MohamedELMACHKOURI,RaduS.STOICA Asymptoticnormalityofkernelestimatesinaregressionmodelforrandomelds

Asymptotic normality of kernel estimates in a regression model for random elds

Mohamed EL MACHKOURI, Radu S. STOICA

Université Lille 1 Laboratoire Paul Painlevé 59655 Villeneuve d'Ascq Cedex France

Résumé

We establish the asymptotic normality of the regression estimator in a xed-design setting when the errors are given by a eld of dependent random variables. The result applies to martingale-dierence or strongly mixing random elds. On this basis, a statistical test that can be applied to image analysis is also presented.

Résumé

Nous établissons la normalité asymptotique de l'estimateur de la ré- gression lorsque la grille est xée et les erreurs sont données par un champ de variables aléatoires dépendantes. Le résultat s'applique pour des champs de type diérence de martingales ou fortement mélangeants.

Dans ce contexte, on présente un test statistique qui peut être utilisé en traitement d'images.

Mathematics Subject Classications (2000): 60G60, 60F05, 62G08 Keywords: Nonparametric regression estimation, asymptotic normality, kernel estimator, strongly mixing random eld.

Our aim in this paper is to establish the asymptotic normality of a regression estimator in a xed-design setting when the errors are given by a stationary eld of random variables which show spatial interaction. LetZ^d, d≥1 denote the integer lattice points in thed-dimensional Euclidean space. By a stationary random eld we mean any family(ε_k)_k∈Zd of real-valued random variables de- ned on a probability space(Ω,F,P)such that for any(k, n)∈Z^d×N^∗and any (i1, ..., in) ∈(Z^d)ⁿ, the random vectors (εi₁, ..., εi_n) and (εi₁+k, ..., εi_n+k) have the same law. The regression model which we are interested in is

Yi=g(i/n) +εi, i∈Λn ={1, ..., n}^d (1) wheregis an unknown smooth function and(εi)_i∈Zdis a zero mean and square- integrable stationary random eld. Let K be a probability kernel dened on R^d and (hn)n≥1 a sequence of positive numbers which converges to zero and which satises (nhn)n≥1 goes to innity. We estimate the function g by the kernel-type estimatorgn dened for anyxin [0,1]^d by

gn(x) = X

i∈Λn

Y_iK

x−i/n hn

X

i∈Λ_n

K

x−i/n h_n

. (2)

In a previous paper, El Machkouri [10] obtained strong convergence of the es- timator g_n(x) with optimal rate. However, most of existing theoretical non- parametric results for dependent random variables pertain to time series (see Bosq [4]) and relatively few generalisations to the spatial domain are available.

Key references on this topic are Biau [2], Carbon et al. [5], Carbon et al. [6], Hallin et al. [12], [13], Tran [26], Tran and Yakowitz [27] and Yao [29] who have investigated nonparametric density estimation for random elds and Altman [1], Biau and Cadre [3], Hallin et al. [14] and Lu and Chen [17], [18] who have studied spatial prediction and spatial regression estimation.

Letµbe the law of the stationary real random eld (εk)_k∈Zd and consider the projectionf fromR^Z

d toRdened byf(ω) =ω₀ and the family of translation operators(T^k)_k∈ZdfromR^Z

dtoR^Z

d dened by(T^k(ω))i=ωi+k for anyk∈Z^d and any ω in R^Z

d. Denote by B the Borel σ-algebra of R. The random eld (f ◦T^k)_k∈Zd dened on the probability space (R^Z

d,B^Zd, µ) is stationary with the same law as (εk)_k∈Zd, hence, without loss of generality, one can suppose that (Ω,F,P) = (R^Z

d,B^Zd, µ)and εk =f◦T^k. An elementA of F is said to be invariant ifT^k(A) =Afor any k∈Z^d. We denote byI theσ-algebra of all measurable invariant sets. On the latticeZ^d we dene the lexicographic order as follows: ifi = (i1, ..., id)andj = (j1, ..., jd)are distinct elements ofZ^d, the notationi <lexj means that either i1< j1 or for somepin{2,3, ..., d},ip< jp

and iq = jq for 1 ≤ q < p. Let the sets {V_i^k;i ∈ Z^d, k ∈ N^∗} be dened as

follows:

V_i¹={j∈Z^d;j <lexi}, and fork≥2

V_i^k=V_i¹∩ {j∈Z^d;|i−j| ≥k} where |i−j|= max

1≤l≤d|il−jl|.

For any subsetΓ ofZ^d deneFΓ=σ(εi;i∈Γ)and set E_|k|(ε_i) =E(ε_i|F_V|k|

i

), k∈V_i¹.

Note that Dedecker [8] established the central limit theorem for any stationary square-integrable random eld(εk)_k∈Zd which satises the condition

X

k∈V₀¹

kεkE_|k|(ε0)k1<∞. (3) A real random eld(Xk)_k∈Zd is said to be a martingale-dierence random eld if for any m in Z^d, E(Xm|σ(Xk;k <lex m) ) = 0 a.s. The condition (3) is satised by martingale-dierence random elds. Nahapetian and Petrosian [21] dened a large class of Gibbs random elds(ξk)_k∈Zd satisfying the stronger martingale-dierence property: E(ξm|σ(ξk;k 6= m) ) = 0 a.s. for any m in Z^d. Moreover, for these models, phase transition may occur (see [19],[20]).

