Nathalie Akakpo
email: nathalie.akakpo@upmc.fr

Claire Lacour
email: claire.lacour@u-psud.fr

Inhomogeneous and Anisotropic Conditional Density Estimation from Dependent Data

Keywords: AMS 2000 subject classifications: 62G05, 62H12, 62M05, 62M09 Conditional density, Model selection, Anisotropy, Dependent Data, Adaptive estimation

The problem of estimating a conditional density is considered. Given a collection of partitions, we propose a procedure that selects from the data the best partition among that collection and then provides the best piecewise polynomial estimator built on that partition. The observations are not supposed to be independent but only β-mixing; in particular, our study includes the estimation of the transition density of a Markov chain. For a well-chosen collection of possibly irregular partitions, we obtain oracle-type inequalities and adaptivity results in the minimax sense over a wide range of possibly anisotropic and inhomogeneous Besov classes. We end with a short simulation study.

Introduction

In this paper, we are concerned with conditional density estimation. Such a model brings more information than the well-studied regression model; for instance, it may reveal multimodality. Yet, references about conditional density estimation are rather scarce, even for nonadaptive procedures. For independent data, we can cite for instance Györfi and Kohler [START_REF] Györfi | Nonparametric estimation of conditional imsart[END_REF] for a histogram based procedure, or Faugeras [START_REF] Faugeras | A product type non-parametric estimator of the conditional density by quantile transform and copula representation[END_REF] for a copula-based kernel estimator. For mixing data, De Gooijer and Zerom [START_REF] De Gooijer | On conditional density estimation[END_REF] and Fan and Yim [START_REF] Fan | A crossvalidation method for estimating conditional densities[END_REF] propose kernel methods. For Markov chains, nonadaptive estimation of the transition density is considered for instance in [START_REF] Roussas | Nonparametric estimation in markov processes[END_REF][START_REF] Birgé | Approximation dans les espaces metriques et theorie de l'estimation[END_REF][START_REF] Doukhan | Estimation de la transition de probabilité d'une chaîne de Markov Doëblin-récurrente. Étude du cas du processus autorégressif général d'ordre 1[END_REF], and we also refer to [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] for a more complete bibliography. But, in order to reach the optimal rate of convergence, those methods require the smoothness of the function to estimate to be known, so as to choose adequately some tuning parameter.

Adaptive estimators of the conditional density have only recently been proposed. For independent data, Efromovich [START_REF] Efromovich | Conditional density estimation in a regression setting[END_REF][START_REF] Efromovich | Oracle inequality for conditional density estimation and an actuarial example[END_REF] and Brunel, Comte and Lacour [START_REF] Brunel | Adaptive Estimation of the Conditional Density in Presence of Censoring[END_REF] give oracle inequalities and adaptivity results in the minimax sense. Efromovich [START_REF] Efromovich | Conditional density estimation in a regression setting[END_REF][START_REF] Efromovich | Oracle inequality for conditional density estimation and an actuarial example[END_REF] uses a Fourier decomposition to build a blockwise-shrinkage Efromovich-Pinsker estimator, whereas Brunel et al. [START_REF] Brunel | Adaptive Estimation of the Conditional Density in Presence of Censoring[END_REF] perform model selection based on a penalized least-squares criterion. Regarding dependent data, Clémençon [START_REF] Clémençon | Adaptive estimation of the transition density of a regular Markov chain[END_REF] and Lacour [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] study adaptive estimators of the conditional density for Markovian observations, the former via wavelet thresholding, and the latter via model selection. Besides, the procedures proposed by [START_REF] Efromovich | Conditional density estimation in a regression setting[END_REF][START_REF] Efromovich | Oracle inequality for conditional density estimation and an actuarial example[END_REF][START_REF] Brunel | Adaptive Estimation of the Conditional Density in Presence of Censoring[END_REF][START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] are all able to adapt to anisotropy; otherwise said, the conditional density to estimate is allowed to have unknown and different degrees of smoothness in each direction.

But the smoothness of the function to estimate may also vary spatially. If the risk of the estimator is measured via some L q -norm, one way to take into account that inhomogeneous behaviour is to consider functions whose smoothness is measured in a L p -norm, with p < q. Among the aforementioned references, only Clémençon [START_REF] Clémençon | Adaptive estimation of the transition density of a regular Markov chain[END_REF] is able to cope with inhomogeneous smoothness. In the imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 simpler framework of density estimation, without conditioning variables, adaptation to inhomogeneity has been studied in the following works. Thresholding methods, in a univariate framework, are proposed by Hall, Kerkyacharian and Picard [START_REF] Hall | Block threshold rules for curve estimation using kernel and wavelet methods[END_REF] for independent data, Clémençon [START_REF] Clémençon | Méthodes d'ondelettes pour la statistique non paramétrique des chaînes de Markov[END_REF] for Markovian data, and Gannaz and Wintenberger [START_REF] Gannaz | Adaptive density estimation under weak dependence[END_REF] for a wide class of weakly dependent data. Piecewise polynomial selection procedures based on a penalized contrast have also been considered, and consist in selecting from the data a best partition and a best piecewise polynomial built on that partition. Thus, Comte and Merlevède [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF] estimate the univariate density of absolutely regular stationary processes, in discrete or continuous time, selecting a best partition among the collection of all the partitions of [0, 1] built on a thin regular grid via a leastsquares criterion. Besides, three papers have lately considered density estimators inspired from the "multiresolution histogram" of Engel [START_REF] Engel | A simple wavelet approach to nonparametric regression from recursive partitioning schemes[END_REF][START_REF] Engel | The multiresolution histogram[END_REF], or the "dyadic CART procedure" of Donoho [START_REF] Donoho | CART and best-ortho-basis: a connection[END_REF]. Willett and Nowak [START_REF] Willett | Multiscale Poisson intensity and density estimation[END_REF] select best piecewise polynomials built on partitions into dyadic cubes via a penalized maximum likelihood contrast. Klemelä [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF] and Blanchard, Schäfer, Rozenholc and Müller [START_REF] Blanchard | Optimal dyadic decision trees[END_REF] select best histograms based on partitions into dyadic rectangles via a penalized criterion based on the L 2 -distance for the first one, and on Kullback-Leibler divergence for the second ones. But all these procedures only reach optimal rates of convergence up to a logarithmic factor, and only [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF] is able to prove adaptivity both to anisotropy and inhomogeneity.

In this paper, we provide an estimator of the conditional density via a piecewise polynomial selection procedure based on an adequate least-squares criterion. To deal with the possible dependence of the observations, we mainly use β-mixing coefficients and their coupling properties. Thus, our dependence assumptions, while being satisfied by a wide class of Markov chains, are not restricted to Markovian assumptions. We first prove nonasymptotic oracle type inequalities fulfilled by any collection of partitions satisfying some mild structural conditions. We then consider the collection of partitions into dyadic rectangles, as [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF] or [START_REF] Blanchard | Optimal dyadic decision trees[END_REF]. We obtain oracle-type inequalities and adaptivity results in the minimax sense, without logarithmic factor, over a wide range of Besov smoothness classes that may contain functions with inhomogeneous and anisotropic smoothness, whether the data are independent or satisfy suitable dependence assumptions. The adaptivity of our procedure greatly relies on the approximation result proved in [START_REF] Akakpo | From adaptive approximation to adaptive estimation for functions with inhomogeneous and anisotropic smoothness[END_REF]. Moreover, determining in practice the penalized estimator based on that collection only requires a computational complexity linear in the size of the sample.

This paper is organized as follows. We begin by describing the framework and the estimation procedure, and we present an evaluation of the risk on one model. This study allows to understand what bound for the L 2 -risk we seek to obtain. The choice of a penalty yielding an oracle-type inequality is the topic of Section 3.1. Section 3.2 is devoted to the collection of partitions into dyadic rectangles, and adaptivity results are proved for an adequate penalty. We show in Section 4 that all these results can be extended to dependent data. In Section 5, the practical implementation of our estimator is explained and some simulations are presented, both for independent and dependent data. Most imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 proofs are deferred to Section 6.

Framework and estimation procedure

In this section we define a contrast and we deduce a collection of estimators ŝm . In order to understand which model m we should choose, we give an evaluation of the risk for each estimator ŝm . This allows us to define the penalized estimator.

Framework and notation

Let {Z i } i∈Z = {(X i , Y i )} i∈Z be a strictly stationary process, where, for all i ∈ Z, X i and Y i take values respectively in [0, 1] d1 and [0, 1] d2 , with d 1 and d 2 positive integers. We assume that the variables (X i ) i∈Z admit a bounded marginal density f with respect to the Lebesgue measure. Given some integer n ≥ 2, our aim is to estimate, on the basis of the observation of (Z 1 , . . . , Z n ), the marginal density s of Y i conditionally to X i . Thus, our parameter of interest s is the realvalued function of d variables, where

d = d 1 + d 2 , such that, for all x ∈ [0, 1] d1 , s(x, .) : [0, 1] d2 → R is the density of Y i conditionally to X i = x.
In particular, if (X i ) i∈Z is a homogeneous Markov chain of order 1, and Y i = X i+1 for all i ∈ Z, then s is the transition density of the chain (X i ) i∈Z .

Let us introduce some standard notation. For any real-valued function t defined and bounded on some set D, we set

ι(t) = inf x∈D |t(x)| and t ∞ = sup x∈D |t(x)|.
We denote by L 2 [0, 1] d1 × [0, 1] d2 the set of all real-valued functions which are square integrable with respect to the Lebesgue measure. Since f is bounded, we can also define on L 2 [0, 1] d1 × [0, 1] d2 the semi-scalar product

t, u f = [0,1] d 1 ×[0,1] d 2 t(x, y)u(x, y)f (x)dxdy
and the associated semi-norm . f .

Contrast and estimator on one model

In order to estimate the conditional density s, we consider the empirical criterion γ described in [START_REF] Brunel | Adaptive Estimation of the Conditional Density in Presence of Censoring[END_REF] and defined on

L 2 [0, 1] d1 × [0, 1] d2 by γ(t) = 1 n n i=1 [0,1] d 2 t 2 (X i , y)dy -2t(X i , Y i ) .
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Due to the nature of the function to estimate, the contrast used here borrows both from the classical regression and density least-squares contrasts. This contrast verifies:

E s [γ(t) -γ(s)] = E s (t 2 -s 2 )(X 1 , y)dy -2(t -s)(X 1 , Y 1 ) = (t 2 -s 2 )(x, y)f (x)dxdy -2 (t -s)(x, y)s(x, y)f (x)dxdy = (t 2 -2ts + s 2 )(x, y)f (x)dxdy = s -t 2 f , so that s minimizes t → E s [γ(t)] over L 2 ([0, 1] d1 × [0, 1] d2
). Thus, a natural way to build an estimator of s consists in minimizing γ over some subset of

L 2 ([0, 1] d1 × [0, 1] d2
), that we choose here as a space of piecewise polynomial functions with degree smaller than a given nonnegative integer r. More precisely, for a partition m of [0, 1] d1 × [0, 1] d2 into rectangles, we denote by S m the space of all real-valued piecewise polynomial functions on [0, 1] d1 × [0, 1] d2 which are polynomial with coordinate degree ≤ r on each rectangle of m. We define a best estimator of s with values in the model S m by setting ŝm = argmin t∈Sm γ(t).

An explicit formula for computing ŝm is given in Section 5.

Risk on one model

In this subsection, we fix some partition m of [0, 1] d1 × [0, 1] d2 into rectangles and give some upper-bound for the risk of ŝm when Z 1 , . . . , Z n are independent.

