\section{Central Limit Theorem for entries of the Oja vector}\label{appendix:entrywise_clt}
We consider the following setup from~\cite{ChernoCLT2015}.
Let $\mathcal{A}^{\text{re}}$ denote the class of all hyperrectangles in $\mathbb{R}^p$. That is, $\mathcal{A}^{\text{re}}$ consists of all sets $A$ of the form:
\begin{equation}
    A = \{w \in \mathbb{R}^p : a_j \leq w_j \leq b_j \text{ for all } j = 1, \dots, p\}
\end{equation}
for some real values $a_j$ and $b_j$ satisfying $-\infty \leq a_j \leq b_j \leq \infty$ for each $j = 1, \dots, p$. 

% Let 
% \begin{equation}
%     L_n := \max_{1 \leq j \leq p} \frac{1}{n} \sum_{i=1}^{n} \mathbb{E}[|X_{ij}|^3],
% \end{equation}
% and for $\phi \geq 1$, define
% \begin{equation}
%     M_{n, X}(\phi) := n^{-1} \sum_{i=1}^{n} \mathbb{E} \left[ \max_{1 \leq j \leq p} |X_{ij}|^3 \mathbf{1} \left\{ \max_{1 \leq j \leq p} |X_{ij}| > \sqrt{n}/(4\phi \log p) \right\} \right].
% \end{equation}
% Similarly, let $M_{n,Y}(\phi)$ be defined analogously with $X_{ij}$ replaced by $Y_{ij}$ in (5), and set
% \begin{equation}
%     M_n(\phi) := M_{n,X}(\phi) + M_{n,Y}(\phi).
% \end{equation}
Consider 
\[
S_n^X  =\frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i.
\]
where $X_i,i\in [n]\in \mathbb{R}^p$ are independent random vectors with $\E[X_{ij}]=0$ and $\E[X_{ij}^2]<\infty$, for $i\in [n], j\in [p]$.
Consider the following  Gaussian approximation to $S_n^X$.
Define the normalized sum for the Gaussian random vectors:
\[
S_n^Y =\frac{1}{\sqrt{n}} \sum_{i=1}^{n} Y_i,
\]
where $Y_1, \dots, Y_n$ be independent mean zero Gaussian random vectors in $\mathbb{R}^p$ such that each $Y_i$ has the same covariance matrix as $X_i$. We are interested in bounding the quantity 
\bas{\rho_{n}\bb{\mathcal{\mathcal{A}^{\text{re}}}} := \sup_{A \in \mathcal{A}^{\text{re}}}\Abs{\Prob\bb{S_{n}^{X} \in A} - \Prob\bb{S_{n}^{Y} \in A}}}
Let $C_n \geq 1$ be a sequence of constants possibly growing to infinity as $n \rightarrow \infty$, and let $b, q > 0$ be some constants. Assume that $X_{i}$ satisfy,
% \begin{theorem}[Theorem 2.1~\cite{ChernoCLT2015}]
   
%  Consider 
% \[
% S_n^X  \frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i.
% \]
% where $X_i,i\in [n]\in \mathbb{R}^p$ are independent random vectors with $\E[X_{ij}]=0$ and $\E[X_{ij}^2]<\infty$, for $i\in [n], j\in [p]$.
% Consider the following  Gaussian approximation to $S_n^X$.
% Define the normalized sum for the Gaussian random vectors:
% \[
% S_n^Y =\frac{1}{\sqrt{n}} \sum_{i=1}^{n} Y_i,
% \]
% where $Y_1, \dots, Y_n$ be independent centered Gaussian random vectors in $\mathbb{R}^p$ such that each $Y_i$ has the same covariance matrix as $X_i$.
%  \textit{Suppose that there exists some constant $b > 0$ such that}
% \begin{equation}
%     n^{-1} \sum_{i=1}^{n} \mathbb{E}[X_{ij}^2] \geq b
% \end{equation}
% \textit{for all $j = 1, \dots, p$. Then there exist constants $K_1, K_2 > 0$ depending only on $b$ such that for every constant $C_n \geq L_n$, we have}
% \begin{equation}
%     \rho_n(\mathcal{A}^{\text{re}}) \leq K_1 \left[ \left( \frac{C_n^2 \log^7 p}{n} \right)^{1/6} + \frac{M_n(\phi_n)}{L_n} \right]
% \end{equation}
% \textit{with}
% \begin{equation}
%     \phi_n := K_2 \left( \frac{L_n^2 \log^4 p}{n} \right)^{-1/6}.
% \end{equation}
% \end{theorem}

