Asymptotic properties of the kernel estimate of spatial conditional mode when the regressor is functional

Abstract

The kernel method estimator of the spatial modal regression for functional regressors is proposed. We establish, under some general mixing conditions, the $L^p$-consistency and the asymptotic normality of the estimator. The performance of the proposed estimator is illustrated in a real data application.

Appendix

We first state the following lemmas which are due to Carbon et al. (1997). They are needed for the convergence of our estimates. Their proofs will then be omitted.

Lemma 7

Suppose $E_1,\dots ,E_r$ be sets containing $m$ sites each with ${\text{ d }ist}(E_i,\ E_j)\ge \gamma$ for all $i\ne j$ where $1\le i\le r$ and $1\le j\le r$. Suppose $Z_1,\dots ,Z_r$ is a sequence of real-valued r.v.’s measurable with respect to $\mathcal {B}(E_1),\dots ,\mathcal {B}(E_r)$, respectively, and $Z_i$ takes values in $[a,b]$. Then there exists a sequence of independent r.v.’s $Z_1^*,\dots ,Z_r^*$ independent of $Z_1,\dots ,Z_r$ such that $Z_i^*$ has the same distribution as $Z_i$ and satisfies

$$\begin{aligned} \sum _{i=1}^r E|Z_i-Z_i^*|\le 2r(b-a)\psi ((r-1)m,m)\varphi (\gamma ). \end{aligned}$$

Lemma 8

• (i) Suppose that (2) holds. Denote by $\mathcal {L}_r(\mathcal {F})$ the class of $\mathcal {F}$-measurable r.v.’s (random variables) $X$ satisfying $\Vert X\Vert _r=(E|X|^r)^{1/r}<\infty$. Suppose $X \in \mathcal {L}_r(\mathcal {B}(E))$ and $Y \in \mathcal {L}_s(\mathcal {B}(E'))$. Assume also that $1\le r,\ s,\ t<\infty$ and $r^{-1}+s^{-1}+t^{-1}=1$. Then

  $$\begin{aligned} |EXY-EXEY|\le C\Vert X\Vert _r \Vert Y\Vert _s \{\psi (\mathrm{Card}(E),\mathrm{Card}(E'))\varphi (\text {dist}(E,E'))\}^{1/t}. \end{aligned}$$

  (10)

• (ii) For r.v.’s bounded with probability 1, the right-hand side of (10) can be replaced by

  $$\begin{aligned} C\psi (\mathrm{Card}(E),\mathrm{Card}(E'))\varphi (\text {dist}(E,E')). \end{aligned}$$

Proof of Lemma 1

We have

$$\begin{aligned} \left\| \widehat{f}^{x\, (1)}_N(\theta (x))-E\left[ \widehat{f}^{x\, (1)}_N(\theta (x))\right] \right\| _p= \frac{1}{\widehat{\mathbf{n }}\, h_H^2EK_\mathbf{1 }}\left\| \sum _\mathbf{i \in \mathbf I _\mathbf n } \theta _\mathbf{i }\right\| _p \end{aligned}$$

where

$$\begin{aligned}&\theta _\mathbf{i }=K_\mathbf i H_\mathbf i ^{(1)}(\theta (x)))-E\left[ K_\mathbf i H_\mathbf i ^{(1)}(\theta (x)))\right] . \\&K_\mathbf i =K(h_K^{-1}d(x,X_\mathbf{i })),\, H_\mathbf i =K(h_H^{-1}(y-Y_\mathbf{i }). \end{aligned}$$

We have $EK_\mathbf 1 =O(\phi _x(h_K)),$ (because of H6), so it remains to show that

$$\begin{aligned} \left\| \sum _\mathbf{i \in \mathbf I _\mathbf n } \theta _\mathbf{i }\right\| _p=O(\sqrt{\widehat{\mathbf{n }}h_H\phi _x(h_K)}). \end{aligned}$$

Clearly, if we show the claimed result for $p=2r$, we get the other lower values of $p$ by the Holder inequality. Thus, it suffices to prove the case $p=2r$. To do that we use the same ideas of Gao et al. (2008) and Ould Abdi et al. (2010b). Indeed, we have

$$\begin{aligned} \left[ \left( \sum _\mathbf{i \in \mathcal {I}_\mathbf{n }}\theta _\mathbf{i }\right) ^{2r}\right] =\sum _\mathbf{i \in \mathcal {I}_\mathbf{n }}E\left[ \theta _\mathbf{i }^{2r}\right] +\sum _{s=1}^{2r-1}\sum _{\nu _0+\nu _1+\dots +\nu _s=2r}V_s(\nu _0,\nu _1,\dots ,\nu _s) \end{aligned}$$

where $\sum _{\nu _0+\nu _1+\dots +\nu _s=2r}$ is the summation over $(\nu _0,\nu _1,\dots ,\nu _s)$ with positive integer components satisfying $\nu _0+\nu _1+\dots +\nu _s=2r$ and

$$\begin{aligned} V_s(\nu _0,\nu _1,\dots ,\nu _s)=\sum _\mathbf{i _0\ne \mathbf i _1\ne \dots \ne \mathbf i _s\in \mathcal {I}_\mathbf{n }}E\left[ \theta _\mathbf{i _0}^{\nu _0}\theta _\mathbf{i _1}^{\nu _1}\dots \theta _\mathbf{i _s}^{\nu _s}\right] . \end{aligned}$$

Firstly, by stationarity, we have

$$\begin{aligned} \sum _\mathbf{i \in I_\mathbf{n }}E\left( \theta _\mathbf{i }\right) ^{2r} \le C\widehat{\mathbf{n }}E\left( \left| \theta _\mathbf{i }\right| \right) ^{2r} \le \widehat{\mathbf{n }}E\left( K_\mathbf i H_\mathbf i ^{(1)}(\theta (x)))\right) ^{2r}\le C\widehat{\mathbf{n }} h_H\phi _x(h_K). \end{aligned}$$

Now, we control the second term $V_s(\nu _0,\nu _1,\dots ,\nu _s)$. For this, we distinguished the two cases $s< r$ and $s\ge r$. The second case can be evaluated by straightforward modification of the proof of Lemma 3.4 of Gao et al. (2008) by taking into account the fact that

$$\begin{aligned} E\left| \theta _\mathbf{i _0}^{\nu _0}\theta _\mathbf{i _2}^{\nu _2}\dots \theta _\mathbf{i _s}^{\nu _s}\right| \le Ch_H^{1+s}\phi _x^{1+v_{s}}(h_K). \end{aligned}$$

It follows that, for a certain a real sequence $P:=P_\mathbf{n }$ we have

$$\begin{aligned} V_s(\nu _0,\nu _1,\dots ,\nu _s) \le C(\widehat{\mathbf{n }})^r\left( P^{Nr}h_H^{1+s}\phi _x(h_K)^{(1+v_{s})}+P^{Nr-1-\delta }\right) . \end{aligned}$$

So, if we take $P=\phi _x(h_K)^{-(1+v_{Nr})/1+\delta )}$, we obtain that

$$\begin{aligned} \displaystyle V_s(\nu _0,\nu _1,\dots ,\nu _s)=O\left( \left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{r}\right) ,\qquad \text{ for }\, \qquad r\le s\le 2r-1. \end{aligned}$$

