Let $X=(X\_1,\ldots,X\_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X\_i$ is normally distributed with variance $R^2/n$, but I'm not quite sure.


Question
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More formaly, if $\Phi$ is the CDF of the standard Guassian $\mathcal N(0, 1)$, what is a good upper bound for the quantity $\alpha\_n := \sup\_{z \in \mathbb R}|P(X\_1 \le nR^{-2}z) - \Phi(z)|$ ?


Observations
============


My wild guess is that $\alpha\_n \le Cn^{-1/2}$ for some absolute constant $C$ independent of $n$ and $R$.



Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere
Without loss of generality, $R=1$. 
Let $Z\_1,\ldots,Z\_n$ be iid standard normal random variables (r.v.'s). Then 
\begin{equation}
 \sqrt n\, X\_1\overset{\text{D}}=\frac{\sqrt n\,Z\_1}{\sqrt{Z\_1^2+\cdots+Z\_n^2}}
 \overset{\text{D}}= 
 \frac{Z\_1+\cdots+Z\_n}{\sqrt{Z\_1^2+\cdots+Z\_n^2}}=:T\_1, 
\end{equation}
where $\overset{\text{D}}=$ denotes the equality 
in distribution. 
By the [top display on page 20](https://arxiv.org/abs/0906.0177) (you may also want to see the [published version](https://projecteuclid.org/euclid.ejs/1460463653)), 
\begin{equation}
 d\_{Ko}(T\_1,Z\_1)\le d\_{Ko}(T,Z\_1)+\frac{0.24}n, 
\end{equation}
where $d\_{Ko}(X,Y):=\sup\_{x\in\mathbb R}|P(X\le x)-P(Y\le x)|$ is the Kolmogorov distance between r.v.'s $X,Y$, and $T$ is a r.v. with the Student distribution $t\_{n-1}$ with $n-1$ degrees of freedom. 


By [Theorem 1.2](https://arxiv.org/abs/1101.3328) (you may also want to see the [published version](https://www.esaim-ps.org/articles/ps/abs/2015/01/ps140014/ps140014.html)), for $n\ge 5$
\begin{equation}
 d\_{Ko}(T,Z\_1)<\frac{0.16}{n-1}, 
\end{equation}
so that
\begin{equation}
 \sup\_{x\in\mathbb R}|P(\sqrt n\,X\_1\le x)-\Phi(x)|
 =d\_{Ko}(T\_1,Z\_1)\le\frac{0.24}n+\frac{0.16}{n-1}\sim\frac{0.4}n. 
\end{equation}


I think the latter constant factor $0.4$ can be improved to about $0.16$ by using directly the method of proof of [Theorem 1.2](https://arxiv.org/abs/1101.3328). 


