# Scalar-on-Function Relative Error Regression for Weak Dependent Case
## Appendix A
We use the second part of (D5) to obtain  
\[\big{|}Cov(\mathrm{Y}_{i},\mathrm{Y}_{j})\big{|} \leq C\big{|}\mathbb{E}\big{[}K_{i}(x)K_{j}(x)\big{]}\big{|}+\big{|} \mathbb{E}[K_{i}(x)]\mathbb{E}\big{[}K_{j}(x)\big{]}\big{|}\] \[\leq C\Big{(}\phi_{x}^{(a+1)/a}(h_{n})+\phi_{x}^{2}(h_{n})\Big{)}.\]  
Therefore,  
\[T_{I}\leq Cm_{n}\phi_{x}^{(a+1)/a}(h_{n}).\] (A5)  
Since the observations are quasi-associated, and the kernel \(K\) is bounded, based on the Lipschitz, we obtain  
\[T_{II} \leq \Big{(}h_{n}^{-1}\mathrm{Lip}(K)\Big{)}^{2}\sum_{i=1}^{n}\sum_{ \begin{subarray}{c}j=1\\ |i-j|>m_{n}\end{subarray} }^{n}\mathrm{Y}_{i,j}\] (A6) \[\leq C\Big{(}(\mu_{n}h)^{-1}\mathrm{Lip}(K)\Big{)}^{2}\sum_{i=1}^{n} \sum_{ \begin{subarray}{c}j=1\\ |i-j|>m_{n}\end{subarray} }^{n}\mathrm{Y}_{i,j}\] \[\leq C\Big{(}(\mu_{n}h)^{-1}\mathrm{Lip}(K)\Big{)}^{2}\sum_{i=1}^{n} \sum_{ \begin{subarray}{c}j=1\\ |i-j|>m_{n}\end{subarray} }^{n}\mathrm{Y}_{i,j}\] \[\leq Cn\Big{(}(\mu_{n}h)^{-1}\mathrm{[PERSON]}(K)\Big{)}^{2}\mathrm{Y}_{m _{n}}\] \[\leq Cn\Big{(}(\mu_{n}h)^{-1}\mathrm{Lip}(K)\Big{)}^{2}e^{-am_{n}}.\]Then, by (A5) and (A6), we obtain  
\[\sum_{i=1}^{n}\sum_{ \begin{subarray}{c}j=1\\ j\
eq i\end{subarray} }^{n}Cov(Y_{i},Y_{j}) \leq C\bigg{(}nm_{n}\phi_{x}^{(a+1)/a}(h_{n})+n\Big{(}(\mu_{n}h)^{-1} \text{Lip}(K)\Big{)}^{2}e^{-am_{n}}\bigg{)}.\]  
\[\text{Putting }m_{n}=\log\bigg{(}\frac{\big{(}(\mu_{n}h)^{-1}\text{ Lip}(K)\big{)}^{2}}{ab_{x}^{(a+1)/a}(h_{n})}\bigg{)},\text{ we obtain}\] \[\frac{1}{n\phi_{x}(h_{n})}\sum_{i=1}^{n}\sum_{ \begin{subarray}{c} j=1\\ j\
eq i\end{subarray} }^{n}Cov(Y_{i},Y_{j}) \to 0,\text{ as }n\to\infty.\] (A7)  
Combining together results (A4) and (A7), we show that  
\[Var\Bigg{(}\sum_{i=1}^{n}Y_{i}\Bigg{)}=O\bigg{(}\frac{1}{n\phi_{x}(h_{n})} \bigg{)}.\] (A8)  
We evaluate the covariance term  
\[Cov(Y_{s_{1}}\dots Y_{s_{u}},Y_{t_{1}}\dots Y_{t_{v}}),\quad(s_{1},\dots,s_{u },t_{1},\dots,t_{v}))\in\mathbb{N}^{u+v}.\]  
To do that, we treat the following cases:  
* The first case is \(t_{1}>s_{u}\); based on the definition of quasi-association, we obtain \[|Cov(Y_{s_{1}}\dots Y_{s_{u}},Y_{t_{1}}\dots Y_{t_{v}})| \leq\bigg{(}\Big{(}(\mu_{n}h)^{-1}\text{Lip}(K)\Big{)}^{2}(n \text{E}[K_{1}(x)])^{-1}\bigg{)}^{2}\] (A9) \[\qquad\bigg{(}\frac{C}{n\mu_{n}\text{E}[K_{1}(x)]}\Big{)}^{u+v-2 }\sum_{i=1}^{u}\sum_{j=1}^{v}Y_{s_{i},t_{j}}\] \[\leq\Big{(}h_{n}^{-1}\text{Lip}(K)\Big{)}^{2}\bigg{(}\frac{C}{n \mu_{n}\text{E}[K_{1}(x)]}\bigg{)}^{u+v}v\Upsilon_{t_{1}-s_{u}}\] \[\leq\Big{(}h_{n}^{-1}\text{Lip}(K)\Big{)}^{2}\bigg{(}\frac{C}{n \mu_{n}\phi_{x}(h_{n})}\bigg{)}^{u+v}vv^{-a(t_{1}-s_{u})}.\]  
On the other hand, we have  
\[|Cov(Y_{s_{1}}\dots Y_{s_{u}},Y_{t_{1}}\dots Y_{t_{v}})| \leq \bigg{(}\frac{C\|K\|_{\infty}}{n\mu_{n}\text{E}[K_{1}(x)]}\bigg{)} ^{u+v-2}\times\] (A10) \[(|\text{E}[Y_{s_{u}}Y_{t_{1}}]|+\text{E}[Y_{s_{u}}|\text{E}[Y_{t_ {1}}])\] \[\leq \bigg{(}\frac{C\|K\|_{\infty}}{n\mu_{n}\text{E}[K_{1}(x)]}\bigg{)} ^{u+v-2}\bigg{(}\frac{C}{n\mu_{n}\text{E}[K_{1}(x)]}\bigg{)}^{2}\times\] \[\Big{(}\phi_{x}^{(a+1)/a}(h_{n})+\phi_{x}^{2}(h_{n})\Big{)}\] \[Cov(Y_{s_{1}}\dots Y_{s_{u}},Y_{t_{1}}\dots Y_{t_{v}}),\quad(s_{1},\dots,s_{u },t_{1},\dots,t_{v}))\in\mathbb{N}^{u+v}.\]0  
Furthermore, taking a \(\frac{1}{2(a+1)}\)-power of (A9) and a \((\frac{2a+1}{a+1})\)-power of (A10), we get for \(1\leq s_{1}\leq\dots\leq s_{u}\leq t_{1}\leq\dots\leq t_{v}\leq n:\)  
\[Cov(Y_{s_{1}}\dots Y_{s_{u}},Y_{t_{1}}\dots Y_{t_{v}}),\quad(s_{1},\dots,s_{u },t_{1},\dots,t_{v}))\in\mathbb{N}^{u+v}.\]1* The second one is where \(t_{1}=s_{u}\).