Next, the case $s>r$ is evaluated by the same way as in Lemma 3.3 in Gao et al. (2008). Indeed, we denote by

$$\begin{aligned} V_{s1}=\sum _\mathbf{i _0\ne \mathbf i _1\ne \cdots \ne \mathbf i _s}\left[ E\left( \prod _{j=0}^s\theta _\mathbf{i _j}^{\nu _j}\right) -\prod _{j=0}^s E \theta _\mathbf{i _j}^{\nu _j}\right] \, \text{ and } \, V_{s2}=\sum _\mathbf{i _0\ne \mathbf i _1\ne \cdots \ne \mathbf i _s}\prod _{j=0}^s E \theta _\mathbf{i _j}^{\nu _j}. \end{aligned}$$

So,

$$\begin{aligned} V_s(\nu _0,\nu _1,\dots ,\nu _s)=V_{s1}+V_{s2}. \end{aligned}$$

It is clear that,

$$\begin{aligned} \displaystyle \left| V_{s2}\right| \le C\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{s+1}. \end{aligned}$$

Furthermore for the term $V_{s1}$, we have

$$\begin{aligned} E\left( \prod _{j=0}^s\theta _\mathbf{i _j}^{\nu _j}\right) -\prod _{j=0}^s E \theta _\mathbf{i _j}^{\nu _j}&= \sum _{l=0}^{s-1}\left( \prod _{j=0}^{l-1} E \theta _\mathbf{i _j}^{\nu _j}\right) \\&\quad \times \left( E\left[ \prod _{j=l}^s\theta _\mathbf{i _j}^{\nu _j}\right] -E\left[ \theta _\mathbf{i _l}^{\nu _l}\right] E\left[ \prod _{j=l+1}^s\theta _\mathbf{i _j}^{\nu _j}\right] \right) \end{aligned}$$

where we define $\prod _{j=l}^s . =1$ if $l>s$. Then we obtain

$$\begin{aligned} \left| V_{s1}\right|&\le \sum _{l=0}^{s-1}\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^l\sum _\mathbf{i _l\ne \cdots \ne \mathbf i _s} \left| E\left[ \prod _{j=l}^s\xi _\mathbf{i _j}^{\nu _j}\right] -E\left[ \xi _\mathbf{i _l}^{\nu _l}\right] E\left[ \prod _{j=l+1}^s\xi _\mathbf{i _j}^{\nu _j}\right] \right| \\&= \sum _{l=0}^{s-1}\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^lV_{ls1}. \end{aligned}$$

We have

$$\begin{aligned} V_{ls1}&= \sum _{0<\text {dist}\left( \{\mathbf{i }_l\},\{\mathbf{i }_{l+1},\dots ,\mathbf{i }_s\}\right) \le P}\left| E\left[ \prod _{j=l}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] -E\left[ \theta _{\mathbf{i }_l}^{\nu _l}\right] E\left[ \prod _{j=l+1}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] \right| \\&\quad +\sum _{0<{\text{ d }ist}\left( \{\mathbf{i }_l\},\{\mathbf{i }_{l+1},\dots ,\mathbf{i }_s\}\right) > P}\left| E\left[ \prod _{j=l}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] -E\left[ \theta _{\mathbf{i }_l}^{\nu _l}\right] E\left[ \prod _{j=l+1}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] \right| \\&:= V_{ls11}+V_{ls12}. \end{aligned}$$

Once again by (H2) we have

$$\begin{aligned} \left| E\left[ \prod _{j=l}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] -E\left[ \theta _{\mathbf{i }_l}^{\nu _l}\right] E\left[ \prod _{j=l+1}^s\theta _{\mathbf{i }_j}^{\nu _j}\right] \right| \le Ch_H^{s-l}\phi _x(h_K)^{1+v_{s+1-l}}. \end{aligned}$$

Thus, we have

$$\begin{aligned} V_{ls11}\displaystyle \le Ch_H^{s-l}\phi _x(h_K)^{1+v_{s+1-l}}\widehat{\mathbf{n }}^{(s-l)}P^N \end{aligned}$$

Since the variables $\theta _{\mathbf{i }}$ are bounded, we have (see Lemma 8)

$$\begin{aligned} V_{ls12} \le C\widehat{\mathbf{n }}^{(s-l)}\sum _{t=P+1}^\infty t^{N-1}\varphi (t). \end{aligned}$$

Combining the upper bounds of $V_{ls11}$ and $V_{ls12}$, we have

$$\begin{aligned}&\left| V_{s1}\right| \displaystyle \le C\sum _{l=0}^{s-1}\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^l\left[ h_H^{s-l}\phi _x(h_K)^{1+v_{s+1-l}}\widehat{\mathbf{n }}^{(s-l)}P^N +\widehat{\mathbf{n }}^{(s-l)}\sum _{t=P+1}^\infty t^{N-1}\varphi (t)\right] \\&\quad \displaystyle \le C\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{(s+1)}\\&\quad \displaystyle \times \sum _{l=0}^{s-1}\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{l-s-1} \left[ h_H^{s-l}\phi _x(h_K)^{1+v_{s+1-l}}\widehat{\mathbf{n }}^{(s-l)}P^N +\widehat{\mathbf{n }}^{(s-l)}\sum _{t=P+1}^\infty t^{N-1}\varphi (t)\right] .\\ \end{aligned}$$

Taking $P=(h_H\phi _x(h_K))^{-1/N}$, we obtain

$$\begin{aligned} \left| V_{s1}\right| \le C\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{(s+1)}. \end{aligned}$$

$\square$

Proof of Lemma 2

It is easy to see that

$$\begin{aligned} E\left[ \widehat{f}^{x\, (1)}_N(\theta (x))\right] -f^{x\, (1)}(\theta (x))&= \frac{1}{h_H^2EK_1}E\left[ K_1 E\left[ H_1^{(1)}(\theta (x))\ | \ X_1\right] \right] \\&\quad -f^{x\, (1)}(\theta (x)). \end{aligned}$$

By an integration by part and the change of variables $\displaystyle t=\frac{y-z}{h_H}$, we have

$$\begin{aligned}&E\left[ \widehat{f}^{x\, (1)}_N(\theta (x))\right] -f^{x\, (1)}(\theta (x))\\&\quad \le \frac{1}{EK_1}\left( EK_1\int H(t) \left| f^{X_1\, (1)}(\theta (x)-h_Ht)-f^{x\, (1)}(\theta (x))\right| {d}t\right) . \end{aligned}$$

Hypotheses (H4) and (H6) allow to get the desired result. $\square$

Proof of Lemma 3

It is clear that, for all $\epsilon <1$, we have

$$\begin{aligned} P\left( \widehat{f}^x_D=0\right) \le P\left( \widehat{f}^x_D\le 1-\epsilon \right) \le P\left( |\widehat{f}^x_D-E[\widehat{f}^x_D]|\ge \epsilon \right) . \end{aligned}$$

