Let $X_i\sim\text{Gamma}(\alpha,p_i),i=1,2,...,n+1$ be independent random variables.
Define $Z_1=\sum_{i=1}^{n+1}X_i$ and $Z_i=\frac{X_i}{\sum_{j=1}^iX_j},\quad i=2,3,...,n+1$. Then show that $Z_1,Z_2,...,Z_{n+1}$ are independently distributed.
The joint density of $(X_1,...,X_{n+1})$ is given by
$$f_{\bf X}(x_1,...,x_{n+1})=\left[\frac{\alpha^{\sum_{i=1}^{n+1}p_i}}{\prod_{i=1}^{n+1}\Gamma(p_i)}\exp\left(-\alpha\sum_{i=1}^{n+1}x_i\right)\prod_{i=1}^{n+1}x_i^{p_i-1}\right]\mathbf I_{x_i>0}\quad,\alpha>0,p_i>0$$
We transform $\mathbf X=(X_1,\cdots,X_{n+1})\mapsto\mathbf Z=(Z_1,\cdots,Z_{n+1})$ such that
$Z_1=\sum_{i=1}^{n+1}X_i$ and $Z_i=\frac{X_i}{\sum_{j=1}^iX_j},\quad i=2,3,...,n+1$
$\implies x_{n+1}=z_1z_{n+1},$
$\qquad x_n=z_1z_n(1-z_{n+1}),$
$\qquad x_{n-1}=z_1z_{n-1}(1-z_n)(1-x_{n+1}),$
$\qquad\vdots$
$\qquad x_3=z_1z_3\prod_{j=4}^{n+1}(1-z_j)$
$\qquad x_2=z_1z_2\prod_{j=3}^{n+1}(1-z_j)$
$\qquad x_1=z_1\prod_{j=2}^{n+1}(1-z_j)$, where $0<z_1<\infty$ and $0<z_i<1,\quad i=2,3,\cdots,n+1$
The Jacobian of the transformation is $J=\dfrac{\partial(x_1,...,x_{n+1})}{\partial(z_1,...,z_{n+1})}=\det\begin{pmatrix}\frac{\partial x_1}{\partial z_1} &\cdots& \frac{\partial x_1}{\partial z_{n+1}} \\ &\ddots\\\frac{\partial x_{n+1}}{\partial z_1} &\cdots& \frac{\partial x_{n+1}}{\partial z_{n+1}}\\\end{pmatrix}$
Performing the operation $R_1'=\sum_1^{n+1}R_i$, we get $J$ as the determinant of
$\begin{pmatrix}1&0&0&\cdots&0&0 \\z_2\prod_3^{n+1}(1-z_j)& z_1\prod_3^{n+1}(1-z_j) \\ z_3\prod_4^{n+1}(1-z_j)&0& z_1\prod_4^{n+1}(1-z_j)\\&&&\ddots\\z_n(1-z_{n+1})&0&0&\cdots&z_1(1-z_{n+1})& -z_1z_n \\z_{n+1}&0&0 &\cdots&0& z_1\\\end{pmatrix}$
which equals $z_1^n(1-z_3)(1-z_4)^2...(1-z_n)^{n-2}(1-z_{n+1})^{n-1}$.
After some simplification we get the joint density of $\bf Z$ as
$f_{\bf Z}(z_1,...,z_{n+1})=\prod_{i=1}^{n+1}f_{Z_i}(z_i)$
where $Z_1\sim \text{Gamma}(\alpha,\sum_1^{n+1}p_i),$
$\qquad Z_2\sim \text{Beta}_1(p_1,p_2),$
$\qquad Z_3\sim \text{Beta}_1(p_3,p_1+p_2),$
$\qquad \vdots$
$\qquad Z_{n+1}\sim \text{Beta}_1(p_{n+1},\sum_1^np_i)$,
with $0<z_1<\infty$ and $0<z_i<1,\quad i=2,3,\cdots,n+1$,
$\alpha>0$ and $p_i>0$ for $i=1,2,...,n+1$.
Needless to say, finding the inverse solutions $x_i$'s and evaluating the Jacobian was cumbersome and time consuming. Besides getting the job done, it also determines the distributions of the $Z_i$'s.
Is there any simpler way to show just the independence of the $Z_i$'s?