\begin{itemize}
    \item[(M.1)] \( n^{-1} \sum_{i=1}^{n} \mathbb{E}[X_{ij}^2] \geq b \) \textit{for all} \( j = 1, \dots, p \),
    \item[(M.2)] \( n^{-1} \sum_{i=1}^{n} \mathbb{E}[|X_{ij}|^{2+k}] \leq C_n^k \) \textit{for all} \( j = 1, \dots, p \) \textit{and} \( k = 1,2 \).
\end{itemize}

Further, the authors consider examples where one of the following conditions also holds:

\begin{itemize}
    \item[(E.1)] \( \mathbb{E}[\exp(|X_{ij}| / C_n)] \leq 2 \) \textit{for all} \( i = 1, \dots, n \) \textit{and} \( j = 1, \dots, p \),
    \item[(E.2)] \( \mathbb{E}[(\max_{1 \leq j \leq p} |X_{ij}| / C_n)^q] \leq 2 \) \textit{for all} \( i = 1, \dots, n \).
\end{itemize}

Let
\[
D_n^{(1)} = \left( \frac{C_n^2 \log^7 (pn)}{n} \right)^{1/6}, \quad
D_{n,q}^{(2)} = \left( \frac{C_n^2 \log^3 (pn)}{n^{1 - 2/q}} \right)^{1/3}.
\]

Now we present Proposition 2.1 \citep{ChernoCLT2015}.

\begin{theorem}[\label{thm:prop2pt1}Proposition 2.1~\citep{ChernoCLT2015}]
    \textit{Suppose that conditions (M.1) and (M.2) are satisfied. Then under (E.1), we have}
\[
\rho_n(\mathcal{A}^{\text{re}}) \leq C D_n^{(1)},
\]
\textit{where the constant} \( C \) \textit{depends only on} \( b \); \textit{while under (E.2), we have}
\[
\rho_n(\mathcal{A}^{\text{re}}) \leq C \{ D_n^{(1)} + D_{n,q}^{(2)} \},
\]
\textit{where the constant} \( C \) \textit{depends only on} \( b \) \textit{and} \( q \).
\end{theorem}


Next, we will need the following result cited by~\citet{chernozhukov2017detailed}.
\begin{theorem}[\label{thm:Nazarov}Nazarov's inequality~\citep{nazarov2003maximal}, Theorem~1 in~\citep{ChernoCLT2015}]
    
Let \( Y = (Y_1, \dots, Y_p)^T \) be a centered Gaussian random vector in \( \mathbb{R}^p \) such that 
\[
\mathbb{E}[Y_j^2] \geq \sigma^2, \quad \text{for all } j = 1, \dots, p,
\]
for some constant \( \sigma > 0 \). Then, for every \( y \in \mathbb{R}^p \) and \( \delta > 0 \),
\[
\Prob(Y \leq y + \delta) - \Prob(Y \leq y) \leq \frac{\delta}{\sigma} (\sqrt{2 \log p} + 2).
\]
Here, for vector $y \in \R^{p}$, $y + \delta$ denotes the vector constructed by adding $\delta$ to each entry of $y$.
\end{theorem}

% \begin{lemma}[Entrywise CLT]
% Assume $\lambda_>1$, and $Mtwo^4n\eta_n^2/(\lambda_1-\lambda_2)\leq 1$. Also assume $(E\|A_n-\Sigma\|^4)^{1/4}\leq \Mfour$. Let $\{X_i\}_{i=1}^n\in \mathbb{R}^d$ be a random mean-zero vector with covariance matrix \( \Sigma \) such that for all vectors \( v \in \mathbb{R}^d \), we have
% \[
% \mathbb{E} \left[ \exp \left( v^T X \right) \right] \leq \exp \left( \frac{\sigma^2 v^T \Sigma v}{2} \right).
% \]