The Markov’s inequality allows to get, for any $p>0$,

$$\begin{aligned} P\left( |\widehat{f}^x_D-E[\widehat{f}^x_D]|\ge \epsilon \right) \le \frac{E\left[ |\widehat{f}^x_D-E[\widehat{f}^x_D]|^p\right] }{\epsilon ^p}. \end{aligned}$$

So

$$\begin{aligned} \left( P\left( \widehat{F}^x_D=0\right) \right) ^{1/p}=O\left( \left\| \widehat{f}^x_D-E[\widehat{f}^x_D]\right\| _p\right) . \end{aligned}$$

The computation of $\displaystyle \left\| \widehat{f}^x_D-E[\widehat{f}^x_D]\right\| _p$ can be done by following the same arguments as those invoked to get (5). This yields the proof. $\square$

Proof of Lemma 4

We start by writing $\forall y\in S$

$$\begin{aligned} E[\widehat{f}^{x\, (1)}_N(y)]&= \frac{1}{E[K_\mathbf{1 }]}E\left[ K_\mathbf{1 }E[h_H^{-2}H_\mathbf{1 }^{\prime }|X]\right] \text{ with } h_H^{-2}E\left[ H_\mathbf{1 }^{\prime }|X\right] \\&= \int _{{\mathrm {I\!R}}}H(t)f^{X\, (1)}(y-h_ {H}t){d}t. \end{aligned}$$

Next, we use a Taylor expansion under (H4), as follows

$$\begin{aligned} h_H^{-2}E\left[ H_\mathbf{1 }^{\prime }|X\right] =f^{X\, (1)}(y)+ \frac{h_H}{2}\left( \int tH(t) {d}t\right) \frac{\partial ^2 f^{X}(y)}{\partial ^2 y} +o(h_H) . \end{aligned}$$

Thus, we get

$$\begin{aligned} E\left[ \widehat{f}^{x\, (1)}_N(y)\right]&= \frac{1}{E[K_\mathbf{1 }]}\left( E\left[ K_\mathbf{1 }f^{X\, (1)}(y)\right] \right. \\&\quad \left. +\,h_H\left( \int tH(t) {d}t\right) E\left[ K_\mathbf{1 }\frac{\partial ^2 f^{X}(y)}{\partial ^2 y}\right] +o(h_H)\right) . \end{aligned}$$

Noting $\psi (\cdot ,y):=\frac{\partial f^{\cdot }(y)}{\partial y }$: since $\Phi (0)=0$, we have

$$\begin{aligned} E\left[ K_\mathbf{1 }\psi (X,y)\right]&= \psi (x,y) E[K_\mathbf{1 }]+E\left[ K_\mathbf{1 }\left( \psi (X,y)-\psi (x,y)\right) \right] \\&= \psi (x,y) E[K_\mathbf{1 }]+E\left[ K_\mathbf{1 }\left( \Phi (d(x,X)\right) \right] \\&= \psi (x,y) E[K_\mathbf{1 }]+\displaystyle \Phi ^{\prime }(0) E\left[ d(x,X)K_\mathbf{1 }\right] + o(E\left[ d(x,X)K_\mathbf{1 }\right] ). \end{aligned}$$

Therefore,

$$\begin{aligned} E\left[ \widehat{f}^{x\, (1)}_N(y)\right]&= f^{x\, (1)}(y)+ \frac{h_H}{2}\frac{\partial ^2 f^x(y)}{\partial y^2}\int tH(t) {d}t+o\left( h_H^2\frac{E\left[ d(x,X)K_\mathbf{1 }\right] }{E[K_\mathbf{1 }]}\right) \\&\quad +\displaystyle \Phi ^{\prime }_0(0)\frac{E\left[ d(x,X)K_\mathbf{1 }\right] }{E[K_\mathbf{1 }]}+o\left( \frac{E\left[ d(x,X)K_\mathbf{1 }\right] }{E[K_\mathbf{1 }]}\right) . \end{aligned}$$

By using the same idea as Ferraty et al. (2007), we obtain that

$$\begin{aligned} \frac{1}{\phi _x(h_K)}E\left[ d(x,X)K_\mathbf{1 }\right] =h_K\left( K(1)-\int _0^1 (sK(s))'\beta _x(s){d}s+ o(1)\right) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\phi _x(h_K)}E\left[ K_\mathbf{1 }\right] =K(1)-\int _0^1K'(s)\beta _x(s)ds +o(1). \end{aligned}$$

So,

$$\begin{aligned} E\left[ \widehat{f}_N^{x\, (1)}(y)\right]&= f^{x\, (1)}(y)+ \frac{h_H}{2}\frac{\partial ^2 f^x(y)}{\partial y^2}\int tH(t) dt\\&\quad +\displaystyle h_K\Phi ^{\prime }(0)\frac{\left( K(1)-\int _0^1 (sK(s))'\beta _x(s)ds\right) }{\left( K(1)-\int _0^1K'(s)\beta _x(s)ds\right) }+o(h_H)+o(h_K) . \end{aligned}$$

In particular for $y=\theta (x)$, we obtain the desired result of the lemma. $\square$

Proof of Lemma 5

For the variance term $\mathrm{Var}[\widehat{f}^x_N(y)]$, we have

$$\begin{aligned} \mathrm{Var}[\widehat{f}^{x\, (1)}_N(y)]=\frac{1}{(h_H^2\widehat{\mathbf{n }} E\left[ K_\mathbf{1 }\right] )^2}\mathrm{Var}\left[ \sum _{\mathbf{i \in I_\mathbf{n }}} D_\mathbf{i } \right] \end{aligned}$$

where

$$\begin{aligned} D_\mathbf{i }=K_\mathbf{i }H_\mathbf{i }^{\prime }-E\left[ K_\mathbf{i }H_\mathbf{i }^{\prime }\right] . \end{aligned}$$

Thus

$$\begin{aligned} \mathrm{Var}[\widehat{f}^{x\, (1)}_N(y)]=\frac{1}{\widehat{\mathbf{n }}(h_H^2E\left[ K_\mathbf{1 }\right] )^2} \mathrm{Var}\left[ D_\mathbf{1 }\right] +\frac{1}{(h_H^2\widehat{\mathbf{n }}E\left[ K_\mathbf{1 }\right] )^2}\sum _{\mathbf{i }\not =\mathbf{j }} \mathrm{Cov}(D_\mathbf{i },D_\mathbf{j }). \end{aligned}$$

Let us calculate the quantity $\mathrm{Var}\left[ D_\mathbf{1 }\right]$. We have:

$$\begin{aligned} \mathrm{Var}\left[ D_\mathbf{1 }\right]&= E\left[ K_\mathbf{1 }^2H_\mathbf{1 }^{\prime ^2}\right] -\left( E\left[ K_\mathbf{1 } H_\mathbf{1 }^{\prime }\right] \right) ^2\\&= E\left[ K_\mathbf{1 }^2\right] \frac{E\left[ K_\mathbf{1 }^2H_\mathbf{1 }^{\prime ^2}\right] }{E\left[ K_\mathbf{1 }^2\right] }-(E\left[ K_\mathbf{1 }\right] )^2\left( \frac{E\left[ K_\mathbf{1 }H_\mathbf{1 }^{\prime }\right] }{E\left[ K_\mathbf{1 }\right] }\right) ^2. \end{aligned}$$