As for all the theorems stated in the sequel, we evaluate that risk in the random semi-norm . n naturally associated to our problem, and defined, for all t ∈ L 2 ([0, 1] d1 × [0, 1] d2 ), by

t 2 n = 1 n n i=1 [0,1] d 2
t 2 (X i , y)dy (remember that our problem is a mixture of regression in the x-direction and of density estimation in the y-direction). However, it is also possible to control the classical L 2 -norm, using a truncated estimator (see, for instance, Corollary 3.2 in Section 3.) Besides, for any partition m ′ of a unit cube into rectangles, we denote by |m ′ | the number of rectangles in m ′ and say that the partition m ′ is regular if all its rectangles have the same dimensions. For the risk of the estimator ŝm , we can prove the following result. Let s m be the orthogonal projection of s on S m for the norm . , and D m denote the dimension of S m , so that D m = (r + 1) d |m|. Assume that s and f are bounded, and that f is also bounded from below by a positive constant. If the variables Z 1 , . . . , Z n are independent, then

E s s -ŝm 2 n ≤ 2 s -s m 2 f + 11 s ∞ D m n + C n ,
where C only depends on r, d, ι(f ), f ∞ , s ∞ .

We recover approximately in the upper-bound stated in Proposition 2.1 the usual decomposition into a squared bias term, of order s-s m 2 f , and a variance term of order s ∞ D m /n, proportional to the dimension of the model S m . A major interest of such a bound is that it allows to understand how to build an optimal estimator from the minimax point of view. Let us first recall that when s belongs to classical classes of functions with isotropic smoothness σ (isotropic Besov classes for instance), a minimax estimator over such a class reaches the estimation rate n -2σ/(2σ+d) . Roughly speaking, when s belongs to a well-chosen class of isotropic functions with smoothness σ measured in a L pnorm with p ≥ 2, the bias term s -s m 2 f is at most of order D -2σ/d m for any regular partition m into cubes. If we knew at least the smoothness parameter σ, we could choose some regular partition m opt (σ) into cubes realizing a good compromise between the bias and the variance terms, i.e. such that D -2σ/d mopt(σ) and D mopt(σ) /n are of the same order. We would then obtain with ŝmopt(σ) an estimator that reaches the optimal estimation rate n -2σ/(2σ+d) whatever p ≥ 2. But when s has isotropic smoothness σ measured in a L p -norm with p < 2, one can only ensure that the bias term s -s m 2 f is at most of order D -2σ/d m for some irregular partition m into cubes that does not only depend on σ, but must be adapted to the inhomogeneity of s over the unit cube (see for instance Section 3.2 and [START_REF] Akakpo | From adaptive approximation to adaptive estimation for functions with inhomogeneous and anisotropic smoothness[END_REF]). Thus, there exists some well-chosen irregular partition m opt (s) into cubes such that D mopt(s) is of order n d/(2σ+d) but that reaches the estimation rate n -2σ/(2σ+d) only at s, and probably not on the whole class of functions with smoothness σ in a L p -norm with p < 2. Last, if s has anisotropic smoothness, similar properties still hold, with partitions into rectangles -regular or not depending on the homogeneity of s-whose dimensions are adapted to the anisotropy of s.

Penalized estimator

We give ourselves a finite collection M of partitions of [0, 1] d1 × [0, 1] d2 into rectangles. The aim is to choose the best estimator among the collection {ŝ m } m∈M without assumption on the smoothness of s. To do so, we use the model selection method introduced by [BBM99] which allows us to select an estimator only from the data, by minimizing a penalized criterion. Thus, we consider the random selection procedure where pen : M → R + is a so-called penalty function that remains to be chosen so that s performs well. The choice of the collection of partitions M is discussed is the next section. The practical implementation of the penalized estimator based on the collection of partitions into dyadic rectangles is described in Section 5.

Main result

In this section, we study the risk of the penalized estimator s for independent data, first with a general collection of partitions, secondly with a relevant choice of collection that ensures the optimal estimation of a possibly inhomogeneous and anisotropic function s.

Oracle inequality

Ideally, we would like to choose a penalty pen such that s is almost as good as the best estimator in the collection {ŝ m } m∈M , in the sense that

E s s -s 2 n ≤ C min m∈M E s s -ŝm 2 n (3.1)
for some positive constant C. Theorem 3.1 below suggests a form of penalty yielding an inequality akin to

E s s -s 2 n ≤ C min m∈M s -s m 2 f + D m n . (3.2)
Yet, as recalled in the previous section, for each m ∈ M, E s s -ŝm 2 n is expected to be of order s -s m 2 f + D m /n. So, Inequality (3.2) is expected to be almost as good as Inequality (3.1). In order to deal with a large collection M that may contain irregular partitions, we only impose a minor structural condition on M. That assumption ensures that all the models are included in a biggest model, without imposing that the models be nested as in [START_REF] Brunel | Adaptive Estimation of the Conditional Density in Presence of Censoring[END_REF]. We also assume that s and f are bounded.

Assumption (P1) All the partitions in the collection M are built on a regular partition m ⋆ of [0, 1] d into cubes such that We establish an oracle type inequality for a very general collection of partitions. Thus we state the following model selection theorem.

|m ⋆ | 2 ≤ n Assumption (B) s ≤ s ∞ < ∞, 0 < ι(f ) ≤ f ≤ f ∞ < ∞ imsart-
Theorem 3.1. Let M be a collection of partitions satisfying Assumption (P1) and {L m } m∈M be a family of reals greater than or equal to 1, that may depend on n, such that

m∈M exp(-L m |m|) ≤ 1. (3.3)
Assume that (Z i ) 1≤i≤n are independent and s, f satisfy Assumption (B). If the penalty satisfies, for all m ∈ M,

pen(m) = κ s ∞ + (2r + 1) d ι(f ) L 2 m D m n
for some large enough positive absolute constant κ, then

E s s -s 2 n ≤ C 1 max m∈M L 2 m min m∈M s -s m 2 f + D m n . (3.4)
where

C 1 is a positive constant that depends on κ, r, d 1 , d 2 , s ∞ , ι(f ), f ∞ .
Theorem 3.1 is only proved in its general version for dependent data (see Theorem 4.1 in Section 4.3).

The penalty contains unknown terms, but in practice, s ∞ and ι(f ) can be replaced with an estimator, as in [START_REF] Birgé | From model selection to adaptive estimation[END_REF] (Proposition 4) for instance, and κ is calibrated via a simulation study. To state a result with the precise replacement, we choose m • 1 and m • 2 regular partitions of [0, 1] d1 and [0, 1] d2 into cubes such that m

• = m • 1 × m • 2 verifies Assumption (P1). We define fm • 1 = arg min t∈F m • 1 n -1 n i=1 [ t 2 -2t(X i )]
, where F m • 1 is the space of all functions on [0, 1] d1 which are polynomial with coordinate degree ≤ r on each rectangle of m • 1 , and estimate ι(f

) by ι(f ) = inf x∈[0,1] d 1 fm • 1 (x).
We also impose Besov-type smoothness assumptions on f and s. For σ = (σ 1 , . . . , σ d ) ∈ (0, r + 1) d , R > 0, p > 0, we refer to [START_REF] Triebel | Theory of function spaces[END_REF] (Chapter 5) for a definition of the anisotropic Besov space B σ pp ′ and the associated norm .|B σ pp ′ , and we introduce the anisotropic Besov balls

B(σ, p, R) = {t : [0, 1] d → R s.t. t|B σ pp ′ ≤ R}, (3.5)
where p ′ = ∞ if 0 < p ≤ 1 or p ≥ 2, and p ′ = p if 1 < p < 2. We recall that, due to the continuous embeddings stated for instance in [START_REF] Triebel | Theory of function spaces[END_REF], B σ p∞ contains all the spaces B σ pp ′ , for p ′ > 0, so our choice of p ′ in the definition of B(σ, p, R) is the less stringent one for 0 < p ≤ 1 or p ≥ 2. Last, we set σ = min 1≤l≤d σ l and denote by H(σ) the harmonic mean of σ 1 , . . . , σ d , i.e.

1 H(σ) = 1 d d l=1 1 σ l .
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Corollary 3.1. Assume that s ∈ B(σ, p, R) and f ∈ B(α, p, R 1 ) with

H(σ) d > 1 p + 1 2 H(σ) σ , H(α) d 1 > 1 p - 1 2 + + H(α) α . Assume that |m • 1 | ≥ ln n and, for all m ∈ M, pen(m) = κ ŝm • ∞ + (2r + 1) d ι(f ) L 2 m D m n
for some large enough positive constant κ. Then, under the assumptions of Theorem 3.1, for n large enough,

E s s -s 2 n ≤ C ′ 1 max m∈M L 2 m min m∈M s -s m 2 + D m n .
where

C ′ 1 is a positive constant that depends on κ, r, d 1 , d 2 , s ∞ , ι(f ), f ∞ .
We omit the proof since it exactly follows the proof of Theorem 12 in [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF]. The smoothness conditions arise from the control of s-s m • ∞ and f -f m • 1 ∞ , for which we use the results of [Aka10] (Lemma 2). It should be noticed that m • may differ from m ⋆ . In particular, it may be chosen less fine than m ⋆ so as to have better estimates of s ∞ and ι(f ).

Let us now comment on Inequality (3.4), which is similar to (3.2), up to the factors C 1 , that does not depend on n, and max m∈M L 2 m . We have already explained that we need irregular partitions to estimate inhomogeneous functions. However, irregular partitions often form a too rich collection. If L m only depends on D m , Condition (3.3) means that L D have to be large enough to balance the number of models of same dimension D. If the number of model for each dimension is high, the L m 's have to be high too. For instance, [START_REF] Birgé | From model selection to adaptive estimation[END_REF] use weights (L m ) m∈M of order log(n) to ensure condition (3.3), which spoils the rates of convergence. We describe in the next section an interesting collection of partitions for which the factor max m∈M L 2 m can be bounded by a constant, although the collection is rich enough to have good approximation qualities with respect to functions of inhomogeneous smoothness.

Let us mention that we can define an estimator s * for which we can control the risk associated to the norm . instead of . n .

Corollary 3.2. Define s * = s1 s ≤n . Then, under assumptions of Theorem 3.1,

E s s -s * 2 ≤ C ′′ 1 max m∈M L 2 m min m∈M s -s m 2 + D m n .
where

C ′′ 1 is a positive real that depends on κ, r, d 1 , d 2 , s ∞ , ι(f ), f ∞ .
The proof exactly follows the proof of Theorem 4 in [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] and then is omitted. (The idea is the following: when s ≤ n then the result is already proved; and

P ( s > n) ≤ n -2 E s 2 ≤ 2n -2 ( s 2 + E s -s 2
) is low enough to become a remainder term.) Then all the following results (Theorems 2-5) can be stated for the L 2 -norm . replacing s by s * = s1 s ≤n .

imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

The penalized estimator based on dyadic partitions

Let us describe the particular collection of partitions that we use here. We call dyadic rectangle of [0, 1] d any set of the form I 1 ×. . .×I d where, for all 1 ≤ l ≤ d,

I l = [0, 2 -j l ] or I l = (k l 2 -j l , (k l + 1)2 -j l ]
with j l ∈ N and k l ∈ {1, . . . , 2 j l -1}. Otherwise said, a dyadic rectangle of [0, 1] d is defined as a product of d dyadic intervals of [0, 1] that may have different lengths. We consider the collection of partitions of [0, 1] d into dyadic rectangles with sidelength ≥ 2 -J⋆ , where J ⋆ is a nonnegative integer chosen according to Proposition 3.1 below. We denote by M rect such a collection of partitions. Let us underline that a partition of M rect may be composed of rectangles with different Lebesgue measures, as illustrated by Figure 1. For such a collection, we obtain as a straightforward consequence of Theorem 3.1 that the estimator s is almost as good as the best estimator in the collection {ŝ m } m∈M rect .