%     Let $\roja=\voja - (v_{1}^{\top}\voja)v_{1}$. Assume $\max((n\eta_n)^{3}\Mfour^2, n\eta_n\sigma^4, M_{d,2}^4/(\lambda_1-\lambda_2)^4)=\tilde{o}(n)$.  Consider the set $J=\{j:  \bar{\mathbb{V}}_{jj}\geq  b\}$ for some $b>0$. Let $p=|J|$.
%     Let $\mathcal{A}^{\text{re}}$ denote the class of all hyperrectangles in $\mathbb{R}^p$. That is, $\mathcal{A}^{\text{re}}$ consists of all sets $A$ of the form:
% \begin{equation}
%     A = \{u \in \mathbb{R}^d : a_j \leq u_j \leq b_j \text{ for all } j = 1, \dots, p\}
% \end{equation}
% for some real values $a_j$ and $b_j$ satisfying $-\infty \leq a_j \leq b_j \leq \infty$ for each $j = 1, \dots, p$.      Let $H_i = \vp\lambp^{n-i-1}\vp^{\top}\bb{A_{i}-\Sigma}v_{1}$. Let $Y_i\in \mathbb{R}^p$ denote independent mean zero normal vectors such that $\E[Y_iY_i^T]=n\eta_n\E[H_i[J]H_i[J]^T]$.
%     We have 
%     \bas{
%     \sup_{A\in \mathcal{A}_{re}}|\Prob((\lambda_1\eta_n)^{-1/2}\roja[J]\in A)-\Prob(n^{-1/2}S_Y\in A)|\leq \max\bb{\bb{\Mfour^2\eta_n\lambda_1}^{1/6},\bb{\frac{\Mtwo^4n\eta_n^2}{\lambda_1(\lambda_1-\lambda_2)}}^{1/6}}
%     }
%     % \bas{
%     % \sup_{A\in \mathcal{A}^{\text{re}}}|\Prob((\lambda_1\eta_n)^{-1/2}\roja[J]\in A)-\Prob(S_Y/\sqrt{n}\in A)|=o(1)
%     % }
% \end{lemma}

Now we are ready to state our main result in Proposition~\ref{prop:clt_appendix},

\begin{proposition}[\label{prop:clt_appendix}CLT for a suitable subset of entries]
Suppose the learning rate $\eta_n$, set according to Lemma~\ref{lemma:learning_rate_choice}, satisfies $\frac{\Mtwo^{2} \lambda_1 \eta_n}{\bb{\eigengap}^2}\leq \frac{C_0 b}{2}$ for some $b>0$ and a small universal constant $C_0$.  Let $\{X_i\}_{i=1}^n\in \mathbb{R}^d$ be i.i.d. mean-zero random vectors with covariance matrix \( \Sigma \) such that for all vectors \( v \in \mathbb{R}^d \), we have
\[
\mathbb{E} \left[ \exp \left( v^T X_1 \right) \right] \leq \exp \left( \frac{\sigma^2 v^T \Sigma v}{2} \right).
\]

