# Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

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

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 $$R$$.

Without loss of generality, $$R=1$$. Let $$Z_1,\ldots,Z_n$$ be iid standard normal random variables (r.v.'s). Then $$$$\sqrt n\, X_1\overset{\text{D}}=\frac{\sqrt n\,Z_1}{\sqrt{Z_1^2+\cdots+Z_n^2}} \overset{\text{D}}= \frac{Z_1+\cdots+Z_n}{\sqrt{Z_1^2+\cdots+Z_n^2}}=:T_1,$$$$ where $$\overset{\text{D}}=$$ denotes the equality in distribution. By the top display on page 20 (you may also want to see the published version), $$$$d_{Ko}(T_1,Z_1)\le d_{Ko}(T,Z_1)+\frac{0.24}n,$$$$ where $$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 (you may also want to see the published version), for $$n\ge 5$$ $$$$d_{Ko}(T,Z_1)<\frac{0.16}{n-1},$$$$ so that $$$$\sup_{x\in\mathbb R}|P(\sqrt n\,X_1\le x)-\Phi(x)| =d_{Ko}(T_1,Z_1)\le\frac{0.24}n+\frac{0.16}{n-1}\sim\frac{0.4}n.$$$$

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.

• OK great. This is better than my imagined $n^{-1/2}$ rate. The references are even a bigger treasure. Thanks! – dohmatob Nov 13 '18 at 14:27
• Thanks. I have added refs. to the published versions of the papers. – Iosif Pinelis Nov 13 '18 at 14:29
• Great. Would you mind throwing in 1 or two details hinting the "hidden" computation "$d_{Ko}(T1, T) \le 0.24/n$" ? I guess this follows from your delta-method, but a sentence saying what's going on (e.g "one can take the function $f=...$", etc.) might be really useful. – dohmatob Nov 13 '18 at 14:36
• No, the inequality $d_{Ko}(T_1,Z_1)\le d_{Ko}(T,Z_1)+\frac{0.24}n$ does not use the delta method results at all. Rather, it follows immediately from elementary formula (4.24) in the delta-method paper, since the cdf's of $T_1$ and $T$ are easy to express in terms of each other. – Iosif Pinelis Nov 13 '18 at 14:42
• Off-topic: I wonder if you would mind helping on this mathoverflow.net/questions/314409/… or this math.stackexchange.com/questions/2976654/…. Thanks in advance. – dohmatob Nov 13 '18 at 14:54

We may assume $$R=1$$. A useful trick is to realize the uniform measure on the unit sphere as the distribution of $$\left(\frac{G_1}{|G|},\dots,\frac{G_n}{|G|} \right),$$ where $$G=(G_1,\dots,G_n)$$ is a Gaussian vector with independant $$N(0,1)$$ coordinates, and $$|G|=\sqrt{G_1^2+\cdots+G_n^2}$$. With this in hand you can now write $$P(X_1 \leq \frac{z}{\sqrt{n}}) = P(G_1 \leq \frac{|G|}{\sqrt{n}} z) \approx P(G_1 \leq z) ,$$ where in the last step you have to argue that $$|G|$$ concentrates around $$\sqrt{n}$$ with fluctuations $$O(1)$$ (google "chi-squared concentration").