Expectile regression for spatial functional data analysis (sFDA)

Abstract

This paper deals with the nonparametric estimation of the expectile regression when the observations are spatially correlated and are of a functional nature. The main findings of this work is the establishment of the almost complete convergence for the proposed estimator under some general mixing conditions. The performance of the proposed estimator is examined by using simulated data. Finally, the studied model is used to evaluate the air quality indicators in northeast China.

Appendix

In order to keep the standard techniques in nonparametric spatial statistics, we will assume for our asymptotic results, without loss of generality, that the set of sites of observations $I_n$ is a rectangular region. Next, we state the following two lemmas that were used by Carbon et al. (1996). These lemmas will be useful to prove the strong convergence of our estimators. Their proofs are obviously omitted.

Lemma 1

Let $E_1,\ldots , E_r$ be r sets containing m sites each with $dist(E_i,\ E_j)\ge \gamma$ for all $i\ne j$ where $1\le i, j\le r$. Assume that $Z_1, \ldots , Z_r$ is a sequence of real-valued random variables which are measurable with respect to $\mathscr {B}(E_1),\ldots ,\mathscr {B}(E_r)$ respectively, and that $Z_i$ takes values in [a, b]. Then, there exists a sequence of independent random variables $Z_1^*,\ldots ,Z_r^*$ which is independent of $Z_1, \ldots ,Z_r$ such that $Z_i^*$ has the same distribution as $Z_i$ and satisfies

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

(6)

Lemma 2

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

$$\begin{aligned} |{\mathbb {E}}[XY]-{\mathbb {E}}[X]{\mathbb {E}}[Y]|\le C\Vert X\Vert _r \Vert Y\Vert _s \{s(Card(E),Card(E'))\varphi (dist(E,E'))\}^{1/t}.\nonumber \\ \end{aligned}$$

(7)

(ii) For r.v.’s bounded with probability 1, the right-hand side of (7) can be replaced by

$$\begin{aligned} Cs(Card(E),Card(E'))\varphi (dist(E,E')). \end{aligned}$$

Proof of Theorem 1 For $\epsilon >0$ small enough, we use the fact that the function $G(x,\cdot )$ is an increasing function and has a strictly positive derivative in the neighbourhood of $\xi _p(x)$, to write that

$$\begin{aligned}&\displaystyle \sum _{{n}} {\mathbb {P}}\Bigg (\Big | \widehat{\xi _p}(x)-\xi _p(x)\Big | >\epsilon \Bigg )\\&\quad \le \displaystyle \sum _{{n}} {\mathbb {P}}\Bigg (\Big |\widehat{G}(x,\xi _p(x)-\epsilon )- G(x,\xi _p(x)-\epsilon )\Big | \ge C\epsilon \Bigg ) \\&\qquad + \displaystyle \sum _{{n}} {\mathbb {P}}\Bigg (\Big | \widehat{G}(x,\xi _p(x)+\epsilon )- G(x,\xi _p(x)+\epsilon )\Big | \ge C \epsilon \Bigg ). \end{aligned}$$

So, it suffices to apply the result of the Proposition 1 to conclude the result of Theorem 1.

Proof of Proposition 1 We decompose $\widehat{G}(x,t)-G(x,t)$ as follows

$$\begin{aligned} \widehat{G}(x,t)-G(x,t)= & {} \frac{\displaystyle \widehat{G}_{1}(x,t)}{\displaystyle \widehat{G}_{2}(x,t)}-\frac{\displaystyle G_{1}(x,t)}{\displaystyle G_{2}(x,t)} \nonumber \\= & {} \frac{\displaystyle 1}{\displaystyle \widehat{G}_{2}(x,t)}\Big [\widehat{G}_{1}(x,t)- G_{1}(x,t)\Big ] \nonumber \\&+\, \frac{\displaystyle G(x,t)}{\displaystyle \widehat{G}_{2}(x,t)}\Big [G_{2}(x,t)- \widehat{G}_{2}(x,t)\Big ]. \end{aligned}$$

(8)

Thus, Proposition 1 as well as Theorem 1’s results are direct consequences of the following intermediate results.

