hence, our adaptive estimator will still be consistent as soon as a > 2d + 1, with possibly a suboptimal rate of convergence. In the simulations, for d = 1, we shall look at the boundary case a = 3.

To conclude this section, let us take a look at some articles that consider the problem of adaptive density estimation in a dependent context. We are going to treat separately the articles which deal with β-mixing sequences, from those which deal with α-mixing sequences, because the case of β-mixing is both simpler from a technical point of view, and more restrictive in terms of examples. Note that in some articles (such as [START_REF] Lerasle | Adaptive density estimation of stationary β-mixing and τ -mixing processes[END_REF][START_REF] Lerasle | Optimal model selection for density estimation of stationary data under various mixing conditions[END_REF], [START_REF] Bertin | Pointwise adaptive estimation of the marginal density of a weakly dependent process[END_REF]) other notions of dependencies are also considered.

As mentioned in the Introduction, the only article that we know of that deals with adaptive estimation in a mixing context and in R d , is that of Bertin et al. [START_REF] Bertin | Adaptive density estimation on bounded domains under mixing conditions[END_REF]. Note that the context and method are quite different from ours (kernel estimation on a bounded domain), and that these authors do not consider the anisotropic case and assume that the β-mixing coefficients decrease at an exponential rate.

All the articles we are going to discuss now only consider the case d = 1. In the β-mixing framework, Tribouley and Viennet (1998) [START_REF] Tribouley | Lp adaptive density estimation in a β mixing framework[END_REF] proposed a wavelet method and Comte and Merlevède (2002) [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] a general model selection procedure (valid for a large class of models) to estimate the density on a compact support. Both adaptive estimators reach the minimax rates of convergence over Besov classes under the condition β(k) = O(k -a ) for a > 4. [START_REF] Comte | Adaptive density deconvolution with dependent inputs[END_REF] [START_REF] Comte | Adaptive density deconvolution with dependent inputs[END_REF] proposed a model selection procedure for projection estimators on the Shannon basis (i.e. the orthonormal basis generated by the sinus-cardinal function), to estimate the density on the whole real line. The results are valid under the condition β(k) = O(k -a ) for a > 4. In the same context, Lerasle [START_REF] Lerasle | Adaptive density estimation of stationary β-mixing and τ -mixing processes[END_REF][START_REF] Lerasle | Optimal model selection for density estimation of stationary data under various mixing conditions[END_REF] gave a general model selection result (valid for a large class of models) under the condition β(k) = O(k -a ) for a > 3. Note that in [START_REF] Lerasle | Optimal model selection for density estimation of stationary data under various mixing conditions[END_REF], the delicate question of the "data-driven" penalty is discussed. Asin and Johannes (2018) [START_REF] Asin | Adaptive nonparametric estimation in the presence of dependence[END_REF] proposed a model selection procedure to estimate the density, when the regularity of f is given by the decrease of the coefficients of the decomposition of f on a fixed orthonormal basis. Here again, the results are valid if β(k) = O(k -a ) for some a > 4 (see the comments on condition (4.6), after their Corollary 4.7). In the α-mixing framework, [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] also study the case where α(k) decreases geometrically (see their Theorem 3.2). Their adaptive estimator reaches the minimax rates of convergence over Besov classes, up to a power of a logarithmic term. In the case where α(k) = O(k -a ) (see their Proposition 3.2), their results hold provided that a > 6, and under an additional regularity assumption on the joint densities g X k ,X , k = .

All previously cited articles give results for the integrated quadratic risk (integrated L p -risk are also considered in [START_REF] Tribouley | Lp adaptive density estimation in a β mixing framework[END_REF]), like the one we are considering here. Let us also mention the article by Bertin and Klutchnikoff (2017) [START_REF] Bertin | Pointwise adaptive estimation of the marginal density of a weakly dependent process[END_REF] which proposes an automatic bandwidth selection for a kernel estimator, and for the pointwise risk. Their result applies to a large class of dependent sequences, in particular to α-mixing sequences for which the mixing coefficient decreases geometrically.

We can see that, whether for β or α mixing sequences, the constraint a > 3 on the mixing speed is always required for the adaptive estimation. This corresponds to the minimum constraint we found for d = 1. Yet there is no heuristic explanation for this, since when the regularity is known, non adaptive estimators attain the minimax i.i.d. rate as soon as α(k) < ∞.

The article closest to ours in the case d = 1 is that of Comte and Merlevède (2002) [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF], although the context and the method of estimation are different. The mixing speed we get to approach the minimax rate is a bit better than theirs (a > 5 instead of a > 6), but we still have a loss in a power of log n. Moreover, we do not need any condition on the joint densities g X k ,X .