Proposition 3.1. The notation is that of Theorem 3.1 and Assumption (B) is supposed to be fulfilled. Let

J ⋆ = max k ∈ N s.t. 2 kd ≤ √ n
and let pen be given on M rect by

pen(m) = κ s ∞ + (2r + 1) d ι(f ) D m n
where κ is some positive absolute constant. If κ is large enough, then

E s s -s 2 n ≤ C 2 min m∈M rect s -s m 2 f + D m n (3.6)
where C 2 is a positive real that depends on κ, r,

d 1 , d 2 , s ∞ , ι(f ), f ∞ .
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Proof. Let D a positive integer. Building a partition of [0, 1] d into D dyadic rectangles amounts to choosing a vector (l 1 , . . . , l D-1 ) ∈ {1, . . . , d} D-1 of cutting directions and growing a binary tree with root corresponding to [0, 1] d and with D leaves. For instance, the partition of [0, 1] 2 represented in Figure 1 can be described by the binary tree structure represented in Figure 2 together with the sequence of cutting directions (2, 1, 2, 2), where 1 stands for a vertical cut, and 2 stands for a horizontal cut. Since the number of binary trees with D leaves

[0, 1] × [0, 1] [0, 1] × 0, 1 2 (2) 0, 1 2 × 0, 1 2 (1) 1 2 , 1 × 0, 1 2 1 2 , 1 × 0, 1 4 (2) 1 2 , 1 × 1 4 , 1 2 [0, 1] × 1 2 , 1 [0, 1] × 1 2 , 3 4 (2) [0, 1] × 3 4 , 1 Fig 2.
Binary tree labeled with the sequence of cutting directions (2, 1, 2, 2) corresponding with the dyadic partition represented in Figure 1.

is given by the Catalan number

1 D 2(D -1) D -1 ≤ 4 D D
(see for instance [START_REF] Stanley | Enumerative combinatorics[END_REF]), the number of such partitions is at most (4d) D . Therefore, Condition (3.3) is fulfilled for weights L m all equal to the same constant, and a possible choice is

L m = log(8d), for all m ∈ M rect .
Inequality (3.6) is then a straightforward consequence of Theorem 3.1.

We are now able to compute estimation rates for the penalized estimator based on the collection M rect over the anisotropic Besov balls defined by (3.5), by combining Proposition 3.1 with the approximation results of [START_REF] Akakpo | From adaptive approximation to adaptive estimation for functions with inhomogeneous and anisotropic smoothness[END_REF] (Proposition 2 and Theorem 2). Let

q(σ, d, p) = σ H(σ) d + 2H(σ) 2H(σ) H(σ) d - 1 p - 1 2 + ,
where (x) + stands for the positive part of a real x. Contrary to [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF], we have chosen a parameter J ⋆ that does not depend on the unknown smoothness of s, hence the factor σ/H(σ) in the above definition. That factor, which is inferior or equal to 1 with equality only in the isotropic case, may be interpreted as an imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

index measuring the lack of isotropy. We assume that q(σ, d, p) > 1, which is equivalent to

H(σ) d >      1 λ -1 2 if p ≥ 2 1 2 1 p -1 + 1 λ + 1 p -1 + 1 λ 2 + 2 1 p -1 2 if 0 < p < 2,
where λ = σ/H(σ). Thus, if q(σ, d, p) > 1, then H(σ)/d > 1/p, so B(σ, p, R) only contains continuous functions which are uniformly bounded by C(σ, r, d, p)R.

Theorem 3.2. The notation is that of Theorem 3.1 and Proposition 3.1, and the assumptions those of Proposition 3.1. Let p > 0 and σ ∈ (0, r + 1)

d such that q(σ, d, p) > 1. If n -1 ≤ R 2 ≤ n q(σ,d,p)-1
, then there exists some positive real C(σ, r, d, p) that only depends on σ, r, d, p such that

sup s∈B(σ,p,R) E s s -s 2 n ≤ C 2 C(σ, r, d, p) f ∞ Rn -H(σ)/d 2d/(d+2H(σ)) .
The rate Rn -H(σ)/d 2d/(d+2H(σ)) is the minimax one given the lower bounds proved in [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] for transition density estimation of a Markov chain. We are able to reach that rate not only for functions with homogeneous smoothness, i.e. for p ≥ 2, as [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF], but also for functions with inhomogeneous smoothness, i.e. for 0 < p < 2, which is impossible with the collection of regular models considered in [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF]. Besides, let us underline that, among the references cited in the introduction, only [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF] can deal simultaneously with anisotropy and inhomogeneous smoothness. Theorem 3.2 improves on [START_REF] Klemelä | Multivariate histograms with data-dependent partitions[END_REF] by allowing to approximately reach the minimax risk up to a factor that does not depend on n and considering smoothness parameters possibly larger than 1.

Dependent data

We now show that the previous results can be extended to dependent variables. The case of a Markov chain is of particular interest: if (X i ) i∈Z is a homogeneous Markov chain of order 1, and Y i = X i+1 for all i ∈ Z, then s is the transition density of the chain (X i ) i∈Z .

Definitions and notation

Let us introduce the notions of dependence used in the sequel. For two sub-σfields A and B of F , the β-mixing (or absolute regularity) coefficient is defined by where the supremum is taken over all real-valued random variables X and Y that are respectively A and B-measurable and square integrable. We recall that β and ρ-mixing are among the weakest forms of mixing conditions, in the sense that both β and ρ-mixing are implied by φ-mixing (uniform mixing) and imply αmixing (see for instance [START_REF] Doukhan | Mixing[END_REF]). Besides, in general, ρ-mixing does not imply β-mixing, and β-mixing does not imply ρ-mixing. In the sequel, the letter θ stands for β or ρ. For all positive integer j, let

β(A, B) = E sup
θ Z j = θ (σ(Z i , i ≤ 0), σ(Z i , i ≥ j)) .
The process (Z i ) i∈Z is said to be θ-mixing when lim j→+∞ θ Z j = 0. In particular, (Z i ) i∈Z is geometrically θ-mixing with rate b, b > 0, if there exists a positive constant a such that, for all positive integer j, θ Z j ≤ a exp(-bj). We shall also use the 2-mixing coefficients θ(σ(Z 0 ), σ(Z j )), that satisfy, for all j ≥ 1,

θ(σ(Z 0 ), σ(Z j )) ≤ θ Z j (4.1) and, if (Z i ) i∈Z is a Markov chain, θ(σ(Z 0 ), σ(Z j )) = θ Z j .

Dependence assumptions

We consider the following dependence assumptions. Except for the last one, they are related to some rate of mixing. In each case, we also define a real ϑ, that may vary according to the dependence assumption, and will appear in the penalty proposed in the following section.

Assumption (Dβ)

The process (Z i ) i∈Z is geometrically β-mixing, with a ≥ 0 and b > 0 such that, for all positive integer j, β Z j ≤ a exp(-bj). Then we denote ϑ = 1 and δ = 1.

Assumption (Dβρ) Assumptions (Dβ) is satisfied and, in addition, the series S ρ := j∈N ρ Z 2 j converges. Then we denote ϑ = 250

∞ j=0 1 + ρ Z ⌊2 j/3 ⌋+1 and δ = 0.
Assumption (Dβ2-ρ) Assumptions (Dβ) is satisfied and, in addition, the series S 2-ρ := j≥1 ρ(σ(Z 0 ), σ(Z j )) converges. Then we denote ϑ = (1 + 2S 2-ρ ) and δ = 0.

Assumption (Dβ cond ) Assumptions (Dβ) is satisfied and, in addition, for all j ≥ 2, Z j is independent of Z 1 conditionally to X j . Then we denote ϑ = 1 and δ = 0.

Note that (Dβρ) is in some sense a weaker assumption than (Dβ2-ρ), since a logarithmic ρ-mixing is sufficient. In particular, if (Z i ) i∈Z is a Markov chain, (Dβ2-ρ) implies (Dβρ) (according to (4.1) and since (Z i ) i∈Z is geometrically ρ-mixing if and only if it is ρ-mixing (cf. [START_REF] Bradley | Basic properties of strong mixing conditions. A survey and some open questions[END_REF], Theorem 3.3)). On the other hand, Assumption (Dβ cond ) does not imply ρ-mixing. For instance, if (X i ) i∈Z is a Markov chain and Y i = X i+1 for all i ∈ Z, then Assumption (Dβ cond ) is satisfied, but (X i ) i∈Z , and therefore (Z i ) i∈Z , can be chosen non mixing (cf. [START_REF] Dedecker | New dependence coefficients. Examples and applications to statistics[END_REF] for instance).

Let us give sufficient conditions for (Z i ) i∈Z to be θ-mixing. First, if (X i ) i∈Z is a strictly stationary θ-mixing process, and

Y i = X i+1 for all i ∈ Z, then (Z i ) i∈Z also is θ-mixing since, for all j ≥ 2, θ Z j = θ X j-1 . Next, if (Z i ) i∈Z is

Main result

All the results of Section 3 can be extended to the case of dependent data, under slightly more restrictive conditions on the thinest partition. Assume that (Z i ) i∈Z satisfies Assumption (Dβ) and s, f satisfy Assumption (B). If the penalty satisfies, for all m ∈ M,

pen(m) = κ ϑ(b -1 log(n)) δ s ∞ + (2r + 1) d b 2 ι(f ) L 2 m D m n
for some large enough positive absolute constant κ (where b, δ and ϑ are defined in the assumptions of dependence), then

E s s -s 2 n ≤ C 3 max m∈M L 2 m min m∈M s -s m 2 f + log δ (n) D m n .
where C 3 is a positive constant that depends on κ, ϑ, δ, a, b, r,

d 1 , d 2 , s ∞ , ι(f ), f ∞ .
Under Assumptions (Dβρ), (Dβ2-ρ), the price to pay for avoiding the logarithmic factor despite the dependence of the data is the presence of the term ϑ in the penalty. For practical purposes, it is necessary to include this term in the constant κ to calibrate. Notice that under Assumption (Dβ cond ), for instance when we estimate the transition density of a Markov chain, the logarithmic factor still disappears and ϑ = 1 so that the penalty is almost as simple as in the independent case. Actually it is possible to consider an arithmetical β-mixing instead of a geometrical one. In this case, it is necessary to slightly strengthen assumption (P2), assuming rather |m * | 2 ≤ n 1-ζ , with ζ a number in (0, 1). Then, if β q ≤ aq -b with b > 5/ζ -2, Theorem 4.1 is still valid in the cases where δ = 0 (ρ-mixing and conditional independence). The penality is identical, except the term b 2 which is removed. The proof is the same as the original statement, see Subsection 6.3, but with q n = ⌊n ξ ⌋ where ξ ∈ ((5 -ζ)/(2 + 2b), ζ/2).

Then, for our penalized estimator based on partitions into dyadic rectangles described in Section 3.2, we can state the following theorem.

Theorem 4.2. The notation is that of Theorems 4.1, Assumption (B) is supposed to be fulfilled. Let

J ⋆ = max k ∈ N s.t. 2 kd ≤ √ n/ log(n) .
Let p > 0 and σ ∈ (0, r + 1) d,p) , then there exists some positive real C(σ, r, d, p) that only depends on σ, r, d, p such that sup s∈B(σ,p,R)

d such that q(σ, d, p) > 1. If log δ (n)/n ≤ R 2 ≤ n q(σ,d,p)-1 log(n) δ-2q(σ,
E s s -s 2 n ≤ C 4 C(σ, r, d, p) f ∞ R n log δ (n) -H(σ)/d 2d/(d+2H(σ))
.

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Thus we recover the same rate of estimation as with independent data (cf. Theorem 3.2) up to a logarithmic factor that disappears under Assumptions (Dβρ), (Dβ2-ρ) or (Dβ cond ).