Given two sub-σ-algebrasU andV, dierent measures of their dependence have been considered in the literature. We are interested by one of them. The strong mixing (orα-mixing) coecient has been introduced by Rosenblatt [25] and is dened by

α(U,V) = sup{|P(U∩V)−P(U)P(V)|, U ∈ U, V ∈ V}.

Denote by ]Γ the cardinality of any subset Γ of Z^d. In the sequel, we shall use the following non-uniform mixing coecients dened for any (k, l, n) in (N^∗∪ {∞})²×Nby

αk,l(n) = sup{α(FΓ₁,FΓ₂), ]Γ1≤k, ]Γ2≤l, ρ(Γ1,Γ2)≥n},

where the distance ρ is dened by ρ(Γ₁,Γ₂) = min{|i−j|, i ∈ Γ₁, j ∈ Γ₂}. We say that the random eld(ε_k)_k∈Zd is strongly mixing (orα-mixing) if there exists a pair(k, l)in (N^∗∪ {∞})²such thatlim_n→∞α_k,l(n) = 0.

The condition (3) is satised by strongly mixing random elds. For example, one can construct stationary Gaussian random elds with a suciently large polynomial decay of correlation such that (5) holds ([9], p. 59, Corollary 2).

2 Main results

First, we recall the concept of stability introduced by Rényi [22].

Denition. Let(Xn)_n≥0 be a sequence of real random variables and letX be dened on some extension of the underlying probability space (Ω,A,P). LetU be a sub-σ-algebra ofA. Then (Xn)n≥0 is said to converge U-stably to X if for any continuous bounded functionϕand any bounded andU-measurable variable Z we have limn→∞E(ϕ(Xn)Z) =E(ϕ(X)Z).

For any B > 0, we denote by C¹(B) the set of real functions f continuously dierentiable on[0,1]^d such that

sup

x∈[0,1]^d

α∈Mmax|Dα(f)(x)| ≤B,

where

Dα(f) = ∂^αˆf

∂x^α₁¹... ∂x^α_d^d and M={α= (αi)i∈N^d; ˆα=

d

X

i=1

αi≤1}.

In the sequel we denotekxk= max_1≤k≤d|x_k|for anyx= (x₁, ..., x_d)∈[0,1]^d. We make the following assumptions on the regression functiongand the prob- ability kernel K:

A1) The probability kernel K fulls R

K(u)du = 1 and R

K²(u)du < ∞. K is also symmetric, non-negative, supported by [−1,1]^d and satises a Lipschitz condition|K(x)−K(y)| ≤rkx−yk for anyx, y∈[−1,1]^d and somer >0. In addition there existsc, C >0such thatc≤K(x)≤Cfor anyx∈[−1,1]^d.

A2) There existsB >0 such thatgbelongs toC¹(B). We consider also the notations:

σ²= Z

R^d

K²(u)du and η= X

k∈Z^d

E(ε₀ε_k|I).

The following proposition (see [10]) gives the convergence ofEg_n(x)to g(x). Proposition 1 Assume that the assumption A2) holds then

sup

x∈[0,1]^d

sup

g∈C¹(B)

|Eg_n(x)−g(x)|=O[h_n].

By proposition 3 in [8], we know that under condition (3), the random variable η belongs toL¹. Our main result is the following.

Main theorem. Ifnh^d+1_n → ∞and the condition(3)holds then for anyk∈N^∗ and any distinct pointsx1, ..., xk in[0,1]^d, the sequence

(nh_n)^d/2







g_n(x₁)−Eg_n(x₁) ...

gn(xk)−Egn(xk)







−−−−−−→L

n→+∞ σ√

η





 τ⁽¹⁾

...

τ^(k)





 (I-stably)

where σ² = R

R^dK²(u)du and (τ⁽ⁱ⁾)_1≤i≤k ∼ N(0,Ik) where Ik is the identity matrix. Moreover,(τ⁽ⁱ⁾)_1≤i≤k is independent ofη=P

k∈Z^dE(ε0εk|I).

As a consequence of this theorem, we obtain the following result for strongly mixing random elds.

Corollary. Let us consider the following assumption X

k∈Z^d

Z α1,∞(|k|) 0

Q²_ε0(u)du <∞ (4) where Q_ε0 denotes the cadlag inverse of the function H_ε0 : t → P(|ε₀|> t). Then(4)implies(3) and also the main theorem.

Remark. Ifε₀is(2 +δ)-integrable for someδ >0then the condition

∞

X

m=1

m^d−1α^δ/(2+δ)_1,∞ (m)<∞ (5) is more restrictive than condition(4).

In order to use the main theorem for establishing condence intervals, one needs to estimateη. It is done by the following result established in [8].

Proposition 2 Assume that the condition (3) holds. For any N ∈ N^∗, set GN ={(i, j)∈Λn×Λn;|i−j| ≤N}. Letρn be a sequence of positive integers satisfying:

n→+∞lim ρ_n= +∞ and lim

n→+∞ρ^3d_nE(ε²₀(1∧n^−dε²₀) = 0 Then

1 n^dmax



1, X

(i,j)∈G_ρn

εiεj





−−−−−−→P

n→+∞ η.

3 Proofs

3.1 Proof of the main theorem

Letxin[0,1]^d andn≥1 be xed. For anyiinΛ_n, denote ai(x) =K

x−i/n h_n

and bi(x) = ai(x) qP

j∈Λna²_j(x) .