Lemma 3

Under the assumptions of Lemma 4 and, in addition, we assume that the assumptions (A2) and (A4) hold, then we have, as ${n} \rightarrow \infty$,

$$\begin{aligned} \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big |\widehat{G}_{1}(x,t)- \mathbb {E}\Big [\widehat{G}_{1}(x,t)\Big ]\Big | =O_{a.co.}\Bigg ( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\Bigg ), \end{aligned}$$

(9)

and

$$\begin{aligned} \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big | \widehat{G}_{2}(x,t)-\mathbb {E}\Big [\widehat{G}_{2}(x,t)\Big ] \Big | =O_{a.co.}\Bigg ( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\Bigg ), \end{aligned}$$

where $O_{a.co.}(\cdot )$ denotes O almost completely.

Lemma 4

Assume that assumptions (A1), (A3) and (A5) are fulfilled. We have, as ${n} \rightarrow \infty$,

$$\begin{aligned} \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big |\mathbb {E}\Big [\widehat{G}_{1}(x,t)\Big ]-G_{1}(x,t)\Big | = O\Big (h_{n}^{k_\mathbf{1}}\Big ), \end{aligned}$$

(10)

and

$$\begin{aligned} \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big |\mathbb {E}\Big [\widehat{G}_{2}(x,t)\Big ]-G_{2}(x,t)\Big | = O\Big (h_{n}^{k_{2}}\Big ). \end{aligned}$$

Lemma 5

Under the assumptions of Lemma 4 and, in addition, we assume that the assumptions (A2) and (A4) are satisfied, we have, as ${n} \rightarrow \infty$,

$$\begin{aligned} \displaystyle \sum _{n}{\mathbb {P}}\Bigg (\inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}\Big | \widehat{G}_{2}(x,t)\Big | \le \epsilon '\Bigg )< \infty ,\quad \text{ for } \text{ certain } \quad \epsilon '>0. \end{aligned}$$

(11)

Then, in order to show the consistency of $\widehat{\xi _p}(x)$, it suffices to show that the estimator $\widehat{G}_{2}(x,t)$ converges almost completely, as ${n} \rightarrow \infty$.

Proof of Lemma 3. Because of the similarity between the first and the second cases we prove only the first one. To do that, we combine the arguments of Laksaci et al. (2009) in FDA to those of Tran (1990) in the spatial model (finite dimension). Indeed, the compactness of $[\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]$ allows to recover it by $d_{n}=O\left( \widehat{n}^{1/2 \varsigma _1}\right)$ intervals whose the subset of their extremities is

$$\begin{aligned} \mathcal {T}_{n}=\left\{ t_j-l_{n},t_j+l_{n},1\le j\le d_{n}\right\} , \end{aligned}$$

(12)

with $l_{n}=O(d_{n}^{-1})$. Next, using the monotony of $\widehat{G}_{1}(x,\cdot )$ and ${\mathbb {E}}[ \widehat{G}_{1}(x,\cdot )]$ to conclude that

$$\begin{aligned}&\sup _{t\in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}\left| \widehat{G}_{1}(x,t) -{\mathbb {E}}\left[ \widehat{G}_{1}(x,t)\right] \right| \\&\qquad \le \max _{1\le j\le d_{n}} \max _{z\in \{t_j-l_{n}, t_j+l_{n}\}}\left| \widehat{G}_{1}(x,z)- {\mathbb {E}}\left[ \widehat{G}_{1}(x,z)\right] \right| +Cl_{n}^{\varsigma _1}. \end{aligned}$$

Evidently, we have

$$\begin{aligned} l_{n}^{\varsigma _l}=o\left( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) . \end{aligned}$$

So, all it remains to prove is that

$$\begin{aligned} \max _{1\le j\le d_{n}}\max _{z\in \{t_j-l_{n},t_j+l_{n}\}}\left| \widehat{G}_{1}(x,z)- {\mathbb {E}}\left[ \widehat{G}_{1}(x,z)\right] \right| =O\left( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) , \; a.co. \end{aligned}$$