So, by using the same arguments as those used in the previous lemma, we get

$$\begin{aligned} \begin{array}{ll} &{}\displaystyle \frac{1}{\phi _x(h_K)}E\left[ K_\mathbf{1 }^2\right] \displaystyle =K^2(1)-\int _0^1 (K^2(s))'\beta _x(s))ds +o(1) \\ &{}\displaystyle \frac{E\left[ K_\mathbf{1 }^2H_\mathbf{1 }^{\prime ^2}\right] }{E\left[ K_\mathbf{1 }^2\right] } \displaystyle = h_Hf^x(y)\int H^{\prime ^2}(t)dt +o(h_H) \\ &{}\displaystyle \frac{E[K_\mathbf{1 }H_\mathbf{1 }^{\prime }]}{E\left[ K_\mathbf{1 }\right] }\displaystyle =h_Hf^x(y)\int H^{\prime }(t)dt+o(h_H) \end{array} \end{aligned}$$

which implies that

$$\begin{aligned} \mathrm{Var}\left[ D_\mathbf{i }\right]&= h_H\phi _x(h_K)f^x(y)\int H^{\prime ^2}(t)dt\left( K^2(1)-\int _0^1 (K^2(s))'\beta _x(s))ds \right) \nonumber \\&\quad +o\left( h_H\phi _x(h_K)\right) . \end{aligned}$$

(11)

Now, let us focus on the covariance term. To do that, we define

$$\begin{aligned} E_1=\{\mathbf{i, j }\in I_{\mathbf{n }}: 0<\left\| \mathbf{i }-\mathbf{j }\right\| \le c_{\mathbf{n }}\}\quad \text{ and } \quad E_2=\{\mathbf{i, j }\in I_{\mathbf{n }}: \left\| \mathbf{i }-\mathbf{j }\right\| > c_{\mathbf{n }}\}. \end{aligned}$$

For all $(\mathbf{i },\mathbf{j })\in E_1^2$ we write

$$\begin{aligned} {\text {Cov}}\left( D_{\mathbf{i }},D_{\mathbf{j }}\right) = E\left[ K_{\mathbf{i }}K_{\mathbf{j }}H_{\mathbf{i }}^{\prime }H_{\mathbf{j }}^{\prime }\right] - \left( E\left[ K_{\mathbf{i }}H_{\mathbf{i }}^{\prime }\right] \right) ^2 \end{aligned}$$

and we use the fact that ${\mathrm {I\!E}}\left[ H_\mathbf{i }^{\prime }H_\mathbf{j }^{\prime }|(X_\mathbf{i },X_\mathbf{j } )\right] =O(h_H^2); \; \forall \; \mathbf i \not =\mathbf j$, ${\mathrm {I\!E}}\left[ H_\mathbf{i }^{\prime }|X_\mathbf{i }\right] =O(h_H); \; \forall \; \mathbf i$ , under (H2) and (H5) we get

$$\begin{aligned} E\left[ K_\mathbf{i }K_\mathbf{j }H_\mathbf{i }^{\prime }H_\mathbf{j }^{\prime }\right] \le Ch_H^2P\left[ (X_\mathbf i ,X_\mathbf j ) \in B(x,h_K)\times B(x,h_K)\right] \end{aligned}$$

and

$$\begin{aligned} E\left[ K_\mathbf{i }H_\mathbf{i }^{\prime }\right] \le C h_H P\left( X_\mathbf i \in B(x,h_K)\right) . \end{aligned}$$

It follows that

$$\begin{aligned} \mathrm{Cov}\left( D_\mathbf{i },D_\mathbf{j }\right) \le Ch_H^2\phi _x(h_K)(\phi _x(h_K)+\phi _x^{v_1}(h_K)) \end{aligned}$$

So

$$\begin{aligned} \displaystyle \sum _{E_1}\mathrm{Cov}\left( D_\mathbf{i },D_\mathbf{j }\right) \le C \widehat{\mathbf{n }}c_\mathbf{n }^N h_H^2\phi _x^{1+v_1}(h_K). \end{aligned}$$

On the other hand, Lemma 8 and $\left| D_\mathbf{i }\right| \le C$, permit to write that for $(\mathbf i ,\mathbf j )\in E_2^2$

$$\begin{aligned} \left| \mathrm{Cov}\left( D_\mathbf{i },D_\mathbf{j }\right) \right| \le C \varphi \left( \Vert \mathbf i -\mathbf j \Vert \right) \end{aligned}$$

and

$$\begin{aligned} \sum _{E_2}Cov\left( D_\mathbf{i },D_\mathbf{j }\right)&\le C \sum _{E_2}\varphi \left( \Vert \mathbf i -\mathbf j \Vert \right) \\&\le C \widehat{\mathbf{n }}\sum _\mathbf{i :\Vert \mathbf i \Vert >c_\mathbf{n }}\varphi \left( \Vert \mathbf i \Vert \right) \\&\le C\widehat{\mathbf{n }}c_\mathbf{n }^{-Na} \sum _\mathbf{i :\Vert \mathbf i \Vert >c_\mathbf{n }}\Vert \mathbf i \Vert ^{Na}\varphi \left( \Vert \mathbf i \Vert \right) . \end{aligned}$$

Finally, we have:

$$\begin{aligned} \sum _{\mathbf{i }\not =\mathbf{j }} \mathrm{Cov}(D_\mathbf{i },D_\mathbf{j })\le \left( C\widehat{\mathbf{n }}c_\mathbf{n }^N h_H^2\phi _x^{1+v_1}(h_K)+C\widehat{\mathbf{n }}c_\mathbf{n }^{-Na}\sum _\mathbf{i : \Vert \mathbf i \Vert >c_\mathbf{n }}\Vert \mathbf i \Vert ^{Na}\varphi \left( \Vert \mathbf i \Vert \right) \right) . \end{aligned}$$

Let $c_\mathbf{n }=(h_H^{2/(a+1)}\phi _x^{1/a}(h_K))^{-1/N}$, then we obtain that

$$\begin{aligned} \sum \mathrm{Cov}\left( D_\mathbf{i },D_\mathbf{j }\right) =o\left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) . \end{aligned}$$

In conclusion, we have

$$\begin{aligned} \mathrm{Var}[\widehat{f}^{x\, (1)}_N(y)]&= \frac{f^x(y)}{\widehat{\mathbf{n }} h_H^3\phi _x(h_K) }\left( \int H^{\prime ^2}(t)dt\right) \left( \frac{\left( K^2(1)-\int _0^1 (K^2(s))'\beta _x(s)ds\right) }{\left( K(1)-\int _0^1K'(s)\beta _x(s)ds\right) ^2} \right) \\&\quad +o\left( \frac{1}{\widehat{\mathbf{n }}h_H^3\phi _x(h_K) }\right) . \end{aligned}$$