Denote also

v_n(x) =

s (nh_n)^d P

i∈Λnai(x)× sP

i∈Λ_na²_i(x) P

i∈Λnai(x).

Without loss of generality, we consider the casek = 2 and we refer tox1 and x2 as xandy. Let λ1 and λ2 be two real numbers such thatλ²₁+λ²₂ = 1and letx, y∈[0,1]^d such thatx6=y. One can notice that

(nhn)^d/2

σ [λ₁(g_n(x)−Eg_n(x)) +λ₂(g_n(y)−Eg_n(y))] = X

i∈Λn

˜

s_i(x, y)ε_i

where˜si(x, y) = (λ1vn(x)bi(x) +λ2vn(y)bi(y))/σ. Lemma 1 Letx, y∈[0,1]^d be xed. Ifnh^d+1_n → ∞ then

n→+∞lim 1 (nhn)^d

X

i∈Λn

a_i(x)a_i(y) =δ_xyσ² (6) and

n→+∞lim 1 (nhn)^d

X

i∈Λn

a_i(x) = 1 (7)

whereδxy equals1 ifx=y and0 ifx6=y.

Proof of Lemma 1. In the sequel, we denoteψ(u) = _h¹d nK

x−u h_n

K

y−u h_n

and In(x, y) =R

[0,1]^dψ(u)du, we have In(x, y) = X

i∈Λn

Z

R_i/n

ψ(u)du

= X

i∈Λn

λ(R_i/n)ψ(c_i)withc_i∈R_i/n

= X

i∈Λn

n^−dψ(ci)

where R_i/n =](i₁−1)/n, i₁/n]×...×](id−1)/n, i_d/n] and λ is the Lebesgue measure onR^d. Letϕx(u) = (x−u)/hn, for anyv in[0,1]^d, we have

d(K◦ϕ_x)(u)(v) = −1 hn

d

X

i=1

v_i

d

X

j=1

∂K

∂uj

(ϕ_x(u)).

Using the assumptions on the kernelK and noting that dψ(u) = 1

h^d_n

d(K◦ϕ_x)(u)×K(ϕ_y(u)) +d(K◦ϕ_y)(u)×K(ϕ_x(u))

we derive that there existsc >0such that sup_u∈[0,1]dkdψ(u)k ≤ch^−(d+1)n . So, it follows that

1 (nhn)^d

X

i∈Λn

ai(x)ai(y)−In(x, y)

=

X

i∈Λn

n^−d(ψ(i/n)−ψ(ci))

≤ sup

u∈[0,1]^d

kdψ(u)k X

i∈Λn

n^−dki/n−c_ik_∞

≤ c h^d+1n

X

i∈Λn

n^−(d+1)

= c

nh^d+1n

−−−−−−→

n→+∞ 0.

Moreover,

In(x, y) = Z

[0,1]^d

1 h^d_nK

x−u hn

K

y−u hn

du

= Z

ϕ_x([0,1]^d)

K(u)K

u+y−x hn

du.

So, by the dominated convergence theorem, we obtain

n→+∞lim In(x, y) =δxyσ²

and consequently (6) holds. The proof of (7) follows the same lines. The proof of Lemma1is complete.

Using Lemma 1 and denotingκ²_xy= (λ₁+λ₂)²δ_xy+ 1−δ_xy, we derive

n→+∞lim X

i∈Λn

˜

s²_i(x, y) =κ²_xy= 1 (since x6=y). So, denoting

si(x, y) = s˜i(x, y) qP

j∈Λns˜²_j(x, y) ,

it suces to prove the convergenceI-stably ofP

i∈Λnsi(x, y)εi to√

ητ0 where τ0 ∼ N(0,2). In fact, we are going to adapt the proof of the central limit theorem by Dedecker [8].

For any i in Z^d, let us dene the tail σ-algebra Fi,−∞ = ∩k∈N^∗F_Vk

i (we are going to note F−∞ in place of F0,−∞) and consider the following proposition established in [8].

Proposition Theσ-algebraI is included in the P-completion ofF−∞. Letf be a one to one map from[1, N]∩N^∗to a nite subset ofZ^d and(ξi)_i∈Zd

a real random eld. For all integerskin [1, N], we denote S_f(k)(ξ) =

k

X

i=1

ξ_f(i) and S^c_f(k)(ξ) =

N

X

i=k

ξ_f(i)

with the convention S_f(0)(ξ) = S_f(N^c ₊₁₎(ξ) = 0. To describe the set Λn = {1, ..., n}^d, we dene the one to one mapf_n from[1, n^d]∩N^∗ toΛ_nby: f_nis the unique function such that for1≤k < l≤n^d, f(k)<_lexf(l). From now on, we consider two independent elds(τ_i⁽¹⁾)_i∈Zd and(τ_i⁽²⁾)_i∈Zd of i.i.d. random vari- ables independent of(ε_i)_i∈Zd andI such thatτ₀⁽¹⁾ and τ₀⁽²⁾ have the standard normal lawN(0,1). We introduce the two sequences of eldsX_i=s_i(x, y)ε_iand γ_i =s_i(x, y)τ_i√