(13)

which requires to show that

$$\begin{aligned} \sum _{n} d_{n}\max _{1\le j\le d_{n}}\max _{z\in \{t_j-l_{n},t_j+l_{n}\}}{\mathrm {I\!P}}\left( \left| \widehat{G}_{1}(x,z)- {\mathbb {E}}\left[ \widehat{G}_{1}(x,z)\right] \right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) <\infty . \end{aligned}$$

So, for all $z=t_j\mp l_{n}$ with $1\le j\le d_{n}$, we must evaluate the quantity

$$\begin{aligned} {\mathrm {I\!P}}\left( \left| \widehat{G}_{1}(x,z)- {\mathbb {E}}\left[ \widehat{G}_{1}(x,z)\right] \right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) . \end{aligned}$$

(14)

For this aim, we use the truncation method, where we define

$$\begin{aligned} \widehat{G^{*}_1}(x,t)=\frac{1}{\widehat{n}\mathbb {E}\Big [K_\mathbf{1}(x)\Big ]} \sum _{\mathbf{i}\in I_n} K\Big (h^{-1}_{n}d(x,X_{\mathbf{i}})\Big )Y_{\mathbf{i}}^*, \end{aligned}$$

such that $Y_{\mathbf{i}}^*= (Y_\mathbf{i}-t)\mathbb {I}_{\big \{Y_\mathbf{i}-t\le 0, \, |Y_\mathbf{i}|< \gamma _{{n}}\big \}}$ and $K_\mathbf{i}(x)=K\Big (h^{-1}_{n}d(x,X_\mathbf{i})\Big )$.

So, the claimed quantity (14) is a consequence of the following three quantities:

$$\begin{aligned}&{\mathrm {I\!P}}\left( \left| \widehat{G^{*}_1}(x,t)- \widehat{G}_1(x,t)\right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) , \end{aligned}$$

(15)

$$\begin{aligned}&\Big |{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]- {\mathbb {E}}[\widehat{G}_1(x,t)]\Big |=o\left( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) , \end{aligned}$$

(16)

and

$$\begin{aligned} {\mathrm {I\!P}}\left( \left| \widehat{G^{*}_1}(x,t) -{\mathbb {E}}\big [\widehat{G^{*}_1}(x,t)\right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) . \end{aligned}$$

(17)

• Concerning the quantity (15). The Markov’s inequality permits to write that,

  $$\begin{aligned} {\mathrm {I\!P}}\Big (\Big |\widehat{G^{*}_1}(x,t)- \widehat{G}_1(x,t)\Big |> \epsilon _0\left( \sqrt{\frac{\log \widehat{n}}{\widehat{n}\,\phi _x(h_{n})}}\right) \Big )\le & {} \sum _{\mathbf{i}\in I_n} {\mathrm {I\!P}}\Big (Y_{\mathbf{i}}>\gamma _{{n}}\Big ) \\\le & {} \widehat{n}\gamma _{{n}}^{-q}{\mathbb {E}}\Big [Y^q\Big ]. \end{aligned}$$

  It follows that

  $$\begin{aligned} {\mathrm {I\!P}}\left( \left| \widehat{G^{*}_1}(x,t)-\widehat{G}_1(x,t)\right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) \le C \widehat{n}\gamma _{{n}}^{-q}. \end{aligned}$$

  (18)

• For the quantity (16). The main tool of this part is the Hölder’s inequality. Indeed, put $\displaystyle \alpha =\frac{q}{2}$ then for $\beta$ such that

  $$ \frac{1}{\alpha }+\frac{1}{\beta }=1$$

  we have

  $$\begin{aligned}&\Big |{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]- {\mathbb {E}}[\widehat{G}_1(x,t)]\Big |\\&\quad \le \frac{C}{ \mathbb {E}\Big [K_\mathbf{1}(x)\Big ]}{\mathbb {E}}\Big [\big |Y^-\big |\mathbb {I}_{\big \{Y\ge \gamma _{{n}}\big \}}K\Big (h^{-1}_{n}d(x,X)\Big )\Big ]\\&\quad \le \frac{\gamma _{{n}}^{-1}}{\mathbb {E}\Big [K_\mathbf{1}(x)\Big ]} {\mathbb {E}}^{1/\alpha }\Big [\big |Y^{2\alpha }\big |\Big ] {\mathbb {E}}^{1/\beta }\Big [K^{\beta }\Big (h^{-1}_{n}d(x,X_1)\Big )\Big ]\\&\quad \le \frac{C\gamma _{{n}}^{-1}}{\mathbb {E}\Big [K_\mathbf{1}(x)\Big ]} {\mathbb {E}}^{1/\beta }\Big [K_\mathbf{1}^{\beta }(x)\Big ].\end{aligned}$$