Let $\roja := \voja - (v_{1}^{\top}\voja)v_{1}$. Consider the set $J := \{j:  \V_{jj}\geq  b\}$, and let $p :=|J|$.
%     Let $\mathcal{A}^{\text{re}}$ denote the class of all hyperrectangles in $\mathbb{R}^p$. That is, $\mathcal{A}^{\text{re}}$ consists of all sets $A$ of the form:
% \begin{equation}
%     A = \{u \in \mathbb{R}^p : a_j \leq u_j \leq b_j \text{ for all } j = 1, \dots, p\}
% \end{equation}
% for some real values $a_j$ and $b_j$ satisfying $-\infty \leq a_j \leq b_j \leq \infty$ for each $j = 1, \dots, p$.     
Let $H_i := \frac{\sign(u_0^Tv_1)}{1+\eta_n\lambda_1} \vp\lambp^{n-i}\vp^{\top}\bb{X_{i}X_{i}^{\top}-\Sigma}v_{1}$. Let $Y_i\in \mathbb{R}^p$ be independent mean zero normal vectors such that $$\E[Y_iY_i^T]=\frac{n\eta_n}{\eigengap}\E[H_i[J]H_i[J]^T].$$ %Define $\rho(\mathcal{A}_{re})=$
    Then, {\small
    \bas{
    &\sup_{A\in \mathcal{A}_{re}}\left|P\bb{\frac{\roja[J]}{\sqrt{\bb{\eigengap}\eta_n}}\in A}-P\bb{\frac{\sum_i Y_i}{\sqrt{n}}\in A}\right|= 
   \tilde{O}\bb{ \max\bb{\bb{\frac{\Mfour}{\eigengap }}^{1/3}n^{-1/6}, \bb{\frac{\Mtwo}{\eigengap}}^{1/2}n^{-1/8}}},
%\max\bb{\bb{\Mfour^2\eta_n\lambda_1}^{1/6},\max\bb{\frac{\sqrt{\log p}}{b},1}\bb{\frac{\Mtwo^4n\eta_n^2}{\lambda_1(\lambda_1-\lambda_2)}}^{1/6}}%\max\bb{\bb{\Mfour^2\eta_n\lambda_1}^{1/6},\bb{\frac{\Mtwo^4n\eta_n^2}{\lambda_1(\lambda_1-\lambda_2)}}^{1/6}}
    }
    }
    % \bas{
    % \sup_{A\in \mathcal{A}^{\text{re}}}|\Prob((\lambda_1\eta_n)^{-1/2}\roja[J]\in A)-\Prob(S_Y/\sqrt{n}\in A)|=o(1)
    % }
    where $\tilde{O}$ hides logarithmic factors in $n$, $p$, and constants depending on $b$.
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{prop:clt_appendix}]
Consider the error decomposition of the Oja vector in Lemma~\ref{lemma:oja_error_decomposition}. 
   We have $\roja = \Psi_{n,1} + \Psi_{n,2} + \Psi_{n,3} + \Psi_{n,4}$, where $\Psi_{n,1}, \Psi_{n,2}, \Psi_{n,3}, \Psi_{n,4}$ are defined in Equation~\eqref{eq:ojadecomp}.
  Let $\rem:=\Psi_{n,2} + \Psi_{n,3} + \Psi_{n,4}$.
   
   For any $\delta \in \bb{0,1}$, $\exists \epsilon > 0$ such that from Lemma~\ref{lemma:oja_error_decomposition_higher_order_tail_bounds} we have,  
   \bas{
   \Prob((\eta_n\bb{\eigengap})^{-1/2}\|\rem\|_{2}\geq \epsilon)\leq \delta
   }
   we will specify $\epsilon$ as needed in the proof.
   %where $\epsilon=\sqrt{\eta_{n}\bb{e_k^{\top}\mathbb{V}e_{k}}\log\bb{\frac{d}{\delta}}} + \rd \eta_n b_k \tilde{O}\bb{\max\bb{\frac{\Mtwo^2}{(\lambda_1-\lambda_2)^2)},\frac{\Mone\sqrt{\lambda_1}}{(\lambda_1-\lambda_2)^{3/2}}}}$.
   % From \rd Lemma B.3 in~\cite{lunde2021bootstrapping}\bk, we show that 
   % \bas{
   % \Prob(\frac{1}{\eta^{1/2}}\|E_{n,4}\|\geq \epsilon)\leq \enfourerror
   % }
   % \rd PS: Shourya, please check
   % Syamantak, what is M and $\sigma$ for us?
   % \bk
  
   % From Lemma~\ref{lemma:en2_concentration_bound}, we have:
   % \bas{
   %  P\bb{\|E_{n,2}\|/\sqrt{\eta} \ge \frac{12b_k \eta^2 \mathcal{M}_2^2 n \log d} {\sqrt{\delta}}} \le \delta.
   %  }

     
%      Let
% \bas{
% S_Y \defeq \frac{1}{\sqrt{n}}\sum_{i=1}^n U_i^n[J]. 
% }

For all $i \in [n]$, let 
\ba{
U_i:=\underbrace{\sqrt{n\eta_n/\bb{\eigengap}}}_{c_n} H_i
}
 We show that $U_1, U_2, \dots, U_n$ satisfy conditions (M.1) and (M.2) with suitable constants. 