Geometrically τ -mixing sequences

To measure the dependence between the variables X i , we consider the τ -mixing coefficients introduced by Dedecker an Prieur [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF]. Let us recall the definition of these coefficients. Definition 3.2. Let (Ω, A, P) be a probability space. Let | • | 2 be the euclidean norm on R d . On

(R d ) k , consider the 1 norm |z| k,1 = |z 1 | 2 + • • • + |z k | 2 . Let Z be an (R d ) k -valued random variable such that E(|Z| k,1 ) < ∞. Let Λ 1 ((R d ) k ) be the space of 1-Lipschitz function from ((R d ) k , | • | k,1 )
to R. The τ -mixing coefficient between a σ-algebra M and the random variable Z is defined by

τ (M, Z) = sup (R d ) k f (x)P Z|M (dx) - (R d ) k f (x)P Z (dx) , f ∈ Λ 1 ((R d ) k ) 1 .
Let now (X i ) i∈Z be a sequence of R d -valued random variables, and let

F k = σ(X i : i ≤ k). Define then τ (n) = sup ≥1 1 sup k+n≤i 1 <•••<i τ (σ(X i : i ≤ k), (X i 1 , . . . , X i )) .
As stated in [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF], the τ -mixing coefficient is a coupling coefficient for the Kantorovich distance. This property has two consequences : one can use the coupling properties to demonstrate very precise deviation inequalities for partial sums, and one can give a bound on these coefficients for many classes of non-mixing processes in the sense of Rosenblatt (functions of i.i.d. sequences, non-irreducible Markov chains, dynamical systems... see again [START_REF] Dedecker | New dependence coefficients. examples and applications to statistics[END_REF]). As a recent exemple, let us mention that this coefficient can be computed for dynamical systems that can be modelled by Young Towers; for instance, if the return time to the base of the tower has an exponential moment, then the τ -mixing coefficient decreases at an exponential rate (see [START_REF] Cuny | Deviation and concentration inequalities for dynamical systems with subexponential decay of correlation[END_REF]).

For τ -mixing sequences whose coefficients decrease at an exponential rate, we consider the estimator f n defined in (2.1) with a slight modification in the power of the logarithm in the threshold, namely we take κ n = 1 + κ log n. Theorem 3.2. Assume that τ (n) ≤ Ka n for some K ≥ 1 and a ∈ (0, 1). There exist constants K 1 , K 2 , K 3 depending on (K, a, d) and K 4 , b ∈ (a, 1) depending on (K, a, d, κ) such that (for n ≥ 2)

E[ f n -f 2 ] ≤ inf m∈[0,n] d ,A∈A K 1 log n f -f A,m 2 + K 2 log n + 2(1 + (κ + √ K 3 ) log n) 2 |det(A)| π d n m 1 • • • m d + 2 7 π d n d-κ 2 K 3 + K 4 b n .
Explicit upper bounds for the constants K 1 , K 2 , K 3 , K 4 and b can be computed from the proof of Theorem 3.2. We infer from Theorem 3.2 that the resulting rates of convergence of f n on the Sobolev class S(A, s, L) defined by (2.4) are minimax optimal up to a (log n) 2 term. Remark 3.2. In case of subexponential decay, that is τ (n) ≤ Ka n γ for some K ≥ 1, a ∈ (0, 1) and γ ∈ (0, 1), one can take κ n = 1 + κ(log n) (1+γ)/2γ . This lead to the upper bound

E[ f n -f 2 ] ≤ inf m∈[0,n] d ,A∈A K 1 (log n) 1 γ f -f A,m 2 + K 2 (log n) 1 γ + 2(1 + (κ + √ K 3 )(log n) (1+γ)/2γ ) 2 |det(A)| π d n m 1 • • • m d + 2 7 π d n d-κ 2 K 3 + K 4 b n γ
for some b ∈ (a, 1). We refer to [START_REF] Cuny | Deviation and concentration inequalities for dynamical systems with subexponential decay of correlation[END_REF] for examples of sequences for which τ (n) decreases at a subexponential rate.

To conclude this section, note that, in the case d = 1, Lerasle [START_REF] Lerasle | Adaptive density estimation of stationary β-mixing and τ -mixing processes[END_REF] gave a model selection result for a wavelet type estimator under the condition τ (k) = O(k -a ) for a > 6 (see also his Remark 4.4 in case a > 4).

Numerical study in the dependent case

α-mixing sequences We illustrate our adaptive procedure in dimension 1. We expect that our empirical procedure to select κ will adapt to the unknown value of C α .

To simulate α-mixing sequences, we use the Markov chain introduced by Doukhan et al. (1994) [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF]. The interest of this chain is first that it is easy to simulate (as many Markov chains for which one can exhibit explicitly the iterated random system), and secondly we can compute exactly its rate of mixing from the parameters of the transition kernel. Since the chain is irreducible, positively recurrent ans aperiodic, it is also β-mixing in the sense of Volkonskii and Rozanov (1959) [START_REF] Volkonskiȋ | Some limit theorems for random functions[END_REF]. Recall that, for such Markov chains, β-mixing is equivalent to α-mixing.

Let us now describe this Markov chain in more details. Let a > 1 (as before a will calibrate the rate of mixing), and let µ and ν be the two probability measures on [0, 1] with respective densities f µ (x) = ax a-1 1 0≤x≤1 and f ν (x) = (a + 1)x a 1 0≤x≤1 .

Let F ν be the cumulative distribution function of ν, and let Y 1 be a random variable with law µ. Let (ε j ) j≥1 = ((U j , V j )) j≥2 be a sequence of i.i.d. random variables with uniform law over [0, 1] 2 and independent of Y 1 . For j ≥ 1 define then

Y j+1 = Y j 1 U j+1 ≥Y j + F -1 ν (V j+1 )1 U j+1 <Y j .
It is proved in [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF] that this chain is strictly stationary, with invariant distribution µ and that the β-mixing coefficients of this chain are exactly of order n -a . Note also that Y a j is uniformly distributed over [0, 1]. To derive from this chain a sequence with invertible cumulative distribution function F , we take X j = F -1 (Y a j ). The sequence (X j ) j≥1 is also a stationary Markov chain (as an invertible function of a stationary Markov chain), and its β-mixing coefficients are such that: there exist two positive constants B > A > 0 such that, for any n ≥ 1,

A n a ≤ β(n) ≤ B n a
(and the same is true for the coefficients α(n) for different constants A, B).