  We deduce that

  $$\begin{aligned} \Big |{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]- {\mathbb {E}}[\widehat{G}_1(x,t)]\Big |\le C\gamma _{{n}}^{-1}\phi _x^{(1-\beta )/\beta }(h_{n}). \end{aligned}$$

  (19)

• About the quantity (17). We use the spatial blocks decomposition which was used by Tran (1990). Indeed, we write

  $$\begin{aligned} \widehat{G^{*}_1}(x,t) -{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]= {1\over \widehat{n}\mathbb {E}\Big [K_\mathbf{1}(x)\Big ]}\sum _{\mathbf{i}\in I_n} K_{\mathbf{i}}(x) Y^*_{\mathbf{i}}-{\mathbb {E}}\Big [K_\mathbf{1}(x)Y^*_{\mathbf{i}}\Big ], \end{aligned}$$

  where $\displaystyle \Lambda _{\mathbf{i}}=K_{\mathbf{i}}(x) Y^*_{\mathbf{i}}-{\mathbb {E}}\Big [K_\mathbf{1}(x)Y^*_{\mathbf{i}}\Big ].$ The spatial blocks decomposition of this sum are defined, for an arbitrary integer sequence $(p_{n})$ by taking

  $$\begin{aligned} U(1,{n},\mathbf{j})= & {} \sum _{{\mathop {k=1,\ldots ,N}\limits ^{i_k=2j_kp_{n}+1}}}^{2j_kp_{n}+p_{n}}\Lambda _{\mathbf {i}},\\ U(2,{n},\mathbf{j})= & {} \displaystyle \sum _{{\mathop {k=1,\ldots ,N-1}\limits ^{i_k=2j_kp_{n}+1}}}^{2j_kp_{n}+p_{n}} \quad \displaystyle \sum _{i_N=2j_Np_{n}+p_{n}+1}^{(j_N+1)p_{n}}\Lambda _{\mathbf {i}},\\ U(3,{n},\mathbf{j})= & {} \displaystyle \sum _{{\mathop {k=1,\ldots ,N-2}\limits ^{i_k=2j_kp_{n}+1}}}^{2j_kp_{n}+p_{n}} \quad \displaystyle \sum _{i_{N-1}=2j_{N-1}p_{n}+p_{n}+1}^{2(j_{N-1}+1)p_{n}} \quad \displaystyle \sum _{i_N=2j_Np_{n}+1}^{2j_Np_{n}+p_{n}}\Lambda _{\mathbf {i}},\\ U(4,{n},\mathbf{j})= & {} \displaystyle \sum _{{\mathop {k=1,\ldots ,N-2}\limits ^{i_k=2j_kp_{n}+1}}}^{2j_kp_{n}} \quad \displaystyle \sum _{i_{N-1}=2j_{N-1}p_{n}+p_{n}+1}^{2(j_{N-1}+1)p_{n}}\quad \displaystyle \sum _{i_N=2j_Np_{n}+p_{n}+1}^{2(j_N+1)p_{n}}\Lambda _{\mathbf {i}}, \end{aligned}$$
  $$\vdots $$

  The last term is

  $$\begin{aligned} U(2^N,{n},\mathbf{j})= & {} \sum _{{\mathop {k=1,\ldots ,N}\limits ^{i_k=2j_kp_{n}+p_{n}+1}}}^{2(j_k+1)p_{n}} \Lambda _{\mathbf {i}}. \end{aligned}$$

  Finally, we get

  $$\begin{aligned} \widehat{G^{*}_1}(x,t) -{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]=\frac{1}{\widehat{n}\mathbb {E}\Big [K_\mathbf{1}(x)\Big ]}\displaystyle \sum _{i=1}^{2^N}T(n,i), \end{aligned}$$

  (20)

  with $\displaystyle T(n,i)=\sum _{\mathbf {j}\in \mathcal {J}}U(i,n,\mathbf {j})$; $\displaystyle \mathcal {J}=\big \{0,\ldots ,r_1-1\big \}\times \cdots \times \big \{0,\ldots ,r_N-1\big \}$ and $\displaystyle r_{{ i}} =2 n_{{ i}}\big /p_{n}$ for $i = 1, \ldots ,N$.

