Take $X$ to be a multivariate random variable with $n$ iid Gaussian components with mean $0$ and constant variance. Then $X/\|X\|_2$ is uniformly distributed on the $n-1$ dimensional sphere $S^{n-1}$.
Now let the components of $X$ have variance $1/n$. The mean and the variance of $\|X\|_2^2$ are then $1$ and $2/n$ respectively, because $\|X\|_2^2$ is $\chi_n^2$ distributed.
If $n$ is large one could approximate $\operatorname{var}(\|X\|_2^2) \approx 0$, so $\|X\|_2^2 \approx 1$ and finally take $X$ as an approximation of the uniform distribution on $S^{n-1}$.
Which literature can I cite, where the approximation of a uniform distribution on a large dimensional sphere by a multivariate Gaussian distribution is shown rigorously? What errors could I expect, when I do this approximation?
This question has been answered at https://mathoverflow.net/questions/315221/distribution-of-the-individual-coordinates-of-a-uniform-random-vector-on-a-high.
It was shown that the convergence of the cdf of the components $X_j$ of $X$ to the cdf $\Phi$ of a Gaussian with variance $1/n$ in the supremum norm on $\mathbb{R}$ is at least as fast as $1/n$. In formula $$ \sup_{x\in\mathbb{R}} |P(X_j \le x) - \Phi(x)| \le \frac{0.24}{n} + \frac{0.16}{n-1} \sim \frac{1}{n}. $$