Remarks on the dependence assumptions

We can wonder if weaker assumptions of dependence could be used. Another assumption of dependence is used for instance by [START_REF] Bosq | Nonparametric statistics for stochastic processes[END_REF] (Theorem 2.1) to prove that, asymptotically, the quadratic risk of kernel density estimators reaches the minimax rate (see also [START_REF] Comte | Adaptive estimation of the stationary density of discrete and continuous time mixing processes[END_REF]). But we can prove (see [START_REF] Akakpo | Estimation adaptative par sélection de partitions en rectangles dyadiques[END_REF]) that this assumption is much stronger than Assumption (Dβ2-ρ), which is enough for obtaining the optimal estimation rate from the minimax point of view.

It is difficult to bound the risk for s under weaker dependence assumptions but it is possible to weaken the assumptions to bound the risk E s -ŝm 2 n for one model. In [START_REF] Akakpo | Estimation adaptative par sélection de partitions en rectangles dyadiques[END_REF], a version of Proposition 2.1 is proved under assumptions of geometrical α-mixing. Actually a sufficient condition to ensure that E s ŝm -s m 2 n is of the same order as in the independent case is that for some constant C and all t ∈ S m , Var

n i=1 t(Z i ) ≤ CnVar (t(Z 1 )) . (4.2)
Assumptions (Dβρ) and (Dβ2-ρ) are optimal for obtaining such an inequality in the following sense. Let us assume that (Z i ) i∈N is a strictly stationary Harris ergodic and reversible Markov chain satisfying (4.2) for all real-valued function t defined on [0, 1] d . Then the chain is variance bounding in the sense of [START_REF] Roberts | Variance bounding Markov chains[END_REF], which implies that there is a spectral gap in L 2 (sf ) := {t : [0, 1] d → R s.t. t, sf = 0 and t sf < ∞} (Theorem 14 in [START_REF] Roberts | Variance bounding Markov chains[END_REF]). This leads to the geometrical ergodicity of the chain (Theorem 2.1 in [START_REF] Roberts | Geometric ergodicity and hybrid Markov chains[END_REF]), which, given the reversibility assumption, implies that the chain is ρ-mixing. As a conclusion, a strictly stationary Harris ergodic and reversible Markov chain (Z i ) i∈Z satisfies (4.2) for all real-valued function t defined on [0, 1] d if and only if it is ρ-mixing.

Implementation and simulations

In order to provide useful characterizations for ŝm and m in practice, we need to introduce some adequate basis of each S m , for m ∈ M. Let (Q j ) j∈N be the orthogonal family of the Legendre polynomials in L 2 ([-1, 1]). For all j ∈ N, we recall that Q j satisfies

Q j ∞ = 1 and Q j 2 = 2 (2j + 1)
.

(5.1)

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For K 1 = d1 i=1 [u i , v i ] rectangle of [0, 1] d1 , k 1 = (k 1 (1), . . . , k 1 (d 1 )) ∈ {0, . . . , r} d1 and x = (x 1 , . . . , x d1 ) ∈ [0, 1] d1 , we set φ K1,k1 (x) = 1 µ d1 (K 1 ) d1 i=1 2k 1 (i) + 1 Q k1(i) 2x i -u i -v i v i -u i 1 K1 (x),
where µ d1 denotes the Lebesgue measure in R d1 . Therefore, for K 1 rectangle in [0, 1] d1 , (φ K1,k1 ) k1∈{0,...,r} d 1 is a basis of the space of piecewise polynomials functions with support K 1 and coordinate degree ≤ r, which is orthonormal for the norm . . For K 2 rectangle in [0, 1] d2 and k 2 ∈ {0, . . . , r} d2 , we define in the same way ψ K2,k2 on [0, 1] d2 . For K rectangle in [0, 1] d , we shall denote by K 1 and K 2 the rectangles in [0, 1] d1 and [0, 1] d2 such that K = K 1 × K 2 . For k ∈ {0, . . . , r} d , we shall denote by k 1 and k 2 the multi-indices in {0, . . . , r} d1 and {0, . . . , r} d2 such that k = (k 1 , k 2 ). For any rectangle K ∈ [0, 1] d and any multi-index k ∈ {0, . . . , r} d , we define Φ K,k by the decomposition of ŝm in the basis (Φ K,k ) K∈m,k∈{0,...,r} d . For all K ∈ m, we define the matrices A K = âK,(k1,k2) (k1,k2)∈{0,...,r} d 1 ×{0,...,r} d 2 ,

Φ K,k (x, y) = φ K1,k1 (x)ψ K2,k2 (y) for z = (x, y) ∈ [0, 1] d1 ×[0, 1] d2 .
Υ K = 1 n n i=1 φ K1,k1 (X i )ψ K2,k2 (Y i ) (k1,k2)∈{0,...,r} d 1 ×{0,...,r} d 2
, and

G K1 = 1 n n i=1 φ K1,k1 (X i )φ K1,l1 (X i ) (k1,l1)∈{0,...,r} d 1 ×{0,...,r} d 1 .
Since φ K1,k1 and φ L1,l1 (resp. ψ K2,k2 and ψ L2,l2 ) have disjoint supports when

K 1 = L 1 (resp. K 2 = L 2
) and (ψ K2,k2 ) k2∈{0,...,r} d 2 is orthonormal, we obtain after some computation that, for all K ∈ m, A K is given by

G K1 A K = Υ K . (5.2)
Let us mention that when r = 0, we can write, for all rectangle K (we do not mention any index k),

ŝm 1 K = 1 µ d2 (K 2 ) n i=1 1 K1 (X i ) n i=1 1 K (Z i ) if some X i ∈ K 1 ,
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 and ŝm 1 K = 0 otherwise, where µ d2 denotes the Lebesgue measure in R d2 .

Thanks to Formula (5.2), one can check that, for all m ∈ M rect ,

γ(ŝ m ) = - K∈m k∈{0,...,r} d (A K ) (k1,k2) (Υ K ) (k1,k2) .
We shall consider a penalty pen of the form

pen(m) = c ŝm • ∞ D m n , (5.3)
where c is some positive constant, as in Theorem 4.1. With such a penalty, m is given by m = argmin

m∈M rect K∈m L(K)
where, for all rectangle K,

L(K) = k∈{0,...,r} d -(A K ) (k1,k2) (Υ K ) (k1,k2) + c ŝm • ∞ n .
That characterization allows to determine m without having to compute all the estimators of the collection {ŝ m } m∈M rect . Indeed, we can for instance adapt to our estimation framework the algorithm proposed by [START_REF] Donoho | CART and best-ortho-basis: a connection[END_REF], which requires a computational complexity of order 2 dJ⋆ . Thus, choosing 2 dJ⋆ at most of order n, which allows for a larger choice of J ⋆ than prescribed by our theoretical results (cf. Proposition 3.1 and Theorem 4.2), the computational complexity is at most linear in the number of observations. Let us also mention that the algorithm proposed by [START_REF] Blanchard | Optimal dyadic decision trees[END_REF] allows for instance the slightly larger choice J ⋆ = ⌊log(n)⌋, that does not depend on d, while keeping an almost linear computational complexity, that is of order nd log d+1 (n).

We propose a simulation study based on the 4 following examples. Example 1.

Y i = 0.5X i + 1 + ǫ i , i = 1, . . . , n,
where (X i ) 1≤i≤n are i.i.d. Gaussian variables with mean 6 and variance 4/3, (ǫ i ) 1≤i≤n are i.i.d. reduced and centered Gaussian variables, independent of the

X i 's. Example 2. Y i = sin(X i ) + (cos(X i ) + 3)ǫ i , i = 1, . . . , n,
where (X i ) 1≤i≤n are i.i.d. uniformly distributed over [-6, 6], (ǫ i ) 1≤i≤n are i.i.d. reduced and centered Gaussian variables, independent of the X i 's. Example 3. Let β(., a, b) be the density of the β distribution with parameters a and b,

Y i = 1 3 (X i + 1) + 1 9 - 1 23 1 2 β(5X i /3, 4, 4) + 1 20 β((5X i -2)/3,
Y i = 1 4 (g(X i ) + 1) + 1 8 ǫ i , i = 1, . . . , n
where (X i ) 1≤i≤n are i.i.d. uniformy distributed in [0, 1], (ǫ i ) 1≤i≤n are i.i.d. Gaussian reduced and centered, independent of the X i 's, and g is the density of

3 4 N 1 + 1 4 N 2
where N 1 is Gaussian with mean 1/2 and standard error 1/6, N 2 is Gaussian with mean 3/4 and standard error 1/18, N 1 and N 2 are independent. Each model is of the form

Y i = µ(X i ) + σ(X i )ǫ i ,
where ǫ i is a reduced and centered Gaussian variable, so the conditional density of Y i given X i is given by

s(x, y) = φ((y -µ(x))/σ(x))/σ(x),
where φ is the density of ǫ 1 . Besides, this allows us to consider Markovian counterparts of Examples 1 to 4, that we will call Example 1 (Markov),. . ., Example 4 (Markov). More precisely, we also estimate the transition density of the Markov chain (X i ) i≥1 that satisfies

X i+1 = µ(X i ) + σ(X i )ǫ i ,
with X 1 that follows the stationary distribution of the chain. Thus, for Example 1 (Markov), X 1 has the same distribution as in Example 1, but in the other examples, the distribution of X 1 differs between the independent and the Markovian cases. In practice, we simulate the chain long enough so that it finally reaches the stationary regime. We estimate s respectively on [4, 8] 2 for Example 1, [-6, 6] 2 for Example 2, and [0, 1] 2 for Examples 3 and 4, both for independent and Markovian data. The four conditional densities are represented on these rectangles in Figure 5. We may say that the first two examples are rather homogeneous functions, whereas the last two are rather inhomogeneous. We implement s for r = 0 and choose the following parameters. The supremum norm of s is estimated by ŝm • , where m • is the regular partition of [0, 1] 2 into cubes with sidelength 2 J• . We select a best partition among those into dyadic rectangles with sidelength ≥ 2 -J⋆ , with 2 J⋆ as close as possible to √ n. For n = 250, we set J • = 2 and J ⋆ = 4, and for n = 1000, we set J • = 3 and J ⋆ = 5. Let us denote by s(c) the penalized estimator obtained with the penalty (5.3) for the penalty constant c. For the sample sizes n = 250 and n = 1000, we give respectively in Tables 1 and2 Results for n = 250 data and 100 simulations. obtained by varying c from 0 to 4 by step 0.1. All these quantities have been estimated over 100 simulations. Besides, for Example 3, we represent in Figure 5 the selected partition for one simulation with 1000 independent data and the penalty constant c = 3. That partitition is both anisotropic and inhomogeneous and well adapted to the function, which illustrates the interest of allowing nonregular and non-isotropic partitions in our selection procedure. Just below, we represent two sections of that conditional density (dark line) together with the corresponding sections of s(3). The closeness between the minimal risk min c E s s -s(c) 2 n and E s [ s -s(3)

the estimated values of ŝm • ∞ , E s [ s -s(3)
n = 250 s ∞ ŝm • ∞ Es [ s -s(3) 2 n ] minc Es s -s(c) 2 n Example 1 0,4 0,
2 n ] indicates that a penalty constant equal to 3 seems to be a good choice. We observe that, for each example, the risks obtained for the independent and the Markovian cases are also close, which tends to confirm Theorem 4.1, otherwise said that the penalty under assumption (Dβ cond ) is not so much affected by the dependency between the data. For Examples 1 and 2, we can compare ourselves with the results of Lacour [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF] in the Markovian case, obtained via regular model selection. We obtain either similar results for Example 2 or even imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 better results for Example 1. Last, let us mention that the performance of s, in practice, might still be improved by a data-driven choice of the penalty constant based on the slope heuristics, as described in [START_REF] Baudry | Slope heuristics: overview and implementation[END_REF] for instance, but this is beyond the scope of the paper.