η whereτ_i=τ_i⁽¹⁾+τ_i⁽²⁾∼ N(0,2). Lethbe any function from RtoR. For0≤k≤l≤n^d+ 1, we introducehk,l(X) =h(S_f(k)(X) +S_f(l)^c (γ)). With the above convention we have that h_k,nd+1(X) = h(S_f(k)(X)) and also h_0,l(X) =h(S_f(l)^c (γ)). In the sequel, we will often writeh_k,l instead ofh_k,l(X) ands_iinstead ofs_i(x, y). We denote byB₁⁴(R)the unit ball ofC_b⁴(R): hbelongs toB₁⁴(R)if and only if it belongs toC⁴(R)and satisesmax_0≤i≤4kh⁽ⁱ⁾k_∞≤1. 3.1.1 Lindeberg's decomposition

LetZ be a I-measurable random variable bounded by 1. It suces to prove that for allhin B₁⁴(R),

n→+∞lim E Zh(S_f(nd)(X))

=E Zh

(λ1τ₀⁽¹⁾+λ2τ₀⁽²⁾)√ η

.

We use Lindeberg's decomposition:

E Zh

h(S_f(nd)(X))−h

(λ1τ₀⁽¹⁾+λ2τ₀⁽²⁾)√ ηi

=E Z

h(S_f(nd)(X))−h S_f(nd)(γ)

=E Z[h_nd,n^d+1−h_0,1]

=

n^d

X

k=1

E(Z[h_k,k+1−h_k−1,k]).

Now,

hk,k+1−h_k−1,k=hk,k+1−h_k−1,k+1+h_k−1,k+1−h_k−1,k. Applying Taylor's formula we get that:

h_k,k+1−h_k−1,k+1=X_f(k)h⁰_k−1,k+1+1

2X_f(k)² h⁰⁰_k−1,k+1+R_k and

h_k−1,k+1−h_k−1,k=−γ_f(k)h⁰_k−1,k+1−1

2γ²_f(k)h⁰⁰_k−1,k+1+rk

where|Rk| ≤X_f(k)² (1∧ |X_f(k)|)and|rk| ≤γ_f(k)² (1∧ |γ_f(k)|). Since(X, τi)_i6=f(k) is independent ofτ_f(k), it follows that

E

Zγf(k)h⁰_k−1,k+1

= 0 and E

Zγ_f(k)² h⁰⁰_k−1,k+1

=E

Zs²_f(k)ηh⁰⁰_k−1,k+1

Hence, we obtain

E Zh

h(Sn(X))−h

(λ1τ₀⁽¹⁾+λ2τ₀⁽²⁾)√ ηi

=

n^d

X

k=1

E(ZX_f(k)h⁰_k−1,k+1) +

n^d

X

k=1

E Z

X_f(k)² −s²_f(k)ηh⁰⁰_k−1,k+1 2

!

+

n^d

X

k=1

E(R_k+r_k).

Arguing as in Rio [24], it is easily proved that

n→+∞lim

n^d

X

k=1

E(|Rk|+|rk|) = 0.

Let us denoteC_N = [−N, N]^d∩Z^dfor any positive integerN. If we deneη_N = P

k∈CN−1E(ε0εk|I), the upper bound E|η −ηN| ≤ 2P

k∈V₀^NE|E(ε0εk|I)| holds. Hence according to condition (3) and the above proposition, we derive lim_N→+∞E|η−η_N|= 0and consequently we have only to show

N→+∞lim lim sup

n→+∞

n^d

X

k=1

E(ZXf(k)h⁰_k−1,k+1) +

+E Z

X_f(k)² −s²_f(k)η_Nh⁰⁰_k−1,k+1 2

!!

= 0.

(8) 3.1.2 First reduction

First, we focus on Pn^d k=1E

ZX_f(k)h⁰_k−1,k+1

. For all N in N^∗ and all inte- gerkin[1, n^d], we dene

E_k^N =f([1, k]∩N^∗)∩V_f(k)^N and S_f(k)^N (X) = X

i∈E^N_k

Xi.

For any functionΨfromRtoR, we deneΨ^N_k−1,l = Ψ(S_f(k)^N (X) +S_f(l)^c (γ))(we shall apply this notation to the successive derivatives of the function h). Our aim is to show that

lim

N→+∞lim sup

n→+∞

n^d

X

k=1

E Z

X_f(k)h⁰_k−1,k+1 −

− X_f(k)

S_f(k−1)(X)−S_f(k)^N (X)

h⁰⁰_k−1,k+1

= 0.

(9)

First, we use the decomposition

X_f(k)h⁰_k−1,k+1=X_f(k)h⁰_k−1,k+1^N +X_f(k)

h⁰_k−1,k+1−h⁰_k−1,k+1^N .

We consider a one to one map m from [1,|E^N_k |]∩N^∗ to E_k^N and such that

|m(i)−f(k)| ≤ |m(i−1)−f(k)|. This choice ofmensures thatS_m(i)(X)and S_m(i−1)(X) areF_V|m(i)−f(k)|

f(k) -measurable. The fact that γ is independent of X together with proposition 3 in [8] imply that

E

ZX_f(k)h⁰

S_f(k+1)^c (γ)

=E h⁰

S_f(k+1)^c (γ)

E ZE X_f(k)|F_−∞

= 0.

Therefore|E

ZX_f(k)h⁰_k−1,k+1^N

|equals

|E_k^N|

X

i=1

E

ZX_f(k)

h⁰

S_m(i)(X) +S_f(k+1)^c (γ)

−h⁰

S_m(i−1)(X) +S_f(k+1)^c (γ) .

