Independence of Gamma and Beta random variables with common term

Question

Given $\textbf{P}$ independent and identically distributed random variables, $X_1, X_2, ..., X_P \sim \Gamma(M,2c)$ how can we prove that:

$$U = X_1 + X_2 + ... + X_P$$

and

$$V = \frac{X_1}{X_1 + X_2 + ... + X_P}$$

are independent?

Where $U \sim \Gamma(MP,2c)$ and $V \sim \beta(M,M(P-1))$.

Answer 1

Just for the record: this is answered in item 25 of http://www.randomservices.org/random/special/Beta.html