The main purpose of this paper is to establish probabilistic asymptotic convergence properties of the power sums D-n,D-k= Sigma Y-n (i=1)i(k) of the Dirichlet distribution Y = (Y-1,Y-2,& centerdot;& centerdot;& centerdot;, Y-n) similar to Dir(a) for positive integers k. We propose stability conditions on the parameters a = (a(1), a(2),& centerdot;& centerdot;& centerdot; , a(n)) is an element of Rn+ such that these power sums satisfy the almost sure convergence n(k-2-delta)D(n,k) -> 0 for all delta > 0 and their distributions are also asymptotically normal as n -> infinity. This extends previously known results for the special case of symmetric Dirichlet distributions, including especially the widely studied case of uniform spacings associated with the classical random division. As an application, we show that the semiperimeters and areas of random cyclic polygons generated from such Dirichlet distributions satisfy analogous probabilistic convergence and asymptotic normality estimates as n -> infinity. Additionally, we also present some extrapolation improvements with faster rates of convergence.