Proofs

Notation and preliminary lemmas

In all the proofs, the letter C denotes a real that may change from line to line. The notation C(θ) means that the real C may depend on θ.

For all t ∈ L 2 ([0, 1] d1 × [0, 1] d2 ) and all z = (x, y) ∈ [0, 1] d1 × [0, 1] d2 , let Γ t (z) = t(x, y) - [0,1] d 2 t(x, u)s(x, u)du, (6.1)
and let ν be the empirical process defined on

L 2 ([0, 1] d1 × [0, 1] d2 ) by ν(t) = 1 n n i=1 Γ t (Z i ). (6.2)
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Table 2 Results for n = 1000 data and 100 simulations.

n = 1000 s ∞ ŝm • ∞ Es [ s -s(3) 2 n ] minc Es s -s(c)
For all i, E s [Γ t (Z i )|X i ] = E s [t(X i , Y i )|X i ] -[0,1] d 2 t(X i , u)s(X i , u)du = 0, so that ν(t) is centered.
We will use several times the following lemma to bound some variance terms.

Lemma 6.1. Let q be a positive integer. For all

t ∈ L 2 ([0, 1] d1 × [0, 1] d2 ), Var s q i=1 Γ t (Z i ) ≤ ϑq 1+δ Var s (Γ t (Z 1 )) (6.3)
where δ and ϑ are defined in Section 4.2 for the dependent case, or δ = 0 and ϑ = 1 when the variables Z i are independent. Besides,

Var s (Γ t (Z 1 )) ≤ E s t 2 (Z 1 ) ≤ min{ s ∞ t 2 f , sf ∞ t 2 }.
Proof: First we use a convexity inequality to write, without further assumption,

Var s q i=1 Γ t (Z i ) ≤ E s q q i=1 Γ 2 t (Z i ) ≤ q 2 Var s (Γ t (Z 1 ))
whereas in the independent case, Var s (

q i=1 Γ t (z i )) = q i=1
Var s (Γ t (Z 1 )) . Now, under Assumption (Dβρ), Lemma 8.15 in [START_REF] Bradley | Introduction to strong mixing conditions[END_REF] provides

Var s q i=1 Γ t (Z i ) ≤ C 1 qVar s (Γ t (Z 1 )) with C 1 = 250 ⌊log 2 q⌋ j=0 1 + ρ Z ⌊2 j/3 ⌋+1 . Next, by stationarity, Var s q i=1 Γ t (z i ) = qVar s (Γ t (Z 1 )) + 2 q-1 j=1 (q -j)Cov s (Γ t (Z 1 ), Γ t (Z j+1 )) .
Under Assumption (Dβ2-ρ), we immediately deduce from the definition of the ρ-mixing coefficients and the stationarity of (Z i ) i∈Z that, for all 1 ≤ j ≤ q -1,

Cov s Γ t (Z 1 ), Γ t (Z j+1 ) ≤ ρ(σ(Z 1 ), σ(Z j+1 ))Var s (Γ t (Z 1 )) .
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Thus,

1 q Var s q i=1 Γ t (Z i ) = Var s Γ t (Z 1 ) + 2 q-1 j=1 1 - j q Cov s Γ t (Z 1 ), Γ t (Z j+1 ) ≤   1 + 2 q-1 j=1 ρ(σ(Z 1 ), σ(Z j+1 ))   Var s (Γ t (Z 1 )) .
Under Assumption (Dβ cond )

Cov s (Γ t (Z 1 ), Γ t (Z j+1 )) = E s [Γ t (Z 1 )E s [Γ t (Z j+1 )|Z 1 , X j+1 ]] = 0,
hence Inequality (6.3) in the last case. Besides,

Var s [Γ t (Z 1 )] = Var s [t(Z 1 )] + E s E s E s [t(Z 1 )] -E s [t(Z 1 )|X 1 ] 2 |X 1 ≤ E s [t 2 (Z 1 )] = [0,1] d 1 [0,1] d 2
t 2 (x, y)s(x, y)f (x)dxdy.

We recall here Bernstein's Inequality for independent random variables (see [Mas07] (Section 2.2.3) for a proof). Lemma 6.2 (Bernstein inequality). Let (W i ) 1≤i≤n be an independent and identically dsitributed sequence, defined on the probability space (Ω, F , P), with values in W. Let n ≥ 1 and g be a real-valued and bounded function defined on W. Let σ 2 g = Var(g(W 1 )). Then, for all x > 0,

P n i=1 (g(W i ) -E[g(W i )]) ≥ 2nσ 2 g x + g ∞ x/3 ≤ 2 exp (-x) .

Proof of Proposition 2.1

Since ŝm = argmin t∈Sm γ(t), we have γ(ŝ m ) ≤ γ(s m ). The contrast γ satisfies, for

all t, u ∈ L 2 ([0, 1] d1 × [0, 1] d2 ), γ(t) -γ(u) = t -s 2 n -u -s 2 n -2ν(t -u),
where ν is defined by (6.2), hence

s -ŝm 2 n ≤ s -s m 2 n + 2ν(ŝ m -s m ). Let χ f (m) = sup t∈Sm t f =1 ν(t),
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 and let θ be some positive constant, to be chosen later, then

2ν(ŝ m -s m ) ≤ 2 ŝm -s m f χ f (m) ≤ 1 θ ŝm -s m 2 f + θχ 2 f (m).
Let us fix η > 1, to be determined later, and define

Ω η (m) = For all t ∈ S m \{0}, t 2 f ≤ η t 2 n . (6.4)
We deduce from the triangle inequality that, on Ω η (m),

2ν(ŝ m -s m ) ≤ η θ ŝm -s m
Let (Φ f λ ) λ∈Λ(m) be a basis of S m orthonormal for . f . Since ν is linear, we deduce from Schwarz Inequality and its equality case that

χ 2 f (m) = λ∈Λ(m) ν 2 Φ f λ , so nE s χ 2 f (m) = 1 n λ∈Λ(m) Var n i=1 Γ Φ f λ (Z i ) .
Since Z 1 , . . . , Z n are independent, we deduce from Lemma 6.1 that

nE s χ 2 f (m) ≤ s ∞ D m , (6.6) hence E s s -ŝm 2 n 1 Ωη(m) ≤ 2 s -s m 2 f + 11 s ∞ D m n .
In order to bound the risk of ŝm on Ω c η (m), we use the following two lemmas, proved just below. Lemma 6.3. Assume that d 1 is positive and that s is bounded. Let m be a partition of [0, 1] d1 × [0, 1] d2 into rectangles built on a regular partition m ⋆ 1 × m ⋆ 2 , where m ⋆ 1 and m ⋆ 2 are regular partitions of [0, 1] d1 and [0, 1] d2 into cubes. Then

s -ŝm 2 n ≤ 2 s 2 ∞ + 2(r + 1) d2 (2r + 1) d2 |m ⋆ 2 |.
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 Lemma 6.4. Let m ⋆ = m ⋆ 1 × m ⋆ 2 , where m ⋆ 1 and m ⋆ 2 are regular partitions of [0, 1] d1 and [0, 1] d2 into cubes. Let η > 1 and Ω η (m ⋆ ) be defined by (6.4). Then there exists an absolute constant C such that

P s (Ω c η (m ⋆ )) ≤ C(r + 1) 2d1 |m ⋆ 1 | exp - ι 2 (f )(1 -1/η) 2 n 3 f ∞ C(r, d 1 )|m ⋆ 1 | . Since m is built on m ⋆ = m ⋆ 1 × m ⋆ 2 , Ω η (m ⋆ ) ⊂ Ω η (m).
Given the conditions on m ⋆ 1 and m ⋆ 2 , we then obtain

E s s -ŝm 2 n 1 Ω c η (m) ≤ 2 s 2 ∞ + (r + 1) d2 (2r + 1) d2 |m ⋆ 2 | P s (Ω c η (m ⋆ )) ≤ C(r, d 1 , s, f )/n,
where C(r, d 1 , s, f ) is a nonnegative real that only depends on r, d 1 , s ∞ , ι(f ) and f ∞ .

Let us end with the proofs of Lemmas 6.3 and 6.4. Proof of Lemma 6.3: We shall use the notation

• . R n defined for v = {v i } 1≤i≤n ∈ R n by v R n = n i=1 v 2 i /n; • for t ∈ L 2 ([0, 1] d1 ×[0, 1] d2 ) and y ∈ [0, 1] d2 , t X (y) = {t(X i , y)} 1≤i≤n ∈ R n ; • V X m (y) = {t X (y), t ∈ S m } and P V X m (y) the orthogonal projection of R n on V X m (y).
For all y ∈ [0, 1] d2 , let us also define the R n -vector

vm (y) =    J∈m2 j∈{0,...,r} d 2 ψ J,j (Y i )ψ J,j (y)    1≤i≤n .
As [Lac07] (Proposition 2.1), we can prove that ŝX m (y) = P V X m (y) (v m (y)) . Using the triangle inequality and the shrinking property of P V X m (y) , we get

s -ŝm 2 n = [0,1] d 2 s X (y) -ŝX m (y) 2 R n dy ≤ 2 [0,1] d 2 s X (y) 2 R n dy + 2 [0,1] d 2 vm (y) 2 R n dy.
From the orthonormality of {ψ J,j } J∈m2,j∈{0,...,r} d 2 , we deduce that

[0,1] d 2 vm (y) 2 R n dy = 1 n n i=1 J∈m2 j∈{0,...,r} d 2 ψ 2 J,j (Y i ).
Now, using (5.1),

ψ J,j 2 ∞ = d2 i=1 (2k 2 (i) + 1) µ d2 (J) ≤ (2r + 1) d2 µ d2 (J) .
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

where A = a = (a K1,k1 ) K1∈m1,k1∈{0,...,r} d 1 s.t.

(φ) ≤ C(r, d 1 )|m ⋆ 1 |. Let x = ι 2 (f )(1 -1/η) 2 3 f ∞ L(φ) and ∆ = K1∈m ⋆ 1 ,k1,l1∈{0,...,r} d 1 |ν ′′ (φ K1,k1 φ K1,l1 )| < 2 f ∞ xV K1,k1,l1 + 1 3 B K1,k1,l1 x .
One can easily check that, on ∆, sup t∈S m ⋆ / t f =1 |ν ′ (t 2 )| ≤ 1 -1/η, so that Ω c η (m) ⊂ ∆ c . Lemma 6.4 then follows from Lemma 6.2.

