k-nearest neighbors prediction and classification for spatial data

Abstract

This paper proposes a spatial k-nearest neighbor method for nonparametric prediction of real-valued spatial data and supervised classification for categorical spatial data. The proposed method is based on a double nearest neighbor rule which combines two kernels to control the distances between observations and locations. It uses a random bandwidth in order to more appropriately fit the distributions of the covariates. The almost complete convergence with rate of the proposed predictor is established and the almost sure convergence of the supervised classification rule was deduced. Finite sample properties are given for two applications of the k-nearest neighbor prediction and classification rule to the soil and the fisheries datasets.

Appendix

We start by the following technical lemmas that are helpful to handle the difficulties induced by the random bandwidth $H_{\textbf{n},x}$ in $r_{\textrm{kNN}}(x)$. They are adaptation of the results given in Collomb (1980) (for independent multivariate data) and their generalized version by Burba et al. (2009), Kudraszow and Vieu (2013) (for independent functional data).

1.1 Technical lemmas

For any random positive variable T, $\textbf{n}\in \mathbb {N}^{*N}$, and $x \in D$, we define

$$\begin{aligned} c_{\textbf{n}}(T)=\frac{\sum _{\textbf{i}\in \mathcal {I}_{\textbf{n}},\mathbf {s_0}\ne \textbf{i}}Y_{\textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{T}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }{\sum _{\textbf{i}\in \mathcal {I}_{{\textbf{n}}},\mathbf {s_0}\ne \textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{T}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }. \end{aligned}$$

Let us set the following sequences, for all $\textbf{n}\in \mathbb {N}^{*N}$

$$\begin{aligned} v_{\textbf{n}}=\left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}, \end{aligned}$$

and for all $\beta \in ]0,1[$ and $x \in D$

$$\begin{aligned} D_{\textbf{n}}^{-}(\beta ,x)=\left( \frac{k_\textbf{n}}{cf(x)k^{'}_{\textbf{n}}}\right) ^{1/d}\beta ^{1/2d}, \qquad D_{\textbf{n}}^{+}(\beta ,x)=\left( \frac{k_\textbf{n}}{cf(x)k^{'}_{\textbf{n}}}\right) ^{1/d}\beta ^{-1/2d}, \end{aligned}$$

(16)

where c is the volume of the unit sphere in $\mathbb {R}^{d}$. It is clear that

$$\begin{aligned} \forall \, \textbf{n}\in \mathbb {N}^{*N}, \forall x \in D \qquad D_{\textbf{n}}^{-}(\beta ,x)\le D_{\textbf{n}}^{+}(\beta ,x). \end{aligned}$$

Lemma 3

If the following conditions are verified:

$(L_{1})$:

$\displaystyle \mathbb {I}_{\left\{ D_{\textbf{n}}^{-} (\beta ,x)\le H_{\textbf{n},x}\le D_{\textbf{n}}^{+}(\beta ,x),\, \forall x \in D \right\} } \;\longrightarrow 1 \quad a.co.$

$(L_{2})$:

$\displaystyle \sup _{x \in D}\left| \frac{\sum _{\textbf{i}\in \mathcal {I}_{\textbf{n}},\mathbf {s_0}\ne \textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{-}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }{\sum _{\textbf{i}\in \mathcal {I}_{\textbf{n}},\mathbf {s_0}\ne \textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{+}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }-\beta \right| \longrightarrow 0\quad a.co.$

$(L_{3})$:

$\displaystyle \sup _{x \in D}\left| c_{\textbf{n}}\left( D_{\textbf{n}}^{-}(\beta ,x)\right) -r(x)\right| \longrightarrow 0\quad a.co.$, $\displaystyle \sup _{x\in D} \left| c_{\textbf{n}}\left( D_{\textbf{n}}^{+}(\beta ,x)\right) -r(x) \right| \longrightarrow 0\quad a.co$,

then we have $\displaystyle \sup _{x \in D} |c_{\textbf{n}}\left( H_{\textbf{n},x}\right) - r(x)|\longrightarrow 0\qquad a.co.$

Lemma 4

Under the following conditions:

$(L_{1})$:

$\displaystyle \mathbb {I}_{\left\{ D_{\textbf{n}}^{-} (\beta ,x)\le H_{\textbf{n},x}\le D_{\textbf{n}}^{+}(\beta ,x),\, \forall x \in D \right\} } \;\longrightarrow 1 \quad a.co.$

$(L_{2}^{'})$:

$\displaystyle \sup _{x \in D}\left| \frac{\sum _{\textbf{i}\in \mathcal {I}_{\textbf{n}},\mathbf {s_0}\ne \textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{-}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }{\sum _{\textbf{i}\in \mathcal {I}_{\textbf{n}},\mathbf {s_0}\ne \textbf{i}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{+}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }-\beta \right| =\mathcal {O}(v_{\textbf{n}}) \qquad a.co.$

$(L_{3}^{'})$:

$\displaystyle \sup _{x \in D}\left| c_{\textbf{n}}\left( D_{\textbf{n}}^{-}(\beta ,x)\right) -r(x)\right| =\mathcal {O}(v_{\textbf{n}}) \qquad a.co$, $\displaystyle \sup _{x\in D} \left| c_{\textbf{n}}\left( D_{\textbf{n}}^{+}(\beta ,x)\right) -r(x) \right| =\mathcal {O}(v_{\textbf{n}})\qquad a.co,$

we have, $\displaystyle \sup _{x \in D}\left| c_{\textbf{n}}\left( H_{\textbf{n},x}\right) - r(x)\right| =\mathcal {O}(v_{\textbf{n}})\qquad a.co.$

The proof of Lemma 4 is similar to that of the Lemma in Collomb (1980) (page 162) and Lemma 3 in Burba et al. (2009) (page 1866). Therefore, it is omitted. Lemma 3 is a particular case of of Lemma 4 when we take $v_{\textbf{n}}=1$.

1.2 Proofs of lemma 1 and lemma 2

Since the proof of Lemma 1 is based on the result of Lemma 3, it is sufficient to check conditions $(L_{1})$, $(L_{2})$ and $(L_{3})$. For the proof of Lemma 2, it suffices to check conditions $(L_{2}')$ and $(L_{3}')$.

To check the condition $(L_1 )$, we need the following two lemmas.