As in Section 2.4 we evaluate the normalized L 2 -risks from 500 independent datasets, for the sample sizes n = 500, 2000 and 5000 and different mixing coefficients a ∈ {3, 6, 10}. Note that a > 3 is the minimal value for which we proved that our estimator is consistent (without being necessarily optimal). It is then interesting to see if, numerically, our procedure still works at the boundary a = 3. If a > 5, Theorem 3.1 shows that our estimator reaches the minimax rate of the i.i.d. case, up to a logarithmic term. We consider two distributions Gamma Γ(3, 2) and the mixture 0.7N (3, 2) + 0.3N (8, 1). On Figures 3 and4, we observe the behavior of 50 adaptive estimators around the true density distribution. We observe that even for values of a smaller than 5 the estimator proposed is relevant and as expected we observe an improvement of the results when a and/or n increase. The risks empirically estimated in Tables 2 and 3 confirm these observations. It is interesting to notice that for fixed values of a the selected values for κ are similar, even when n increases, and that numerically we can proceed without knowing the constant C α .

τ -mixing sequence We consider a stationary Markov chain (X j ) simulated according to the following auto-regressive mechanism: where (ε j ) j≥1 are i.i.d., with common distribution B 1 2 , and independent of X 0 . The X i 's are uniformly distributed over [0, 1]. To generate another stationary Markov chain with invariant cumulative distribution function F , we apply F -1 to this sequence. It is well known that the Markov chain (X j ) j≥0 is not α-mixing in the sense of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF] (see for instance [START_REF] Andrews | Non-strong mixing autoregressive processes[END_REF]). However, one can prove that other dependence coefficients, such as the τ -dependence coefficients defined in Definition 3.2, converge to 0 at an exponential rate (an easy argument shows that it is still the case for the sequence F -1 (X i ) for quantile functions F -1 that are Lipschitz on any interval I ε = [ε, 1 -ε], ε > 0, provided the Lipschitz constant on I ε does not grows faster than Cε -γ for some C > 0, γ > 0). (5.40 × 10 -3 ) (1.39 × 10 -3 ) (0.67×10 -3 ) κ n 0.12 0.12 0.13 (0.06) (0.05) (0.05)

X 0 ∼ U [0, 1] X j = 1 2 (X j-1 + ε j ) , j ≥ 1,
n = 500 n = 2000 n = 5000

Proofs

4.1 Proof of Theorem 2.1

Preliminaries

We start with two technical lemmas, an Hoeffding inequality for complex valued random variable and a lemma that enables to control variance terms in the proof of Theorem 2.1.

Lemma 4.1. Let b > 0. For all u ∈ R d , d ≥ 1, the following inequality holds P | ϕ X,n (u) -ϕ X (u)| ≥ b √ log n √ n ≤ 4n -b 2 4 .
Proof of Lemma 4.1. We apply the Hoeffding inequality to the centered variable

ϕ X,n (u) -ϕ X (u) = 1 n n j=1 (cos( u, X j ) -Re(ϕ X (u))) + i 1 n n j=1 (sin( u, X j ) -Im(ϕ X (u))) .
By standard arguments, we first note that, for two real-valued random variables A, B and x > 0,

P A 2 + B 2 ≥ x ≤ P(|A| ≥ x/ √ 2) + P(|B| ≥ x/ √ 2) . (4.1)
Using (4.1), we get

P | ϕ X,n (u) -ϕ X (u)| ≥ b √ log n √ n ≤P   1 n n j=1 (cos( u, X j ) -Re(ϕ X (u))) ≥ b √ log n √ 2n   + P   1 n n j=1 (sin( u, X j ) -Im(ϕ X (u))) ≥ b √ log n √ 2n   ≤4 exp - b 2 log n 4 . Lemma 4.2. Let m = (m 1 , . . . , m d ) ∈ (0, n] d .
For any κ > 0 and any A ∈ A, the following inequality holds

1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ 5 + (1 + (κ + 2) √ log n) 2 |det(A)| π d n m 1 • • • m d . (4.2)
Proof of Lemma 4.2. First note that

E | ϕ X,n (u) -ϕ X (u)| 2 = E | ϕ X,n (u) -ϕ X (u)| 2 1 | ϕ X,n (u)|≥ κn √ n + |ϕ X (u)| 2 P | ϕ X,n (u)| < κ n √ n . (4.
3) The first term in the right hand side is bounded by 1 n . Recall that κ n = 1 + κ √ log n, we decompose the second term on the set {u, |ϕ

X (u)| < 1+(κ+2) √ log n √ n
} and its complementary where we derive from the triangle inequality that

1 | ϕ X,n (u)|< 1+κ √ log n √ n 1 |ϕ X (u)|> 1+(κ+2) √ log n √ n ≤ 1 | ϕ X,n (u)-ϕ X (u)|≥ 2 √ log n √ n . This leads to |ϕ X (u)| 2 P | ϕ X,n (u)| < κ n √ n ≤ (1 + (κ + 2) √ log n) 2 n + P | ϕ X,n (u) -ϕ X (u)| ≥ 2 √ log n √ n ≤ (1 + (κ + 2) √ log n) 2 n + 4 n ≤ 4 + (1 + (κ + 2) √ log n) 2 n , (4.4)
where we used Lemma 4.1 with b = 2. To conclude the proof, it suffices to note that

