# Uniform distribution on a simplex

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?

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 $$Exp(1)$$.
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 $$1 \leq p \leq \infty$$.