\subsubsection{$\Ezero{n}$ Tail Bound}\label{sec:en0_concentration}

% \begin{lemma}\label{lemma:en0_concentration_bound} Let $\Ezero{n}$ be defined as in Lemma~\ref{lemma:oja_error_decomposition} for $\voja$ defined in \eqref{eq:voja_def}. Let $\left\{\Ezero{n}^{(i)}\right\}_{i \in [m]}$ and $\left\{\voja^{(i)}\right\}_{i \in [m]}$ be $m$ iid instances of $\Ezero{n}$ and $\voja$ respectively. For any $k \in [d]$ and  all $i \in [m]$, define 
% \bas{
%     \Delta_{i,k} := \bb{e_{k}^{\top}\Ezero{n}^{(i)}}^{2}
% }
% Then, for any $\delta \in \bb{0,1}$ and $k \in [d]$,
% \bas{
% P\bb{\sum_{i\in [m]}\frac{\Delta_{i,k}}{m} \leq \bb{e_{k}^{\top}v_1}^{2}\frac{C^{2}m^{6}\log^{2}\bb{\frac{1}{\delta}}}{\delta^{6}}\frac{\eta_{n}^{2}\Mtwo^{4}}{(\eigengap)^{2}}} \ge 1-\delta,
% }
% where $b_k := \norm{\vp^{\top}e_k}_{2}$ and $C$ is some universal constant.
% \end{lemma}
% \begin{proof}
%     Note that
%     \ba{
%     \sum_{i\in[m]}\frac{\Delta_{i,k}}{m} = \bb{e_{k}^{\top}v_1}^{2} \sum_{i \in [m]}\frac{\bb{v_1^{\top} \voja^{(i)}}^2}{m} \label{eq:Delta_bound}
%     }
%     For any $i \in [m]$, define the event
%     \bas{
%     \mathcal{E}_i := \left\{\bb{v_{1}^{\top}\voja^{(i)}}^{2} \geq 1 - \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{n}\Mtwo^{2}}{(\eigengap)}\right\}
%     }
%     for a universal constant $C > 0$. Then, using Corollary 1 from \cite{lunde2021bootstrapping}, $\forall i \in [m]$, $\Prob\bb{\mathcal{E}_i} \geq 1-\delta$.
%     \bas{
%         \min\left\{|v_1^{\top}(\voja^{(2i)} - v_{1}^{\top}\voja^{(2i-1)})|, |v_1^{\top}(\voja^{(2i)} + v_{1}^{\top}\voja^{(2i-1)})|\right\} &= \Abs{\Abs{v_1^{\top}(\voja^{(2i)}} - \Abs{v_{1}^{\top}\voja^{(2i-1)}}} \\
%         &\leq 1 - \sqrt{1 - \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{n}\Mtwo^{2}}{(\eigengap)}} \\
%         &\leq 1 - \bb{1 - \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{n}\Mtwo^{2}}{(\eigengap)}} \\
%         &\leq  \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{n}\Mtwo^{2}}{(\eigengap)}
%     }
%     Conditioned on $\mathcal{E} := \bigcap_{i \in [m]}\mathcal{E}_i$, it follows that
%     \bas{
%     \sum_{i\in[m/2]}\frac{\Delta_{i,k}}{m} \leq \frac{C^{2}\log^{2}\bb{\frac{1}{\delta}}}{\delta^{6}}\frac{\eta_{n}^{2}\Mtwo^{4}}{(\eigengap)^{2}} 
%     }
%     By a union bound, $\Prob\bb{\mathcal{E}} \geq 1-m\delta$. The result follows by replacing $\delta$ with $\frac{\delta}{m}$. 
% \end{proof}

\begin{lemma}\label{lemma:en0_tail_bound} Let $\Ezero{n}$ be defined as in Lemma~\ref{lemma:oja_error_decomposition} for $\voja$ defined in \eqref{eq:voja_def}. Let $\left\{\Ezero{n}^{(i)}\right\}_{i \in [m]}$ and $\left\{\voja^{(i)}\right\}_{i \in [m]}$ be $m$ iid instances of $\Ezero{n}$ and $\voja$ respectively. Then, for any $k \in [d]$,
\bas{
P\bb{\sum_{i\in [m]}\frac{\bb{e_k^{\top} \Ezero{n}^{(i)}}^2}{m} \le \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{N}\Mtwo^{2}}{(\eigengap)}} \ge 1-\delta.
}
% Then, for any $\delta \in \bb{0,1}$ and $k \in [d]$,
% \bas{
% P\bb{\sum_{i\in [m]}\frac{\bb{e_k^{\top} \Ezero{n}^{(i)}}^2}{m} \leq \bb{e_{k}^{\top}v_1}^{2}\frac{C^{2}m^{6}\log^{2}\bb{\frac{1}{\delta}}}{\delta^{6}}\frac{\eta_{n}^{2}\Mtwo^{4}}{(\eigengap)^{2}}} \ge 1-\delta,
% }
% where $b_k := \norm{\vp^{\top}e_k}_{2}$ and $C$ is some universal constant.
\end{lemma}
\begin{proof}
    For any $i \in [m]$,
    \bas{
    \Abs{e_k^{\top} \Ezero{n}^{(i)}} &= \Abs{e_k^{\top}\bb{v_1v_1^{\top} - \vmain \vmain^{\top}} \voja^{(i)}} \le \norm{v_1v_1^{\top} - \vmain \vmain^{\top}} = \sqrt{2}\Abs{\sin\bb{\vmain, v_1}}.
    }
     The result now follows from Corollary 1 of \cite{lunde2021bootstrapping}, which states that with probability at least $1-\delta$,
    \bas{
    \sin^2 \bb{\vmain, v_1} \le \frac{C\log\bb{\frac{1}{\delta}}}{\delta^{3}}\frac{\eta_{N}\Mtwo^{2}}{(\eigengap)}.
    }
    for some universal constant $C > 0$. 
\end{proof}