1 A([-m,m]) (u)du = |det(A)|m 1 • • • m d . (4.5)

Proof of Theorem 2.1

Let m = (m 1 , . . . , m d ) ∈ (0, n] d . We first note that

E[ f n -f 2 ] ≤ 2 f A,m -f 2 +2E[ f n -f A,m 2 ] = 2 (2π) d A([-m,m]) c |ϕ X (u)| 2 du+2E[ f n -f A,m 2 ].
The first term is a bias term and we treat the second variance term using the Parseval equality to get that

2E[ f n -f A,m 2 ] = 2 (2π) d R d E | ϕ X (u)1 [-n,n] d (u) -ϕ X (u)1 A([-m,m]) (u)| 2 du = 2 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du + 2 (2π) d [-n,n] d \A([-m,m]) E | ϕ X,n (u)| 2 du. (4.6)
The first term is a variance term which is controlled with (4.2) for κ > 0. In the sequel we focus on the second term. To that end notice that

| ϕ X,n | 2 ≤ 2|ϕ X | 2 + 2| ϕ X,n -ϕ X | 2 .
From which we get

2 (2π) d [-n,n] d \A([-m,m]) E | ϕ X,n (u)| 2 du ≤ 4 (2π) d [-n,n] d \A([-m,m]) |ϕ X (u)| 2 du + 4 (2π) d [-n,n] d \A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ 4 (2π) d A([-m,m]) c |ϕ X (u)| 2 du + 4 (2π) d [-n,n] d \A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du.
The first term is a bias term, and we treat the last one by considering the set {u, |ϕ X (u)| > n -1/2 } and its complementary. On the set {u, |ϕ X (u)| > n -1/2 }, we derive that

E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + 1 n ≤ 2|ϕ X (u)| 2 . ( 4.7)
Consequently, we recover a bias term since

4 (2π) d [-n,n] d \A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 1 {|ϕ X (u)|>n -1/2 } du ≤ 8 (2π) d A([-m,m]) c |ϕ X (u)| 2 du.
Next on the complementary, using that | ϕ X,n (u) -ϕ X (u)| 2 ≤ 4 and the definition of ϕ X,n , we derive that

4 (2π) d E [-n,n] d \A([-m,m]) | ϕ X,n (u) -ϕ X (u)| 2 1 {|ϕ X (u)|≤n -1/2 } du ≤ 4 (2π) d [-n,n] d \A([-m,m]) |ϕ X (u)| 2 du+ 16 (2π) d [-n,n] d \A([-m,m]) P(| ϕ X,n (u)| ≥ κ n n -1/2 )1 {|ϕ X (u)|≤n -1/2 } du ≤ 4 (2π) d A([-m,m]) c |ϕ X (u)| 2 du + 16 (2π) d [-n,n] d P(| ϕ X,n (u) -ϕ X (u)| > κ(log n/n) 1/2 )du ≤ 4 (2π) d A([-m,m]) c |ϕ X (u)| 2 du + 64 π d n d-κ 2 /4 . (4.8)
The last inequality is a direct consequence of Lemma 4.1 with b = κ. Gathering all terms it follows that for all m ∈ (0, n] d ,

E[ f n -f 2 ] ≤ 18 f -f A,m 2 + 10 + 2(1 + (κ + 2) √ log n) 2 |det(A)| π d n m 1 • • • m d + 64 π d n d-κ 2 /4
. Taking the infimum over m completes the proof.

Proof of Equation (2.3)

Firstly, note that an inductive argument allows to show that the density of V d = U 1 . . . U d , d ≥ 1, where (U j ) 1≤j≤d are i.i.d. U([0, 1]) random variables, is given by

g d (z) = (-1) d (log z) d-1 (d -1)! 1 [0,1] (z).
Secondly, we compute the quantity in (2.3) using the latter result and a change of variables

Dn du = [-n,n] d 1 |u 1 ...u d |≤n du 1 . . . du d = 2 d [0,n] d 1 |u 1 ...u d |≤n du 1 . . . du d = 2 d n d [0,1] d 1 |u 1 ...u d |≤n 1-d du 1 . . . du d = (2n) d P(V d ≤ n 1-d ) = (2n) d n 1-d 0 (-1) d (log z) d-1 (d -1)! 1 [0,1] (z)dz = (2n) d (d -1)! ∞ log(n d-1 ) z d-1 e -z dz = (2n) d (d -1)! Γ(d, log(n d-1 ))
where we made the change of variable z ← -log(z) and Γ(s, x) is the incomplete Gamma function. Using the asymptotic expansion Γ(s, x) , define X = (bX 1 , aX 1 + bX 2 ) where X 1 and X 2 are independent, X 1 ∼ Γ(α + 1 2 , 1) and X 2 ∼ Γ(β + 1 2 , 1), where 0 < β < α. Denote by f the density of X; it is straightforward to check that

∼ x→∞ x s-1 e -x we get Dn du ∼ n→∞ (2n) d (d -1)! (log(n d-1 )) d-1 e -log(n d-1 ) = 2 d (d -1) d-1 (d -1)! log(n) d-1 n.
|Ff (u)| 2 = 1 (1 + |bu 1 + au 2 | 2 ) α+ 1 2 1 (1 + |bu 2 | 2 ) β+ 1 2 .
We can give a lower bound of the bias term, let m 1 ≥ 1 and m 2 ≥ 1,

