4
$\begingroup$

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

$\endgroup$
6
$\begingroup$

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 $Exp(1)$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Great, thanks a lot for the comment and the reference! $\endgroup$ – Gianfranco Aug 3 at 15:46
2
$\begingroup$

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 $1 \leq p \leq \infty$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks you very much for your interesting answer $\endgroup$ – Gianfranco Aug 4 at 6:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.