Proof of Theorem 4.1

Let us fix m ∈ M. We also fix η ≥ 1 and θ 1 > 0, to be determined at the end of the proof. By definition of m and ŝm , γ(s) + pen( m) ≤ γ(ŝ m ) + pen(m) ≤ γ(s m ) + pen(m). (6.8)

Using the same arguments as in the proof of Proposition 2.1, we deduce from (6.8) that s -

s 2 n ≤ s -s m 2 n + pen(m) + 2ν(s -s m ) -pen( m). As s -s m ∈ S m + S m ⊂ S m ⋆ ,
we obtain in the same way as Inequality (6.5) that, on the set Ω η (m ⋆ ) defined as in (6.4),

2ν(s -s m ) ≤ 2η θ 1 s -s 2 n + 2η θ 1 s -s m 2 n + θ 1 χ 2 f (m, m) with χ f (m, m ′ ) = sup t∈Sm+S m ′ t f =1 |ν(t)|.
imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Consequently, provided θ 1 > 2η,

1 - 2η θ 1 s -s 2 n 1 Ωη (m ⋆ ) ≤ 1 + 2η θ 1 s -s m 2 n 1 Ωη (m ⋆ ) + pen(m)
+θ 1 χ 2 f (m, m) -pen( m). (6.9)

To pursue the proof, we have to control the term χ 2 f (m, m). Since the data are β-mixing, we can introduce blockwise independent data. More precisely, let q n = ⌈3b -1 log(n)⌉ (where b is defined in Assumption (Dβ)) and let (d n , r n ) be the unique couple of nonnegative integers such that n = d n q n + r n and 0 ≤ r n < q n . For the sake of simplicity, we assume in the sequel that r n = 0 and d n = 2p n > 0, but the other cases can be treated in a similar way. For l = 0, . . . , p n -1, let us set

A l = {Z i } 2lqn+1≤i≤(2l+1)qn and B l = {Z i } (2l+1)qn+1≤i≤(2l+2)qn .
As recalled for instance in [START_REF] Viennet | Inequalities for absolutely regular sequences: application to density estimation[END_REF] (proof of Proposition 5.1), we can build, for l = 0, . . . , p n -1,

A • l = {Z • i } 2lqn+1≤i≤(2l+1)qn and B • l = {Z • i } (2l+1)qn+1≤i≤(2l+2)qn
such that, for all l = 0, . . . , p n -1,

• A l , A • l , B l and B • l have the same distribution; • P s (A l = A • l ) ≤ β Z qn and P s (B l = B • l ) ≤ β Z qn ; • (A • l )
0≤l≤pn-1 are independent random variables, and so are (B • l ) 0≤l≤pn-1 . We set

Ω • = n i=1 {Z • i = Z i }.
The proof of Theorem 4.1 heavily relies on the following concentration inequality satisfied by the random variables χ 2 f (m, m ′ ), for m, m ′ partition built on m ⋆ . The proof of that proposition is deferred to Section 6.4. Proposition 6.1. Under the assumptions of Theorem 4.1, there exists a positive constant C such that

m ′ ∈M E s [χ 2 f (m, m ′ ) -pen(m) -pen(m ′ )] + 1 Ω• ≤ C log δ (n) n ,
where [x] + denotes the positive part of a real x and C depends on ϑ, s ∞ , r, d, ι(f ), b.

We shall first bound the quadratic risk of s on Ω η (m ⋆ ) ∩ Ω • . Combining (6.9) and Proposition 6.1

1 - 2η θ 1 s -s 2 n 1 Ωη (m ⋆ )∩Ω• ≤ 1 + 2η θ 1 s -s m 2 n 1 Ωη (m ⋆ ) + 2pen(m) +θ 1 [χ 2 f (m, m) -pen(m) -pen( m)] + 1 Ω• imsart-
- 2η θ 1 E s s -s 2 n 1 Ωη (m ⋆ )∩Ω• ≤ 1 + 2η θ 1 E s s -s m 2 n + 2pen(m) + θ 1 C log δ (n) n . ( 6
(m ⋆ ) ∪ Ω c • . A straightforward upper-bound for the P s -measure of Ω c η (m ⋆ ) ∪ Ω c • is P s (Ω c η (m ⋆ ) ∪ Ω c • ) ≤ P s (Ω c • ) + P s (Ω c η (m ⋆ ) ∩ Ω • ).
One easily deduces from one of the properties of the A • l 's and B • l 's that

P s (Ω c • ) ≤ 2p n β Z qn = n q n β Z qn . (6.11)
In order to bound P s (Ω c η (m ⋆ ) ∩ Ω • ), we follow the proof of Lemma 6.4. Thus there exists some constant C(r, d 1 ) that only depends on d 1 and r such that

P s (Ω c η (m ⋆ ) ∩ Ω • ) ≤ 4(r + 1) 2d1 |m ⋆ 1 | exp -C(r, d 1 ) ι 2 (f )(1 -1/η) 2 f ∞ n q n |m ⋆ 1 | . (6.12) where m ⋆ 1 and m ⋆ 2 are the partition of [0, 1] d1 and [0, 1] d2 such that m ⋆ = m ⋆ 1 × m ⋆ 2 .
Combining Inequalities (6.11) and (6.12) with Lemma 6.3 then provides for

E s s -s 2 n 1 Ω c η (m ⋆ )∪Ω c • ∪Ω c T the upper-bound C(r, d 1 , d 2 )|m ⋆ 2 | n q n β Z qn + |m ⋆ 1 | exp -C(η, r, d 1 , ι(f ), f ∞ ) n q n |m ⋆ 1 | . (6.13)
Last, let us choose η = 7/6, θ 1 = 7.

Under Assumption (P2) on m ⋆ 1 and m ⋆ 2 , we deduce from (6.10) and (6.13) that

E s s -s 2 n ≤3 s -s m 2 f + pen(m) + C(ϑ, s ∞ , r, d 1 , d 2 , ι(f ), f ∞ , b, a) log δ (n) n .
Theorem 4.1 then follows by taking the minimum over m ∈ M. Notice that when the data are independent, we can take ϑ = 1, q n = 1, β qn = 0 and δ = 0. Then Proposition 6.1 and the rest of the proof are valid under Assumption (P1).

imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Proof of Proposition 6.1

We recall that Γ is given by (6.1) and we define on L 2 ([0, 1] d1 × [0, 1] d2 ), for all m, m ′ ∈ M and j = 1, 2, Since A • 0 , . . . , A • pn-1 are independent and identically distributed on Ω • , we deduce from Lemma 6.1 that sup

t∈Sm+S m ′ t f =1 pn-1 l=0 Var s   1 n (2l+1)qn i=2lqn+1 Γ t (Z • i )1 Ω•   ≤ V 2 n
and also, by using the same arguments as for (6.6), that

E 2 s χ • f,(1) (m, m ′ )1 Ω• ≤ V 2 (D m + D m ′ ) n .
If t ∈ S m +S m ′ and t f = 1, then by developing t in the basis (Φ K,k ) K∈m ⋆ ,k∈{0,...,r} d and using Schwarz Inequality, we get and, from the hypotheses on (A • l ) 0≤l≤pn-1 and (B • l ) 0≤l≤pn-1 , χ • f,(1) (m, m ′ )1 Ω• and χ • f,(2) (m, m ′ )1 Ω• are identically distributed. Thus, denoting by ε some positive constant, applying Talagrand's inequality (as stated for instance in [START_REF] Massart | Concentration inequalities and model selection[END_REF], Inequality (5.50)) to each χ • f,(j) (m, m ′ )1 Ω• , we deduce that, for all x > 0, there exists an event Ω m,m ′ (x) such that P s (Ω c m,m ′ (x)) ≤ 2 exp(-x) and over which

t 2 ∞ ≤ max
√ nχ f (m, m ′ )1 Ω• ≤ 2 5 2 V D m + D m ′ + V √ 2x + B x √ n .
Let u > 0, then on Ω c m,m ′ (|m

′ |L m ′ + u), √ nχ f (m, m ′ )1 Ω• ≤ 2 5 2 V D m + D m ′ + V 2D m ′ L m ′ + B |m ′ |L m ′ √ n + 2 V √ 2u + B u √ n
Since q n = ⌈3b -1 log(n)⌉ ≤ 6b -1 log(n) and |m ⋆ | ≤ √ n/ log(n) ,

B|m ′ | ≤ B |m ⋆ | |m ′ | ≤ 12 n(2r + 1) d D m ′ b 2 ι(f ) .
Therefore, still on Ω c m,m ′ (|m ′ |L m ′ + u),

√ nχ f (m, m ′ )1 Ω• ≤ 2 5 2 V D m + D m ′ + V 2D m ′ L m ′ + 12 (2r + 1) d D m ′ b 2 ι(f ) L m ′ + 4 max V √ 2u, B u √ n so that nχ 2 f (m, m ′ )1 Ω• ≤ 8 5 2 V D m + D m ′ + V 2D m ′ L m ′ + 12 (2r + 1) d D m ′ b 2 ι(f ) L m ′ 2 + 32 max 2V 2 u, B 2 u 2 n .
As L m ′ ≥ 1 for all m ′ ∈ M, choosing pen such that for all m ′ ∈ M pen(m ′ )1 Last, we recall that Fubini's Theorem yields, for any random variable Z, Assume that s ∈ B(σ, p, R). Then, for all k ∈ N, there exists some partition m k of [0, 1] d that only contains dyadic rectangles with edge-length at least 2 -Jσ/σ l in the l-th direction, l = 1, . . . , d, such that |m k | ≤ C(d, p, σ)2 kd and s -s m k 2 ≤ C(d, p, r, σ)R 2 2 -2Jd(H(σ)/d+1/2-1/p)σ/H(σ) + 2 -2kH(σ) .

E ([Z] + ) =
Under the assumptions of Theorem 3.2, we set δ = µ = 0, whereas under the assumptions of Theorem 4.2, δ is given in Section 4.2 and µ = 1. Let us fix R, σ, p > 0 satisfying the assumptions of Theorems 3.2 or 4.2, and s ∈ B(σ, p, R). Since, σ ≤ σ l for all l = 1, . . . , d, Theorem 6.1 applied with J = J ⋆ provides partitions (m k ) k∈N that all belong to M rect . Thus, with τ = H(σ)/d -(1/p -1/2) + , we obtain min ≤ C(d, p, r, σ) R 2 2 -2J⋆dτ σ/H(σ) + R(n log -δ (n)) -H(σ)/d 2d/(d+2H(σ)) .

Last, since 2 dJ⋆ ≤ √ n/ log µ (n), it is enough to choose R 2 ≤ n q(σ,d,p)-1 (log(n)) δ-2µq(σ,d,p) so that the first term in the upper-bound is smaller than the second.

  m = argmin m∈M {γ(ŝ m ) + pen(m)} imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 and the penalized estimator s = ŝ m,

Fig 1 .

 1 Fig 1. A partition of [0, 1] 2 into dyadic rectangles.

B∈B

  |P(B|A) -P(B)| , and the ρ-mixing (or maximal correlation) coefficient by ρ(A, B) = sup X,Y |Cov(X, Y )| Var(X)Var(Y ) imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Assumption ( P2 )

 P2 All the partitions in the collection M are built on a regular partition m ⋆ of [0, 1] d into cubes such that |m ⋆ | 2 ≤ n log 2 (n) By comparison with Theorem 3.1, a logarithmic factor then appears in the penalty (and then in the rate of estimation) under the sole condition of β-mixing but this term disappears under Assumption (Dβρ), (Dβ2-ρ) or (Dβ cond ), imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011 hence the factor log δ (n) with δ ∈ {0, 1}. Let us first present the oracle type inequality. Theorem 4.1. Let M be a collection of partitions satisfying Assumption (P2) and {L m } m∈M be a family of reals, greater than or equal to 1, such that m∈M exp(-L m |m|) ≤ 1.

Fig 3 .

 3 Fig 3. Level lines of the conditional densities to estimate.

Fig 4 .

 4 Fig 4. Top left: Level lines of the the conditional density s for Example 4. Top right: selected partition for c = 3 and n = 1000. Bottom : two sections of s (dark line) together with the corresponding sections of s(3) (light line).

  K1∈m1 k1∈{0,...,r} d 1 a 2 K1,k1 = 1 . Let us introduce B = (B K1,k1,l1 ) K1∈m1,k1,l1∈{0,...,r} d 1 and V = (V K1,k1,l1 ) K1∈m1,k1,l1∈{0,...,r} d 1 defined respectively by B K1,k1,l1 = φ K1,k1 φ K1,l1 ∞ and V K1,k1,l1 = φ K1,k1 φ K1,l1 . Let us set ρ(B) = sup a∈A K1∈m1 k1,l1∈{0,...,r} d 1 |a K1,k1 ||a K1,l1 |B K1,k1,l1 , define ρ(V ) in the same way, and set L(φ) = max{ ρ2 (V ), ρ(B)}. Then, Schwarz Inequality and the properties of the family (φ K1,k1 ) K1∈m1,k1∈{0,...,r} d 1 recalled in Section 6.1 provide L

  .10) Let us now bound the quadratic risk of s on Ω c η

  i=(2l+j-1)qn +1 Γ t (Z • i ) and χ • f,(j) (m, m ′ ) = sup t∈Sm+S m ′ t f =1 ν • (j) (t).We setV = ϑq δ n s ∞ /2 and B = 2q n (2r + 1) d D m ⋆ ι(f ) .