B f (m 1 , m 2 ) ≥ |Ff (u)| 2 1 u 2 >m 2 du 1 du 2 + |Ff (u)| 2 1 u 2 <-m 1 1 u 1 >m 1 du 1 du 2 ≥ c α ∞ m 2 1 (1 + b 2 u 2 2 ) β+ 1 2 du 2 + |Ff (u)| 2 1 u 2 <-m 1 1 u 1 >m 1 du 1 du 2 ≥ C 1 m 2β 2 + |Ff (u)| 2 1 u 2 <-m 1 1 u 1 >m 1 du 1 du 2
where we used that (1 + b 2 u 2 2 ) ≤ (1 + bu 2 ) 2 and where we defined c α = R (1 + z 2 ) -(α+ 1 2 ) dz and

C 1 = c α (2βb(1 + b) 2β ) if m 2 ≥ 1. Now for the second term, |Ff (u)| 2 1 u 2 <-m 1 1 u 1 >m 1 du 1 du 2 = -m 1 -∞ 1 (1 + |bu 2 | 2 ) β+ 1 2 1 (1 + |bu 1 + au 2 | 2 ) α+ 1 2 1 u 1 >m 1 du 1 du 2 = -m 1 -∞ 1 (1 + |bu 2 | 2 ) β+ 1 2 1 (1 + z 2 ) α+ 1 2 1 z>am 1 +by dz du 2 ≥ -m 1 -∞ 1 (1 + |bu 2 | 2 ) β+ 1 2 1 (1 + z 2 ) α+ 1 2 1 z>0 dz du 2 = c α 2 -m 1 -∞ 1 (1 + |bu 2 | 2 ) β+ 1 2 du 2 ≥ C 1 2m 2β 1
where we used that am 1 + by < 0 for all y < -m 1 as a -b < -b 2 < 0 and m 1 ≥ 1. Gathering all terms we derive that

B f (m 1 , m 2 ) ≥ c α 2bβ(1 + b) 2β 1 m 2β 2 + 1 2m 2β 1 . (4.9)
Now consider the matrix A a,b which is such that

|Ff (A a,b u)| 2 = 1 (1 + |u 1 | 2 ) α+ 1 2 1 (1 + |u 2 | 2 ) β+ 1 2 .
The associated bias term can be bounded as follows

B f A a,b (m 1 , m 2 ) ≤ |Ff (A a,b u)| 2 1 |u 1 |>m 1 du 1 du 2 + |Ff (A a,b u)| 2 1 |u 2 |>m 2 du 1 du 2 = c β ∞ m 1 1 (1 + u 2 1 ) α+ 1 2 du 1 + c α ∞ m 2 1 (1 + u 2 2 ) β+ 1 2 du 2 ≤ c β ∞ m 1 1 u 2α+1 1 du 1 + c α ∞ m 2 1 u 2β+1 2 du 2 = c β 2α 1 m 2α 1 + c α 2β 1 m 2β 2 , (4.10)
where

c β = R (1 + z 2 ) -(β+ 1
2 ) dz. Gathering (4.9) and (4.10) we get for α > β the announced result (2.6)

if m 1 ≥ 1 ∨ C 1 α,β,b , m 2 ≥ 1 ∨ C 2 α,
E | ϕ X,n (u) -ϕ X (u)| 2 ≤ (2C α,n -1) n .
Proof of Lemma 4.3. Note first that

Var(e i u,X 1 ) = E(e i u,X 1 e -i u,X 1 ) -E(e i u,X 1 )E(e -i u,X 1 ) = 1 -|ϕ X (u)| 2 .

Using the stationarity of the sequence (X i ), we get for the variance

E | ϕ X,n (u) -ϕ X (u)| 2 ≤ 1 -|ϕ X (u)| 2 n + 2 n n k=2
|Cov (e i u,X 1 , e i u,X k )|. (4.11) We now apply Ibragimov's inequality [START_REF] Ibragimov | Some limit theorems for stationary processes[END_REF] : Let X, Y be two real-valued random variables that are almost surely bounded, and denote by X ∞ the essential supremum norm of |X|. Then, it holds

|Cov (X, Y )| ≤ 2α(σ(X), σ(Y )) X ∞ Y ∞ . (4.12)
For complex-valued random variables, whose modulus are almost surely bounded, one can easily prove that (see for instance [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF], Chapter 1, Exercice 7)

|Cov (X, Y )| ≤ 4α(σ(X), σ(Y )) X ∞ Y ∞ . (4.13)
Using the inequality (4.13) and that |e i u,X k | < 1, we get

n k=2 |Cov (e i u,X 1 , e i u,X k )| ≤ 4 n k=2 α(k -1).
It follows from (4.11) that,

E | ϕ X,n (u) -ϕ X (u)| 2 ≤ 1 n + 8 n n-1 k=1 α(k) = (2C α,n -1) n ,
and Lemma 4.3 is proved.

Our second lemma is a Fuk-Nagaev inequality stated by Rio [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF]. We need a version of this inequality which was proposed as an exercise in [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF] (Exercise 1, Chapter 6). Lemma 4.4. Let (Y i ) i>0 be a sequence of real-valued random variables such that Y i ∞ ≤ 1/2, and let (α(n)) n≥0 be the sequence of strong mixing coefficients of Definition 3.

