$E_1, E_2 \dots E_n$ be i.i.d exponential with density $p_x = e^{-x}$
For $k=1,2,3, \dots n+1$:
$S_k = E_1 + \dots + E_k$
Show that $( \frac{S_1}{S_{n+1}}, \dots , \frac{S_n}{S_{n+1}}) \xrightarrow{D} (U_{(1)}, \dots U_{(n)})$
where $U_{(i)}$ is the $i^{th}$ order uniform statistic of $\mathcal{U}[0,1]$ variables.
Attempt:
$f_{U_{(1)}\dots U_{(n)}}(u_1, \dots, u_n) = n! f_U(u_1)\dots f_U(u_n) = n!$
I am not able to deduce the distribution of $\frac{S_{k}}{S_{n+1}}$
Keywords: "Ratio Distribution"
Define $Z_{(k,n+1)}=S_k/S_{n+1}$, then on the support of $\{0\leq z_1\leq \ldots\leq z_n\leq 1\}$ we have for the joint distribution:
$${\quad f_{Z_{(1,n+1)},Z_{(2,n+1)},\ldots,Z_{(n,n+1)}}(z_1,z_2,\ldots, z_n)\\= \int_{\Bbb R^+} \begin{Vmatrix}\dfrac{\partial (tz_1, tz_2,\ldots,tz_n,t)}{\partial (z_1,z_2,\ldots,z_n, t)}\end{Vmatrix}\cdot f_{S_1,S_2,\ldots, S_n, S_{n+1}}(tz_1,\ldots, tz_n,\; t) \mathrm d t \\ = \int_{\Bbb R^+} t^n\cdot f_{E_1,E_2,\ldots,E_n, E_{n+1}}(tz_1, t(z_2-z_1),\ldots, t(z_n-z_{n-1}), t(1-z_n))\mathrm d t \\ \vdots \\ = n! }$$
Similarly, though a little more involved, now show that for any particular $k\in\{1,....,n\}$ that $Z_{(k,n+1)}\sim U_{(k,n)}$ . That is, that the marginal distributions are equal too.