For (M.1), using equation \eqref{eq:ojadecomp}, 

\ba{ \sum_{i=1}^{n}H_{i} = \Psi_{n,1}.
\label{eq:psin1}}


By Lemma~\ref{lemma:second_moment_matrix} (equation \eqref{eq:variance_diff_bound}), there exists a universal constant $C_0$ such that
\bas{
\Abs{e_j^{\top} \bb{ \frac{\eta_n}{\eigengap} \sum_{i=1}^{n} \E\bbb{H_iH_i^{\top}} - \V} e_j} \leq \frac{\eta_n \lambda_1 \Mtwo^2}{C_0 \bb{\eigengap}^2} \le \frac{b}{2} \le \frac{\V_{jj}}{2}.
}
for all $j \in J$, where the last two inequalities follow by assumption and definition of $J$. This implies for all $j\in J$,
\bas{
\frac{\eta_n}{\eigengap} \sum_{i}^{n} \E\bbb{H_{ij}^2} \ge \V_{jj}/2 \ge b/2 \iff \frac{1}{n} \sum_i \E[U_{ij}^2] \geq \V_{jj}/2\geq b/2
}
 
% We will now use the fact that $|X_{ij}|\leq M_d \sqrt{n\eta_n} $ 
To show (M.2), by Lyapunov's inequality and Assumption~\ref{assumption:bounded_moments}:
    \bas{
    \E\bbb{\|U_{ij}^{2+k}\|_{2}} = \E \bbb{c_n^{2+k}|H_{ij}|^{2+k}}&\leq 2 (c_n\Mfour)^{2+k}
    }
    for $k \in \{1,2\}$, where $C_n:=2c_n\Mfour$.
    
    We now check condition E.1. Now note that for any unit vector $u \in \R^{d}$, $u^T H_i$ is subexponential with parameter  $\sigma^2\lambda_1$  (Proposition 2.7.1. of~\citep{vershynin2018high}). Hence, there exists a constant $C > 0$ such that 
    \bas{
    \E [\exp(|H_{ij}|/C\lambda_1 \sigma^2)] \leq 2
    }
    Therefore, 
    \bas{
    \E [\exp(|U_{ij}|/C\lambda_1 c_n\sigma^2)]\leq 2.
    }
    
    % \bas{
    % \E[\exp(|X_{ij}|/C_n)]\leq \E[\exp(1/8)]\leq 2, \ \ i\in [n], j\in J
    % }
    %Now consider convex sets of the form
    Now we set $C_n :=\max(2c_n\Mfour,C\lambda_1 c_n\sigma^2)$.
    
    Using Eq~\ref{eq:psin1},
    \bas{\frac{1}{\sqrt{(\eigengap)\eta_n}}\Psi_{n,1}[J]=\sqrt{\eta_n/\bb{\eigengap}}\sum_i H_i[J]=\frac{1}{\sqrt{n}}\sum_iU_i[J],}
     the random variables $U_{i}[J], i \in [n]$ satisfy conditions (M.1), (M.2) and (E.1). By Theorem~\ref{thm:prop2pt1},
    \bas{
    \rho(\mathcal{A}^{\text{re}})\leq C \left( \frac{C_n^2 \log^7 (pn)}{n} \right)^{1/6}
    }

%    Let $P\left(\eta_n^{-1/2}\|E_{n,2}+E_{n,3}+E_{n,4}\|\geq \epsilon\right)\leq \delta $.
Recall from the statement of the proposition that $Y_i,i\in [n]$ are mean zero independent Gaussian vectors in $ \mathbb{R}^{p}$ with the same covariance structure as $U_i[J]$, i.e, $\E\bbb{Y_{i}Y_{i}^{\top}} = \E\bbb{U_i[J]U_i[J]^{\top}}$.

Let $S_W$ be the random variable $\sum_i W_i$ for any collection $W$ of $n$ random variables $W_1, W_2, \dots, W_n$. Consider the vector $S_{W}[J]$ to be the projection of $W$ on the set $J$, defined as $e_{i}^{\top}S_{W}[J] = e_{i}^{\top}S_{W}$ for $i \in J$.

