# Scalar-on-Function Relative Error Regression for Weak Dependent Case
## Appendix A  
In this appendix, we briefly give the proof of preliminary results; the proofs of Lemmas 3 and 4 are omitted, as they can be obtained straightforwardly through the adaptation of the proof of [PERSON] et al. [17]. Proof of Lemma 1.: Clearly, the proof of both terms is very similar. So, we will focus only in the first one. In fact, the difficulty in this kind of proof comes from the fact that the quantity \(Y_{i}^{-1}\) is not bounded. So, to deal with this problem, the truncation method is used to define  
\[\widetilde{R}_{N}^{*}(x)=\frac{1}{n\operatorname{\mathbb{E}}[K_{1}(x)]}\sum_{ i=1}^{n}K\Big{(}h_{n}^{-1}\|x-X_{i}\|\Big{)}Y_{i}^{-1}\mathds{1}_{|Y_{i}|>\mu_{ n}}\ \ \text{with}\ \mu_{n}=n^{-\
icefrac{{\delta}}{{\delta}}}.\]  
Then, the desired result is a consequence of  
\[\Big{|}\operatorname{\mathbb{E}}[\widetilde{R}_{N}^{*}(x)]-\operatorname{ \mathbb{E}}[\widetilde{R}_{N}(x)]\Big{|}=O\Bigg{(}\sqrt{\frac{\log n}{n^{1- \
icefrac{{\delta}}{{\delta}}}\phi_{x}(h_{n})}}\Bigg{)}, \tag{11}\]  
\[\Big{|}\widetilde{R}_{N}^{*}(x)-\widetilde{R}_{N}(x)\Big{|}=O_{a.co.}\Bigg{(} \sqrt{\frac{\log n}{n^{1-\
icefrac{{\delta}}{{\delta}}}\phi_{x}(h_{n})}} \Bigg{)}, \tag{12}\]  
and  
\[\Big{|}\widetilde{R}_{N}^{*}(x)-\operatorname{\mathbb{E}}[\widetilde{R}_{N}^ {*}(x)]\Big{|}=O_{a.co.}\Bigg{(}\sqrt{\frac{\log n}{n^{1-\
icefrac{{\delta}} {{\delta}}}\phi_{x}(h_{n})}}\Bigg{)}. \tag{13}\]  
We start by proving (13). For this, we write  
\[\widetilde{R}_{N}^{*}(x)-\operatorname{\mathbb{E}}\Big{[}\widetilde{R}_{N}^ {*}(x)\Big{]}=\sum_{i=1}^{n}\operatorname{Y}_{i}\ \ \ \ \text{where}\ \ \ \operatorname{Y}_{i}=\frac{1}{n \operatorname{\mathbb{E}}[K_{1}(x)]}\chi(X_{i},Y_{i}),\]  
with  
\[\chi(z,w)=w^{-1}K(h_{n}^{-1}\|x-z\|))\mathds{1}_{|w|>\mu_{n}}-\operatorname{ \mathbb{E}}\Big{[}K_{1}(x)Y^{-1}\mathds{1}_{|Y_{1}|>\mu_{n}}\Big{]},\,z\in \mathcal{H},\,w\in\mathbb{R}.\]  
Observe that,  
\[\|\chi\|_{\infty}\leq C\mu_{n}^{-1}\|K\|_{\infty}\ \ \text{and}\ \ \operatorname{ Lip}(\chi)\leq\leq C\mu_{n}^{-1}h_{n}^{-1}\operatorname{Lip}(K).\]  
The key tool for proving (13) is the application of [PERSON] and Newmann's inequality (see [21], p. 2). We apply this inequality on \(\operatorname{Y}_{i}\). It requires evaluating asymptotically two quantities: \(Var(\sum_{i=1}^{n}\operatorname{Y}_{i})\) and \(Cov(\operatorname{Y}_{s_{1}}\dots\operatorname{Y}_{s_{u}},\operatorname{Y}_{ t_{1}}\dots\operatorname{Y}_{t_{v}})\), for all \((s_{1},\dots,s_{u})\in\mathbb{N}^{u}\) and \((t_{1},\dots,t_{v})\in\mathbb{N}^{v}\). Note that the above formula has two terms. For the \(Var(\mathrm{Y}_{1})\) and under (D5), we obtain  
\[\mathbb{E}\Big{[}Y^{-2}\mathbf{1}_{|Y_{1}|>\mu_{n}}K_{1}^{2}(x) \Big{]} \leq \mathbb{E}\Big{[}K_{1}^{2}(x)\mathbb{E}\Big{[}Y_{1}^{-2}|X_{1} \Big{]}\Big{]}\] \[\leq \mathbb{CE}[K_{1}^{2}].\]  
Then, we use  
\[\mathbb{E}\Big{[}K_{1}^{j}(x)\Big{]}=O(\phi_{x}(h_{n}))\]  
to deduce that  
\[\Big{|}\operatorname{\mathbb{E}}[\widetilde{R}_{N}^{*}(x)]-\operatorname{ \mathbb{E}}[\widetilde{R}_{N}(x)]\Big{|}=O\Bigg{(}\sqrt{\frac{\log n}{n^{1- \
icefrac{{\delta}}{{\delta}}}\phi_{x}(h_{n})}}\Bigg{)}, \tag{11}\]0 (A4)  
Now, we need to examine the covariance term. To do that, we use the techniques of Massry to obtain the decomposition:  
\[\sum_{i=1}^{n}\sum_{ \begin{subarray}{c}j=1\\ j\
eq i\end{subarray} }^{n}Cov(\mathrm{Y}_{i},\mathrm{Y}_{j}) = \sum_{i=1}^{n}\sum_{ \begin{subarray}{c}j=1\\ 0<|i-j|\leq m_{n}\end{subarray} }^{n}Cov(\mathrm{Y}_{i},\mathrm{Y}_{j})\] \[+\sum_{i=1}^{n}\sum_{ \begin{subarray}{c}j=1\\ |i-j|>m_{n}\end{subarray} }^{n}Cov(\mathrm{Y}_{i},\mathrm{Y}_{j})\] \[=: T_{I}+T_{II}.\]  
Note that \((m_{n})\) is a positive sequence of real number integers, which tends to infinity as \(n\to\infty\).