Independence of Gamma and Beta random variables with common term

Given $P$ independent and identically distributed random variables, $X_1, X_2, ..., X_P \sim \text{Gamma}(M,2c)$ 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, when $U \sim \text{Gamma}(MP,2c)$ and $V \sim \text{Beta}(M,M(P-1))$.

1 Answer

In Devroye's Non-uniform random variate generation, Chapter 5 starts with two theorems on Uniform spacings (pp.207-208):

Theorem 2.1 - Given an iid Uniform sample $U_1,\ldots,U_n$, the uniform spacing statistics$$S_1=U_{(1)}\,,\quad S_i=U_{(i)}-U_{(i-1)}\,,\quad i=2,\ldots n\,,\quad S_{n+1}=1-U_{(n)}$$associated with the order statistics are such that $(S_1,\ldots,S_n)$ is uniformly distributed over the simplex$$A_n=\left\{(x_1,\ldots,x_n);x_i\ge 0\,,\,\sum_{i=1}^n x_i\le 1\right\}$$

and

Theorem 2.2 - $(S_1,\ldots,S_{n+1})$ is distributed as $$\dfrac{E_1}{\sum_{i=1}^{n+1} E_i},\ldots,\dfrac{E_{n+1}}{\sum_{i=1}^{n+1} E_i}$$where $E_1,\ldots,E_{n+1}$ is an iid sequence of exponential random variables. Furthermore, if $G_{n+1}$ is independent of $(S_1,\ldots,S_{n+1})$ and is Gamma$(n+1,1)$ distributed, then $$S_1G_{n+1},\ldots,S_{n+1}G_{n+1}$$is distributed as $E_1,\ldots,E_{n+1}$