Recently I come up with an embarrassingly easy question. It should be known or elementary but I am still not able to find either a correct answer or references:


"Consider a random vector $x=(x\_1,...,x\_n)$ in the simplex $0\le x\_i, x\_1+..+x\_n=1$. It is easy to show that each $x\_i$ has beta distribution $B(1,n-1)$. It can be also checked that the expected value of $|x|\_2$ is of order $n^{-1/2}$. 


I am wondering if there is any concentration result saying that there exists a sufficiently large constant $C$ such that $|x|\_2 \le Cn^{-1/2}$ with high probability, say $1-n^{-3}$? "


Thanks.



Random vector of fixed entry-sum 
For the expected norm see equation (19) in [this paper](http://arxiv.org/abs/math/0505618) (or equation (12) in the published version of that paper), which, after some renormalizing, states that
$$
\mathbb{E} x\_1^{r\_1} \cdots x\_n^{r\_n} = \frac{(n-1)! r\_1! \cdots r\_n!}{(r+n-1)!},
$$
where $r\_i \ge 0$ and $r = r\_1 + \cdots r\_n$. So in particular (if you can trust my arithmetic)
$$
\mathbb{E} ||x||^2 = \frac{2}{n+1}, \quad \mathbb{E} ||x||^4 = \frac{4(n+5)}{(n+1)(n+2)(n+3)}.
$$
From these standard Hölder's inequality estimates give that $\mathbb{E} ||x||$ is of order $n^{-1/2}$.


For concentration of the norm, there are general concentration results for convex bodies that apply, in particular "Borell's lemma" which gives much sharper concentration than you asked for — see [this answer to another question](https://mathoverflow.net/questions/67521/concentration-of-measure-for-arbitrary-convex-bodies/67770#67770).


Other relevant results are in [this famous paper](http://stat-reports.lib.berkeley.edu/accessPages/85.html) of Diaconis and Freedman.


