Dirichlet sample by normalising Gamma RVs I know that if you sample $K$ random variables $(X_1, X_2, \dots, X_K)$ from Gamma distributions using shape parameters $(\alpha_1, \alpha_2, \dots \alpha_K)$ and a scale parameter $\theta = 1$ such that $X_i \sim \Gamma(\alpha_i,\theta) = \Gamma(\alpha_i,1)$ then $\left(\frac{X_1}{\sum_{i = 1}^K X_i}, \frac{X_2}{\sum_{i = 1}^K X_i}, \dots, \frac{X_K}{\sum_{i = 1}^K X_i}\right) \sim \textrm{Dir}(\alpha_1, \alpha_2, \dots, \alpha_K)$ where $\textrm{Dir}(\alpha_1, \alpha_2, \dots, \alpha_K)$ is a Dirichlet distribution with a concentration parameter $(\alpha_1, \alpha_2, \dots, \alpha_K)$.
My question is does this result hold for any scale parameter $\theta > 0$? The proofs I've read (e.g. http://mayagupta.org/publications/FrigyikKapilaGuptaIntroToDirichlet.pdf), which use the change-of-variables formula, seem to set $\theta = 1$.
 A: The result for scale 1 straightforwardly implies the generalized result for arbitrary scale $\theta>0$ as follows. Let
\begin{equation}
X_k \sim \Gamma(\alpha_i,\theta),~i\in \{1,\ldots,K\},
\end{equation}
and the $X_i$s be mutually independent. Now, let us define variables $(Z_1,\ldots,Z_k)$ by scaling the $X$s:
\begin{equation}
Z_i = \frac{X_i}{\theta},~i\in \{1,\ldots,K\}.
\end{equation}
The $Z_i$s are mutually independent and  $Z_i\sim \Gamma(\alpha_i,1)$. Then, we express
\begin{equation}
\left(\frac{X_1}{\sum_{i = 1}^K X_i}, \frac{X_2}{\sum_{i = 1}^K X_i}, \dots, \frac{X_K}{\sum_{i = 1}^K X_i}\right)
\end{equation}
in terms of the $Z_i$s:
\begin{equation}
=\left(\frac{Z_1\theta}{\sum_{i = 1}^K( Z_i\theta)}, \frac{Z_2\theta}{\sum_{i = 1}^K (Z_i\theta)}, \dots, \frac{Z_K\theta}{\sum_{i = 1}^K (Z_i\theta)}\right).
\end{equation}
Divide both numerator and denominator by $\theta$:
\begin{equation}
=\left(\frac{Z_1}{\sum_{i = 1}^K Z_i}, \frac{Z_2}{\sum_{i = 1}^K Z_i}, \dots, \frac{Z_K}{\sum_{i = 1}^K Z_i}\right).
\end{equation}
By construction the $Z_i$s have $\Gamma(\alpha_i,1)$ distributions, and thus the last expression has the $\mathrm{Dir}(\alpha_1,\alpha_2,\ldots,\alpha_K)$ distribution. That is, it has been shown that 
\begin{equation}
\left(\frac{X_1}{\sum_{i = 1}^K X_i}, \frac{X_2}{\sum_{i = 1}^K X_i}, \dots, \frac{X_K}{\sum_{i = 1}^K X_i}\right) \sim \mathrm{Dir}(\alpha_1,\alpha_2,\ldots,\alpha_K).
\end{equation}