Recall that 
\bas{
e_i^T\roja:= e_{i}^{\top}\bb{\sum_{j=1}^n\eta_{n}H_j + \rem}.
}
Let $A :=\{u\in \mathbb{R}^{p}|u_i\in [a_i,b_i], i\in J\}$. Let $A_{\epsilon}^+ :=\{X|X_i\in [a_i-\epsilon,b_i+\epsilon], i\in [p]\}$ and $A_{\epsilon}^- := \{X|X_i\in [a_i+\epsilon,b_i-\epsilon], i\in J\}$. 

Let $S_R[J] := \sum_{i \in J}e_i^T\roja$. Then, we have $S_R[J]=\eta_n  S_H[J]+\rem[J]$. 

We will use the following identity for vectors $G_1,G_2\in \mathbb{R}^p$.  
\bas{
\Prob(G_1\in A_{\epsilon}^-, \|G_2\|\leq \epsilon)\leq \Prob(G_1+G_2\in A,\|G_2\|\leq \epsilon)\leq P(G_1\in A_{\epsilon}^+, \|G_2\|\leq \epsilon)
}

So, 
\bas{
\Prob(G_1+G_2\in A)&\leq \Prob(G_1\in A_{\epsilon}^+, \|V\|\leq \epsilon)+P(\|V\|\geq \epsilon)\\
\Prob(G_1+G_2\in A)&\geq P(G_1\in A_{\epsilon}^-, \|G_2\|\leq \epsilon)
}

Using $G_1=S_U[J]/\sqrt{n}$ and $G_2=(\eta_n \bb{\eigengap})^{-1/2}\rem$, we have:
\bas{
&\Prob((\bb{\eigengap}\eta_n)^{-1/2}\roja[J]\in A)-\Prob(S_Y/\sqrt{n}\in A)\\
   &\leq \Prob((\bb{\eigengap}\eta_n)^{-1/2}\roja[J] \in A, (\eta_n\bb{\eigengap})^{-1/2}\|\rem\|\leq \epsilon)+\Prob((\eta_n \bb{\eigengap})^{-1/2}\|\rem\|_{2}\geq \epsilon) \\
   & \;\;\;\; -\Prob(S_Y/\sqrt{n}\in A)\\
   &\leq \Prob(S_U[J]/\sqrt{n}\in A_\epsilon^+)+\Prob((\eta_n \bb{\eigengap})^{-1/2}\|\rem\|_{2}\geq \epsilon)-\Prob(S_Y/\sqrt{n}\in A) =: \gamma_A.
}
Note that $\gamma_A$ can be written as
\bas{
   \gamma_A \leq |\Prob(S_U[J]/\sqrt{n}\in A_\epsilon^+)-\Prob(S_Y/\sqrt{n}\in A_\epsilon^+)|+|\Prob(S_Y/\sqrt{n}\in A_\epsilon^+)-\Prob(S_Y/\sqrt{n}\in A)|+\Prob((\eta_n \bb{\eigengap})^{-1/2}\|\rem\|\geq \epsilon).
    }
Similarly,
\bas{
   &\Prob((\bb{\eigengap} \eta_n)^{-1/2}\roja[J]\in A)-\Prob(S_Y/\sqrt{n}\in A) \ge \omega_{A},
   }
   where
   \bas{
   \omega_{A} &:= \Prob(S_U[J]/\sqrt{n}\in A_\epsilon^-, (\eta_n \bb{\eigengap})^{-1/2}\|\rem\|\geq \epsilon)-\Prob(S_Y/\sqrt{n}\in A)\\
   &\geq \Prob(S_U[J]/\sqrt{n}\in A_\epsilon^-)-\Prob((\eta_n \bb{\eigengap})^{-1/2}\|\rem\|\geq \epsilon)-\Prob(S_Y/\sqrt{n}\in A_\epsilon^-)+\Prob(S_Y/\sqrt{n}\in A_\epsilon^-)-\Prob(S_Y/\sqrt{n}\in A)
}
Therefore, we have by Theorem~\ref{thm:prop2pt1} that for some constant $C'$ that depends only on $b$,
    \ba{
    \sup_{A\in \mathcal{A}_{re}}|\gamma_{A}|\leq  C' \left( \frac{C_n^2 \log^7 (pn)}{n} \right)^{1/6}+\Abs{\Prob(S_Y/\sqrt{n}\in A_\epsilon^+)-\Prob(S_Y/\sqrt{n}\in A)}+\delta\label{eq:gamma}
    }