1. Let S k = k i=1 (Y i -E(Y i )) and s 2 n be such that s 2 n ≥ n i=1 n j=1 |Cov (Y i , Y j )|. ( 4

.14)

Then for any λ ≥ s n , it holds

P sup k∈[1,n] |S k | ≥ 4λ ≤ 4 exp - λ 2 log 2 2s 2 n + 4nλ -1 α s 2 n λ .
Proof of Lemma 4.4. We start from Theorem 6.2 Inequality (6.4) in [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF], we take r = λ 2

s 2 n and Q = 1 (since Y i -E(Y i ) ∞ ≤ 1). Using Rio's notations let H λ r = α λ r = α s 2 n λ .
We obtain that

P sup k∈[1,n] |S k | ≥ 4λ ≤ 4 1 + λ 2 rs 2 n -r 2 + 4nλ -1 H(λ/r) 0 Q(u) du ≤ 4 1 + λ 2 λ 2 s 2 n s 2 n -λ 2 2s 2 n + 4nλ -1 α([s 2 n /λ]) 0 1 du ≤ 4 × 2 -λ 2 2s 2 n + 4nλ -1 α s 2 n λ ≤ 4 exp - λ 2 log 2 2s 2 n + 4nλ -1 α s 2 n λ .
and Lemma 4.4 is proved.

We apply Lemma 4.4 to | ϕ X,n (u) -ϕ X (u)|.

Lemma 4.5. Recall that C α,n = 1 + 4 n-1 k=1 α(k). For any b > 0, it holds

P | ϕ X,n (u) -ϕ X (u)| ≥ b √ log n √ n ≤ 8n -b 2 /93Cα,n + 64 √ 2n b √ log n α 2 √ 2nC α,n b √ log n .
Proof of Lemma 4.5. Using (4.1), we infer that

P (| ϕ X,n (u) -ϕ X (u)| > x) ≤ P   1 n n j=1 (cos( u, X j ) -Re(ϕ X (u))) ≥ x √ 2   + P   1 n n j=1 (sin( u, X j ) -Im(ϕ X (u))) ≥ x √ 2   . Taking λ = b √ n log n/8 √ 2, one has P | ϕ X,n (u) -ϕ X (u)| > b √ log n √ n ≤ P   1 2 n j=1 (cos( u, X j ) -Re(ϕ X (u))) ≥ 4λ   + P   1 2 n j=1 (sin( u, X j ) -Im(ϕ X (u))) ≥ 4λ   .
We can now apply Lemma 4.4 to the variables Y j = cos( u, X j )/2 (or Y j = sin( u, X j )/2), whose absolute values are bounded by 1/2. Note that, by (4.12), one has

n i=1 n j=1 |Cov (Y i , Y j )| ≤ n 4 1 + 4 n-1 k=1 α(k) = n 4 C α,n .
Hence, one can take s 2 n = nC α,n /4 in (4.14). It follows that

P | ϕ X,n (u) -ϕ X (u)| > b √ log n √ n ≤ 8 exp - b 2 log n 93C α,n + 64 √ 2n b √ log n α 2 √ 2nC α,n b √ log n ,
proving the lemma (the constant 93 comes from 64/ log 2 < 93).

Using Lemmas 4.3 and 4.5, we can now prove the last lemma of this subsection:

Lemma 4.6. Let m = (m 1 , . . . , m d ) ∈ (0, n] d . For any κ > 0 and any A ∈ A, the following inequality holds

1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d (2C α,n + 7) + (1 + (κ + 93C α,n ) √ log n) 2 |det(A)| π d n + 64 √ 2n (2d+1)/2 π d 93C α,n √ log n α 2 2nC α,n √ 93 √ log n . Proof of Lemma 4.6. Considering the set {u, |ϕ X (u)| < 1+(κ+ √ 93Cα,n) √ log n √ n
}, proceeding as for (4.3), (4.4), and using Lemma 4.3 and (4.5), we get

1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d 2C α,n -1) + (1 + (κ + 93C α,n ) √ log n) 2 |det(A)| π d n + 1 (2π) d A([-m,m]) P | ϕ X,n (u) -ϕ X (u)| ≥ 93C α,n √ log n √ n du .
This together with Lemma 4.5 (with b = 93C α,n ) yield

1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d (2C α,n + 7) + (1 + (κ + 93C α,n ) √ log n) 2 |det(A)| π d n + m 1 • • • m d |det(A)| 64 √ 2n π d 93C α,n √ log n α 2
(u)| > n -1/2 }, E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + (2C α,n -1) n ≤ 2C α,n |ϕ X (u)| 2 .
The third change concerns Inequality (4.8), which becomes (applying Lemma 4.5 with b = κ)

4 (2π) d E [-n,n] d \A([-m,m]) | ϕ X,n (u) -ϕ X (u)| 2 1 {|ϕ X (u)|≤n -1/2 } du ≤ 4 (2π) d u∈A([m,m]) c |ϕ X (u)| 2 du + 2 7 n d π d n -κ 2 /93Cα,n + 2 10 √ 2n (2d+1)/2 π d κ √ log n α 2 √ 2nC α,n κ √ log n .
4.6 Proof of Theorem 3.2

Preliminaries

We state the modifications of Lemmas 4.3-4.5 for τ -mixing sequences whose coefficients decrease at an exponential rate.

Lemma 4.7. Assume that τ (n) ≤ Ka n for some K ≥ 1 and a ∈ (0, 1). The following inequality holds

E | ϕ X,n (u) -ϕ X (u)| 2 ≤ C 1 (a, K) n + 2 log(n) n log(1/a) , with C 1 (a, K) = 1 + 2 1 -a + 2 log(2K √ d) log(1/a)
.