Since S_m(i)(X) and S_m(i−1)(X) are F_V|m(i)−f(k)|

f(k) -measurable, we can take the conditional expectation ofX_f(k) with respect toF_V|m(i)−f(k)|

f(k) in the right hand side of the above equation. On the other hand the function h⁰ is 1-Lipschitz, hence

|h⁰

S_m(i)(X) +S^c_f(k+1)(γ)

−h⁰

S_m(i−1)(X) +S^c_f(k+1)(γ)

| ≤ |Xm(i)|.

Consequently, the term

E

ZX_f(k)

h⁰

S_m(i)(X) +S_f(k+1)^c (γ)

−h⁰

S_m(i−1)(X) +S_f(k+1)^c (γ) is bounded by

E|X_m(i)E|m(i)−f(k)| X_f(k)

| and

|E

ZX_f(k)h⁰_k−1,k+1^N

| ≤

|E^N_k|

X

i=1

E|Xm(i)E|m(i)−f(k)|(X_f(k))|.

Hence,

n^d

X

k=1

E

ZX_f(k)h⁰_k−1,k+1^N

≤

n^d

X

k=1

|sf(k)|

|E_k^N|

X

i=1

|sm(i)|E|εm(i)E|m(i)−f(k)|(ε_f(k))|

≤

n^d

X

k=1

|s_f(k)| X

j∈V₀^N

|s_j+f(k)|E|εjE_|j|(ε0)|

≤A X

j∈V₀^N

kεjE_|j|(ε0)k1<+∞ (A∈R^∗+)

where (by Lemma 1) we used the fact that sup

i∈Λn

|si|=O 1

(nhn)^d/2

(10)

and X

i∈Λn

|si|=O

(nhn)^d/2

. (11)

Since (3) is satised, this last term is as small as we wish by choosing N large enough. Applying again Taylor's formula, it remains to consider

X_f(k)(h⁰_k−1,k+1−h⁰_k−1,k+1^N ) =X_f(k)(S_f(k−1)(X)−S_f(k)^N (X))h⁰⁰_k−1,k+1+R⁰_k, where|R⁰_k| ≤2|Xf(k)(S_f(k−1)(X)−S_f(k)^N (X))(1∧ |S_f(k−1)(X)−S_f(k)^N (X)|)|. It follows that

n^d

X

k=1

E|R⁰_k| ≤2





n^d

X

k=1

|s_f(k)|



E |ε0| X

i∈Λ_N

|si||εi|

!

1∧ X

i∈Λ_N

|si||εi|

!!

≤2A E |ε0| X

i∈ΛN

|εi|

!

1∧ X

i∈ΛN

|si||εi|

!!

(A∈R^∗+).

Keeping in mind thats_i →0 as n→ ∞ and applying the dominated conver- gence theorem, this last term converges to zero asn tends to innity and (9) follows.

3.1.3 The second order terms It remains to control

W₁=E



Z

n^d

X

k=1

h⁰⁰_k−1,k+1 X_f(k)²

2 +X_f(k)

S_f(k−1)(X)−S_f(k)^N (X)

−s²_f(k)η_N 2

!

. We consider the following sets: (12)

Λ^N_n ={i∈Λn;d(i, ∂Λn)≥N} and I_n^N ={1≤i≤n^d;f(i)∈Λ^N_n}, and the functionΨfromR^Z

d toRsuch that Ψ(ε) =ε²₀+ X

i∈V₀¹∩CN−1

2ε0εi.

Forkin [1, n^d], we setD^N_k =ηN −Ψ◦T^f(k)(ε). By denition ofΨand of the setI_n^N, we have for anykinI_n^N

Ψ◦T^f(k)(ε) =ε²_f(k)+ 2ε_f(k)(S_f(k−1)(ε)−S_f(k)^N (ε)).

Therefore forkin I_n^N

s²_f(k)D^N_k =s²_f(k)η_N −X_f(k)² −2X_f(k)(S_f(k−1)(X)−S_f(k)^N (X)).

Sincelim_n→+∞n^−d|I_n^N|= 1, it remains to prove that

lim

N→+∞lim sup

n→+∞

E



Z

n^d

X

k=1

s²_f(k)h⁰⁰_k−1,k+1D^N_k



= 0. (13) 3.1.4 Conditional expectation with respect to the tailσ-algebra

Now, we are going to replace D_k^N by E D_k^N|F_f(k),−∞. We introduce the expression

H_n^N =

n^d

X

k=1

E

s²_f(k)Zh⁰⁰_k−1,k+1[Ψ◦T^f(k)(ε)−E(Ψ◦T^f(k)(ε)|F_f(k),−∞)]

.

For sake of brevity, we have writtenh⁰⁰_k−1,k+1instead ofh⁰⁰_k−1,k+1(X). Using the stationarity of the eld we get that

H_n^N =

n^d

X

k=1

E

s²_f(k)Z(h⁰⁰_k−1,k+1◦T^−f(k))(X)[Ψ(ε)−E(Ψ(ε)|F_−∞)]

.

For any positive integerp, we decomposeH_n^N in two parts

H_n^N =

n^d

X

k=1

J_k¹(p) +

n^d

X

k=1

J_k²(p),

where

J_k¹(p) =E

s²_f(k)Z(h⁰⁰_k−1,k+1^p ◦T^−f(k))[Ψ(ε)−E(Ψ(ε)|F_−∞)]

andJ_k²(p)equals to E

s²_f(k)Z[h⁰⁰_k−1,k+1◦T^−f(k)−h⁰⁰_k−1,k+1^p ◦T^−f(k)](X)[Ψ(ε)−E(Ψ(ε)|F−∞)]

.