    
    Similarly, 
    \ba{\label{eq:omega}
    \sup_{A\in \mathcal{A}_{re}}|\omega_{A}|\leq  C' \left( \frac{C_n^2 \log^7 (pn)}{n} \right)^{1/6}+\Abs{\Prob(S_Y/\sqrt{n}\in A_\epsilon^-)-\Prob(S_Y/\sqrt{n}\in A)}+\delta
    }
    For $\Prob(S_Y/\sqrt{n}\in A_\epsilon^+)-\Prob(S_Y/\sqrt{n}\in A)$, we will use Nazarov's inequality (Lemma~\ref{thm:Nazarov}):
    \ba{\label{eq:aplusnazarov}
   \Abs{\Prob(S_Y/\sqrt{n}\in A_\epsilon^+)-\Prob(S_Y/\sqrt{n}\in A)}\leq \frac{\sqrt{2}\epsilon}{b^{1/2}}(\sqrt{2\log p}+2)
    }

For bounding the terms concerning $A_{\epsilon}^-$, we need to be a little careful because if $b_i-a_i \leq  2\epsilon$, then $A_{\epsilon}^-$ has measure zero under the Gaussian distribution. If $A_{\epsilon}^-$ is nonempty, then we have the same bound as Eq~\ref{eq:aplusnazarov}.
However, in case that is not true, note that there must be some $i\in [p]$ such that $b_i-a_i \leq 2\epsilon$.
Hence
\ba{\label{eq:aminusnazarovempty}
   \Abs{\Prob(S_Y/\sqrt{n}\in A_\epsilon^-)-\Prob(S_Y/\sqrt{n}\in A)}&=\Prob(S_Y/\sqrt{n}\in A)\notag\\
   \rd &=\Prob(S_Y[i]/\sqrt{n}\in [a_i,b_i])\notag \\
   &\leq \frac{2\epsilon}{\sqrt{\pi}b^{1/2}}\bk
    }
    So overall,
   \ba{\label{eq:aminusnazarov}
   \Abs{\Prob(S_Y/\sqrt{n}\in A_\epsilon^-)-\Prob(S_Y/\sqrt{n}\in A)}&=\Prob(S_Y/\sqrt{n}\in A)\notag\\
   \rd &=\Prob(S_Y[i]/\sqrt{n}\in [a_i,b_i])\notag \\
   &\leq \max\bb{\frac{2\epsilon}{\sqrt{\pi}b^{1/2}},\frac{\sqrt{2}\epsilon}{b^{1/2}}(\sqrt{2\log p}+2)\bk
    }
    }
    Putting Eqs~\ref{eq:gamma},~\ref{eq:omega}, ~\ref{eq:aplusnazarov} and~\ref{eq:aminusnazarov} together, we have, for some absolute constant $C_1$:
    \ba{
    &\sup_{A\in \mathcal{A}_{re}}|\Prob((\bb{\eigengap}\eta_n)^{-1/2}\roja[J]\in A)-\Prob(n^{-1/2}S_Y\in A)|\leq \max(\sup_{A\in \mathcal{A}_{re}}|\gamma_{A}|,\sup_{A\in \mathcal{A}_{re}}|\omega_{A}|) \notag \\
    &\lesssim \left(\frac{C_n^2 \log^7 (pn)}{n} \right)^{1/6}+\frac{C_1\epsilon}{b^{1/2}}\sqrt{\log p}+\delta  \label{eq:clt_bound_1}
    }
We invoke Lemma A.2.3 in~\cite{kumarsarkar2024markovoja} to see that:  $\Mfour \leq \lambda_1+\sigma^2\tr{\Sigma}$. Therefore, for some constant $C'' > 0$,
\bas{
C_n=\max(2c_n\Mfour,C\lambda_{1}c_n\sigma^2) \leq C''\sqrt{\frac{n\eta_n}{\bb{\eigengap}}}\Mfour
}
From Lemma~\ref{lemma:oja_error_decomposition_higher_order_tail_bounds} and the assumption on the learning rate (Lemma~\ref{lemma:learning_rate_choice}),
\ba{
\sqrt{\eta_{n}\bb{\eigengap}}\epsilon \lesssim \frac{ \eta_{n}^2n \Mtwo^2 \log d} {\sqrt{\delta}} + \frac{\sqrt{s_n}\eta_n\sqrt{n}\Mtwo\log\bb{d}}{\sqrt{\delta}}
         + \frac{\log\bb{\frac{1}{\delta}}}{\delta^{3}}\bb{ \frac{\sqrt{\eta_{n}^{3}n}\Mtwo^{2}\log\bb{d}}{\sqrt{\lambda_{1}-\lambda_{2}}}} \label{eq:Sv_tail_bound}  \bk
}