Lemma 5

(Ibragimov and Linnik (1971) or Deo (1973))

• i) We assume that the mixing condition (5) is satisfied. We denote by $\mathcal {L}_{r}\left( \mathcal {F}\right)$ the class of $\mathcal {F}-$mesurable random variables X satisfying

  $$\begin{aligned} \Vert X\Vert _{r}:=\left( E\left( |X|^{r}\right) \right) ^{1/r}< \infty . \end{aligned}$$

  Let $X\in \mathcal {L}_{r}\left( \mathcal {B}(E)\right)$, $Y\in \mathcal {L}_{s}\left( \mathcal {B}(E')\right)$ and $1\le r,s,t\le \infty$ such that $\frac{1}{r}+\frac{1}{s}+\frac{1}{t}=1$, then

  $$\begin{aligned} |\textrm{Cov}(X,Y)|\le \Vert X\Vert _{r}\Vert Y\Vert _{s}\left\{ \psi \left( \mathrm {Card(E),Card(E^{'})}\right) \varphi \left( \textrm{dist}(E,E^{'})\right) \right\} ^{1/t}. \end{aligned}$$

  (17)

• ii) For random variables X, Y bounded with probability 1, we have

  $$\begin{aligned} |\textrm{Cov}(X,Y)|\le C\psi \left( \mathrm {Card(E),Card(E^{'})}\right) \varphi \left( \textrm{dist}(E,E^{'})\right) . \end{aligned}$$

  (18)

Lemma 6

Under assumptions of Theorem 1, we have

$$\begin{aligned} S_{\textbf{n}}+R_{\textbf{n}}=\mathcal {O}\left( k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) , \end{aligned}$$

where

$$\begin{aligned} S_{\textbf{n}}= \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\textrm{Var}\left( \Lambda _{\textbf{i}}\right) \; \textrm{and} \quad R_{\textbf{n}}=\sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\sum _{\underset{\textbf{j}\ne \textbf{i}}{\textbf{j} \in \mathcal {V}_{\mathbf {s_0}}}}\left| \textrm{Cov}\left( \Lambda _{\textbf{i}},\Lambda _{\textbf{j}}\right) \right| , \\ \Lambda _{\textbf{i}}=\mathbb {I}_{B(x,D_{\textbf{n}})}(X_{\textbf{i}}), \quad \textbf{i}\in \mathcal {I}_{\textbf{n}}, \quad \delta _{\textbf{n}}=\mathbb {P}\left( \Vert X-x\Vert <D_{\textbf{n}}\right) ,\quad D_{\textbf{n}}^{d}=\mathcal {O}\left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) , \end{aligned}$$

$B(x,\varepsilon )$ denotes the closed ball of  $\mathbb {R}^{d}$ with center x and radius $\varepsilon$.

1.3 Proof of lemma 6

Let $\delta _{\textbf{n},\textbf{i}}=\mathbb {P}\left( \Vert X_{\textbf{i}}-x\Vert < D_{\textbf{n}}\right)$, we can deduce that

$$\begin{aligned} S_{\textbf{n}}=\sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}Var\left( \Lambda _{\textbf{i}}\right) =\sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\delta _{\textbf{n},\textbf{i}}(1-\delta _{\textbf{n},\textbf{i}})=\mathcal {O}\left( k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) , \end{aligned}$$

by the following results.

Firstly, under the Lipschitz condition of f (assumption (H1)), we have

$$\begin{aligned} \delta _{\textbf{n}}= & {} \;\mathbb {P}\left( \Vert X-x\Vert < D_{\textbf{n}}\right) \nonumber \\= & {} \;f(x)\int _{B(x,D_{\textbf{n}})}du+\int _{B(x,D_{\textbf{n}})} (f(u)-f(x))du\nonumber \\= & {} \;c f(x) D_{\textbf{n}}^{d}+ \mathcal {O}\left( D_{\textbf{n}}^{d+1} \right) . \end{aligned}$$

(19)

Secondly

$$\begin{aligned} \delta _{\textbf{n},\textbf{i}}-\delta _{\textbf{n}}= & {} \int _{B(x,D_{\textbf{n}})}\left( f_{\textbf{i}}(u)-f(u)\right) (u)du \nonumber \\= & {} \sup _u\left| f_{\textbf{i}}(u)-f(u)\right| \mathcal {O}\left( \frac{k_{\textbf{n}}}{k^{'}_{\textbf{n}}}\right) . \end{aligned}$$

(20)

Thus, the local stationarity assumption ($H_8$) implies

$$\begin{aligned} \sum _{\textbf{i}\in \mathcal {V}_{\mathbf {s_0}}}\left( \delta _{\textbf{n},\textbf{i}}-\delta _{\textbf{n}}\right) =o(k_{\textbf{n}}). \end{aligned}$$

(21)

Now for $\displaystyle R_{\textbf{n}}$, it should be noted that by (H5) and for each $\textbf{j}\ne \textbf{i}$

$$\begin{aligned} \left| \textrm{Cov}\left( \Lambda _{\textbf{i}},\Lambda _{\textbf{j}}\right) \right|= & {} \left| \mathbb {P}\left( \Vert X_{\textbf{i}}-x\Vert< D_{\textbf{n}},\Vert X_{\textbf{j}}-x\Vert< D_{\textbf{n}}\right) \right. \nonumber \\{} & {} \quad - \left. \mathbb {P}\left( \Vert X_{\textbf{i}}-x \Vert< D_{\textbf{n}}\right) \mathbb {P}\left( \Vert X_{\textbf{j}}-x \Vert < D_{\textbf{n}}\right) \right| \nonumber \\{} & {} \le \int _{B\left( x,D_{\textbf{n}}\right) \times B\left( x,D_{\textbf{n}}\right) } \left| f_{X_{\textbf{i}}X_{\textbf{j}}}(u,v)-f_{\textbf{i}}(u)f_{\textbf{j}}(v)\right| dudv \nonumber \\{} & {} \le C D_{\textbf{n}}^{2d} \le C \delta _\textbf{n}^2, \end{aligned}$$

(22)

since by (19)

$$\begin{aligned} \frac{D_{\textbf{n}}^{d}}{\delta _{\textbf{n}}}\rightarrow \frac{1}{cf(x)},\qquad \textrm{as}\quad \textbf{n}\rightarrow \infty . \end{aligned}$$

Using Lemma 5 and (19), we can write for $\displaystyle r=s= 4$

$$\begin{aligned} \left| \textrm{Cov}\left( \Lambda _{\textbf{i}},\Lambda _{\textbf{j}}\right) \right|\le & {} C\left[ E\left( \Lambda _{\textbf{i}}^{4}\right) E\left( \Lambda _{\textbf{j}}^{4}\right) \right] ^{1/4}\left( \psi (1,1)\varphi \left( \Vert \textbf{i}-\textbf{j}\Vert \right) \right) ^{1/2}\nonumber \\\le & {} C\delta _{\textbf{n}}^{1/2}\varphi \left( \Vert \textbf{i}-\textbf{j}\Vert \right) ^{1/2}. \end{aligned}$$

(23)