From the denition ofh⁰⁰_k−1,k+1^p , we infer that the variableh⁰⁰_k−1,k+1^p ◦T^−f(k)(X) is F_V^p

0-measurable. Therefore, we can take the conditional expectation of Ψ(ε)−E(Ψ(ε)|F_−∞) with respect to F_V^p

0 in the expression of J_k¹(p). Now, the backward martingale limit theorem implies that

p→+∞lim E|E(Ψ(ε)|F_V^p

0)−E(Ψ(ε)|F−∞)|= 0 and consequently

p→+∞lim lim sup

n→+∞

n^d

X

k=1

J_k¹(p)

= 0.

On the other hand

n^d

X

k=1

J_k²(p)

≤E



2∧X

|i|<p

s²_f(i)|εi|



|Ψ(ε)−E(Ψ(ε)|F_−∞)|

.

Hence, applying the dominated convergence theorem, we conclude that H_n^N tends to zero asntends to innity. It remains to consider

W2=E



Z

n^d

X

k=1

h⁰⁰_k−1,k+1s²_f(k)E(D^N_k |F_f(k),−∞)



.

3.1.5 Truncation

For any integerkin [1, n^d]and anyM in R⁺ we introduce the two sets B_k^N(M) =E(D^N_k|F_f(k),−∞)11_{|ηN−E(Ψ◦T}f(k)(ε)|Ff(k),−∞)|≤M

and

B^N_k(M) =E(D^N_k |F_f(k),−∞)−B_k^N(M).

The stationarity of the eld ensures thatE|B^N_k(M)|=E|B^N₁ (M)|for any kin [1, n^d]. Now, applying the dominated convergence theorem, we have

M→+∞lim E|B^N₁(M)|= 0.

It follows that

Mlim→+∞

n^d

X

k=1

E

h⁰⁰_k−1,k+1s²_f(k)B^N_k(M)

= 0.

Therefore instead ofW2 it remains to consider

W3=E



Z

n^d

X

k=1

h⁰⁰_k−1,k+1s²_f(k)B_k^N(M)



.

3.1.6 An ergodic lemma

The next result is the central point of the proof.

Lemma 2 For allM in R⁺, we introduce

βN(M) =E [ηN−E(Ψ(ε)|F−∞)]11_{|ηN−E(Ψ(ε)|F−∞)|≤M} I

. Then

lim

M→+∞β_N(M) = 0 a.s. and lim

n→+∞E

β_N(M)−

n^d

X

k=1

s²_f(k)B_k^N(M)

= 0.

Proof of Lemma2. Let

u(ε) = [ηN−E(Ψ(ε)|F−∞)]11_{|ηN−E(Ψ(ε)|F−∞)|≤M}.

Using the functionu, we writeβN(M) =E(u(ε)|I). The fact thatβN(M)tends to zero asM tends to innity follows from the dominated convergence theorem.

In fact

M→∞lim u(ε) =η_N −E(Ψ(ε)|F_−∞)

andu(ε)is bounded by|ηN−E(Ψ(ε)|F_−∞)|which belongs toL¹. This implies that

M→∞lim βN(M) =E ηN −E(Ψ(ε)|F_−∞)

I a.s.

SinceI is included in theP-completion ofF_−∞(see the above proposition) and keeping in mind thatηN isI-measurable, it follows that

M→∞lim βN(M) =ηN−E(Ψ(ε)|I) a.s.

By stationarity of the random eld, we know that E(ε₀ε_k|I) = E(ε₀ε_−k|I) which implies that

E(Ψ(ε)|I) = X

k∈CN−1

E(ε₀ε_k|I) =η_N and the result follows.

We are going to prove the second point of Lemma2. First note that Bk(M) = [ηN−E(Ψ◦T^f(k)(ε)|F_f(k),−∞)]₁1_{|ηN−E(Ψ◦T}f(k)(ε)|Ff(k),−∞)|≤M

=u◦T^f(k)(ε).

Consequently

n^d

X

k=1

s²_f(k)B_k^N(M) = X

i∈Λn

s²_iu◦Tⁱ(ε).

Finally, the proof of lemma 2 is completed by the following lemma which is proved in Section 5.

Lemma 3

n→∞lim

X

i∈Λ_n

s²_iu◦Tⁱ(ε)−E(u(ε)|I) ₂

= 0.

As a direct application of lemma 2, we see that

E



Z

n^d

X

k=1

h⁰⁰_k−1,k+1s²_f(k)βN(M)





≤E|βN(M)|

is as small as we wish by choosingMlarge enough. So instead ofW3we consider

W4=E



Z

n^d

X

k=1

h⁰⁰_k−1,k+1s²_f(k)[B_k^N(M)−βN(M)]



.

3.1.7 Abel transformation

In order to controlW4, we use the Abel transformation:

W4=E ^nd

X

k=1 k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]

!

Z(h⁰⁰_k−1,k+1−h⁰⁰_k,k+2)

+E



Zh⁰⁰_nd,n^d+2 n^d

X

k=1

s²_f(k)[B_k^N(M)−βN(M)]



.