Furthermore, from (20) we get, for all $\eta >0$

$$\begin{aligned} {\mathrm {I\!P}}\left( \Big |\widehat{G^{*}_1}(x,t) -{\mathbb {E}}[\widehat{G^{*}_1}(x,t)]\Big |\ge \eta \right) \le 2^{N}\max _{i=1,\ldots , 2^N}{\mathrm {I\!P}}\Big ( T(n,i)\ge \eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]\Big ). \end{aligned}$$

Now, the claimed result is based on the evaluation of the quantities

$$\begin{aligned} \displaystyle {\mathrm {I\!P}}\Big ( T(n,i)\ge \eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]\Big ),\; \hbox { for all } i=1,\ldots , 2^N . \end{aligned}$$

Once again, we only treated the case $i=1$. The other cases are similar. Indeed, the main step of the proof is the application of Lemma 1 on the random variables $U(1,n,\mathbf {j})$ for $\mathbf {j}\in \mathcal {J}$. Precisely, we enumerate these variables in the arbitrary way $Z_1,\ldots , Z_M$ with $M=\prod _{k=1}^N r_k=2^{-N}\widehat{n}p_{n}^{-N}\le \widehat{n}p_{n}^{-N}$ where for each $Z_{{j}}$, there exists certain $\mathbf {j}$ in $\mathcal {J}$ such that

$$\begin{aligned} Z_{{j}}=\sum _{\mathbf {i}\in I(1,n,\mathbf {j})}\Lambda _{\mathbf {i}}, \end{aligned}$$

where $\displaystyle I(1,n,\mathbf {j})=\Big \{\mathbf {i}:2j_{k}p_{n}+1\le i_k\le 2j_{k}p_{n}+p_{n} \hbox { for } k=1,\ldots ,N\Big \}.$

Clearly the sets $\displaystyle I(1,n,\mathbf {j})$ contain $p_{n}^N$ sites and are far apart by distant of $p_{n}$ at least. Moreover, under Assumption (A4), we have

$$\begin{aligned} K\Big (h^{-1}_{n}d(x,X_{\mathbf{i}})\Big )Y_{\mathbf{i}}^*\displaystyle \le C \gamma _{{n}}. \end{aligned}$$

Therefore, the Lemma 1 gives M independent random variables $Z_1^*,\ldots ,Z_M^*$ which are identically distributed as $Z_j$ for $j=1,\ldots , M$ and such that

$$\begin{aligned} \sum _{j=1}^r {\mathbb {E}}\big |Z_{\mathbf {j}}-Z_{\mathbf {j}}^{*}\big |\le 2C \gamma _{{n}} Mp_{n}^N\psi \Big ((M-1)p_{n}^N,p_{n}^N\Big )\varphi (p_{n}). \end{aligned}$$

(21)

Therefore, we have

$$\begin{aligned} {\mathrm {I\!P}}\Big (T(n,i)\ge \eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]\Big )\le B_1(n)+B_2(n), \end{aligned}$$

where

$$\begin{aligned} \displaystyle B_1(n)={\mathrm {I\!P}}\Bigg (\Big |\sum _{j=1}^MZ_{\mathbf {j}}^{*}\Big |\ge \frac{M\eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]}{2M}\Bigg ), \end{aligned}$$

and

$$\begin{aligned} B_2(n)={\mathrm {I\!P}}\Bigg (\sum _{j=1}^M\big |Z_{\mathbf {j}}-Z_{\mathbf {j}}^{*}\big |\ge \frac{\eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]}{2}\Bigg ). \end{aligned}$$

