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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.

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For the expected norm see equation (19) in this paper (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.

Other relevant results are in this famous paper of Diaconis and Freedman.

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  • $\begingroup$ Thanks a lot for the references. When I try to apply "Borell's lemma", where P is the given simplex and A is a sphere of radius $O(n^{1/2})$ centered at $(1/n,\dots,1/n)$, I got a bound like $P(|x|^2 \ge C/n) = O(\exp(-C))$. Although this is good, it is still not anything of order $n^{-3}$. Did I miss something? $\endgroup$
    – user21174
    Commented Feb 6, 2012 at 23:36
  • $\begingroup$ No, I was being silly. You want Paouris's theorem: math.tamu.edu/~grigoris/BNest_cor.pdf $\endgroup$
    – Mark Meckes
    Commented Feb 7, 2012 at 3:00
  • $\begingroup$ Thanks Mark! It looks like we may want to apply Theorem 1.4 in there, and I sort of understand it. However, to make it rigorous we need lots of calculations including finding the appropriate linear transform T that maps the region |x_1|+\dots+|x_n|\le 1 onto an isotropic body. Are there easier ways or this is just standard? $\endgroup$
    – user21174
    Commented Feb 7, 2012 at 10:48
  • $\begingroup$ This is pretty standard. One way to do it is written out quite explicitly in the paper of mine which I linked to in the answer. In any case, by symmetry considerations, once you translate T so that its center is at the origin, what you get is isotropic up to scaling in the hyperplane $x_1 + \cdots + x_n = 0$; the right rescaling (ou need to blow up by a factor of order n I think off the top of my head) follows from the calculations in my answer above. $\endgroup$
    – Mark Meckes
    Commented Feb 7, 2012 at 13:57
  • $\begingroup$ Oh I see now, thanks. $\endgroup$
    – user21174
    Commented Feb 7, 2012 at 16:45

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