The result of the lemma correspond to the case $y=\theta (x).$ $\square$

Proof of Lemma 6

We set

$$\begin{aligned} S_{\mathbf{n }}= \sum _{{\mathbf{i }\in \mathbf I _\mathbf{n }}}\Lambda _{\mathbf{i }} \end{aligned}$$

where

$$\begin{aligned} \Lambda _\mathbf{i }:=\frac{\sqrt{h_H\phi _x(h_K)}}{h_H E[K_{\mathbf{1 }}]}D_\mathbf{i }. \end{aligned}$$

(12)

Clearly, we have

$$\begin{aligned} \sqrt{\widehat{\mathbf{n }}h_H^3\phi _x(h_K)}\left[ \sigma (x)\right] ^{-1}\left( \widehat{f}^{x\, (1)}_N(y)-E\widehat{f}^{x\, (1)}_N(y)\right) =\left( \widehat{\mathbf{n }}(\sigma ^2(x))\right) ^{-1/2}S_{\mathbf{n }}. \end{aligned}$$

So, to prove the asymptotic normality of $\left( \widehat{\mathbf{n }}(\sigma (x))^2\right) ^{-1/2}S_{\mathbf{n }}$, it suffices to show the proof of this lemma. The latter is shown by the blocking method, where the random variables $\Lambda _{\mathbf{i }}$ are grouped into blocks of different sizes defined as follows

$$\begin{aligned} W(1,\mathbf{n },x,\mathbf{j })=\sum _{{i_k=j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+1,\; k=1,\dots ,N}}^{j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\Lambda _{\mathbf{i }} , \end{aligned}$$

$$\begin{aligned} W(2,\mathbf{n },x,\mathbf{j })=\sum _{{i_k=j_k(p_{\mathbf{n }} +q_{\mathbf{n }})+1,\;k=1,\dots ,N-1}}^{j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\quad \sum _{i_N=j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1}^{(j_N+1)(p_{\mathbf{n }}+q_{\mathbf{n }})}\Lambda _{\mathbf{i }} , \end{aligned}$$

$$\begin{aligned} W(3,\mathbf{n },x,\mathbf{j })\!=\!\sum _{{i_k=j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+1,\; k=1,\dots ,N-2}}^{j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\quad \sum _{i_{N-1}=j_{N-1}(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1}^{(j_{N-1}+1)(p_{\mathbf{n }} +q_{\mathbf{n }})}\quad \sum _{i_N=j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+1}^{j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\Lambda _{\mathbf{i }} , \end{aligned}$$

$$\begin{aligned} W(4,\mathbf{n },x,\mathbf{j })=\sum _{{i_k=j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+1,\; k=1,\dots ,N-2}}^{j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\quad \sum _{i_{N-1} =j_{N-1}(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1}^{(j_{N-1}+1)(p_{\mathbf{n }}+q_{\mathbf{n }})} \quad \sum _{i_N=j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1}^{(j_N+1)(p_{\mathbf{n }}+q_{\mathbf{n }})}\Lambda _{\mathbf{i }} , \end{aligned}$$

and so on. The last two terms are

$$\begin{aligned} W(2^{N-1},\mathbf{n },x,\mathbf{j })=\sum _{{i_k=j_k(p_{\mathbf{n }}+q_{\mathbf{n }}) +p_{\mathbf{n }}+1,\;k=1,\dots ,N-1}}^{(j_k+1)(p_{\mathbf{n }}+q_{\mathbf{n }})}\quad \sum _{i_N=j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+1}^{j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}}\Lambda _{\mathbf{i }} , \end{aligned}$$

$$\begin{aligned} W(2^N,\mathbf{n },x,\mathbf{j })=\sum _{{i_k=j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1, \;k=1,\dots ,N}}^{(j_k+1)(p_{\mathbf{n }}+q_{\mathbf{n }})}\Lambda _{\mathbf{i }} \end{aligned}$$

where $q_{\mathbf{n }}=o\left( \left[ \widehat{\mathbf{n }} (h_H\phi _x(h_K))^{(1+2N)}\right] ^{1/(2N)}\right)$ and $p_{\mathbf{n }}=\left[ (\widehat{\mathbf{n }} h_H\phi _x(h_K))^{1/(2N)}/s_{\mathbf{n }}\right]$ with

$s_{\mathbf{n }}=o\left( \left[ \widehat{\mathbf{n }} (h_H\phi _x(h_K))^{(1+2N)}\right] ^{1/(2N)}q_{\mathbf{n }}^{-1}\right)$. Because of (H11) all the sequences $q_{\mathbf{n }}$, $p_{\mathbf{n }}$ and $s_{\mathbf{n }}$ tend to infinity.

Now, Define for each $i=1,\dots ,2^N$,

$$\begin{aligned} T(\mathbf{n },x,i)=\sum _{\mathbf{j }\in \mathcal {J}}W(i,\mathbf{n },x,\mathbf{j }). \end{aligned}$$

where $\mathcal {J}=\{0,\dots ,r_1-1\}\times \cdots \times \{0,\dots ,r_N-1\}$ with $r_k=n_k(p_{\mathbf{n }}+q_{\mathbf{n }})^{-N}.$ So, we have

$$\begin{aligned}&\left( \widehat{\mathbf{n }}(\sigma ^2(x))\right) ^{-1/2}S_{\mathbf{n }}\\&\quad =\left[ \sqrt{\widehat{\mathbf{n }}}\sigma (x)\right] ^{-1}\left( T\left( \mathbf{n },x,1\right) +\sum _{i=2}^{2N}T\left( \mathbf{n },x,i\right) \right) . \end{aligned}$$

Thus, the proof of the asymptotic normality of $\left( \widehat{\mathbf{n }}(\sigma ^2(x))\right) ^{-1/2}S_{\mathbf{n }}$ is reduced to the proofs of the following results

$$\begin{aligned} Q_1\equiv \Big |E\left[ \exp \big [iuT(\mathbf n ,x,1)\big ] \right] -\prod _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1} E\left[ \exp \big [iuW(1,\mathbf n ,x,\mathbf j )\big ]\right] \Big |\rightarrow 0 \end{aligned}$$

(13)

$$\begin{aligned} Q_2\equiv \widehat{\mathbf{n }}^{-1}E\Big [\sum _{i=2}^{2^N}T(\mathbf n ,x,i)\Big ]^2\rightarrow 0 \end{aligned}$$

(14)

$$\begin{aligned} Q_3\equiv \widehat{\mathbf{n }}^{-1}\sum _\mathbf{j \in \mathcal {J}}E\Big [W(1,\mathbf n ,x,\mathbf j )\Big ]^2\rightarrow \sigma ^2(x) \end{aligned}$$

(15)

$$\begin{aligned} Q_4\equiv \widehat{\mathbf{n }}^{-1}\sum _\mathbf{j \in \mathcal {J}}E\Big [(W(1,\mathbf n ,x,\mathbf j ))^2\mathbf 1 _{\{|W(1,\mathbf n ,x,\mathbf j )|>\epsilon \left( \sigma ^2(x) \widehat{\mathbf{n }}\right) ^{1/2}\}}\Big ]\rightarrow 0 \quad \text{ for } \text{ all } \epsilon >0. \end{aligned}$$

