Classes of power semicircle laws that are randomly weighted average distributions.

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized byn, gives the distributions of the average ofnindependent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of sizen−1, for eachn. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions. [ABSTRACT FROM AUTHOR]