% \bas{
%     \frac{ \eta_{n}^2n \Mtwo^2 \log d} {\sqrt{\delta}} + \frac{\sqrt{s_n}\eta_n\sqrt{n}\Mtwo\log\bb{d}}{\sqrt{\delta}}  + \bb{\frac{\sqrt{s_n\log\bb{\frac{1}{\delta}}}}{\delta^{2}} + \frac{1}{\delta^{\frac{3}{2}}}}\bb{\sqrt{d}\exp\bb{-\eta_{n}n\bb{\lambda_{1}-\lambda_{2}}} + \frac{\sqrt{\eta_{n}^{3}n}\Mtwo^{2}\log\bb{d}}{\sqrt{\lambda_{1}-\lambda_{2}}}} 
% }

   % If $\delta> 1/n$, $\epsilon=\tilde{O}\left(\frac{\max(M_{d,2},1)^2}{\sqrt{n}(\lambda_1-\lambda_2)^2\delta^{3/2}}\right)$.
    Substituting the bound on $\eps$ from equation~\eqref{eq:Sv_tail_bound} into equation~\eqref{eq:clt_bound_1} and optimizing over $\delta$ yields
    % \bas{
    % \delta=\max\bb{\Mtwo^{4/3}\eta_n n^{2/3}, \bb{\frac{n\eta_n^2\Mtwo^4}{\lambda_1-\lambda_2}}^{1/3},\frac{(\rd \bk \Mtwo)^{2/3}n\eta_n^3}{(\lambda_1-\lambda_2)^{2/9}}, \bb{\frac{\eta_n^2 n\Mtwo^4}{\lambda_1-\lambda_2}}^{1/5} }
    % }
     \ba{
\delta%=\tilde{O}\bb{\max\bb{\frac{\Mtwo^{4}\eta_n^3 n^{2}}{\lambda_1}}^{1/3}, \bb{\frac{\eta_n^2 n \Mtwo^4}{\lambda_1(\lambda_1-\lambda_2)}}^{1/3}, \bb{\frac{\eta_n^2 n\Mtwo^4}{\lambda_1(\lambda_1-\lambda_2})}^{1/6} }\\
=\tilde{O}\bb{\bb{\frac{\log p}{b}}^{1/8} \sqrt{\frac{\Mtwo}{\eigengap}}n^{-1/8}}. \label{eq:delta_bound}
    }
    % $\epsilon=\tilde{O}\left(\frac{\max(M_{d,2},1)^2}{\sqrt{n}(\lambda_1-\lambda_2)^2}\right)^{2/5}$.
    % We assume $\max((n\eta_n)^{3}\Mfour^2, n\eta_n\sigma^4, M_{d,2}^4/(\lambda_1-\lambda_2)^4)=\tilde{o}(n)$.\bk
    % Note that:
    % \bas{
    % \sup_{A\in \mathcal{A}_{re}}|\rho(A)|\leq 
    % }
    Substituting the choice of $\delta$ from equation~\eqref{eq:delta_bound} in~\eqref{eq:clt_bound_1}, we conclude
    \bas{
    & \sup_{A\in \mathcal{A}^{\text{re}}}|\Prob((\bb{\eigengap}\eta_n)^{-1/2}\roja[J]\in A)-\Prob(n^{-1/2}S_Y\in A)| \\
    & \quad\quad = \tilde{O}\bb{ \max\bb{\bb{\frac{\Mfour}{\eigengap }}^{1/3}n^{-1/6}, \bb{\frac{\Mtwo}{\eigengap}}^{1/2}n^{-1/8}}}
    }
\end{proof}