Uniform distribution on a simplex

Question

In a context where I try to estimate some combinatorial sums, I'm faced with vector random variables $(x_1,...,x_n)$ uniformly distributed with $n \rightarrow \infty$. I want to know if the components have to behave in a wellknown fashion. I recently read the following. (Sourav Chatterjee math summer school 2012)

" Classical example: Uniform distribution on the simplex $\{ (x_1,...,x_n) \ | \ \sum\limits_{k=1}^{n} x_k =1 \}$

In this example, it is known that for n large, the coordinates behave like i.i.d. $Exp(1)$ random variables "

Where can I find a proof of that result or related?

Thanks for any answer/pointer

Answer 1

Let $(X_1,\dots,X_n)$ be a random point uniformly distributed on your simplex. Then it is well known (cf. e.g. Remark 1.3 and formula (2.4)) that $(X_1,\dots,X_n)$ equals $$\frac{(Z_1,\dots,Z_n)}{Z_1+\dots+Z_n}$$ in distribution, where $Z_1,\dots,Z_n$ are iid $Exp(1)$ random variables. So, each $X_i$ equals $$\frac{Z_1}{Z_1+\dots+Z_n}=\frac{Z_1}n\Big/\frac{Z_1+\dots+Z_n}n$$ in distribution. Also, $\frac{Z_1+\dots+Z_n}n\to1$ almost surely and hence in distribution (as $n\to\infty$), by the strong law of large numbers. Thus, for each $i$, the distribution of $nX_i$ (not of $X_i$) goes to $Exp(1)$.

Answer 2

The simplex is, of course, the first quadrant of the $\ell_1$ sphere. This delightful article gives a simple formula for uniform sampling on the $\ell_p$ sphere for any $1 \leq p \leq \infty$.