On the asymptotic distribution of randomly weighted averages of random vectors

Abstract

Let X1,X2,…,Xn be a sequence of mutually independent continuous random vectors with respective positive definite covariance matrices Σ1,Σ2,…,Σn. The main purpose of this article is to derive the asymptotic distribution of Sn:n, a randomly weighted average of the sequence X1,X2,…,Xn, as n→∞. The random weights are the cuts of (0, 1) by an increasing ordered array of the ordered statistics of n independent and identically uniformly distributed random variables. We prove that under certain assumptions on the covariance matrices, n(Sn:n−μ) converges in distribution to the multivariate normal distribution with zero mean and covariance matrix 2Σ, where Σ is the limit of the expression in terms of Σis. Finally, we give an application for the multivariate randomly weighted averages in modeling (height, general intelligence) parents’ gene compositions given to their offspring. The simulation results which are based on three distributions (Bivariate t, Dirichlet and Normal) which confirm the main theorem of the paper.