(16)

For (13), we numerate the $M=\prod _{k=1}^N r_k={\widehat{\mathbf{n }}}(p_\mathbf{n }+q_\mathbf{n })^{-N}\le \widehat{\mathbf{n }}p_\mathbf{n }^{-N}$ random variables $W(1,\mathbf n ,x,\mathbf j ),\; \mathbf j \in \mathcal {J}$ in the arbitrary way $\tilde{U_1},\dots ,\tilde{U_M}$. For $\mathbf j \in \mathcal {J}$, let

$$\begin{aligned} I(1,\mathbf n ,x,\mathbf j )=\left\{ \mathbf i :j_k(p_\mathbf{n }+q_\mathbf{n })+1\le i_k\le j_k(p_\mathbf{n }+q_\mathbf{n })+p_\mathbf{n }; \quad k=1,\dots N\right\} \end{aligned}$$

then we have $W(1,\mathbf n ,x,\mathbf j )=\sum \nolimits _\mathbf{i \in I(1,\mathbf n ,x,\mathbf j )}\Lambda _\mathbf{i }$. Noting that each of the sets of sites $I(1,\mathbf n ,x,\mathbf j )$ contains $p_\mathbf{n }^N$, these sets are distant of $p_\mathbf{n }$ at least. Further, we apply the lemma of Volkonski and Rozanov (1959) to the variables $\left( \exp (iu\tilde{U}_1),\dots ,\exp (iu\tilde{U}_M)\right)$. Since $\displaystyle \Big |\prod \nolimits _{s=j+1}^M\exp [iu\tilde{U}_s]\Big |\le 1$, then:

$$\begin{aligned} Q_1&= \Big |E[\exp \big [iuT(\mathbf n ,x,1)\big ]]-\prod _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1} E[\exp \big [iuW(1,\mathbf n ,x,\mathbf j )\big ]]\Big |\\&= \left| E\prod _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1}\exp \left[ iuW(1,\mathbf n ,x,\mathbf j )\right] -\prod _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1} E\exp \big [iuW(1,\mathbf n ,x,\mathbf j )\big ]\right| \\&\le \sum _{k=1}^{M-1}\sum _{j=k+1}^M\Bigg |E\left( \exp [iu\tilde{U}_k]-1\right) \left( \exp [iu\tilde{U}_j]-1\right) \prod _{s=j+1}^M\exp [iu\tilde{U}_s]\\&\quad - E\left( \exp [iu\tilde{U}_k]-1\right) E\left( \exp [iu\tilde{U}_j]-1\right) \prod _{s=j+1}^M \exp [iu\tilde{U}_s] \Bigg |\\&= \sum _{k=1}^{M-1}\sum _{j=k+1}^M\Big |E\left( \exp [iu\tilde{U}_k]-1\right) \left( \exp [iu\tilde{U}_j]-1\right) \\&\quad -E\left( \exp [iu\tilde{U}_k]-1\right) E\left( \exp [iu\tilde{U}_j]-1\right) \Big |\times \Big |\prod _{s=j+1}^M\exp [iu\tilde{U}_s]\Big |\\&\le \sum _{k=1}^{M-1}\sum _{j=k+1}^M\Big |E\left( \exp [iu\tilde{U}_k]-1\right) \left( \exp [iu\tilde{U}_j]-1\right) \\&\quad -E\left( \exp [iu\tilde{U}_k]-1\right) E\left( \exp [iu\tilde{U}_j]-1\right) \Big |. \end{aligned}$$

Let $\tilde{I}_j$ be the set of sites among the $I(1,\mathbf n ,x,\mathbf j )$ such that $\tilde{U}_j= \sum \nolimits _\mathbf{i \in \tilde{I}_{j}}D_\mathbf{i }$. The Lemma of Carbon et al. (1997) and assumption (2), lead:

$$\begin{aligned}&\Big |E\left( \exp [iu\tilde{U}_k]-1\right) \left( \exp [iu\tilde{U}_j]-1\right) -E\left( \exp [iu\tilde{U}_k]-1\right) E\left( \exp [iu\tilde{U}_j]-1\right) \Big |\\&\quad \le C\varphi \left( d(\tilde{I}_j,\tilde{I}_k)\right) p_\mathbf{n }^N. \end{aligned}$$

Then

$$\begin{aligned} Q_1&\le Cp_\mathbf{n }^N \sum _{k=1}^{M-1}\sum _{j=k+1}^M \varphi \left( d(\tilde{I}_j,\tilde{I}_k)\right) \\&\le Cp_\mathbf{n }^NM\sum _{k=2}^M \varphi \left( d(\tilde{I}_1,\tilde{I}_k)\right) \\&\le Cp_\mathbf{n }^NM\sum _{i=1}^{\infty }\sum _{k: iq_\mathbf{n }\le d(\tilde{I}_1,\tilde{I}_k)<(i+1)q_\mathbf{n }}\varphi \left( d(\tilde{I}_1,\tilde{I}_k)\right) \\&\le Cp_\mathbf{n }^NM\sum _{i=1}^{\infty }i^{N-1}\varphi (iq_\mathbf{n }) \\&\le C\widehat{\mathbf{n }}q_\mathbf{n }^{-\delta }\sum _{i=1}^{\infty }i^{N-1-\delta }, \end{aligned}$$

by (H3). This last tends to zero by the fact that $\widehat{\mathbf{n }}q_\mathbf{n }^{-\delta }\rightarrow 0$ (see (H11)). Proof of (14): We have

$$\begin{aligned} Q_2&\equiv \widehat{\mathbf{n }}^{-1}E\left[ \sum _{i=2}^{2^N}T(\mathbf n ,x,i)\right] ^2\\&= \widehat{\mathbf{n }}^{-1}\left( \sum _{i=2}^{2^N}E\left[ T(\mathbf n ,x,i)\right] ^2+\sum _{\begin{array}{c} i,j=2,\dots ,2^N\\ i\ne j \end{array}}E\left[ T(\mathbf n ,x,i)\right] \left[ T(\mathbf n ,x,j)\right] \right) . \end{aligned}$$

By Cauchy–Schwartz inequality, we get:

$$\begin{aligned}&\forall ~2\le i\le 2^N : \widehat{\mathbf{n }}^{-1}E\left[ T(\mathbf n ,x,i)\right] \left[ T(\mathbf n ,x,j)\right] \\&\quad \le \left( \widehat{\mathbf{n }}^{-1}E\left[ T(\mathbf n ,x,i)\right] ^2\right) ^{1/2}\left( \widehat{\mathbf{n }}^{-1}E\left[ T(\mathbf n ,x,j)\right] ^2\right) ^{1/2}. \end{aligned}$$

Then, it suffices to prove that

$$\begin{aligned} \widehat{\mathbf{n }}^{-1}E\left[ T(\mathbf n ,x,i)\right] ^2\rightarrow 0~~;\forall ~ 2\le i\le 2^N. \end{aligned}$$

