On a class of recursive estimators for spatially dependent observations

: We investigate the mean squared error and the asymptotic nor- mality for a class of recursive kernel estimators based on a sample of spatially dependent observations. Our main result provides suﬃcient condi- tions for a spatial version of a recursive estimator introduced by Hall and Patil (1994) to satisfy a central limit theorem. The results are stated for strongly mixing random ﬁelds in the sense of Rosenblatt (1956) and for weakly dependent random ﬁelds in the sense of Wu (2005). and asymptotic normality under weak and strong dependence conditions. The ﬁrst studies


Introduction
In statistics, the nonparametric estimation of probability density functions of continuous random variables is a basic and central problem. From a given sample of observations, the main goal for a practitionner is to understand the mecanism from which the observations have been generated. In the last several decades, this question has attracted much attention among statisticians since it is of considerable interest in many applied fields such as forecasting, computer vision and machine learning. Among the plethora of nonparametric density estimators is the kernel density estimator introduced by Parzen [31] and Rosenblatt [34] which received considerable attention in nonparametric estimation for time series. More precisely, if (X 1 , ..., X n ) is a sample (observations) drawn from some univariate distribution with an unknown probability density f with respect to the Lebesgue measure on R then the Parzen-Rosenblatt density estimator of f is defined for any positive integer n and any x in R by where K is a density function and the bandwidth h n is a positive parameter which converges to zero such that nh n goes to infinity. The bandwidth h n is the most dominant parameter in the kernel density estimator since it controls its amount of smoothness. In fact, if h n is small then the variance of the estimator is large while the bias is small. This leads to a nonsmooth estimated density.
On the other, if h n is large then the estimated density will be much smoother (small variance) but with a large bias leading to an unsatisfactory estimation. So, in practice, a trade-of between the variance and the bias must be found and the number of publications which are devoted to this crucial question in the literature is very extensive and is still a subject of many works in the statistic community (see for example [8], [17], [22], [38]). From a theoretical but also practical point of view, it is important to investigate asymptotic properties of density estimators when the number n of observations goes to infinity. For example, the consistency and the asymptotic normality of the estimator are very important in order to get pointwise estimation and confidence intervals for the target density f . In his seminal paper, Parzen [31] proved that when the observations (X 1 , ..., X n ) are i.i.d. and the bandwidth h n goes to zero such that nh n goes to infinity then (nh n ) 1/2 (f P R n (x 0 ) − E[f P R n (x 0 )]) converges in distribution to the normal law with zero mean and variance f (x 0 ) R K 2 (t)dt as n goes to infinity and this result was extended by Wu and Mielniczuk [44] for causal linear processes with i.i.d. innovations and by Dedecker and Merlevède [11] for strongly mixing sequences. Previously, Bosq, Merlevède and Peligrad [6] established a central limit theorem for the kernel density estimator f n when the sequence (X i ) i∈Z is assumed to be strongly mixing but the bandwidth parameter h n is assumed to satisfy h n ≥ Cn −1/3 log n (for some positive constant C) which is stronger than the bandwidth parameter assumption in [11], [31] and [44].
In many situations, practicians are also interested by the relationship between some predictors and a response. This is a natural question and a very important task in statistics. The objective is to find a relation between a pair of random variables X (predictor) and Y (response) using a given sample (X i , Y i ) 1 i n drawn from the unknown law of (X, Y ). A very popular tool to handle this problem is the kernel regression estimator introduced by Nadaraya [30] and Watson [41]. More formally, let N be a positive integer and assume that (X i , Y i ) 1 i n are identically distributed R N × R-valued sequence of random variables such that Y i = R(X i , η i ) where R is an unknown functional and (η i ) i∈Z are i.i.d. R Nvalued random variables with zero mean and finite variance and independent of (X i ) i∈Z . Let f be the marginal density function of X 0 . If r is the (unknown) regression function defined for any x in R N by r(x) = E [R(x, η 0 )] if f (x) = 0 and r(x) = E[Y 0 ] if f (x) = 0 then the Nadaraya-Watson regression estimator r NW n of r is defined for any x in R N by (1. 