Firstly, the asymptotic evaluation of the term $B_2(n)$ is based on the Markov inequality and (21), which allow to write that

$$\begin{aligned} B_2(n)\le 2M\gamma _{{n}}p_{n}^N\Big (\eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]\Big )^{-1}\psi \Big ((M-1)p_{n}^N,p_{n}^N\Big )\varphi (p_{n}). \end{aligned}$$

Next, by using the facts that

$$\begin{aligned}&\mathbb {E}\big [K_\mathbf{1}(x)\big ]\le C \phi _x(h_{n}),\\&\displaystyle \widehat{n}=2^NMp_{n}^N, \end{aligned}$$

and

$$\begin{aligned} \psi \Big ((M-1)p_{n}^N,p_{n}^N\Big )\le p_{n}^N, \end{aligned}$$

we get for $\eta =\displaystyle \eta _0\sqrt{\frac{\log \widehat{n}}{\widehat{n}\, \phi _x(h_{n})}}$, that

$$\begin{aligned} \displaystyle B_2(n)\le \widehat{n} \gamma _{{n}} p_{n}^N\Big ( \log \widehat{n}\Big )^{-1/2}\Big (\widehat{n}\phi _x(h_{n})\Big )^{-1/2}\varphi (p_{n}). \end{aligned}$$

So, for the particular choice of $p_{n}= C\,\Bigg (\displaystyle \frac{\widehat{n}\phi _x(h_{n})}{\log \widehat{n}\gamma _{{n}}^2}\Bigg )^{1/2N}$, we get

$$\begin{aligned} B_2 (n)\le \widehat{n}^{1-\delta /2N}\gamma _{\mathbf{n}}^{\delta /N}\phi _x^{-\delta /2N}(h_{n})\log ^{\delta /2N} \widehat{n} <\infty . \end{aligned}$$

(22)

Secondly, the asymptotic evaluation of the $B_1(n)$ is based on the Bernstein inequality. This last allows to write that

$$\begin{aligned} B_1 (n)\le 2\exp \left( -\frac{\Big (\eta \widehat{n}\mathbb {E}\big [K_\mathbf{1}(x)\big ]\Big )^2}{ M\, var\big [Z_1^{*}\big ]+C \eta \gamma _{{n}}p_{n}^N\, \widehat{n}\, \mathbb {E}\big [K_\mathbf{1}(x)\big ] }\right) . \end{aligned}$$

(23)

Now, the rest of the proof is based on the asymptotic evaluation of the variance term

$$\begin{aligned} \, var\big [Z_1^{*}\big ]= \, var\left[ \sum _{\mathbf {i}\in I(1,n,\mathbf {1})}\Lambda _{\mathbf {i}}\right] \, +\sum _{\mathbf {i}\ne \mathbf {j}\in I(1,n,\mathbf {1})}\Big |\, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\Big |. \end{aligned}$$

To do that we evaluate separately the variance and the covariance terms under Assumption (A2). Indeed, on one hand, under the first part of Assumption (A2), we have

$$\begin{aligned} \, var\Big [\Lambda _{\mathbf{i}}\Big ]\le & {} C\; {\mathbb {E}}\Big [K^2_{\mathbf{i}} Y^{*2}_{\mathbf{i}}\Big ] \le C\; {\mathbb {E}}\Big [K^2_{\mathbf{i}} Y^{2}_{\mathbf{i}}\Big ]\\\le & {} C\;{\mathbb {E}}\Big [K^2_{\mathbf{i}} {\mathbb {E}}\Big [Y^{2}_{\mathbf{i}}\big |X_{\mathbf{i}}\Big ] \Big ] \le C\phi _x(h_{n}). \end{aligned}$$

It follows that

$$\begin{aligned} \sum _{\mathbf {i}\in I(1,n,\mathbf {1})} \, var\Big [ \Lambda _{\mathbf {i}}\Big ]=O\Big (p_{n}^N\phi _x(h_{n})\Big ). \end{aligned}$$

