\newpage \begin{appendices} \counterwithin{lstfloat}{section} \section{Proofs for sone Lemmata in ...} In the following there will be proofs for some important Lemmata in Section~\ref{sec:theo38}. Further proofs not discussed here can be found in \textcite{heiss2019} The proves in this section are based on \textcite{heiss2019}. Slight alterations have been made to accommodate for not splitting $f$ into $f_+$ and $f_-$. \begin{Theorem}[Proof of Lemma~\ref{theo38}] \end{Theorem} \begin{Lemma}[$\frac{w^{*,\tilde{\lambda}}_k}{v_k}\approx\mathcal{O}(\frac{1}{n})$] For any $\lambda > 0$ and training data $(x_i^{\text{train}}, y_i^{\text{train}}) \in \mathbb{R}^2, \, i \in \left\{1,\dots,N\right\}$, we have \[ \max_{k \in \left\{1,\dots,n\right\}} \frac{w^{*, \tilde{\lambda}}_k}{v_k} = \po_{n\to\infty} \] \end{Lemma} \begin{Proof}[Proof of Lemma~\ref{lem:s3}] \[ \sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k) h_{k,n} = \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}} \left(\sum_{\substack{k \in \kappa \\ \xi_k \in [\delta l , \delta(l+1))}} \varphi(\xi_k, v_k) h_{k,n}\right) \approx \] \[ \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}} \left(\sum_{\substack{k \in \kappa \\ \xi_k \in [\delta l , \delta(l+1))}} \left(\varphi(\delta l, v_k) \frac{1}{n g_\xi (\delta l)} \pm \frac{\varepsilon}{n}\right) \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l, \delta(l+1))\right\}}}{\abs{\left\{m \in \kappa : \xi_m \in [\delta l, \delta(l+1))\right\}}}\right) \] \[ \approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}} \left(\frac{\sum_{\substack{k \in \kappa \\ \xi_k \in [\delta l , \delta(l+1))}}\varphi(\delta l, v_k)}{\abs{\left\{m \in \kappa : \xi_m \in [\delta l, \delta(l+1))\right\}}} \frac{\abs{\left\{m \in \kappa : \xi_m \in [\delta l, \delta(l+1))\right\}}}{n g_\xi (\delta l)}\right) \pm \varepsilon \] The amount of kinks in a given interval of length $\delta$ follows a binomial distribution, \[ \mathbb{E} \left[\abs{\left\{m \in \kappa : \xi_m \in [\delta l, \delta(l+1))\right\}}\right] = n \int_{\delta l}^{\delta(l+1)}g_\xi (x) dx \approx n (\delta g_\xi(\delta l) \pm \delta \tilde{\varepsilon}), \] for any $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$, since $g_\xi$ is uniformly continuous on its support by Assumption.. As the distribution of $v$ is continuous as well we get that $\mathcal{L}(v_k) = \mathcal{L} v| \xi = \delta l) \forall k \in \kappa : \xi_k \in [\delta l, \delta(l+1))$ for $\delta \leq \delta(\varepsilon, \tilde{\varepsilon})$. Thus we get with the law of large numbers \begin{align*} &\sum_{k \in \kappa : \xi_k < T} \varphi(\xi_k, v_k) h_{k,n} \approx\\ &\approx \sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}\left(\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \stackrel{\mathbb{P}}{\pm}\right) \delta \left(1 \pm \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon \\ &\approx \left(\sum_{\substack{l \in \mathbb{Z} \\ [\delta