2) The literature on the asymptotic properties of r NW n for time series is very expansive. One can refer to Lu and Cheng [24], Masry and Fan [26], Robinson [33], Roussas [36] and many references therein. Kernel nonparametric methods are still very popular and fairly well established in the statistical community but despite their power, the data streams problem, which refers to data sets that continuously and rapidly grow over time, present new challenges. In order to handle such data sets, several recursive versions of the Parzen-Rosenblatt estimator (1.1) have been introduced (see for example [3], [12], [20], [43], [47]). For example, if (w k ) k 1 is a nonincreasing sequence of positive real numbers satisfying k 1 w k = ∞ and (h k ) k 1 is a sequence of positive real numbers going to 0 as n goes to infinity (bandwidth parameters) then the resursive kernel density estimator f HP n of Hall and Patil [20] is defined by This estimator is recursive in the sense that it satisfies f HP n+1 (x) = (1 − γ n+1 )f HP n (x) + γ n+1fn+1 (x) (1.4) where γ n := wn . Such a property endows recursive estimators with a decisive computational advantage because they can be easily updated as new data items arrive over time. More precisely, in order to obtain the estimation f HP n+1 (x) at time n + 1, using the recursive equation (1.4), it is sufficient to combine the estimation f HP n (x) at time n (which is known at time n + 1) with the estimationf n+1 (x) at time n + 1 based on the single observation X n+1 . In fact, a non-recursive estimator must be fully recomputed whenever a new observation is collected. This clearly represents a drawback in a data stream context compared to the recursive approach. The class (1.3) contains the recursive estimators introduced by Wolverton and Wagner [43] and Deheuvels [12] but also a renormalized version of the one introduced by Wegman and Davies [42] and another class of estimators introduced by Amiri [3]. It contains also the (non-recursive) Parzen-Rosenblatt estimator (1.1) when h i = h n and w i = 1 for any 1 i n. In this work, our aim is to investigate asymptotic properties for a spatial version of the Hall and Patil estimator (1.3) in terms of mean squared error and asymptotic normality under weak and strong dependence conditions. The first studies that focused on a recursive version of the Parzen-Rosenblatt estimator were presented by [12], [43] and [47] and later by [20], [25], [29] and many others. Actually, many papers in the literature are devoted to the asymptotic properties of recursive kernel density and regression estimators for i.i.d. observations. There are also some published papers on the asymptotic properties of recursive kernel density and regression estimators for dependent (weakly dependent and strongly mixing) data. One can refer for example to [4], [7], [18], [19], [25], [36], [39], [40] and others.
In our context, we deal with spatial data which is modelized using finite realizations of dependent random fields indexed by Z d where d is a positive integer. More precisely, let N be a positive integer and let (Ω, F, P) be a probability space. We consider a stationary R N -valued random field (X k ) k∈Z d such that the law μ 0 of X 0 is absolutely continuous with respect to the Lebesgue measure λ N on R N and we denote by f the probability density function of μ 0 with respect to λ N . Given two sub-σ-algebras U and V of F, recall that the α-mixing coefficient introduced by Rosenblatt [35] is defined by Let p be fixed in [1, +∞]. The strong mixing coefficients (α 1,p (n)) n 0 associated to (X k ) k∈Z d are defined by where |Γ| is the number of elements in Γ, the collection F Γ is the σ-algebra σ(X k ; k ∈ Γ) and the distance ρ is defined for any subsets Γ 1 and Γ 2 of Z d by We say that the random field (X k ) k∈Z d is strongly mixing if lim n→∞ α 1,p (n) = 0. Moreover, we are going to consider also Bernoulli fields defined for any k ∈ Z d by R m -valued random variables and m is a positive integer. The class of random fields that (1.5) represents is huge and it includes many commonly used linear and nonlinear processes (see Wu [46] for a review). Let (ε k ) k∈Z d be an i.i.d. copy of (ε k ) k∈Z d and let X * k be the coupled version of X k defined by Note that X * k is obtained from X k by replacing ε 0 by its copy ε 0 . For any positive integer and any R -valued random variable Z ∈ L p (Ω, F, P) with p > 0, we denote is the Euclidian norm on R . Following Wu [45] and El Machkouri et al. [16], we define the physical dependence measure coefficient δ k,p := X k − X * k p as soon as X k is p-integrable for p 2. Physical dependence measure should be seen as a measure of the dependence of the function G (defined in (1.5)) in the coordinate zero. In some sense, it quantifies the degree of dependence of outputs on inputs in physical systems and provide a natural framework for a limit theory for stationary random fields (see [16]). In particular, it gives mild and easily verifiable conditions (see condition (H2)(ii) below) because it is directly related to the data-generating mechanism.

