Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex

A weighted Average of n independent continuous random variables X1,...,Xn with random proportions obtained by cutting a simplex by hyper-planes is introduced. A formula between the Stieltjes transforms of the distribution functions of the weighted average and X1,...,Xn is established. The Johnson and Kotz [Johnson, N.L., Kotz, S., 1990. Randomly weighted averages: Some aspects and extensions. Amer. Statist. 44, 245-249] question on the Cauchy distribution characterization of Van Assche [Van Assche, W., 1987. A random variable uniformly distributed between two independent random variables. Sankhya Ser. A 49, 207-211] is answered for n>=2.