On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach

In this article we employ certain techniques in divided differences   to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables X1,…,XmX1,…,Xm to the generalized Stieltjes transforms of the distribution functions F1,…,FmF1,…,Fm; Xi∼Fi,i=1,…,m. The random weights are assumed to be cuts of [0,1][0,1] by m−1m−1 ordered statistics of independent and identically uniformly distributed random variables U1,…,UnU1,…,Un on [0,1][0,1]; m≤nm≤n. Soltani and Homei (2009) treated the case m=nm=n using the Schwartz distribution theory. We identified fairly large classes of randomly weighted average distributions by their generalized Stieltjes transforms; in particular including the uniform, Wigner and certain power semicircle distributions.