Let $q_{\textbf{n}}$ be a sequence of real numbers defined by $\displaystyle q_{\textbf{n}}^{N}=\mathcal {O}\left( \frac{k^{'}_{\textbf{n}}}{k_\textbf{n}}\right)$. Using the later, we define $\displaystyle S=\{ \textbf{i},\textbf{j}\in \mathcal {V}_{\mathbf {s_0}},\; 0<\Vert \textbf{i}-\textbf{j}\Vert \le q_{\textbf{n}}\}$ and $S^{c}$ its complementary in $\mathcal {V}_{\mathbf {s_0}}$, and rewrite

$$\begin{aligned} R_{\textbf{n}}=\sum _{\textbf{i},\textbf{j}\in S}\left| \textrm{Cov}\left( \Lambda _{\textbf{i}},\Lambda _{\textbf{j}}\right) \right| +\sum _{\textbf{i},\textbf{j}\in S^{c}}\left| \textrm{Cov}\left( \Lambda _{\textbf{i}},\Lambda _{\textbf{j}}\right) \right| = R_{\textbf{n}}^{(1)}+R_{\textbf{n}}^{(2)}. \end{aligned}$$

Firstly, according to the definitions of $q_{\textbf{n}}$ and S, and equation (22), we have

$$\begin{aligned} R_{\textbf{n}}^{(1)}\le & {} \sum _{\textbf{i},\textbf{j}\in S}C\delta _{\textbf{n}}^{2} \le C\delta _{\textbf{n}}^{2}k^{'}_{\textbf{n}}q_{\textbf{n}}^{N} =\mathcal {O}\left( k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) , \end{aligned}$$

since $\displaystyle \delta _{\textbf{n}}=\mathcal {O}(q_{\textbf{n}}^{-N})$ by (19).

Secondly, by (6) and (23), we get

$$\begin{aligned} R_{\textbf{n}}^{(2)}\le & {} \;C\delta _{\textbf{n}}^{1/2}\sum _{\textbf{i},\textbf{j}\in S^{c}}\varphi \left( \Vert \textbf{i}-\textbf{j}\Vert \right) ^{1/2} =\;C\delta _{\textbf{n}}^{1/2}k^{'}_{\textbf{n}}\sum _{\textbf{i}\in S^{c}}\varphi \left( \Vert \textbf{i}\Vert \right) ^{1/2}\\= & {} C\delta _{\textbf{n}}k^{'}_{\textbf{n}}\delta _{\textbf{n}}^{-1/2}\sum _{\textbf{i}\in S^{c}}\varphi \left( \Vert \textbf{i}\Vert \right) ^{1/2} \le \;C\delta _{\textbf{n}}k^{'}_{\textbf{n}}\left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{-1/2}\sum _{\textbf{i}\in S^{c}}\varphi \left( \Vert \textbf{i}\Vert \right) ^{1/2}\\\le & {} C\delta _{\textbf{n}}k^{'}_{\textbf{n}}\sum _{\textbf{i}\in S^{c}}\Vert \textbf{i}\Vert ^{(N-\theta )/2} =\mathcal {O}\left( \delta _{\textbf{n}}k^{'}_{\textbf{n}}\right) , \end{aligned}$$

because under assumptions (H6) and (H7), we have $\theta >(1+\frac{2\gamma }{\gamma -\tilde{\gamma }})N$, thus

$$\begin{aligned} \sum _{\textbf{i}\in S^{c}}\Vert \textbf{i}\Vert ^{(N-\theta )/2}\le k^{'}_{\textbf{n}}q_{\textbf{n}}^{^{(N-\theta )/2}} = o(1). \end{aligned}$$

Finally, the result follows:

$$\begin{aligned} R_{\textbf{n}}=\mathcal {O}\left( k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) \; \textrm{and}\; S_{\textbf{n}}+R_{\textbf{n}}=\mathcal {O}\left( k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) . \end{aligned}$$

1.4 Verification of $(L_{1})$

Let $\varepsilon _\textbf{n}=\frac{1}{2} \varepsilon _0\left( k_\textbf{n}/k^{'}_{\textbf{n}}\right) ^{1/d}$ with $\varepsilon _{0}>0$ and let $N_{\varepsilon _\textbf{n}}=\mathcal {O}(\varepsilon _{\textbf{n}}^{-d})$ be a positive integer. Since D is compact, one can cover it by $N_{\varepsilon _\textbf{n}}$ closed balls in $\mathbb {R}^d$ of centers $x_i \in D, \; i=1,\ldots , N_{\varepsilon _\textbf{n}}$ and radius $\varepsilon _\textbf{n}$. Let us show that

$$\begin{aligned} \mathbb {I}_{\left\{ D_{\textbf{n}}^{-}(\beta , x) \le H_{\textbf{n},x}\le D_{\textbf{n}}^{+}(\beta , x),\, \forall x \in D\right\} }\longrightarrow 1 \qquad a.co, \end{aligned}$$

which can be written as, $\forall \; \eta > 0$,

$$\begin{aligned} \sum _{\textbf{n}\in \mathbb {N}^{*N}}\mathbb {P}(\mid \mathbb {I}_{\left\{ D_{\textbf{n}}^{-}(\beta , x) \le H_{\textbf{n},x}\le D_{\textbf{n}}^{+}(\beta , x),\, \forall x \in D\right\} } -1\mid > \eta ) < \infty . \end{aligned}$$

We have

$$\begin{aligned} \,&\mathbb {P}(\mid \mathbb {I}_{\left\{ D_{\textbf{n}}^{-}(\beta , x) \le H_{\textbf{n},x}\le D_{\textbf{n}}^{+}(\beta , x),\, \forall x \in D\right\} } -1 \mid> \eta ) \nonumber \\&\le \mathbb {P}\left( \sup _{x \in D}\left( H_{\textbf{n},x} - D_{\textbf{n}}^{-}(\beta ,x)\right)<0 \right) + \mathbb {P}\left( \inf _{x \in D} \left( H_{\textbf{n},x} -D_{\textbf{n}}^{+}(\beta ,x) \right)> 0\right) \nonumber \\&\le \mathbb {P}\left( \max _{1\le i \le N_{\varepsilon _\textbf{n}}} \left( H_{\textbf{n},x_i} - D_{\textbf{n}}^{-}(\beta ,x_i)\right)<2\varepsilon _\textbf{n} \right) + \mathbb {P}\left( \min _{1\le i \le N_{\varepsilon _\textbf{n}}}\left( H_{\textbf{n},x_i} -D_{\textbf{n}}^{+}(\beta ,x_i) \right)> -2\varepsilon _\textbf{n}\right) \nonumber \\&\le N_{\varepsilon _\textbf{n}} \max _{1\le i \le N_{\varepsilon _\textbf{n}}} \mathbb {P}\left( H_{\textbf{n},x_i} < D_{\textbf{n}}^{-}(\beta ,x_i)+ 2\varepsilon _\textbf{n} \right) \nonumber \\&\qquad + N_{\varepsilon _\textbf{n}} \max _{1\le i \le N_{\varepsilon _\textbf{n}}} \mathbb {P}\left( H_{\textbf{n},x_i} > D_{\textbf{n}}^{+}(\beta ,x_i) -2\varepsilon _\textbf{n}\right) . \end{aligned}$$

(24)