Proof of Lemma 4. Using the inequality (4.15) and that |e i u,X k | < 1, we get

n k=2
|Cov (e i u,X Proof of Lemma 4.8. The proof is the same as that of Lemma 4.4 by using Theorem 2 in [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] instead of Theorem 6.2 in [START_REF] Rio | Asymptotic Theory of Weakly Dependent Random Processes[END_REF].

We now apply Lemma 4.8 to | ϕ X,n (u) -ϕ X (u)|.

Lemma 4.9. Assume that τ (n) ≤ Ka n for some K ≥ 1 and a ∈ (0, 1). There exists a positive constant C 2 (a, K) such that, for any b > 0, proving the lemma (the constant 578 comes from 400/ log 2 < 578).

P | ϕ X,n (u) -ϕ X (u)| > b log n √ n ≤ 8
Using Lemmas 4.7 and 4.9, we can now prove the last lemma of this subsection.

Figure 1 :

 1 Figure 1: Representation of the set {| ϕX,n| ≥ κnn -1/2 } on a grid of R 2 for the GB example for Σ 2 defined in Section 2.4, n = 10 4 , κ = 0.1 (left) and κ = 1 (center). Yellow pixels corresponds to areas where {| ϕX,n| ≥ κnn -1/2 }. The last image (right) represents κ → χ({| ϕX,n| ≥ κnn -1/2 }).

Figure 2 :

 2 Figure 2: Representation for n = 10 4 of the estimated Mix-NN (left), the true density (center) and the difference of both estimated and true density (right).

Figure 3 :

 3 Figure 3: Estimation of f ∼ Γ(3, 2) (red) and the adaptive estimator (green) from 50 Monte Carlo iterations for different values of a and n.

Figure 4 :

 4 Figure 4: Estimation of f ∼ 0.7N (3, 2) + 0.3N (8, 1) (red) and the adaptive estimator (green) from 50 Monte Carlo iterations for different values of a and n.

Figure 5 :

 5 Figure 5: Estimation of f ∼ Γ(3, 2) (red) and the adaptive estimator (green) from 50 Monte Carlo iterations for different values of n.

4. 4

 4 Proof of Equation (2.6) in Example 2.1 Let d = 2, b > 1 and 0 < a < b(1 -b)

  β,b which are positive constants C j α,β,b , j = 1, 2 ensuring that the right hand side of (4.9) is larger than (4.10).

4. 5

 5 Proof of Theorem 3.

1

 1

4. 5 . 1

 51 PreliminariesFirst, we prove several lemmas that are useful for the proof of Theorem 3.1.

Lemma 4 . 3 .

 43 Let C α,n = 1 + 4 n-1 k=1 α(k).The following inequality holds

7 .

 7 We start from(4.11) and use the following covariance inequalities. Let X, Y be two real-valued random variables (with X almost surely bounded) and denote by X ∞ the essential supremum norm of |X| . Then|Cov (X, Y )| ≤ E(|X(E(Y |X) -E(Y ))|) ≤ X ∞ τ (σ(X), Y ) .Similarly, for two complex-valued random variables X, Y (withY = Y 1 + iY 2 and X ∞ < ∞), one has |Cov (X, Y )| ≤ X ∞ (τ (σ(X), Y 1 ) + τ (σ(X), Y 2 )) .(4.15)Now, for the variableY k = cos( u, X k ) (or Y k = sin( u, X k )), we easily see that (for |u| ≤ n) τ (σ(X 1 ), Y k ) ≤ |u| 2 τ (k -1) ≤ √ dnτ (k -1) .

≤ C 2

 2 exp -b 2 log n 578C 2 (a, K) + n 3/2 80 √ 2dK b log n a [C 2 (a,K)10 √ 2n/b] .Proof of Lemma 4.9. Arguing as in Lemma 4.5 and taking λ = b √ n log n/10 √ 2, one hasP | ϕ X,n (u) -ϕ X (u)u, X j ) -Re(ϕ X (u))) ≥ 5λ u, X j ) -Im(ϕ X (u))) ≥ 5λ   .We can now apply Lemma 4.4 to the variables Y j = cos( u, X j )/2 (or Y j = sin( u, X j )/2), whose absolute values are bounded by 1/2. Proceeding as in the proof of Lemma 4.7, we get (a, K)n log n .Hence, one can take s 2 n = C 2 (a, K)n log n in (4.14). It follows thatP | ϕ X,n (u) -ϕ X (u)