Now E



Zh⁰⁰_nd,n^d+2 n^d

X

k=1

s²_f(k)[B^N_k (M)−βN(M)]





≤E

βN(M)−

n^d

X

k=1

s²_f(k)B_k^N(M) .

Then applying lemma2, we obtain

n→+∞lim E



Zh⁰⁰_nd,n^d+2 n^d

X

k=1

s²_f(k)[B_k^N(M)−βN(M)]





= 0.

Therefore it remains to prove that for any positive integerN and any positive realM,

n→+∞lim E ^nd

X

k=1 k

X

i=1

s²_f(i)[B^N_i (M)−βN(M)]

!

Z(h⁰⁰_k−1,k+1−h⁰⁰_k,k+2)

= 0.

3.1.8 Last reductions

We are going to nish the proof. We use the same decomposition as before:

h⁰⁰_k,k+2−h⁰⁰_k−1,k+1=h⁰⁰_k,k+2−h⁰⁰_k,k+1+h⁰⁰_k,k+1−h⁰⁰_k−1,k+1. Applying Taylor's formula

h⁰⁰_k,k+2−h⁰⁰_k,k+1=−γf(k+1)h⁰⁰⁰_k,k+2+t_k and

h⁰⁰_k,k+1−h⁰⁰_k−1,k+1=X_f(k)h⁰⁰⁰_k−1,k+1+T_k

where |tk| ≤ γ_f(k+1)² and |Tk| ≤ X_f(k)² . To examine the remainder terms, we consider:

E





n^d

X

k=1

s²_f(k)

k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]

! Zε²_f(k)



.

The denition ofB_i^N(M)and ofβN(M)enables us to write for all integer kin [1, n^d],

k

X

i=1

s²_f(i)|B_i^N(M)−βN(M)| ≤2M.

Therefore

E

n^d

X

k=1 k

X

i=1

s²_f(i)[B_i^N(M)−βn(M)]

!

s²_f(k)Zε²_f(k)11_|εf(k)|>K

≤2M E ε²₀11_|ε0|>K and applying the dominated convergence theorem this last term is as small as we wish by choosingK large enough. Now, for all K inR⁺, Lemma2 ensures that

n→+∞lim E





n^d

X

k=1

s²_f(k)

k

X

i=1

s²_f(i)[B^N_i (M)−βN(M)]

!

Zε²_f(k)11_|εf(k)|≤K



= 0.

So, we have proved that

n→+∞lim E





n^d

X

k=1 k

X

i=1

s²_f(i)[B^N_i (M)−βN(M)]

! ZTk



= 0.

In the same way, we obtain that

n→+∞lim E





n^d

X

k=1 k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]

! Ztk



= 0.

Moreover since(ε,(τ_i)_i6=f(k+1))is independent ofτ_f(k+1) we have

E

k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]γf(k+1)Zh⁰⁰⁰_k,k+2

!

= 0.

Finally, it remains to consider

W5=E ^nd

X

k=1 k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]

!

ZX_f(k)h⁰⁰⁰_k−1,k+1

.

Letp be a xed positive integer. Since h⁰⁰⁰ is 1-Lipschitz, we have the upper bound|h⁰⁰⁰_k−1,k+1−h⁰⁰⁰_k−1,k+1^p | ≤ |S_f(k−1)(X)−S^p_f(k)(X)|. Now, we can apply the same truncation argument as before: rst we choose the level of our truncation by applying the dominated convergence theorem and then we use Lemma2. So, it follows that

n→+∞lim E ^nd

X

k=1 k

X

i=1

s²_f(i)[B^N_i (M)−βN(M)]

!

ZXf(k)(h⁰⁰⁰_k−1,k+1−h⁰⁰⁰_k−1,k+1^p )

= 0.

Therefore, to prove our theorem it is enough to show that

p→+∞lim lim sup

n→+∞

E ^nd

X

k=1 k

X

i=1

s²_f(i)[B_i^N(M)−βN(M)]

!

ZXf(k)h⁰⁰⁰_k−1,k+1^p

= 0.

We consider a one to one mapmfrom[1,|E^p_k|]∩N^∗toE_k^pand such that|m(i)(14)− f(k)| ≤ |m(i−1)−f(k)|. Now, we use the same argument as before:

h⁰⁰⁰_k−1,k+1^p −h⁰⁰⁰(S_f(k)^c (γ))

=

|E_k^p|

X

i=1

h⁰⁰⁰(S_m(i)(X) +S_f(k)^c (γ))−h⁰⁰⁰(S_m(i−1)(X) +S_f(k)^c (γ))

≤

|E_k^p|

X

i=1

|Xm(i)|.

Here recall thatB_i^N(M)isF_f(i),−∞-measurable andβN(M)isI-measurable.

We haveE(εf(k)|I) = 0, E(εf(k)|F_f(k),−∞) = 0and E(εf(k)|F_f(i),−∞) = 0for any positive integerisuch that i < k. Consequently, for any positive integeri such thati≤k, we have

E

s²_f(i)[B_i^N(M)−βN(M)]Zs_f(k)ε_f(k)h⁰⁰⁰(S^c_f(k)(γ))

= 0.

Therefore using the conditional expectation, we nd

E ^nd

X

k=1 k

X

i=1

s²_f(i)[B^N_i (M)−βN(M)]

!