Main results
Let Λ 0 = ∅, s 0 = 0 ∈ Z d and Λ n = {s 1 , ..., s n } ⊂ Z d for n 1. Let (w sn ) n 1 and (h sn ) n 1 be two nonincreasing sequences of positive real numbers such that (w sn h −N sn ) n 1 is nondecreasing, h sn goes to 0 as n goes to infinity and n 1 w sn = ∞. Let also K : R N → R + be a function (called a kernel) such that R N K(t)dt = 1 and sup x∈R N K(x) < ∞. Assume that K is Lipschitz and satis- R N -valued random variables with zero mean and finite variance and independent of (X k ) k∈Z d and consider the regression model for any v ∈ R N and any 1 j n. One can notice that if Φ(u) = 1 for any u ∈ R then f n,Φ reduces to the spatial version f n,1 of the recursive kernel density estimator of f introduced by Hall and Patil [20] and defined for any x ∈ R N by Moreover, for particular choices of the weights (w sn ) n 1 , the estimator (2.2) reduces to the recursive estimators introduced by [3], [4], [12] or [43]. In particular, one can check that f n,Φ satisfies the following recursive equation 3) is the spatial version of the recursive equation (1.4). It lays emphasis on that the update of f n,Φ at time n can be done from f n−1,Φ at time n − 1 and the new single observation X sn . This is a definitive advantage over the spatial version of the non-recursive Parzen-Rosenblatt estimator f P R n defined by (1.1) since it is necessary to consider the whole sample (X s1 , ..., X sn ) in order to compute f P R n at any time n. In this work, we consider also the following class of spatial semi-recursive kernel regression estimator r n,Φ of r Φ defined for any x in R N by which contains the first two semi-recursive kernel regression estimators introduced by Ahmad and Lin [2] and Devroye and Wagner [13] for time series (i.e. for d = 1) but also the class of semi-recursive kernel regression estimators considered by Amiri [4]. Since r n,Φ is defined from f n,Φ and f n,1 , it inherits the good properties in term of computation time of the recursive estimators f n,Φ and f n,1 and consequently, in a data stream setting, it has a decisive advantage over the spatial version of the non-recursive Nadaraya-Watson estimator defined by (1.2). Now, we are going to present our main contributions. For j ∈ {2, 4}, we adopt the notation and for any sequences (a n ) n 1 and (b n ) n 1 of real positive numbers, we denote a n b n if and only if there exists κ > 0 (not depending on n) such that a n κb n . Recall that (w sn ) n 1 and (h sn ) n 1 are two nonincreasing sequences of positive real numbers such that (w sn h −N sn ) n 1 is nondecreasing, h sn goes to 0 as n goes to infinity and n 1 w sn = ∞ and keep in mind that K : R N → R + is a function (kernel) such that R N K(t)dt = 1 and |K| ∞ := sup t∈R N K(t) < ∞.
For any integer n 1 and any (p, q) ∈ Z 2 , we denote also h p si w q si and we consider the following assumptions: sn . Moreover, one of the following condition holds: (i) (X k ) k∈Z d is strongly mixing and k∈Z d |k| (ii) The function f Φ is twice differentiable with bounded second derivatives.
(iii) For any k ∈ Z d \{0}, the law of (X 0 , X k ) is absolutely continuous with respect to the Lebesgue measure on R N × R N and there exists Assumptions (H1) and (H3) are classical in the context of recursive kernel estimators (see [4], [25], [28], [42] and many others). In (H2), we assume that the bandwidth parameter h sn satisfies a condition sligthly stronger than the usual minimal condition assumed in the non recursive i.i.d. setting (i.e. nh N sn → ∞). However, this fact seems to be inherent to the case of recursive estimators since a condition like nh N (1+ε) sn → ∞ for some ε > 0 is assumed in many contributions for dependent data (see for example [4], [1], [25], [28] or [42]).
For any x in R N , we denote Our first result gives the asymptotic variance of the estimator f n,Φ defined by (2.1).

Proposition 1. Assume that (H1) and (H3) hold and there exists
nw 2 sn and one of the following conditions is satisfied: (i) (X k ) k∈Z d is strongly mixing and k∈Z d |k| where ν 2 (θ) and ν 4 (θ) are defined by (2.5) then for any x ∈ R, where σ 2 Φ (x) is defined by (2.6). We obtain also the convergence to zero of the mean square error of f n,Φ .
Then, for any The main contribution of this paper is the following central limit theorem.