Lemma 4 . 10 . 10 ., 10 √

 4101010 Let m = (m 1 , . . . , m d ) ∈ (0, n] d , and assume that τ (n) ≤ Ka n for some K ≥ 1 and a ∈ (0, 1). For any κ > 0 and any A ∈ A, the following inequality holds1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d 8 + C 1 (a, K) + 2 log n/ log(1/a) + (1 + (κ + 578C 2 (a, K)) log n) 2 |det(A)| π d n + n (2d+3)/2 80 √ dK π d 289C 2 (a, K) log n a [ Considering the set {u, |ϕ X (u)| < 1+(κ+ √ 578C 2 (a,K)) log n √ n}, proceeding as for (4.3), (4.4), and using Lemma 4.7, we get1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d C 1 (a, K) + 2 log n/ log(1/a) + (1 + (κ + 578C 2 (a, K)) log n) 2 |det(A)| π d n + 1 (2π) d A([-m,m]) P | ϕ X,n (u) -ϕ X (u)| ≥ 578C 2 (a, K) log n √ n du .This together with Lemma 4.9 (with b = 578C 2 (a, K)) yield1 (2π) d A([-m,m]) E | ϕ X,n (u) -ϕ X (u)| 2 du ≤ m 1 • • • m d 8 + C 1 (a, K) + 2 log n/ log(1/a) + (1 + (κ + 578C 2 (a, K)) log n) 2 |det(A)| π d n + m 1 • • • m d |det(A)| n 3/2 80 √ dK π d 289C 2 (a, K) log n a [and Lemma 4.10 is proved sincem 1 • • • m d |det(A)| ≤ n d .4.6.2 End of the proof of Theorem 3.2We follow exactly the proof of Theorem 2.1 (withκ n = 1 + κ log n),we only indicate the changes resulting from the application of Lemma 4.9 and Lemma 4.10 instead (respectively) of Lemma 4.1 and Lemma 4.2. The first change is that the first term in (4.6) is now controlled via Lemma 4.10 instead of Lemma 4.2. The second change concerns Inequality (4.7), which becomes (applying Lemma 4.7) : on the set {u, |ϕ X(u)| > n -1/2 }, E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + E[| ϕ X,n (u) -ϕ X (u)| 2 ] ≤ |ϕ X (u)| 2 + C 1 (a, K) n + 2 log(n) n log(1/a) ≤ 1 + C 1 (a, K) + 2 log(n) log(1/a) |ϕ X (u)| 2 .The third change concerns Inequality (4.8), which becomes (applying Lemma 4.9 with b = κ)4 (2π) d E [-n,n] d \A([-m,m]) | ϕ X,n (u) -ϕ X (u)|2 1 {|ϕ X (u)|≤n -1/2 } du ≤ 4 (2π) d u∈A([m,m]) c |ϕ X (u)| 2 du + 2 7 n d π d n -κ 2 /578C 2 (a,K) + n (2d+3)/2 2 8 √ 2d5K π d κ log n a [C 2 (a,K)2n/κ] .

Table 1 :

 1 Evaluated empirical risks multiplied by 100 and adaptive κn, the associated squared variances are given in parenthesis.

	Case		N			Mix-NN		GB
	n	10 3	10 4	10 5	10 3	10 4	10 5	10 3	10 4	10 5
	100×Risk √ 100× V(Risk) κ n	1.04 0.88 (0.19) (0.15) 0.12 0.81	1.32 10 -2 0.79 (0.13)	3.02 1.06 (0.09) (0.09) 0.39 1.01	4.70 10 -2 0.94 (0.09)	5.96 1.09 (0.08) (0.09) (0.15) 1.65 0.27 1.01 1.00

(0.56) (0.05) (0.58 × 10 -2 ) (0.59) (0.07) (1.05 × 10 -2 ) (1.47) (0.24) (0.08)

Table 2 :

 2 Computations via 500 Monte Carlo iterations of the L 2 risks for the Gamma distribution Γ(3, 2).

Table 3 :

 3 Computations by 500 Monte Carlo iterations of the L 2 risks for the distribution 0.7N (3, 2)+0.3N (8, 1).

Table 4 :

 4 Computations via 500 Monte Carlo iterations of the L 2 risks for a Markov chain generated from the

non mixing auto-regressive model with f ∼ Γ(3, 2).

  We follow exactly the proof of Theorem 2.1 and only indicate the changes resulting from the application of Lemma 4.5 and Lemma 4.6 instead (respectively) of Lemma 4.1 and Lemma 4.2.

	√	2nC α,n 93 √ log n	,
	4.5.2 End of the proof of Theorem 3.1
	The first change is that the first term in (4.6) is now controlled via Lemma 4.6 instead of
	Lemma 4.2.
	The second change concerns Inequality (4.7), which becomes (applying Lemma (4.3)) : on
	the set {u, |ϕ X

and Lemma 4.6 is proved (noting that m 1 • • • m d |det(A)| ≤ n d for the last term).

  1 , e i u,X k )| ≤ 2 Our second lemma is a Fuk-Nagaev inequality. Lemma 4.8. Let (Y i ) i>0 be a sequence of real-valued random variables such that Y i ∞ ≤ 1/2 and let (τ Y (n)) n≥0 be the sequence of τ -mixing coefficients of Definition 3.2. Define S k = k i=1 (Y i -E(Y i )) and let s 2 n be as in(4.14). Then for any λ ≥ s n ,

		P sup k∈[1,n]	|S k | ≥ 5λ ≤ 4 exp -	λ 2 log 2 2s 2 n	+ 4nλ -1 τ Y	s 2 n λ	.
												n		√
													min(2	dnτ (k -1), 1).
												k=2
	It follows from (4.11) that,
	E | ϕ X,n (u) -ϕ X (u)| 2 ≤	1 n	+	2 n	∞ k=1	min(2 √	dnτ (k), 1) ≤	1 n	+	2 n	k=1 ∞	min(2 √	dnKa k , 1) .
	Now,	∞ k=1	min(2 √	dnKa k , 1) ≤	1 1 -a	+	log(2K log(1/a) √ d)	+	log(n) log(1/a)	,
	completing the proof of Lemma 4.7

Acknowledgments. The second author was partially supported by the french ANR project "Géométrie dans les données : inférence statistique & partitionnement -GeoDSIC"