ZX_f(k)h⁰⁰⁰_k−1,k+1^p

≤2M

n^d

X

k=1

|s_f(k)|

|E_k^p|

X

i=1

|s_m(i)|E|ε_m(i)E|m(i)−f(k)|(ε_f(k))|

= 2M

n^d

X

k=1

|sf(k)| X

j∈V₀^p

|sj+f(k)|E|εjE_|j|(ε0)|

≤2AM X

j∈V₀^p

E|εjE_|j|(ε0)| (A∈R^∗+) by(10)and(11).

Since (3) is realised the last term is as small as we wish by choosing p large enough, henceW4 is handled. Finally, the main theorem is proved.

3.2 Proof of the corollary

As observed in [8], the proof of the corollary is a direct consequence of Theorem 1.1 in Rio [23]. In fact, for anykin V₀¹, we have

E|εkE_|k|(ε₀)|=Cov

|εk|

11_{E|k|(ε0)≥0}−11_{E|k|(ε0)≤0} , ε₀

≤4

Z α1,∞(|k|) 0

Q²_ε0(u)du.

The proof of the corollary is complete.

4 Application

The direct consequence of our result is that it allows the construction of statis- tical tests able to quantify the estimation error. For this purpose, we show the construction of such a test that can be used in image denoising [11, 16, 28]. In the context given by the model (1), let us consider the following situation : a true imageg is aected by a correlated additive noise , that gives Y for the observed image.

For the original function the classical Lena image is used. This image is a gray level image with pixels values in the interval [0,255]. The size of the image is 256×256pixels. The correlated noise we consider is a Gaussian eld(εk)_k∈Z² built using an exponential covariance function

C(k) =E(ε₀ε_k) =Cst×exp{−|k|

a }.

The choice of such random eld ensures the validity of the projective crite- rion (3)(see [9], p.59 Corollary 2). There exist several methods for simulating such a random eld, here we have opted for the spectral method [15]. In order to obtain an important visual eect of how the noise aects the original image Cst was set to200 anda= 1. The noisy image is obtained by adding pixel by pixel the original image to the simulated noise. The estimator of the original image is computed using the Epanechnikov kernel

K(x) = 3

8(1− |x|²)I{|x|≤1}, x= (x1, x2)∈R².

In order to compute the expectation of the estimated function, several realisa- tion of the noisy image are needed. Here we have considered 50such images, constructed by adding the original Lena image with a noise realisation. Us- ing (2), for each noisy image, an estimate gn of the original function g was computed using the kernel K dened above. The expectation E(gn) is com- puted by just taking the pixel by pixel arithmetical means corresponding to the

images previously restored.

Clearly, it is now possible to estimate the dierencegn−E(gn). Following our theoretical result, the normalised square of this dierence follows a χ² distri- bution with one degree of freedom. Since this quantity is observable,p-values pixel by pixel can be computed.

The original image and the realisation of a noisy image are shown in Figure 1a and 1b, respectively. It can be noticed that in the dirty picture, spots are formed, due to the noise correlation. The expectation of the estimated origi- nal images in Figure 1c exhibits almost no such spots. Furthermore, the visual quality of this restored image is close to the original. A more quantitative eval- uation of this result is given by the image ofp-values of the proposed statistical test given in Figure 1d. The light-coloured pixels representp-values close to1, whereas the dark-coloured pixels indicate values close to 0. We have counted 83%of the pixels for which we have obtained ap-value higher than0.01. This ratio is quite a reliable indicator concerning the restored image. Together with the visual analysis of the results, it provides a detailed description of the ob- tained result. We conclude that, under these considerations, the theoretical results developed in this paper may be used as a basis for the development of practical tools in image analysis.

5 Annexe

In this section we prove lemma3.

Proof of Lemma 3. In fact, for any u in L², we can write u = w+E(u|I) wherew=u−E(u|I), hence it suces to prove that

n→+∞lim

X

k∈Λn

s²_kw◦T^k ₂

= 0.

Let us consider the transformationsT1, T2, ..., Td dened byT₁ⁱ=T^(i,...,0), T₂ⁱ =T(0,i,...,0), ..., T_dⁱ=T^(0,...,i)for any integeri. It is well known (cf. [7]) that the space

H =

h1−h1◦T1−(h2−h2◦T2)−...−(hd−hd◦Td) ; h1, .., hd∈L² is dense in the spaceG={g∈L²;E(g|I) = 0}. Let ε >0 be xed, there exist h₁, ..., h_d ∈L² such that

w−[(h₁−h₁◦T₁)−...−(h_d−h_d◦T_d)]

₂≤ε. So, we derive

X

k∈Λn

s²_kw◦T^k 2

≤ε+

d

X

j=1

X

k∈Λn

s²_k(h_j◦T^k−h_j◦T^k(j)) 2

wherek(j) = (k1, ..., kj−1, kj+ 1, kj+1, ..., kd). Using Lemma 1 and keeping in mind thatK is a bounded and Lipschitzian kernel, one can check that

s²_(k

1,..,k_d)=O 1

(nh_n)^d

and s²_(k

1,..,k_j,..,k_d)−s²_(k

1,..,k_j−1,..,kd)=O 1

(nh_n)^d+1

a) b)

c) d)

Figure 1: Results of the image restoration procedure : a) original Lena image, b) realisation of a noisy image, c) expectation of the restored images, d) obtained p−values as a gray level image (white pixels represent values close to1, whereas black pixels indicate values close to0).