TITLE: PDF of distance from origin to simplex in high dimension
QUESTION [8 upvotes]: Let $\Delta^{n-1}$ be the standard simplex in $n$ dimensions:
$$ \Delta^{n-1} = \{ \mathbf{x}\in \mathbb{R}^n: \sum_i x_i=1 , \mathbf{x}\geq0\} $$
And assuming that we are uniformly sampling points $\mathbf{X}$ from this simplex, then the Euclidean distance $|\mathbf{X}|$ can also be considered a random variable. In the limit of $n\rightarrow\infty$, does the probability distribution of $|\mathbf{X}|$ approach some limit distribution that can be written in closed form?

REPLY [2 votes]: A simple way to get the expected value $\mathbb{E}[\lVert X\rVert^2]$, without relying on the Dirichlet distribution (what we are considering here being a spacial case of it, the symmetric Dirichlet distribution with $\alpha=1$). This is only computing this expectation, not addressing the full question).
It is well-known that sampling $X$ uniformly from the probability simplex is equivalent to sampling $n-1$ i.i.d. uniform r.v.'s $U_1,\dots, U_{n-1}$, sorting them (adding $U_0=0$ and $U_n=1$), and looking at the differences. From this, we get that $X_1$ is distributed as $\min(U_1,U_2,\dots,U_{n-1})$ , from which its pdf is easily seen to be
$$
f(x) = (n-1)(1-x)^{n-2}, \qquad x\in[0,1]
$$
But we have, by symmetry and linearity,
$$
\mathbb{E}[\lVert X\rVert^2] = \sum_{i=1}^n \mathbb{E}[X_i^2] = n \mathbb{E}[X_1^2] 
$$
from which
$$
\mathbb{E}[\lVert X\rVert^2] = n \int_0^1  (n-1)x^2(1-x)^{n-2} dx = \boxed{\frac{2}{n+1}}
$$