Let us evaluate the first term in the right-hand side of (24), without ambiguity we ignore the i index in $x_i$. As justified in the following

$$\begin{aligned} \mathbb {P}\left( H_{\textbf{n},x} <D_{\textbf{n}}^{-}(\beta ,x)+2\varepsilon _\textbf{n} \right)\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\mathbb {I}_{B(x,D_{\textbf{n}}^{-}(\beta ,x)+2\varepsilon _\textbf{n})}(X_{\textbf{i}}) >k_\textbf{n}\right) \end{aligned}$$

(25)

$$\begin{aligned}\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\xi _{\textbf{i}} >k_\textbf{n}-k^{'}_{\textbf{n}}\delta _{\textbf{n}}\right) \end{aligned}$$

(26)

$$\begin{aligned}\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\xi _{\textbf{i}} >Ck_\textbf{n}(1-\beta ^{1/2})\right) := P_{1,\textbf{n}}, \end{aligned}$$

(27)

where $\displaystyle \xi _{\textbf{i}}=\Lambda _\textbf{i}-\delta _{\textbf{n},\textbf{i}}$ is centered and $\Lambda _\textbf{i}$ is defined in Lemma 6 when we replace $D_{\textbf{n}}$ by $D_{\textbf{n}}^{-}+2\varepsilon _\textbf{n}$. From (25), we get (26) by (21) while result (26) permits to get (27) by the help of the following.

Actually, according to the definition of $D_{\textbf{n}}^{-}$ in (16) and replacing $D_{\textbf{n}}$ by $D_{\textbf{n}}^{-}+2\varepsilon _\textbf{n}$ in (19), we get

$$\begin{aligned} k^{'}_{\textbf{n}}\delta _{\textbf{n}}-k_\textbf{n}\left( \varepsilon _0 (cf(x))^{1/d} + \beta ^{1/2d}\right) ^{d}=o(k_\textbf{n}), \end{aligned}$$

(28)

therefore, for all $\varepsilon _{1}>0$,

$$\begin{aligned} k_\textbf{n} - k^{'}_{\textbf{n}}\delta _{\textbf{n}}> k_\textbf{n}\left( 1-\left( \varepsilon _0 (c f(x))^{1/d} + \beta ^{1/2d}\right) ^{d} - \varepsilon _1\right) . \end{aligned}$$

Then, for $\varepsilon _1$ and $\varepsilon _0$ very small such that $1-\left( \varepsilon _0 (cf(x))^{1/d} + \beta ^{1/2d}\right) ^{d} - \varepsilon _1>0$, we can find some constant $C>0$ such that

$$\begin{aligned} k_\textbf{n}-k^{'}_{\textbf{n}}\delta _{\textbf{n}}>C k_\textbf{n}(1-\beta ^{1/2}). \end{aligned}$$

(29)

For the second term in the right-hand side of (24),

$$\begin{aligned} \mathbb {P}\left( H_{\textbf{n},x}>D_{\textbf{n}}^{+}(\beta ,x)-2\varepsilon _\textbf{n} \right)\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\mathbb {I}_{B(x,D_{\textbf{n}}^{+}(\beta ,x)-2\varepsilon _\textbf{n})}(X_{\textbf{i}})<k_\textbf{n}\right) \end{aligned}$$

(30)

$$\begin{aligned}\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\Delta _{\textbf{i}} >k^{'}_{\textbf{n}}\delta _{\textbf{n}}-k_\textbf{n}\right) \end{aligned}$$

(31)

$$\begin{aligned}\le & {} \mathbb {P}\left( \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\Delta _{\textbf{i}} >C k_\textbf{n}\left( \beta ^{-1/2}-1\right) \right) :=P_{2,\textbf{n}}, \end{aligned}$$

(32)

where $\displaystyle \Delta _{\textbf{i}}=\delta _{\textbf{n},\textbf{i}}-\Lambda _{\textbf{i}}$ is centered and $\Lambda _\textbf{i}$ is defined in Lemma 6 replacing $D_{\textbf{n}}$ by $D_{\textbf{n}}^{+} - 2\varepsilon _\textbf{n}$. Result (31) is obtained by (21) while that of (32) is obtained by replacing $D_{\textbf{n}}$ in (19) by $D_{\textbf{n}}^{+}-2\varepsilon _\textbf{n}$. Then, we get

$$\begin{aligned} k^{'}_{\textbf{n}}\delta _\textbf{n}- k_\textbf{n} \left( \beta ^{-1/2d}-\varepsilon _0(cf(x))^{1/d}\right) ^{d}=o(k_\textbf{n}). \end{aligned}$$

(33)

Thus for all $\varepsilon _{2}>0$, it is easy to see that

$$\begin{aligned} k^{'}_{\textbf{n}}\delta _{\textbf{n}}-k_\textbf{n}>k_\textbf{n}\left( \left( \beta ^{-1/2d}-\varepsilon _0 (cf(x))^{1/d}\right) ^{d}-1-\varepsilon _{2}\right) , \end{aligned}$$

so for $\varepsilon _{2}$ and $\varepsilon _0$ small enough such that $\left( \left( \beta ^{-1/2d}-\varepsilon _0 (cf(x))^{1/d}\right) ^{d}-1-\varepsilon _{2}\right) >0$, there exists $C>0$ such that

$$\begin{aligned} k^{'}_{\textbf{n}}\delta _{\textbf{n}}-k_\textbf{n}>C k_\textbf{n}\left( \beta ^{-1/2}-1\right) . \end{aligned}$$

(34)

Now, it suffices to prove that

$$\begin{aligned} \sum _{\textbf{n}\in \mathbb {N}^{*N}}N_{\varepsilon _{\textbf{n}}}P_{1,\textbf{n}}<\infty \quad \textrm{and}\quad \sum _{\textbf{n}\in \mathbb {N}^{*N}}N_{\varepsilon _{\textbf{n}}}P_{2,\textbf{n}}<\infty . \end{aligned}$$

1.4.1 Let us consider $P_{1,\textbf{n}}$

This proof is based on the classical spatial block decomposition of the sum on $\xi _{\textbf{i}}$ in $\mathcal {V}_{\mathbf {s_0}}$ similarly to (Tran 1990, page 44). Let $\mathcal {G}_\textbf{n}\subset \mathcal {I}_{\textbf{n}}$ be the smallest rectangular grid of center $\mathbf {s_0}$ containing $\mathcal {V}_{\mathbf {s_0}}$. Without loss of generality, we assume that $\mathcal {G}_\textbf{n}$ is defined via some $\textbf{k}=(k_1,\ldots ,k_N)$ where $1\le k_j\le n_j, j=1,\ldots ,N$. However, by construction $\mathcal {G}_\textbf{n}$ is of cardinal $\hat{\textbf{k}}=k_1\times \cdots \times k_N$ satisfying $k^{'}_{\textbf{n}}=\mathcal {O}(\hat{\textbf{k}})$. In addition, we assume that $k_{l}=2bq_{l} \; ,\; l=1,\ldots ,N$, where $q_{l}$ and b are positive integers. Then the decomposition can be presented as follows.