Theorem 1. If (H1), (H2) and (H3) hold then for any
where σ 2 Φ (x) is defined by (2.6). One can notice that Theorem 1 is an extension of Theorem 1 in [1] where the case of strongly mixing time series is considered. In fact, with our notations, if d = 1 and Φ(u) = 1 for any u ∈ R then f n,Φ reduces to the recursive kernel density estimator f n,1 introduced by Hall and Patil [20]. In this case, we have ν 2 (θ) = ν 4 (θ) = 1 and (H2)(ii) holds as soon as k>0 = ∞ which are exactly the conditions imposed in Theorem 1 in [1]. Using Theorem 1, we derive the asymptotic normality for the recursive estimator r n,Φ defined by (2.4).

Theorem 2. Assume that (H1), (H2) and (H3) hold. If f is Lipschitz and twice differentiable with bounded second derivatives then for any
Theorem 2 is also an extension of Theorem 2.1 in [36] where the asymptotic normality of the semi-recursive kernel regression estimator for time series (i.e. d = 1) introduced by Ahmad and Lin [2] is obtained under more restrictive conditions on the bandwidth parameter and the strong mixing coefficients. Using Theorem 2 and Proposition 2, the condition nh N +4 sn → 0 can be imposed for the control of the bias of the estimator and leads immediately to the following result.

Preliminary lemmas
This section is devoted to the presentation of several technical lemmas and propositions which are key tools in the proof of the main contributions in section 4. For any real x, we also define x = x + 1, where x is the largest integer less or equal than x. Lemma 1. Let (a k ) k∈Z d be a family of real numbers such that a sn goes to some value a ∈ R as n goes to infinity.
Proof of Lemma 1. For any positive integers i and n, we denote The proof Lemma 1 is complete.
The following lemma will be usefull in order to compute the asymptotic variance of the estimator f n,Φ (see Propositions 1 and 4).
Proof of Lemma 2. Let x ∈ R N and let n be a positive integer. It is obvious that By Theorem 1A in [31], we derive The proof of Lemma 2 is complete.

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For any ∈ {1, 2}, any 1 i n and any sequence (m n ) n 1 of positive integers, we define In the other part, using (H3)(iii), for any 1 i, j n such that i = j we have The proof of Lemma 3 is complete.
Proof of Lemma 4. Let i and j be two positive integers such that i = j and Note that the second term of the right hand side of (3.5) can be dealt with using Lemma 3. Therefore, we focus on the first part of the right hand side only. Let L > 1 be fixed then Using Cauchy-Schwarz's inequality, we obtain Optimizing (3.6) with respect to L, we derive Notice that if 1 N and nh N sn → ∞ then m d n = o (n). Proof of Lemma 5. Let 1 , 2 and r be positive constants such that r > 2 / 1 and let (m n ) n 1 be the sequence defined for any integer n 1 by Since v n → ∞, we have m n → ∞ as n goes to infinity. Moreover, Since v n m n , we have The proof of Lemma 5 is complete.
h N si and we obtain Δ si . The proof of Lemma 6 is complete.

Proposition 3. Let M be a positive integer and let
for some θ ∈]0, +∞] then for any positive integer n and any family (c k ) k∈Λn of real numbers and any (p, q) ∈ [2, +∞[×]0, +∞] such that p + q 2 + θ, we have Proof of Proposition 3. Let M be a positive integer and let x in R N and 1 i n be fixed.
We follow the same lines as in the proof of Proposition 1 in [16]. Let 2 p < 2 + θ and denote by H i the measurable function such that Let τ be a bijection from Z to Z d and in Z be fixed. We define the projection operator P by for any integrable function f , where F = σ ε τ (j) ; j ≤ . One can notice that the operator P depends on the bijection τ . The proof of the following technical result is postponned to section 5.
Using Lemma 7, we obtain So, for any L > 0, we get the bound Optimizing this last inequality in L, we get the following bound Now, we are going to obtain another bound for P W i p . Let 0 and i 1 be two integers. We denote by Γ i, the set of all k in Z d such that |s i − k| = and we define On the lattice Z d we define the lexicographic order as follows: if u = (u 1 , . . . , u d ) and v = (v 1 , . . . , v d ) are distinct elements of Z d , the notation u < lex v means that either u 1 < v 1 or for some k in {2, . . . , d}, u k < v k and u = v for 1 < k. We consider the bijection Given that (D i, ) 1 is a martingale difference sequence with respect to the filtration (G i, ) 1 , we apply Burkholder's inequality ( [10], remark 6, page 85) and we obtain Arguing as before, for any L > 0, we derive Optimizing this last inequality on L and noting that s i − τ i ( ) = −τ 0 ( ), we obtain Consequently, we get (3.12) Since ( n i=1 c si P W i ) ∈Z is a martingale difference sequence with respect to the filtration (F ) ∈Z , the Burkholder inequality (see [10], remark 6, page 85) implies (3.13) Moreover, by the Cauchy-Schwarz inequality, we have (3.14) Now, keeping in mind that P is defined from the bijection τ and using (3.10) and (3.12), we have Similarly, we have also Combining (3.13) and (3.14) with the last two bound above, we get