We will prove this for $i=2$, the case where $i\ne 2$ is similar. We have $T(\mathbf n ,x,2)=\sum \nolimits _\mathbf{j \in \mathcal {J}}W(2,\mathbf n ,x,\mathbf j )= \sum \nolimits _{j=1}^M \hat{U}_j$, where we enumerate the $W(2,\mathbf n ,x,\mathbf j )$ in the arbitrary way $\hat{U}_1,\dots ,\hat{U}_M$. Then:

$$\begin{aligned} E\left[ T(\mathbf n ,x,2)\right] ^2&= \sum _{i=1}^M \mathrm{Var}\left( \hat{U}_i\right) +\sum _{i=1}^M\sum _{\begin{array}{c} j=1\\ i\ne j \end{array}}^M \mathrm{Cov}\left( \hat{U}_i,\hat{U}_j\right) \\&= A_1+A_2. \end{aligned}$$

The stationarity of the process $\left( X_\mathbf{i },Y_\mathbf{i }\right) _\mathbf{i \in \mathbb {Z}^N}$, implies that:

$$\begin{aligned} \mathrm{Var}\left( \hat{U}_i\right)&= \mathrm{Var}\left( \sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N-1 \end{array}}^{p_\mathbf{n }}\sum _{i_N=1}^{q_\mathbf{n }}\Lambda _\mathbf{i }\right) ^2\nonumber \\&= p_\mathbf{n }^{N-1}q_\mathbf{n } ~ \mathrm{Var} \left[ \Lambda _\mathbf{i }\right] \nonumber \\&\quad +\sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N-1 \end{array}}^{p_\mathbf{n }}\sum _{i_N=1}^{q_\mathbf{n }} \sum _{\begin{array}{c} j_k=1\\ k=1,\dots ,N-1\\ ~~\\ \mathbf i \ne \mathbf j \end{array}}^{p_\mathbf{n }}\sum _{j_N=1}^{q_\mathbf{n }} E[\Lambda _\mathbf{i }\Lambda _\mathbf{j }]. \end{aligned}$$

We proved above that $\mathrm{Var}\left[ \Lambda _\mathbf{i } \right] <C$ (see 11). By Lemma 8, we have:

$$\begin{aligned} \left| E[\Lambda _\mathbf{i }\Lambda _\mathbf{j }]\right| \le C(h_H\phi _x(h_K))^{-1}\varphi \left( \Vert \mathbf i -\mathbf j \Vert \right) . \end{aligned}$$

(17)

Then, we deduce that

$$\begin{aligned} \mathrm{Var}\left[ \hat{U}_i\right]&\le Cp_\mathbf{n }^{N-1}q_\mathbf{n }\left( 1+(h_H\phi _x(h_K))^{-1}\sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N-1 \end{array}}^{p_\mathbf{n }} \sum _{i_N=1}^{q_\mathbf{n }}\left( \varphi \left( \left\| \mathbf i \right\| \right) \right) \right) \nonumber \\&\le Cp_\mathbf{n }^{N-1}q_\mathbf{n } (h_H\phi _x(h_K))^{-1}\sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N-1 \end{array}}^{p_\mathbf{n }}\sum _{i_N=1}^{q_\mathbf{n }} \left( \varphi (\left\| \mathbf i \right\| )\right) . \end{aligned}$$

Consequently, we have:

$$\begin{aligned} A_1\le CMp_\mathbf{n }^{N-1}q_\mathbf{n } (h_H\phi _x(h_K))^{-1}\sum _{i=1}^\infty i^{N-1}\left( \varphi (i)\right) . \end{aligned}$$

Let

$$\begin{aligned} I\left( 2,\mathbf{n },x,\mathbf{j }\right)&= \Big \{\mathbf{i }:j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+1\le i_k\le j_k(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }},~~1\le k\le N-1;\\&\quad +j_N(p_{\mathbf{n }}+q_{\mathbf{n }})+p_{\mathbf{n }}+1\le i_N\le (j_N+1)(p_{\mathbf{n }}+q_{\mathbf{n }})\Big \}. \end{aligned}$$

The  variable $U\left( 2,\mathbf n ,x,\mathbf j \right)$ is the sum of the $\Lambda _\mathbf{i }$ such that $\mathbf i$ is in $I\left( 2,\mathbf n ,x,\mathbf j \right)$. Since $p_\mathbf{n }>q_\mathbf{n }$, if $\mathbf i$ and $\mathbf i'$ are, respectively, in the two different sets $I\left( 2,\mathbf n ,x,\mathbf j \right)$ and $I\left( 2,\mathbf n ,x,\mathbf j' \right)$; then $i_k\ne i'_k$ for un certain $k$ such that $1\le k\le N$ and $\Vert \mathbf i -\mathbf i{'} \Vert >q_\mathbf{n }$.

By using the definition of $A_2$, the stationarity of the process and (17), we have:

$$\begin{aligned} A_2\le \displaystyle \sum _{\begin{array}{c} j_k=1\\ k=1,\dots ,N \end{array}}^{n_k}\displaystyle \sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N\\ \Vert \mathbf i -\mathbf j \Vert >q_\mathbf{n } \end{array}}^{n_k}E[\Lambda _\mathbf{i }\Lambda _\mathbf{j }]\le C (h_H\phi _x(h_K))^{-1}\widehat{\mathbf{n }}\displaystyle \sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N\\ \Vert \mathbf i \Vert >q_\mathbf{n } \end{array}}^{n_k}\left( \varphi (\Vert \mathbf i \Vert )\right) \end{aligned}$$

and

$$\begin{aligned} A_2\le C(h_H\phi _x(h_K))^{-1}\widehat{\mathbf{n }}\sum _{i=q_\mathbf{n }}^\infty i^{N-1}\left( \varphi (i)\right) . \end{aligned}$$

We deduce that:

$$\begin{aligned} \widehat{\mathbf{n }}^{-1}E\left[ T(\mathbf n ,x,2)\right] ^2&\le CMp_\mathbf{n }^{N-1}q_\mathbf{n }\widehat{\mathbf{n }} ^{-1} (h_H\phi _x(h_K))^{-1}\sum _{i=1}^\infty i^{N-1-\delta }\nonumber \\&\quad +C(h_H\phi _x(h_K))^{-1}\sum _{i=q_\mathbf{n }}^\infty i^{N-1-\delta }. \end{aligned}$$

From $(p_\mathbf{n }+q_\mathbf{n })^{-N}p_\mathbf{n }^{N-1}q_\mathbf{n }=(p_\mathbf{n }+q_\mathbf{n })^{-N}p_\mathbf{n }^N \left( \frac{q_\mathbf{n }}{p_\mathbf{n }}\right) \le \frac{q_\mathbf{n }}{p_\mathbf{n }}$, we get:

$$\begin{aligned} CMp_\mathbf{n }^{N-1}q_\mathbf{n }\widehat{\mathbf{n }}^{-1} (h_H\phi _x(h_K))^{-1}&= \widehat{\mathbf{n }}(p_\mathbf{n }+q_\mathbf{n })^{-N}p_\mathbf{n }^{N-1}q_\mathbf{n } \widehat{\mathbf{n }}^{-1} (h_H\phi _x(h_K))^{-1}\nonumber \\&\le \left( \frac{q_\mathbf{n }}{p_\mathbf{n }}\right) (h_H\phi _x(h_K))^{-1}\nonumber \\&= q_\mathbf{n }s_\mathbf{n }\left( \widehat{\mathbf{n }}(h_H\phi _x(h_K))\right) ^{\frac{-1}{2N}}(h_H\phi _x(h_K))^{-1}\nonumber \\&= q_\mathbf{n }s_\mathbf{n }\left( \widehat{\mathbf{n }}(h_H\phi _x(h_K))^{(1+2N)}\right) ^{\frac{-1}{2N}}. \end{aligned}$$