$$\begin{aligned} U(1,\textbf{k},\textbf{j})= & {} \sum _{\underset{k=1,\ldots ,N.}{i_{l}=2j_{l}b+1,}}^{(2j_{l}+1)b}\xi _{\textbf{i}} \\ U(2,\textbf{k},\textbf{j})= & {} \sum _{\underset{l=1,\ldots ,N-1.}{i_{l}=2j_{l}b+1,}}^{(2j_{l}+1)b} \; \; \sum _{i_{N}=(2j_{N}+1)b+1,}^{2(j_{N}+1)b}\xi _{\textbf{i}} \\ U(3,\textbf{k},\textbf{j})= & {} \sum _{\underset{l=1,\ldots ,N-2.}{i_{l}=2j_{l}b+1,}}^{(2j_{l}+1)b}\; \; \sum _{i_{N-1}=(2j_{N-1}+1)b+1,}^{2(j_{N-1}+1)b} \; \; \sum _{i_{N}=2j_{N}b+1,}^{(2j_{N}+1)b}\xi _{\textbf{i}} \\ U(4,\textbf{k},\textbf{j})= & {} \sum _{\underset{l=1,\ldots ,N-2.}{i_{l}=2j_{l}b+1,}}^{(2j_{l}+1)b}\; \; \sum _{i_{N-1}=(2j_{N-1}+1)b+1,}^{2(j_{N-1}+1)b} \; \; \sum _{i_{N}=(2j_{N}+1)b+1,}^{2(j_{N}+1)b}\xi _{\textbf{i}}\\ ... \end{aligned}$$

Note that

$$\begin{aligned} U(2^{N-1},\textbf{k},\textbf{j})=\sum _{\underset{l=1,\ldots ,N-1.}{i_{l}=(2j_{l}+1)b+1,}}^{2(j_{l}+1)b} \; \; \sum _{i_{N}=2j_{N}b+1,}^{(2j_{N}+1)b}\xi _{\textbf{i}} \end{aligned}$$

and that

$$\begin{aligned} U(2^{N},\textbf{k},\textbf{j})=\sum _{\underset{l=1,\ldots ,N.}{i_{l}=(2j_{l}+1)b+1,}}^{2(j_{l}+1)b}\xi _{\textbf{i}}. \end{aligned}$$

For each integer $1\le l \le 2^{N}$, let

$$\begin{aligned} T(\textbf{k},l)=\sum _{\underset{l=1,\ldots ,N}{j_{l}=0}}^{q_{l}-1} U(l,\textbf{k},\textbf{j}). \end{aligned}$$

Therefore, we have

$$\begin{aligned} \sum _{\textbf{i} \in \mathcal {V}_{\mathbf {s_0}}}\xi _{\textbf{i}}=\sum _{l=1}^{2^{N}}T(\textbf{k},l). \end{aligned}$$

(35)

It follows that

$$\begin{aligned} P_{1,\textbf{n}}= & {} \mathbb {P}\left( \sum _{l=1}^{2^{N}}T(\textbf{k},l)>Ck_\textbf{n}(1-\sqrt{\beta })\right) \le 2^{N}\mathbb {P}\left( \mid T(\textbf{k},1)\mid >\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N}}\right) . \end{aligned}$$

We enumerate in an arbitrary manner the $\hat{\textbf{q}}=q_{1}\times \ldots \times q_{N}$ terms $\displaystyle U(1,\textbf{k},\textbf{j})$ of the sum $T(\textbf{k},1)$ and denote them $\displaystyle W_{1},\ldots ,W_{\hat{\textbf{q}}}$. Notice that, $\displaystyle U(1,\textbf{k},\textbf{j})$ is measurable with respect to the field generated by the $Z_{\textbf{i}}$ with $\textbf{i}\in \textbf{I}(\textbf{k},\textbf{j})=\{ \textbf{i}\in \mathcal {G}_{\textbf{n}} \; | \;2j_{l}b+1\le i_{l}\le (2j_{l}+1)b ,\;l=1,\ldots ,N\}$, the set $\textbf{I}(\textbf{k},\textbf{j})$ contains $b^{N}$ sites and $\textrm{dist}(\textbf{I}(\textbf{k},\textbf{j}),\textbf{I}(\textbf{k},\textbf{j}^{'}))>b$. In addition, we have $\mid W_{l}\mid \le b^{N}.$

According to Lemma 4.5 of Carbon et al. (1997), one can find a sequence of independent random variables $\displaystyle W_{1}^{*},\ldots ,W_{\hat{\textbf{q}}}^{*}$ where $W_{l}$ has the same distribution as $W_{l}^{*}$ and:

$$\begin{aligned} \sum _{l=1}^{\hat{\textbf{q}}}\mathbb {E}(\mid W_{l}-W_{l}^{*}\mid )\le & {} 4\hat{\textbf{q}}b^{N}\psi ((\hat{\textbf{q}}-1)b^{N},b^{N})\varphi (b). \end{aligned}$$

Then, we can write

$$\begin{aligned} P_{1,\textbf{n}}\le & {} 2^{N}\mathbb {P}\left( \mid T(\textbf{n},1)\mid>\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N}}\right) \\\le & {} 2^{N}\mathbb {P}\left( \mid \sum _{l=1}^{\hat{\textbf{q}}}W_{l}\mid>\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N}}\right) \\\le & {} 2^{N}\mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}-W_{l}^{*}\mid>\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \\{} & {} \quad +2^{N}\mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}^{*}\mid >\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) . \end{aligned}$$

Let $P_{11,\textbf{n}}=\mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}-W_{l}^{*}\mid >\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right)$ and $P_{12,\textbf{n}}=\mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}^{*}\mid >\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) .$ It suffices to show that $\displaystyle \sum _{\textbf{n}\in \mathbb {N}^{*N}}P_{11,\textbf{n}}<\infty$ and $\displaystyle \sum _{\textbf{n}\in \mathbb {N}^{*N}}P_{12,\textbf{n}}<\infty$.

1.4.2 Let us consider first $P_{11,\textbf{n}}$

Using Markov’s inequality, we get

$$\begin{aligned} P_{11,\textbf{n}}= & {} \mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}-W_{l}^{*}\mid >\frac{C k_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \\\le & {} \frac{2^{N+3}}{C k_\textbf{n}(1-\sqrt{\beta })}\hat{\textbf{q}}b^{N}\psi ((\hat{\textbf{q}}-1)b^{N},b^{N})\varphi (b) \\\le & {} \frac{C}{k_\textbf{n}(1-\sqrt{\beta })}k^{'}_{\textbf{n}}\psi ((\hat{\textbf{q}}-1)b^{N},b^{N})\varphi (b), \end{aligned}$$

because $\hat{\textbf{k}}=2^N\hat{\textbf{q}}b^{N}$ by definition and $k^{'}_{\textbf{n}}=\mathcal {O}(\hat{\textbf{k}})$

Let us consider that

$$\begin{aligned} b^{N}=\mathcal {O}\left( \hat{\textbf{n}}^{2(1-s(1-\tilde{\gamma }))/a}\right) , \end{aligned}$$