Concerning the covariance term, we use the second part of Assumption (A2), for all $\mathbf {i}\not =\mathbf {j}$

$$\begin{aligned} \, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\le & {} C\;{\mathbb {E}}\Big [K_{\mathbf{i}} |Y^{*}_{\mathbf{i}} |K_{\mathbf{j}} |Y^{*}_{\mathbf{j}}|\Big ]\le C\;{\mathbb {E}}\Big [K_{\mathbf{i}} K_{\mathbf{j}} |Y_{\mathbf{i}} Y_{\mathbf{j}}|\Big ]\\\le & {} C\; {\mathbb {E}}\Big [K_{\mathbf{i}} K_{\mathbf{j}} {\mathbb {E}}\Big [|Y_{\mathbf{i}} Y_{\mathbf{j}}||X_{\mathbf{i}} X_{\mathbf{j}} \Big ]\Big ]\le C\;{\mathbb {E}}\Big [K_{\mathbf{i}} K_{\mathbf{j}} \Big ]\le C\;\phi _x^{(a+1)/a}(h_{n}). \end{aligned}$$

Furthermore, since ${\mathbb {E}}\Big [Y^{p}_{\mathbf{i}}|X_{\mathbf{i}}\Big ]<\infty$ then, for all $\mathbf {i}\not =\mathbf {j}$

$$\begin{aligned} \, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\le & {} \Vert \Lambda _{\mathbf {i}}\Vert ^{2}_p\varphi ^{1-2/p}(\Vert \mathbf {i}-\mathbf {j}\Vert ) \le C\; \Vert K_{\mathbf{i}} Y^{*}_{\mathbf{i}}\Vert ^{2}_p\varphi ^{1-2/p}(\Vert \mathbf {i}-\mathbf {j}\Vert )\\\le & {} C\; \Vert K_{\mathbf{i}} Y_{\mathbf{i}}\Vert ^{2}_p\varphi ^{1-2/p}(\Vert \mathbf {i}-\mathbf {j}\Vert )\le C\; \Vert K_{\mathbf{i}} \Vert ^{2}_p\varphi ^{1-2/p}(\Vert \mathbf {i}-\mathbf {j}\Vert )\\\le & {} C\;\phi _x^{2/p}(h_{n})\varphi ^{1-2/p}(\Vert \mathbf {i}-\mathbf {j}\Vert )).\end{aligned}$$

Now, we use the same technique as in Masry (2005). The letter is based on the following decomposition

$$\begin{aligned}&\displaystyle \sum _{\mathbf {i}\ne \mathbf {j}\in I(1,n,\mathbf {1})}\Big |\, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\Big |\\\le & {} \displaystyle \sum _{\left\{ \mathbf {i},\mathbf {j}\in I(1,n,\mathbf {1})\,\Big \Vert \mathbf {i}-\mathbf {j}\Big \Vert \le c_{n}\right\} }\Big |\, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\Big | +\displaystyle \sum _{\left\{ \mathbf {i},\mathbf {j}\in I(1,n,\mathbf {1})\,\Big \Vert \mathbf {i}-\mathbf {j}\Big \Vert > c_{n}\right\} }\Big |\, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\Big |\\\le & {} C\; p_{n}^N\phi _x(h_{n})\Big (c_{n}^{N}\phi _x(h_{n})^{1/a} \displaystyle +c_{n}^{-Na}\phi _x^{2/p-1}(h_{n})\sum _{\mathbf {i}:\Vert \mathbf {i}\Vert \ge c_{n}}\Big \Vert \mathbf {i}\Big \Vert ^{Na}\varphi ^{1-2/p}\Big (\Big \Vert \mathbf {i}\Big \Vert \Big )\Big ), \end{aligned}$$

where $(c_{{n}})$ is an arbitrary sequence of real numbers which tends to $+\infty$. In particular, for $c_{{n}}= \phi _x(h_{n})^{2/Np(a+1)-1/Na}$ we have

$$\begin{aligned} \sum _{\mathbf {i}\ne \mathbf {j}\in I(1,n,\mathbf {1})}\Big |\, cov(\Lambda _{\mathbf {i}},\Lambda _{\mathbf {j}})\Big | \le Cp_{n}^N\phi _x(h_{n}). \end{aligned}$$

We deduce that

$$\begin{aligned} \, var\Big [\sum _{\mathbf {i}\in I(1,n,\mathbf {1})}\Lambda _{\mathbf {i}}\Big ]=O\Big (p_{n}^N\phi _x(h_{n})\Big ). \end{aligned}$$