By the definitions of $q_\mathbf{n }$ and $s_\mathbf{n }$, this last term converges to $\rightarrow 0$. Moreover, we have:

$$\begin{aligned}&C(h_H\phi _x(h_K))^{-1}\sum _{i=q_\mathbf{n }}^\infty i^{N-1-\delta }\le C(h_H\phi _x(h_K))^{-1}\int _{q_\mathbf{n }}^\infty t^{N-1-\delta }dt\\&\quad =C(h_H\phi _x(h_K))^{-1}q_\mathbf{n }^{N-\delta }. \end{aligned}$$

This last term converges to zero by (H11) and ends the proof of (14). $\square$

For (15): Let us use the following decomposition of small and big blocks

$$\begin{aligned} S'_\mathbf{n }=T\left( \mathbf n ,x,1\right) ,\qquad S''_\mathbf{n }=\sum _{i=2}^{2^N}T\left( \mathbf n ,x,i\right) . \end{aligned}$$

Then, we can write:

$$\begin{aligned} \widehat{\mathbf{n }}^{-1}E \left[ S'_\mathbf{n }\right] ^2=\widehat{\mathbf{n }}^{-1}E[ S_\mathbf{n }^2]+\widehat{\mathbf{n }}^{-1} E \left[ S''_\mathbf{n }\right] ^2-2\widehat{\mathbf{n }}^{-1}E [S_\mathbf{n }S''_\mathbf{n }]. \end{aligned}$$

Lemma 5 and (14) imply, respectively, that $\widehat{\mathbf{n }}^{-1}E \left( S_\mathbf{n }\right) ^2= \widehat{\mathbf{n }}^{-1}\mathrm{Var} \left( S_\mathbf{n }\right) \rightarrow \sigma ^2(x)$ and $\widehat{\mathbf{n }}^{-1}E \left[ S''_\mathbf{n }\right] ^2\rightarrow 0$.

Then, to show that $\widehat{\mathbf{n }}^{-1}E \left[ S'_\mathbf{n }\right] ^2\rightarrow \sigma ^2(x)$, it suffices to remark that $\widehat{\mathbf{n }}^{-1}E[ S_\mathbf{n }S''_\mathbf{n }]\rightarrow 0$ because, by Cauchy–Schwartz’s inequality, we can write:

$$\begin{aligned} \left| \widehat{\mathbf{n }}^{-1}E[ S_\mathbf{n }S''_\mathbf{n }]\right| \le \widehat{\mathbf{n }}^{-1}E \left| S_\mathbf{n }S''_\mathbf{n }\right| \le \left( \widehat{\mathbf{n }}^{-1}E[ S_\mathbf{n }^2]\right) ^{1/2}\left( \widehat{\mathbf{n }}^{-1}E[ {S''_\mathbf{n }}^2]\right) ^{1/2}. \end{aligned}$$

Recall that $T\left( \mathbf n ,x,1\right) =\sum _\mathbf{j \in \mathcal {J}}W\left( 1,\mathbf n ,x,\mathbf j \right) ,$ so

$$\begin{aligned} \widehat{\mathbf{n }}^{-1}E \left( S'_\mathbf{n }\right) ^2&= \widehat{\mathbf{n }}^{-1}\sum _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1}E\left[ W\left( 1,\mathbf n ,x,\mathbf j \right) \right] ^2\\&\quad +\widehat{\mathbf{n }}^{-1} \times \displaystyle \sum _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1}\displaystyle \sum _{\begin{array}{c} i_k=0\\ k=1,\dots ,N\\ i_k\ne j_k~\mathrm {for some k} \end{array}}^{r_k-1}\mathrm{Cov}(W(1,\mathbf n ,x,\mathbf j ),W(1,\mathbf n ,x,\mathbf i )) \end{aligned}$$

By similar arguments used above for $A_2$, this last term is not greater than

$$\begin{aligned}&C(h_H\phi _x(h_K))^{-1}\displaystyle \sum _{\begin{array}{c} i_k=1\\ k=1,\dots ,N\\ \Vert \mathbf i \Vert >q_\mathbf{n } \end{array}}^{r_k-1 }\left( \varphi (\Vert \mathbf i \Vert )\right) \nonumber \\&\qquad \le C(h_H\phi _x(h_K))^{-1}\sum _{i=q_\mathbf{n }}^\infty i^{N-1}\left( \varphi (i)\right) \le C(h_H\phi _x(h_K))^{-1}q_\mathbf{n }^{N-\delta }\rightarrow 0. \end{aligned}$$

So $Q_3\rightarrow \sigma ^2(x)$. This ends the proof. Proof of 16: Since $\left| \Lambda _\mathbf{i }\right| \le C(h_H\phi _x(h_K))^{-1/2}$, we have: $\left| W\left( 1,\mathbf n ,x,\mathbf j \right) \right| \le Cp_\mathbf{n }^N(h_H\phi _x(h_K))^{-1/2}$. Then we deduce that

$$\begin{aligned} Q_4\le Cp_\mathbf{n }^{2N}(h_H\phi _x(h_K))^{-1}\widehat{\mathbf{n }}^{-1}\sum _{\begin{array}{c} j_k=0\\ k=1,\dots ,N \end{array}}^{r_k-1}P\left[ \left| W\left( 1,\mathbf n ,x,\mathbf j \right) \right| >\epsilon \left( \sigma ^2(x)\widehat{\mathbf{n }}\right) ^{1/2}\right] . \end{aligned}$$

We have $\left| W\left( 1,\mathbf n ,x,\mathbf j \right) \right| /\left( \left( \sigma ^2(x)\widehat{\mathbf{n }}\right) ^{1/2}\right) \le Cp_\mathbf{n }^N\left( \widehat{\mathbf{n }}h_H\phi _x(h_K) \right) ^{-1/2}=C\left( s_\mathbf{n }\right) ^{-N}\rightarrow 0$, because $p_\mathbf{n }=\left[ \left( \widehat{\mathbf{n }}h_H\phi _x(h_K)\right) ^{1/(2N)}/s_\mathbf{n }\right]$ and $s_\mathbf{n }\rightarrow \infty$.

So, for all $\epsilon$ and $\mathbf j \in \mathcal {J}$; if $\widehat{\mathbf{n }}$ is great enough, then $P\left[ W\left( 1,\mathbf n ,x,\mathbf j \right) >\epsilon \left( \sigma ^2(x)\widehat{\mathbf{n }}\right) ^{1/2}\right] =0$. Then $Q_4=0$ for $\widehat{\mathbf{n }}$ great enough. This yields the proof.