(36)

where $a=2+(2+s(2-\tilde{\gamma }))d+s(4+2\tilde{\beta }+2\gamma -3\tilde{\gamma })$. Under the assumption on the function $\psi (n,m)$, we distinguish the following two cases:

Case 1

$$\begin{aligned}{} & {} \psi (n,m)\le C \min (n,m)\, \text {with}\, (1-s(1-\tilde{\gamma }))\theta \\{} & {} >N\left\{ (2+s(2-\tilde{\gamma }))d+2s(2+\gamma -\tilde{\gamma })\right\} , \\{} & {} \text { and }\, 2<s<\frac{1}{1-\tilde{\gamma }}. \end{aligned}$$

In this case, we have

$$\begin{aligned} P_{11,\textbf{n}}\le & {} C\frac{k^{'}_{\textbf{n}}}{k_{\textbf{n}}}b^{N}\varphi (b) \le C \frac{k^{'}_{\textbf{n}}}{k_{\textbf{n}}}b^{N-\theta }. \end{aligned}$$

Then by using (36) and the definition of $N_{\varepsilon _\textbf{n}}$, we have

$$\begin{aligned} N_{\varepsilon _\textbf{n}}P_{11,\textbf{n}}\le C \hat{\textbf{n}}^{-2\left( 1-\frac{3+s(2\tilde{\beta }-1)}{a}\right) }. \end{aligned}$$

One can show that $a>2(3+s(2\tilde{\beta }-1))$ and then $\displaystyle \sum _{\textbf{n}\in \mathbb {N}^{*N}}N_{\varepsilon _\textbf{n}}P_{11,\textbf{n}}<\infty$.

Case 2

$$\begin{aligned}{} & {} \psi (n,m)\le C(n+m+1)^{\tilde{\beta }}\,\text {with}\, (1-s(1-\tilde{\gamma }))\theta \\{} & {} >N\left\{ 2+(2+s(2-\tilde{\gamma }))d+s(4+2\tilde{\beta }+2\gamma -3\tilde{\gamma })\right\} \text {and } \end{aligned}$$

$2<s<\frac{1}{1-\tilde{\gamma }}$. In this case, we have

$$\begin{aligned} P_{11,\textbf{n}}\le & {} C\frac{k^{'}_{\textbf{n}}}{k_\textbf{n}}(k^{'}_{\textbf{n}}b^{N})^{\tilde{\beta } }\varphi (b) \le C\frac{k_{\textbf{n}}^{'}}{k_\textbf{n}} k_{\textbf{n}}^{'\tilde{\beta }}b^{-\theta }\le C \hat{\textbf{n}}^{-(2-\gamma \tilde{\beta })}. \end{aligned}$$

Then, it follows that $\displaystyle \sum _{\textbf{n}\in \mathbb {N}^{N}}N_{\varepsilon _\textbf{n}}P_{11,\textbf{n}}<\infty$ when $\tilde{\beta }<1/\gamma$.

1.4.3 Let us consider $P_{12,\textbf{n}}$

Applying Markov’s inequality, we have for $t>0$:

$$\begin{aligned} P_{12,\textbf{n}}= & {} \mathbb {P}\left( \sum _{l=1}^{\hat{\textbf{q}}}\mid W_{l}^{*}\mid >\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \\\le & {} \exp \left( -t\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \mathbb {E}\left( \exp \left( t\sum _{l=1}^{\hat{\textbf{q}}}W_{l}^{*}\right) \right) \\\le & {} \exp \left( -t\frac{Ck_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \prod _{l=1}^{\hat{\textbf{q}}}\mathbb {E}\left( \exp \left( tW_{l}^{*}\right) \right) , \end{aligned}$$

since the variables $W_{1}^{*},\ldots ,W_{\hat{\textbf{q}}}^{*}$ are independent.

Let $r>0$, for $\displaystyle t=\frac{r\log (\hat{\textbf{n}})}{k_\textbf{n}}$, $l=1,\ldots ,\hat{\textbf{q}}$, by using (36), we can easily get

$$\begin{aligned} t\mid W_{l}^{*}\mid\le & {} \frac{r\log (\hat{\textbf{n}})}{k_\textbf{n}} b^{N} \le C\frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\hat{\textbf{n}}^{2(1-s(1\tilde{\gamma }))/a}\\\le & {} C \frac{\log (\hat{\textbf{n}})}{\hat{\textbf{n}}^{\tilde{a} /a}}, \end{aligned}$$

where $\tilde{a} = a\tilde{\gamma } -2(1-s(1-\tilde{\gamma }))>0$ and $\tilde{a}>0$. However, we have $t\mid W_{l}^{*}\mid <1$ for $\textbf{n}$ large enough.

So, $\exp \left( t W_{l}^{*}\right) \le 1+tW_{l}^{*}+t^{2} W_{l}^{*2}$ then

$$\begin{aligned} \mathbb {E}\left( \exp \left( t W_{l}^{*}\right) \right) \le 1+\mathbb {E}\left( t^{2} W_{l}^{*2}\right) \le \exp \left( \mathbb {E}\left( t^{2} W_{l}^{*2}\right) \right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \prod _{l=1}^{\hat{\textbf{q}}}\mathbb {E}\left( \exp \left( tW_{l}^{*}\right) \right) \le \exp \left( t^{2}\sum _{l=1}^{\hat{\textbf{q}}}\mathbb {E}\left( W_{l}^{*2}\right) \right) . \end{aligned}$$

As $W_{l}^{*}$ and $W_{l}$ have the same distribution, we have

$$\begin{aligned} \sum _{l=1}^{\hat{\textbf{q}}}\mathbb {E}\left( (W_{l}^{*})^{2}\right) =Var\left( \sum _{l=1}^{\hat{\textbf{q}}} W_{l}^{*}\right) =Var\left( \sum _{l=1}^{\hat{\textbf{q}}}W_{l}\right) \le S_{\textbf{n}}+R_{\textbf{n}}. \end{aligned}$$

From Lemma 6, we obtain

$$\begin{aligned} \prod _{l=1}^{\hat{\textbf{q}}}\mathbb {E}\left( \exp \left( t W_{l}^{*}\right) \right)\le & {} \exp \left( Ct^{2}k_\textbf{n}\right) \le \exp \left( Cr^{2}\frac{\log (\hat{\textbf{n}})^{2}}{k_\textbf{n}}\right) \longrightarrow 1, \end{aligned}$$

because $\log (\hat{\textbf{n}})^{2}/k_\textbf{n} \rightarrow 0$ as $\textbf{n}\rightarrow \infty .$

Then, we deduce that

$$\begin{aligned} P_{12,\textbf{n}}\le & {} C\exp \left( -t\frac{C k_\textbf{n}(1-\sqrt{\beta })}{2^{N+1}}\right) \\\le & {} C\exp \left( -\frac{r C(1-\sqrt{\beta })}{2^{N+1}}\log (\hat{\textbf{n}})\right) \le C\hat{\textbf{n}}^{-\frac{rC(1-\sqrt{\beta })}{2^{N+1}}}. \end{aligned}$$