M. El Machkouri and L. Reding
The proof of Proposition 3 is complete.

Proofs of the main results
Now, we denote by V(Z) the variance of any square-integrable R-valued random variable Z and we consider a sequence (m n ) n 1 of positive integers. For any x ∈ R and any ∈ {1, 2}, denote where Φ : R → R is a measurable function and H i,mn = σ(η si , ε si−k ; |k| m n ). First, we note that Proposition 1 is a particular case of the following result.

Proposition 4. Assume that (H1) and (H3)(iii) hold. Let
nw 2 sn and one of the following conditions is satisfied: (ii) (X k ) k∈Z d is of the form (1.5) and k∈Z d |k| whereγ is defined bỹ Proof of Proposition 4. Let x ∈ R N and let (p, q) ∈ {1, 2} 2 be fixed. Using the notations (3.2) and (4.1), for ∈ {p, q}, we have Keeping in mind that A n,0,1 = (nw sn ) −1 n i=1 w si , we get So, we obtain Moreover, for any 1 i n, Using Lemma 2 and Lemma 3, we derive So, using Lemma 1 and (H1), we derive Now, we are going to prove that Since γ 1, using the inequality 2ab a 2 + b 2 , we have Using (4.6) and keeping in mind that h N (1−τ ) sn n i=1 w 2 si nw 2 sn , we get lim n→∞ E 1,n = 0. Now, we are going to control the term E 2,n when (X k ) k∈Z d is assumed to be strongly mixing. Using Rio's inequality ( [32], Theorem 1.1), we have for any 1 i < j n, u} for any u ∈ [0, 1] and any ∈ {i, j}.

(see Lemma 3) and (H1), we have also
where H i,mn = σ(η si , ε si−k ; |k| m n ) for any 1 i n and any ∈ {p, q} and noting that Δ Using (3.12), we obtain (4.9) Consequently, using Lemma 4, we obtain for any i = j, where γ is defined by (3.4). Since γ 1 and using the inequality 2ab a 2 + b 2 , we derive

Using (H1) and keeping in mind that
(4.10) Consequently, we derive Similarly, one can notice that Now, we have to control the last term We are going to prove that Keeping in mind (3.12) and arguing as in (4.9), we have Using (H1), we obtain Using (4.4), we obtain (4.11). Now, arguing as in (4.10), we get So, (4.12) holds.
Finally, if Φ p ∞ < ∞ and Φ q ∞ < ∞ thenγ = 1 and the proof follows exactly the same lines as above. The proof of Proposition 4 is complete.
Proof of Proposition 2. Let x ∈ R d and let n be a positive integer. We have Moreover, Using Taylor's formula, we derive 4+N . The proof of Proposition 2 is complete.
Proof of Theorem 1. We are going to split the proof in two parts. In the first part, we deal simultaneously with the strong mixing case and the weakly mixing case (see (4.15) below) whereas, in the second part, the two dependence conditions are investigated separately.

First part
Let n be a positive integer and x ∈ R N be fixed. One can notice that where H i,mn = σ(η si , ε si−k ; |k| m n ).
Note that U si and U sj are independent if |s i − s j | > 2m n . Using (H1) and Proposition 3, we derive (4.14) From now on, we denote and it suffices to prove the asymptotic normality of the partial sums n −1/2 n i=1 Z si as n goes to infinity. Let (ξ k ) k∈Z d be independent normal random variables independent of (X k ) k∈Z d and (η k ) k∈Z d and such that E[ξ k ] = 0 and . Let 1 i n and define T si = n −1/2 Z si and Ξ si = n −1/2 ξ si . One can notice that n i=1 Ξ si is a gaussian random variable with zero mean. If (X k ) k∈Z d is strongly mixing then Z si = U si and Keeping in mind (2.6) and (H1) and using Lemma 1, Lemma 2 and Lemma 3, we get If (X k ) k∈Z d is of the form (1.5) then Z si = U si and applying (3.12), we get Using (H1), we derive So, we get also (4.16) when (X k ) k∈Z d is of the form (1.5). Let ψ be any measurable function from R to R. For any 1 i j n, we introduce the notation Let h : R → R be a three times continuously differentiable function such that max 0 i 3 h (i)