Finally, we replace $var\Big [Z_1^{*}\Big ]=O\Big (p_{n}^N\phi _x(h_{n})\Big )$ in (23) to show that

$$\begin{aligned} B_1 (n)\le \exp \Big (-C(\eta _0)\log \widehat{n} \Big ). \end{aligned}$$

(24)

We combine equations (14), (19), (22) and (24) together with Assumption (H5) to conclude that

$$\begin{aligned} \sum _{n} d_{n}\max _{1\le j\le d_{n}}\max _{z\in \{t_j-l_{n},t_j+l_{n}\}}{\mathrm {I\!P}}\left( \left| \widehat{G}_{1}(x,z)- {\mathbb {E}}\left[ \widehat{G}_{1}(x,z)\right] \right| >{\eta }\sqrt{\frac{\log \widehat{n}}{\widehat{n}\phi _{x}(h_{n})}}\right) <\infty , \end{aligned}$$

which complete the proof of this lemma.

Proof of Lemma 4 Again, we only prove the first result and the second one can be obtained by the same manner. the proof is based on the stationarity of the process, which allows to write that

$$\begin{aligned} \mathbb {E}\Big [\widehat{G}_{1}(x,t)\Big ]-G_{1}(x,t) = \frac{\displaystyle 1}{\displaystyle \mathbb {E}\Big [K_\mathbf{1}(x)\Big ]} \Big \{\mathbb {E}\Big [K_\mathbf{1}(x)\Big (G_{1}({X_{1}},t)-G_{1}(x,t)\Big )\Big ]\Big \}. \end{aligned}$$

Now, by Assumption (A4), we get

$$\begin{aligned} \mathbb {E}\Big [K_\mathbf{1}(x)\Big (G_{1}(X_{1},t)-G_{1}(x,t)\Big )\Big ]= & {} \displaystyle \mathbb {E}\Big [K_\mathbf{1}(x)\Big (G_{1}(X,t)-G_{1}(x,t)\Big )\mathbb {I}_{B(x,h_{n})}\Big ]. \end{aligned}$$

Furthermore, Assumption (A3) allows to get

$$\begin{aligned}&\sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big |\mathbb {E}\Big [\widehat{G}_{1}(x,t)\Big ]-G_{1}(x,t)\Big |\\&\quad = \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \frac{\displaystyle 1}{\displaystyle \mathbb {E}\Big [K_\mathbf{1}(x)\Big ]}\Big |\mathbb {E}\Big [K_\mathbf{1}(x)\Big (G_{1}(X_{1},t)-G_{1}(x,t)\Big )\mathbb {I}_{B(x,h_{n})}\Big ]\Big | \\&\quad \le \displaystyle C\; h_{n}^{k_{ 1}}, \end{aligned}$$

which easily implies that

$$\begin{aligned} \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} \Big |\mathbb {E}\Big [\widehat{G}_{1}(x,t)\Big ]-G_{1}(x,t)\Big | = O\Big (h_{n}^{k_{ 1}}\Big ). \end{aligned}$$

(25)

Proof of Lemma 5. Observe that

$$\begin{aligned}&{\mathrm {I\!P}}\left( \inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}\Big | \widehat{G}_{2}(x,t)\Big | \le \frac{1}{2}\inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}G_2(x,t)\right) \\&\le {\mathrm {I\!P}}\left( \sup _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}\Big | \widehat{G}_{2}(x,t)\!-\!G_{2}(x,t)\Big | \!>\! \frac{1}{2}\inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}G_2(x,t)\!\right) . \end{aligned}$$

By applying Lemma 3’s result for $\displaystyle \epsilon _0=\frac{1}{2}\inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}G_2(x,t)>0,$ we obtain

$$\begin{aligned} \displaystyle \sum _{n\ge 1}{\mathrm {I\!P}}\left( \inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]}\Big | \widehat{G}_{2}(x,t)\Big | \le \frac{1}{2}\inf _{t \in [\xi _p(x)-\epsilon _0, \, \xi _p(x)+\epsilon _0]} G_2(x,t)\right) < \infty , \end{aligned}$$

hence this lemma’s proof is complete.