Then, we have

$$\begin{aligned} N_{\varepsilon _\textbf{n}}P_{12,\textbf{n}}< C\hat{\textbf{n}}^{\gamma -\tilde{\gamma }-\frac{rC(1-\sqrt{\beta })}{2^{N+1}}}. \end{aligned}$$

Therefore, for some $r>0$ such that $\displaystyle \frac{rC(1-\sqrt{\beta })}{2^{N+1}}\tilde{\gamma }-\gamma >1$, we get

$$\begin{aligned} \sum _{\textbf{n}\in \mathbb {N}^{N}}N_{\varepsilon _\textbf{n}} P_{12,\textbf{n}}<\infty . \end{aligned}$$

By combining the two results on $P_{11,\textbf{n}}$ and $P_{12,\textbf{n}}$, we get $\displaystyle \sum _{\textbf{n}\in \mathbb {N}^{N}} N_{\varepsilon _\textbf{n}}P_{1,\textbf{n}}<\infty$.

Using similar arguments, note that $\sum _{\textbf{n}\in \mathbb {N}^{N}} N_{\varepsilon _\textbf{n}}P_{2,\textbf{n}}<\infty$.

Now the check of conditions $(L_2)$, $(L_3)$, $(L^{'}_{2})$ and $(L^{'}_{3})$ is based on Theorem 3.1 in Dabo-Niang et al. (2016). We need to show that $D_{\textbf{n}}^{-}(\beta ,x)$, $D_{\textbf{n}}^{+}(\beta ,x)$ satisfy assumptions (H6) and (H7) used by these authors for all $(\beta ,x) \in ]0,1[ \times D$. This is proved in the following lemmas where without ambiguity $D_{\textbf{n}}$ will denote $D_{\textbf{n}}^{-}(\beta ,x)$ or $D_{\textbf{n}}^{+}(\beta ,x)$.

Lemma 7

Under assumption (H2) and (H6) on $\psi (.)$, we have

$$\begin{aligned} \hat{\textbf{n}}D_{\textbf{n}}^{d\theta _{0}}h_{\textbf{n},\textbf{s}_0}^{N\theta _{1}}\log (\hat{\textbf{n}})^{-\theta _{2}}u_{\textbf{n}}^{-\theta _{3}} \rightarrow \infty \end{aligned}$$

with

$$\begin{aligned} \theta _{0}=\frac{s(\theta +N(d+2))}{\theta -N(s(d+4)+2d)};\qquad \theta _{1}=\frac{s(\theta +Nd)}{\theta -N(s(d+4)+2d)}, \\ \theta _{2}=\frac{s(\theta -N(d+2))}{\theta -N(s(d+4)+2d)};\qquad \theta _{3}=\frac{2(\theta +N(d+s))}{\theta -N(s(d+4)+2d)}, \end{aligned}$$

and $\displaystyle u_{\textbf{n}}=\prod _{i=1}^{N}\left( \log (\log (n_{i}))\right) ^{1+\varepsilon }\log (n_{i})$ for all $\varepsilon >0$.

1.5 Proof of lemma 7

By the definition of $D_{\textbf{n}}$ in Lemma 6, hypotheses (H2) and (H6), we have

$$\begin{aligned} \hat{\textbf{n}}D_{\textbf{n}}^{d\theta _{0}}h_{\textbf{n},\textbf{s}_{0}}^{N\theta _{1}}\log (\hat{\textbf{n}})^{-\theta _{2}}u_{\textbf{n}}^{-\theta _{3}}\ge & {} C\hat{\textbf{n}}\left( \frac{k_{\textbf{n}}}{k_{\textbf{n}}^{'}}\right) ^{\theta _0}\left( \frac{k^{'}_\textbf{n}}{\hat{\textbf{n}}}\right) ^{\theta _1} \log (\hat{\textbf{n}})^{-\theta _2}u_{\textbf{n}}^{-\theta _3}\\\ge & {} C \frac{\hat{\textbf{n}}^{1 -(\gamma -\tilde{\gamma })\theta _0-(1-\gamma )\theta _1 }}{\log (\hat{\textbf{n}})^{\theta _2}u_{\textbf{n}}^{\theta _3}}. \end{aligned}$$

Note that $u_{\textbf{n}}\le \log (\tilde{n})^{N(2+\varepsilon )} \Rightarrow \frac{1}{u_{\textbf{n}}^{\theta _3}}\ge \frac{1}{\log (\tilde{n})^{(2+\varepsilon )N\theta _3}}$, where $\displaystyle \tilde{n}=\max _{k=1,\ldots ,N}n_{k}$, and

$$\begin{aligned} \log (\hat{\textbf{n}})\le C\log (\tilde{n})\Rightarrow \frac{1}{\log (\hat{\textbf{n}})^{\theta _2}}\ge C\frac{1}{\log (\tilde{n})^{\theta _2}}. \end{aligned}$$

Since $\displaystyle \frac{n_{k}}{n_{i}}\le C, \; \forall \; 1\le k,i \le N$, we deduce that $\displaystyle \hat{\textbf{n}}\ge C \tilde{n}^{N}$ and

$$\begin{aligned} \hat{\textbf{n}}D_{\textbf{n}}^{d\theta _{0}}h_{\textbf{n},\textbf{s}_{0}}^{N\theta _{1}}\log (\hat{\textbf{n}})^{-\theta _{2}}u_{\textbf{n}}^{-\theta _{3}}\ge & {} C \frac{\tilde{n}^{N\left( 1 -(\gamma -\tilde{\gamma })\theta _0-(1-\gamma )\theta _1 \right) }}{\log (\tilde{n})^{\theta _2+N\theta _3(\varepsilon +2)}} \qquad \rightarrow +\infty \end{aligned}$$

because $(1-s(1-\tilde{\gamma }))\theta >N\left( (2+s(2-\tilde{\gamma }))d+2s(2+\gamma -\tilde{\gamma })\right)$.

Lemma 8

Under assumption (H2) and (H7) on $\psi (.)$, we have

$$\begin{aligned} \hat{\textbf{n}}D_{\textbf{n}}^{d\theta _{0}^{'}}h_{\textbf{n},\textbf{s}_0}^{N\theta _{1}^{'}}\log (\hat{\textbf{n}})^{-\theta _{2}^{'}}u_{\textbf{n}}^{-\theta _{3}^{'}} \rightarrow \infty , \end{aligned}$$

with

$$\begin{aligned} \theta _{0}^{'}=\frac{s(\theta +N(d+3))}{\theta -N\left( s(d+3+2\tilde{\beta })+2(d+1)\right) };\quad \theta _{1}^{'}=\frac{s(\theta +N(d+1))}{\theta -N\left( s(d+3+2\tilde{\beta })+2(d+1)\right) } \\ \theta _{2}^{'}=\frac{s(\theta -N(d+1))}{\theta -N\left( s(d+3+2\tilde{\beta })+2(d+1)\right) }; \quad \theta _{3}^{'}=\frac{2\left( \theta +N(s+d+1)\right) }{\theta -N\left( s(d+3+2\tilde{\beta })+2(d+1)\right) }. \end{aligned}$$

The proof of this lemma is the same as the one of Lemma 7 and is omitted.