l, \delta (l+1)) \in [C_{g_\xi}^l,\min\{C_{g_\xi}^u, T \}]}}\mathbb{E}[\phi(\xi, v)|\xi=\delta l] \delta \stackrel{\mathbb{P}}{\pm}\tilde{\tilde{\varepsilon}} \abs{C_{g_\xi}^u - C_{g_\xi}^l} \right)\\ &\phantom{\approx}\cdot \left(1 \pm \frac{\tilde{\varepsilon}}{g_\xi(\delta l)}\right) \pm \varepsilon \end{align*} \end{Proof} \begin{Lemma}[($L(f_n) \to L(f)$), Heiss, Teichmann, and Wutte (2019, Lemma A.11)] For any data $(x_i^{\text{train}}, y_i^{\text{train}}) \in \mathbb{R}^2, i \in \left\{1,\dots,N\right\}$, let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions that converges point-wise in probability to a function $f : \mathbb{R}\to\mathbb{R}$, then the loss $L$ of $f_n$ converges is probability to $L(f)$ as $n$ tends to infinity, \[ \plimn L(f_n) = L(f). \] \proof Vgl. ... \end{Lemma} \begin{Proof}[Step 2] We start by showing that \[ \plimn \tilde{\lambda} \norm{\tilde{w}}_2^2 = \lambda g(0) \left(\int \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)} dx\right) \] With the definitions of $\tilde{w}$, $\tilde{\lambda}$ and $h$ we have \begin{align*} \tilde{\lambda} \norm{\tilde{w}}_2^2 &= \tilde{\lambda} \sum_{k \in \kappa}\left(f_g^{*,\lambda''}(\xi_k) \frac{h_k v_k}{\mathbb{E}v^2|\xi = \xi_k]}\right)^2\\ &= \tilde{\lambda} \sum_{k \in \kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k) \frac{h_k v_k^2}{\mathbb{E}v^2|\xi = \xi_k]}\right) h_k\\ & = \lambda g(0) \sum_{k \in \kappa}\left(\left(f_g^{*,\lambda''}\right)^2(\xi_k)\frac{v_k^2}{g_\xi(\xi_k)\mathbb{E} [v^2|\xi=\xi_k]}\right)h_k. \end{align*} By using Lemma~\ref{lem} with $\phi(x,y) = \left(f_g^{*,\lambda''}\right)^2(x)\frac{y^2}{g_\xi(\xi)\mathbb{E}[v^2|\xi=y]}$ this converges to \begin{align*} &\plimn \tilde{\lambda}\norm{\tilde{w}}_2^2 = \\ &=\lambda g_\xi(0)\mathbb{E}[v^2|\xi=0]\int_{\supp{g_\xi}}\mathbb{E}\left[ \left(f_g^{*,\lambda''}\right)^2(\xi)\frac{v^2}{ g_\xi(\xi)\mathbb{E}[v^2|\xi=x]^2}\Big{|} \xi = x\right]dx\\ &=\lambda g_\xi(0) \mathbb{E}[v^2|\xi=0] \int_{\supp{g_xi}} \frac{\left(f_g^{*,\lambda''}\right)^2 (x)}{g_\xi(x) \mathbb{E}[v^2|\xi=x]} dx \\ &=\lambda g(0) \int_{\supp{g_\xi}} \frac{\left(f_g^{*,\lambda''}\right)^2}{g(x)}dx. \end{align*} \end{Proof} \begin{Lemma}[Heiss, Teichmann, and Wutte (2019, Lemma A.13)] Using the notation of Definition .. and ... the following statement holds: $\forall \varepsilon \in \mathbb{R}_{>0} : \exists \delta \in \mathbb{R}_{>0} : \forall \omega \in \Omega : \forall l, l' \in \left\{1,\dots,N\right\} : \forall n \in \mathbb{N}$ \[ \left(\abs{\xi_l(\omega) - \xi_{l'}(\omega)} < \delta \angle \text{sign}(v_l(\omega)) = \text{sign}(v_{l'}(\omega))\right) \implies \abs{\frac{w_l^{*, \tilde{\lambda}}(\omega)}{v_l(\omega)} - \frac{w_{l'}^{*, \tilde{\lambda}}(\omega)}{v_{l'}(\omega)}} < \frac{\varepsilon}{n}, \] if we assume that $v_k$ is never zero. \proof given in .. \end{Lemma} \input{Appendix_code.tex} \end{appendices} %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: