\begin{align*}
&\mathbb{P}\left(\frac{1}{N}||U \left( I - \left(I-D\right)^m\right) U^\top \varepsilon^N ||^2_2 > \xi \right)
\\
&\leq \mathbb{P}\biggl(\left(\frac{1}{N}||U \left( I - \left(I-D\right)^m\right) U^\top \varepsilon^N ||^2_2 > \xi\right) \cap
\left\{ \mathbf{\widetilde{X}}^N \in \mathcal{B}_N \right\}\biggr)
+ \mathbb{P}\left(\left\{\mathbf{\widetilde{X}}^N \notin \mathcal{B}_N \right\} \right)
\end{align*}