M. El Machkouri and L. Reding
Using Lindeberg's idea [23] (see also [9]), we have Applying Taylor's formula, we get Since, for any 1 i n, the random variable ξ si is gaussian with zero mean and variance E[Z 2 si ], we have Moreover, since (w sn h −N sn ) n 1 is nondeacreasing, we get Using (H1) and E[Δ 2 si ] 1 (see Lemma 3), we get Using (H1), we get For any 1 i < j n and any function ψ from R to R, we define also Using Taylor's formula, we have In order to obtain (4.18), we have to prove and

Second part
First, we assume that (X k ) k∈Z d is strongly mixing. We are going to prove (4.20).

Lemma 8. It holds that sup
Consequently, using (H1), we obtain Combining (4.24), (4.25) and (4.26), we obtain (4.21) and (4.22). Now, it suffices to prove (4.23). Let β 1 be a positive integer. For any 1 j n, the notation E β [Z sj ] will stand for the conditional expectation of Z sj with respect to the σ-algebra σ (Z s ; < j and |s − s j | β). Then, where The next result can be found in [27].

Lemma 9.
Let U and V be two σ-algebras and let X be a random variable which is measurable with respect to U.
Since (X k ) k∈Z d is strongly mixing, we have Z si = U si for any 1 i n. If Φ ∞ = ∞ then using Lemma 9 with p = 1 and r = (2 + θ)/2, we derive From Lemma 6, we have Consequently, using (H1), we get Finally, keeping in mind that ν 2 (θ) 1,∞ (β). (4.28) Consequently, since k∈Z d |k| Keeping in mind that we obtain Let L > 0, then Since Z si = U si for any 1 i n, we derive Arguing as in (4.26) and Lemma 9, we derive Recursive estimators for random fields 4613 Using (4.17), (4.28) and (H1), we get Moreover, If Φ ∞ = ∞ then using (H1) and (4.17), and (w sn h −N sn ) n 1 is nondecreasing) and consequently Finally, it means that Since ν 4 (θ) 1, we have Consequently, denoting ×ε n ε n and finally, Optimizing in L, we get So, we obtain (4.23). Now, we assume that (X k ) k∈Z d is of the form (1.5). As before, we have to prove (4.20), (4.21), (4.22) and (4.23). Now, we have Z si = U si for any 1 i n. Moreover, U si ans U sj are independent as soon as |s i − s j | > M n where M n = 2m n . So, (4.20) holds since Arguing as in (4.24) and (4.25), we have The proof of the following lemma is postponed to section 5.

Lemma 10. For any positive integer n,
Consequently, we obtain (4.21) and (4.22). In order to finish the proof, it suffices to prove (4.23). Let L > 0 be fixed. Keeping in mind that U si and U sj are independent if |s i − s j | > M n , we have +o(1).
Using Lemma 10, we obtain (4.23). The proof of Theorem 1 is complete.
In the proof of Theorem 1, the asymptotic normality of the estimator f n,Φ is obtained using the Lindeberg's method based on the stability of the standard normal law. This approach seems to be superior to the so-called Bernstein's method (see for example [5] and [21]) since it allows us to obtain mild conditions on the weak and strong dependent coefficients of the considered random field. This fact is of theoretical importance and has already been observed in [1], [14] and [15]. where Φ(x) = λ 1 Φ(x) + λ 2 for any x in R.
h N sn . Consequently, using Theorem 1, we get the result. The proof of Theorem 2 is complete.
One can notice that the asymptotic normality of the regression estimator r n,Φ obtained in the proof of Theorem 2 is a direct consequence of Theorem 1.
In some sense, it means that Theorem 1 is quite a deep result since it contains both the asymptotic normality of the kernel density estimator f n,1 and that of the regression estimator r n,1 .
Proof of Theorem 3. Let x ∈ R N such that f (x) > 0. Then, according to Theorem 2, we have . Applying Proposition 2, we have Recall that r Φ (x) = fΦ(x) f (x) . Then, for n sufficiently large, we have Finally, using Slutsky's lemma, we obtain The proof of Theorem 3 is complete.