In this paper it is shown that the distribution of the product of independently distributed beta random variables can be represented as a mixture of beta distributions. This result is then applied to derive the distributions of Wilks' statistic in the central and the noncentral linear cases. In the noncentral linear case the results of computations agree with those of Gupta (1971).
"Central and Noncentral Distributions of Wilks' Statistic in Manova as Mixtures of Incomplete Beta Functions." Ann. Statist. 3 (2) 467 - 472, March, 1975. https://doi.org/10.1214/aos/1176343073