1.6 Verification of $(L_{2})$

Let

$$\begin{aligned} f_{\textbf{n}}\left( x,D_{\textbf{n}}^{-}(\beta ,x)\right) =\frac{1}{\hat{\textbf{n}}h_{\textbf{n},\mathbf {s_0}}^{N}\left( D_{\textbf{n}}^{-}(\beta ,x)\right) ^{d}}\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\, \textbf{i}\ne \mathbf {s_0}} K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{-}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\textbf{s}_{0}-\textbf{i}}{\textbf{n}}\right\| \right) , \end{aligned}$$

and

$$\begin{aligned} f_{\textbf{n}}\left( x,D_{\textbf{n}}^{+}(\beta ,x)\right) =\frac{1}{\hat{\textbf{n}}h_{\textbf{n},\mathbf {s_0}}^{N}\left( D_{\textbf{n}}^{+}(\beta ,x)\right) ^{d}}\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\,\textbf{i}\ne \mathbf {s_0}} K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{+}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) . \end{aligned}$$

Under the hypotheses of Lemma 1 and the results of Lemma 7 and Lemma 8 (Dabo-Niang et al. 2016, pages 447-448), we have

$$\begin{aligned} \sup _{x \in D} \left| f_{\textbf{n}}\left( x,D_{\textbf{n}}^{-}(\beta ,x)\right) -f(x)\right| \longrightarrow 0 \qquad a.co. \\ \sup _{x \in D}\left| f_{\textbf{n}}\left( x,D_{\textbf{n}}^{+}(\beta ,x)\right) -f(x)\right| \longrightarrow 0 \qquad a.co, \end{aligned}$$

then,

$$\begin{aligned} \sup _{x \in D} \left| \frac{\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\,\textbf{i}\ne \mathbf {s_0}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{-}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }{\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\,\textbf{i}\ne \mathbf {s_0}} K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{+}(\beta ,x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }-\beta \right| =\beta \sup _{x \in D} \left| \frac{f_{\textbf{n}}\left( x,D_{\textbf{n}}^{-}(\beta ,x)\right) }{f_{\textbf{n}}\left( x,D_{\textbf{n}}^{+}(\beta ,x)\right) } - 1\right| \rightarrow 0 \quad a.co. \end{aligned}$$

1.7 Verification of $(L_{3})$

Under assumptions of Lemmas 1 and the results of Lemmas 7 and 8, we have by the Theorem 3.1 in Dabo-Niang et al. (2016)

$$\begin{aligned} \sup _{x \in D} |c_{\textbf{n}}\left( D_{\textbf{n}}^{-}(\beta ,x)\right) -r(x)|\rightarrow 0 \qquad a.co \qquad \textrm{and} \qquad \sup _{x \in D}|c_{\textbf{n}}\left( D_{\textbf{n}}^{+}(\beta ,x)\right) -r(x)|\rightarrow 0 \quad a.co. \end{aligned}$$

1.8 Proof of lemma 2

The proof of this lemma is based on the results of Lemma 4. It suffices to check the conditions $(L_{2}')$ and $(L_{3}')$. Clearly, similar arguments as those involved to prove $(L_2)$ and $(L_3)$ can be used to obtain the requested conditions.

1.9 Verification of $(L_{2}')$

Under assumptions of Corollary 1 and Lemmas 7, 8, we have

$$\begin{aligned} \sup _{x\in D}\left| f_{\textbf{n}}\left( x,D_{\textbf{n}}^{-}(\beta ,x)\right) -f(x) \right|= & {} \mathcal {O}\left( D_{\textbf{n}}^{-}(\beta ,x)\right) +\mathcal {O}\left( \left( \frac{\log (\hat{\textbf{n}})}{\hat{\textbf{n}}(D_{\textbf{n}}^{-}(\beta ,x))^{d}h_{\textbf{n},\mathbf {s_0}}^{N}}\right) ^{1/2}\right) \; a.co.\\= & {} \mathcal {O}\left( \left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}\right) \; a.co.,\\ \sup _{x\in D}\left| f_{\textbf{n}}\left( x,D_{\textbf{n}}^{+}(\beta ,x)\right) -f(x)\right|= & {} \mathcal {O}\left( D_{\textbf{n}}^{+}(\beta ,x)\right) +\mathcal {O}\left( \left( \frac{\log (\hat{\textbf{n}})}{\hat{\textbf{n}}(D_{\textbf{n}}^{+}(\beta ,x))^{d}h_{\textbf{n},\mathbf {s_0}}^{N}}\right) ^{1/2}\right) \;a.co.\\= & {} \mathcal {O}\left( \left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}\right) . \; a.co. \end{aligned}$$

We conclude that

$$\begin{aligned} \sup _{x\in D}\left| \frac{\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\,\textbf{i}\ne \mathbf {s_0}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{-}(\beta , x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }{\sum _{\textbf{i}\in \mathcal {I}_\textbf{n},\, \textbf{i}\ne \mathbf {s_0}}K_{1}\left( \frac{x-X_{\textbf{i}}}{D_{\textbf{n}}^{+}(\beta , x)}\right) K_{2}\left( h_{\textbf{n},\mathbf {s_0}}^{-1}\left\| \frac{\mathbf {s_0}-\textbf{i}}{\textbf{n}}\right\| \right) }-\beta \right| \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ =\beta \sup _{x\in D}\left| \frac{f_{\textbf{n}}\left( x,D_{\textbf{n}}^{-}(\beta , x)\right) }{f_{\textbf{n}}\left( x,D_{\textbf{n}}^{+}(\beta , x)\right) }-1 \right| =\mathcal {O}\left( \left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}\right) \, a.co. \end{aligned}$$

1.10 Verification of $(L_{3}')$

It is relatively easy to deduce from Lemmas 7 and 8 (see Remark 4 in Dabo-Niang et al. 2016) that

$$\begin{aligned} \sup _{x\in D} \left| c_{\textbf{n}}\left( D_{\textbf{n}}^{-}(\beta , x)\right) -r(x)\right| =\mathcal {O}\left( \left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}\right) \; a.co. \\ \sup _{x\in D} \left| c_{\textbf{n}}\left( D_{\textbf{n}}^{+}(\beta , x)\right) -r(x)\right| =\mathcal {O}\left( \left( \frac{k_\textbf{n}}{k^{'}_{\textbf{n}}}\right) ^{1/d}+\left( \frac{\log (\hat{\textbf{n}})}{k_\textbf{n}}\right) ^{1/2}\right) \, a.co. \end{aligned}$$

This yields the proof.