≤

  ..,r} d | t, Φ K,k | Φ K,k ∞ t 2 (r + 1) d (2r + 1) d |m ⋆ 1) d (2r + 1) d |m ⋆ | = (2r + 1) d D m ⋆ ι(f ) , As ν = ν • = ν • (1) + ν • (2) on Ω • , we have χ f (m, m ′ )1 Ω• ≤ χ • f,(1) (m, m ′ )1 Ω• + χ • f,(2) (m, m ′ )1 Ω• ,imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

∞ 0 P

 0 ([Z] + ≥ z)dz = ∞ 0 P(Z ≥ z)dz.Therefore, we obtain by integrating the previous inequalityE s [χ 2 f (m, m ′ ) -pen(m) -pen(m ′ )] + 1 Ω• ≤ 32 n e -|m ′ |L m ′ 2V 2 + 4B 2 n -1 and since m ′ ∈M e -|m ′ |L m ′ ≤ 1, we conclude that m ′ ∈M E s [χ 2 f (m, m ′ ) -pen(m) -pen(m ′ )] + 1 Ω• ≤ C(ϑ, s ∞ , r, d, ι(f ), b) log δ (n) n .6.5. Proof of Theorems 3.2 and 4.2It is sufficient to use the following theorem, proved in [Aka10] (Proposition 2 and Theorem 2).Theorem 6.1. Let J ∈ N, R > 0, σ ∈ (0, r + 1) d and p > 0 such that H(σ)/d > max{1/p -1/2, 0}.

  a strictly stationary Harris ergodic Markov chain (aperiodic, irreducible, positive Harris recurrent), then (Z i ) i∈Z is geometrically βmixing, i.e. Assumption (Dβ) is verified, if and only if it is geometrically ergodic (cf.[START_REF] Bradley | Basic properties of strong mixing conditions. A survey and some open questions[END_REF], Theorem 3.7). In the sequel, we will mainly be concerned with mixing assumptions possibly involving ρ-mixing and β-mixing at the same time. Under adequate hypotheses, Markov chains (always assumed to be homogeneous of order 1) provide examples of such processes:

• if (Z i ) i∈Z is a strictly stationary Harris ergodic Markov chain that is also reversible and geometrically ergodic, then (Z i ) i∈Z is both geometrically ρ-mixing and geometrically β-mixing

(cf. [Jon04], Theorem 2); • if (Z i ) i∈Z is a

strictly stationary, ergodic and aperiodic Markov chain satisfying the Doeblin condition, then (Z i ) i∈Z is uniformly ergodic, hence both geometrically ρ-mixing and geometrically β-mixing (cf. [Bra05], 119-121, or [MT93], Section 16.2). We refer to [DG83, Mok90, DT93, Dou94, AN98] for examples of stationary processes that are geometrically β-mixing or both geometrically β and ρ-mixing among commonly used time series such as nonlinear ARMA or nonlinear ARCH models.

  ) 1≤i≤n are i.i.d. uniformy distributed in [0, 1], (ǫ i ) 1≤i≤n are i.i.d. reduced and centered Gaussian variables, independent of the X i 's.

	400, 400) ǫ Example 4.

i imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

where (X i

Table 1

 1

  Ω• ≥ 32 D m ′ L 2 ′ ) -pen(m) -pen(m ′ ))1 Ω• ≥ 32 max 2V 2 u n , B 2 u 2 n 2 ≤ 2e -|m ′ |L m ′ -u .

	n	m ′	((5/2 +	√ 2) 2 /2)ϑq δ n s ∞ + 144 2 (2r + 1) d b 2 ι(f )	,
	we obtain
	P s (χ 2 f (m, m

imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

  With k ⋆ = max{k ∈ N s.t. 2 kd log δ (n)/n ≤ R 2 2 -2kH(σ) }, which is well-defined for R 2 ≥ log δ (n)/n, we obtain

	min m∈M rect	s -s m	2 + log δ (n)	|m| n
	m∈M rect	s -s m	2 + log δ (n)	|m| n
	≤ inf k∈N	s -s m k		2 + log δ (n)	|m k | n
	≤ C(d, p, r, σ) R 2 2 -2J⋆dτ σ/H(σ) + inf k∈N	R 2 2 -2kH(σ) + log δ (n)	2 kd n	.

imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

n ] and min c E s s -s(c) 2 n where the minimum is imsart-ejs ver. 2010/09/07 file: RevdenscondinhomogeneEJS.tex date: October 14, 2011

Acknowledgements

We are grateful to Cécile Durot for initiating this project and proofreading earlier versions of this paper. We also thank Yves Rozenhloc for his advice on the implementation of the penalized estimator and Jérôme Dedecker for his explanations on mixing.

Then, by grouping the ψ J,j having the same support, we get J∈m2 j∈{0,...,r} d 2

hence Lemma 6.3.

Proof of Lemma 6.4: The proof follows almost the same lines as the proof of Proposition 8 in [START_REF] Lacour | Adaptive estimation of the transition density of a Markov chain[END_REF]. Let ν ′ be the centered empirical process defined for

By construction of (Φ K,k ) K∈m ⋆ ,k∈{0,...,r} d , for all K, L ∈ m ⋆ and k, l ∈ {0, . . . , r} d , and all i ∈ {1, . . . , n},

= 1 K1=L1 1 (K2,k2)=(L2,l2) φ K1,k1 φ L1,l1 (X i ). (6.7)

Let t ∈ S m ⋆ \{0}, and for K 1 ∈ m ⋆ 1 and k 1 ∈ {0, . . . , r} d1 , let

It follows from (6.7) and Schwarz inequality that

where ν ′′ is the centered empirical process defined on L 2 ([0, 1] d1 ) by


Laurent Arnaud

Michel Gay

Jean-Marc Barnola

Paul Duval

Ilaging of firn and bubbly ice in coaxial relected light: a new technique for the characterization of these porous .edia

ABSTRAC.

A new technique for characterizing the structure offi n and bubbly ice is presented. This technique, based on observation of etched (sublimation) surfaces in coax ial reflected light, enables one to see simultaneously the pore network of the fin or bubbles in the ice and the crystal boundaries. At the same time, the main stages of image proces sing used to transform the initial photographs into clean binary images are described.

INTRODUCTION

Glaciological, physicochemical and mechanical studies of snow, urn and ice increasingly require structural character ization of the diTerent porous media involved. Stereology and image analysis have helped studies of structure and morphology to evolve from descriptive and qualitative observations to more objective, quantitative and systematic characterizations. The dificulty of obtaining numerous quality data is illustrated in a recent report by Shabtaie and Bentley (1994) in which information obtained "manu ally" by Fuehs (1959) is still used to describe the structure of fi n. To describe the structural change of this porous med ium with density, a large number of images must be obtained and must show the crystals, their boundaries and the pore network. The majority of studies requiring struc tural analysis cannot be satisfied by a uniquely descriptive image of porosity morphologies; the contacts between crys tals m ust also be taken into account. For example, it is across these contacts that the stresses are transmitted through the material, grain growth is directly linked to the grain-bound ary area and most physical phenomena in polycrystals must take into account the eTects of grain boundaries.

The technique used by Fuchs (1959) and more recently by [START_REF] Alley | Texture of polar fin for remote sensing[END_REF][START_REF] Alley | Concening the deposition and diagenesis of strata in polar fin[END_REF], which requires an image taken in trans mitted light to characterize grain structure and another in relected light to characterize pore structure, is difficult to implement. Using reconstructed images by Fuchs (1959) or [START_REF] Alley | Texture of polar fin for remote sensing[END_REF][START_REF] Alley | Concening the deposition and diagenesis of strata in polar fin[END_REF] and thin-section photographs by [START_REF] Gow | Time-temperature dependence of sintering in perennial iso thermal snow-packs[END_REF][START_REF] Gow | Restraints on thin section analysis of grain growth in un strained polycrystalline ice[END_REF] as a reference, we have developed a one-step photographic technique to observe the pore structure and grain boundaries. This technique can be used quickly in the field, as it requires neither impregnation (except for very low-cohesion samples) nor the preparation of thin sec tions. Furthermore, the resulting photograph represents the actual two-dimensional structure of the section (stereologic analysis) and includes a suficient number of grains to be statistically representative of the studied medium.

In glaciology, the most widely used technique for obtain ing two-dimensional images is based on the observation of thin sections in pLain polarized transmitted light and between crossed polarizers (Bader and others, 1939;Kinosi ta and Wakahama, 1959;[START_REF] Gow | On the rates of growth of grains and crystals in South Polar fim[END_REF][START_REF] Gow | Time-temperature dependence of sintering in perennial iso thermal snow-packs[END_REF]. Because of dif f er ences in orientation, the ice crystals in the thin sections show dierent colours due to double refraction when placed between crossed polarizers. This observation technique can be used to characterize crystal size, orientation and shape. However, it does not allow quantitatively accurate visuali zation of pores and the image is not truly two-dimensional information on the media. The image obtained in trans mitted light is built up from information taken across the whole thickness of the section. This is a major problem, if the stereological results are to be used to assign parameters to the material structure for modelling [START_REF] Underwood | Jlallilalive stereology[END_REF][START_REF] Kry | Quantitative stereological analysis of grain bonds in snow[END_REF]Coster and Chermant, 1989). Most models assume that the initial image is purely two-dimensional and that it comes from a random section of the medium under investi gation. Also, this technique, simple and quick to use for ice, becomes more diicult to use for snow and firn. In the case onow-cohesion snow, it is necessary to strengthen the struc ture by illing the pores with a supercooled liquid and freez ing it at a low temperature before thinning down the section (Kinosita and Wakahama, 1959;[START_REF] Gow | On the rates of growth of grains and crystals in South Polar fim[END_REF][START_REF] Narita | Specific surface of deposited snow. I[END_REF][START_REF] Narita | Specific surface of deposited snow[END_REF][START_REF] Alley | Densification and recrystallization offin at Dome . East Antarctica[END_REF][START_REF] Perla | Preparation of section planes in snow specimens[END_REF]Good, 1983). Apart from the problem that impregnation can change the structu re of the medium and that the material used may not spread uni formly through the pores [START_REF] Perla | Preparation of section planes in snow specimens[END_REF], the technique is long and complicated. Even so, impregnation is the only techni que that can be used to analyse the structure of low-density samples (less than 0.4-0.5 Mg m 3). But this technique can not be used with samples without cohesion between grains (or aggregates of grains). So, for these samples (density lower than ",0.2 �1g m 3), the whole structure cannot be di rectly analysed. The most interesting way to study the upper part of the snow cover is to collect grains (or aggregates of grains), to store them in liquid iso-octane to stop all meta morphism and to analyse the shape and size of grains by op tical microscopy [START_REF] Brun | An eicient method for a delayed and accu rate characterization of snow grains from natural snowpacks[END_REF].

Another technique of ten used is observation in annular reflected light [START_REF] Kry | Quantitative stereological analysis of grain bonds in snow[END_REF][START_REF] Perla | Preparation of section planes in snow specimens[END_REF]Good, 1983). This method is used on impregnated samples and allows visuali-zation of the pore structure. The contrast between ice and porcs is improved by dyeing the ice or the impregnating li quids. The image obtained represents the truly two-dimen sional surface and shows the ice and pore phases but not the grain boundaries.

To summarize, if we wished to obtain all the informa tion, two techniques would have to be used: thin-section observations in transmitted light only for grain boundaries and impregnated thick-section observations in annular refl ected light for void shapes.

