I am looking for a reference for the following.

Let $X_i, \dots, X_m$ be independent $\chi^2$ with $v_i$ degrees of freedom. Then $$\frac{X_1}{X_1 + X_2}, \frac{X_1 + X_2}{X_1 + X_2 + X_3}, \dots, \frac{X_1 + \dots + X_{m-1}}{X_1 + \dots + X_{m}} \quad \text{ and } \quad X_1 + \dots + X_{m}$$ are mutually independent.

This results is almost stated in Johnson et al. (1995, p. 212), except the independence of the sum $X_1 + \dots + X_{m}$ is not given for the general case. They do state it for $m=2$, unfortunately without source.

Thanks for your help!

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions Volume 2 (2nd ed.).


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