Some authors [START_REF] Alley | Texture of polar fin for remote sensing[END_REF][START_REF] Alley | Concening the deposition and diagenesis of strata in polar fin[END_REF]Fuchs, 1959;Narita and others, 1978) have used both of these techniques successively on the same sample and then superimposed the two images. In this way, they have been able to obtain all of the informa tion on the grains and pore structures. The use of both tech niques is more comprehensive but also more difficult to implement. The use of annular light necessitates the use of impregnated samples, leading to the problems mentioned above. In fact, due to the high energy at the grain boundary, it is possible o enhance the contrast of these boundaries by "etching" the ice surface by dissolving or subliming the sur face (Cow, 1987).

A method of preparing etched-section images of ice (from close-of density to 0.9 Mg m 3) has recently been des cribed by Nashida (1996). This method is based on obser vations in reflected light of ice surfaces rubbed with alcohol.

The same idea of preferential etching of the crystal boundaries was used to develop our technicques. By etching only using sublimation, we are not obliged to attack directly the sample surface either chemically or mechanically. More over, our soft etching enables us to work with a low-density samplc of in (up to 0.5 Mg m 3). Contrast is also obtained by observing the etched surface by using the coaxial epi scopy technique instead of annular light.

To extract the structural parameters from the primary pictures, specific image-processing tools were developed. This last stage of the characterization of the structure en ables us to obtain binary images. Utilizing these binary images, many structural parameters can be calculated to prO\'ide data for dif f erent cation (Anaud and others, 1998) and pore closure, grain growth and interactions with densification processes, and the e\'olution of structural parameters with cli matic condi tions.

COAXIAL EPISCOPY

The coaxial episcopy technique has long been used in me tallurgy. The technique uses relected light to observe relief on flat, reflecting surfaces. The samples are polished and then generally subjected to chemical etching to highlight the diff erences in composition, phase and orientation of the crystals. The differences in contrast are caused by difer ences in the amount of reflected light received by the lens. These diferences arc related either o diferent levels of light absorption or, as in our case, to the different orientations of the illuminated surfaces (Fig. I).

Using this principle, we cut sections of about I cm in thickness from the in cores. These sections are then care fully polished using a microtome. The samples were at tached to glass slides for microtoming with very littlc water displayed on the slides. This stage cannot modify the struc ture of the samplc surface, because we work with thick samples and there is not sufficient water o percolate through the pores to the surface.

At this stage, the ice and pores can be clearly observed in reflected light using a Leiea MZ8 binocular microscope fitted with coaxial episcopic light (Fig. 2). The observed con trast is caused by differences in refl ection between the "po lished" fl at areas (ice) and the eoncave areas (pores). The parallel straight lines arc left by the blade of the microtome. The large number of these striations is not specific to our blade, but it is due to the sensitivity of our observation technique (coaxial episcopy) which enhances relief of the surface.

Unlike the annular light technique, the coaxial episcopy

I Imm I
Fi. 2. Black and white photograph of a thick section using coaxial eJiscopy bfo re sublimation ( Km 105). technique does not require the sample to be impregnated. The density must be higher than 0.4 Mg m -3. Below this limit, the grains are pulled out during sample microtoming. This limit is much less critical than that for transmitted light techniques which require thinning of the sections to a thickness in the order of only a few grains.

The main limitation of the coaxial episcopy technique is the small observation area. The maximum area is linked to the area of uniform light, which is 10 x 7 mm 2 . To compen sate for this problem, we have taken several adjacent photo graphs to build a suficiently large observational area (40 x 40 mm 2 or more). This area contains a suicient num- Ar naud and others: Imaging of jirn and bubbly ice in coaxial rflected light ber of grains to be statistically representative of the studied medium.

SAMP LE SUBLIMATION

In a similar way to surface etching in metallurgy, the subli mation of sections of ice or fi rn allows one to examine grain boundaries [START_REF] Gow | Time-temperature dependence of sintering in perennial iso thermal snow-packs[END_REF][START_REF] Gow | Restraints on thin section analysis of grain growth in un strained polycrystalline ice[END_REF][START_REF] Nishida | Three-dimensional observations of ice crystal characteristics in polar ice sheels[END_REF]. The sublimation takes place preferentially on the highly disor dered zones of the polycrystal. The grain boundaries will appear as grooves. As shown in Figure 3, the grooves appear as black lines in coaxial episcopy (the contrast is of the same origin as for the pores). By adjusting the pressure and temp erature conditions during sublimation, the grain bound aries can be made to appear without altering the pore data (size and shape).

The optimal sublimation conditions are diicult to fore see and tuning is required before each new set of measure ments. For example, the photographs presented in this paper were obtained in the fi eld after sublimation of samples placed in a plastic sheath at a temperature T = -12°C and atmospheric pressure (63 5 mbar for Vostok Station) for 12 hours. Better control of the surface-sublimation phenom ena would greatly improve our method but this will require a better understanding of the phenomena. Our objective is to be able to calculate sublimation time as a function of pres sure and temperature, so that grain boundaries can be etched without altering the size and the shape of the ice phase. Firn is the best medium for this observational technique; no impregnation is required, work is carried out on thick sections and the pores and crystals are observed with one camera sh ot. For ice analysis, this technique saves time by working with thick sections polished only on one face and provides an image of the truly two-dimensional surface.

It is also possible to use this technique using thin sec tions. A test was therefore conducted to compare the image obtained with our technique to that obtained by the classic transmitted-light technique (Fig. 4a andb), thereby check ing that sublimation does not alter the information concern ing the size and shape of grains. As examples, the structural evolution of fi rn as a func tion of density (or depth) at Km 105 is given in Figure 3 and firn images from the three studied sites (Vostok, Km 105 and Km 200) are compared at the same density in Fig ure 5. The photographs in both Figures 3 and5 represent only half the size of one negative area.

By using many adjacent images (nine-25 photographs per level), a suiciently large image is obtained for each of the density levels (e.g. Fig. 6). These images are statistically representative of our two-dimensional medium, providing the basic data required for the systematic analysis of the evolution of fin by image processing. In this study, we have chosen to emphasize the real quality of untreated photo graphs before all numerical treatment, but we should also like to describe briefl y the main stages of image-processing used to obtain clean binary images.

IMAGE PRO CESSING

The initial negatives obtained at Vostok were digitized with a video camera and transferred to a personal computer (Fig. 7a). Then, the gray-level images were cleaned and con verted into a binary image using tools developed on the IPS (Image Processing System) on the Linux system.

As a irst stage, the image-processing allows removal of the straight lines left by the blade of the microtome, lines due to the mosaic reconstructions and all the small surface imperfections due to sublimation. Figure 7 shows the main stages of image-processing. To remove the straight lines (Fig. 7a), an adapter low fi lter on the Fast l�urier Tr ansform of the gray-level image (Fig. 7b) is used to obtain the filtered image (Fig. 7c). From these filtered gray-level images, two stages based on morphological operations are necessary to obtain the final binary images: one for the porosities and the other one for the crystals. Concening the treatment of the porosity and the erasure of small surface imperfections due to sublimation, morphological operations (Euclidean erosion and geodesic dilation) are used. These operations enable us to obtain a binary image composed with two phases: porosity and ice. The second stage consists of mor phological transformations of the gray-level image (Fig. 7c). First, the morphological "top-hat" transformation (Coster and Chermant, 1989) enables to extract grain boundaries. Secondly, from the last image of grain bound aries, a skeleton transformation reduces the width of the boundary to one pixel. This image of the skeleton of grain boundaries is added to the binary image of the porosity and ice to obtain the final clean binary images (Fig. 7d).

Image-processing also provides the various and numer ous structural parameters useful for modelling. These images are presently analysed with two objectives: the char acterization of structural changes (size and shape of crystals and porosities network) as a function of densit y and the in n uence of cl i matic conditions on these changes (average annual temperature and accumulation). As an example of an analysed parameter, Figure 8 shows the e\'olution of the crystal-size distributions in the Vostok fin and ice duri ng the densification and grain-grow th processes.

CON CLUSI ON

The structural analysis of fin and bubbly iee requires char acterization of ice crystals, pore network and interfaces. The proposed method consists of subliming polished samples in order to re\'eal the grain boundaries, before observing them by coaxial episcopy. Observation in coaxial reflection shows relief differences on our sample surfaces, which makes it pos sible to see the pores and grain boundaries "etched" by sub limation. This procedure simultaneously gives images of the crystal (grain boundaries) and of the porous network that are truly representative of a two-dimensional section of the fin. In addition, this technique requires neither impregna tion nor thin sections, so it is easy to use in the field and in troduces no measurement bias. Another advantage is the short time required to obtain a structural image for a given density, making it possible to collect successive images throughout a progressively thinning section (serial-section ing technique). The study of these images would give a real three-dimensional structure of fin pores and the real size and shape of ice crystals. This work was initiated to obtain images of the structure of rin and ice, in order to extract parameters required by our various models (densiication, pore closure and dif f usion pores). In addition to the issues concening fin densifica tion and the observation of pores and grains, this technique could be applied to the characterization of ice cracks.

  . Diagram describing the origin of contrast due to dier ences m surface orientation.

cFigure 3 .Fig. 4 .

 34 Figure 3. Structural evolution f firn as afunction f densiy at Km 105. (a) Depth: 9 m and relative density: 0.62 Mg m -3 ; (b) Depth: 20 m and relative density: 0.66 M g m -3; (c) Depth: 61 m and relative density: 0.89 M g m -3 ; ( d) Depth: 95 m and relative density: 0.97 Mg m -3.

c

  Fig. 5. Firn structures of three diJerent sites (Km 105. Km 200 and Vastok) at a relative density ab out 0.73 Mg m -3. ( a) Km 105; depth: 28 m; (b) Km 200; depth: 31 m; (c) Vastok ( Bh?): 45 m.

  5mm

Fig. 6 .

 6 Fig. 6. Image reconstructed jrom nine adjacent photographs ( Km 105), constituting the basic data jor image processing (initial mosaic).

  Examples of sublimed saIllples observed by coaxial episcopy This technique has actually been tested in the laboratory and in the field, on non-impregnated samples of fi rn and ice, over a range of densities from 0.5 to 0.91 Mg m 3. A set of measurements was made at ostok (East Antarctica) during the winter of 1995-96. It was possible to record, in the field, images of samples during drilling of a 250 m drill core (Bh7) and also from two other cores stored at Vostok but originating from the Mirny-Vostok traverse at points located 105 and 200 km from Miny. In 1 month, wc were able to use our technique on 40 levels of the Vostok core (Bh7) between 7 and 250 m (between 0.5 and 0.91 Mg m �3), on 17 levels of the Km 105 a c core between 5 and 100 m and on 17 levels of the Km 200 core over the same depth range.

Fig. .

 . Fig. . Main stages f image processingfrom the initial gray-Level image to thefinal cleaned binay image. (a) Initial mosaic digitalized from six adjacent negatives; (b) Fast Fourier Traniformation of initial mosaic ( a); (c) Filtered image obtained with an adapter 10wJilter on the Fast Fourier Tranorm of the gray-level image (b); (d) Final binary image.

  Evolution f the distributions qf the relative crystal size during crystal growth and densJication at Vastok with R: crystal equivalent radius (rR2 is the two-dimensional crystal area).

AC KNOWLEDGEMENTS

This work was supported by PNEDC (P rogramme National d' E tude du Climat), CNRS and by the CEC (Commission of Eropean Communities) Environment program.

We acknowledge the Russian Antarctic Expeditions, the Arctic and Antarctic Research Institute, the U. S. National Science Foundation (Division of Polar Programs) and IFRTP (Institut Frant; ais pour la Recherche et la Te chnolo gie Polaires) for logistic support at Vostok Station. We are also grateful to M. Creseveur from the LGGE for his advice and assistance.

The LGGE has an agreement with the Universit€Joseph Fourier de Grenoble.