The tail behavior of randomly weighted sums of dependent random variables

Abstract

Consider dependent random variables X1, Xd with a common distribution function F and denote by ωF the right endpoint of the support of F. Let Θ1 Θd be non-negative random variables, independent of X = (X1, Xd) and satisfying certain moment conditions if necessary. Under the assumption that X is in the maximum domain of attraction of a multivariate extreme value distribution, we establish the asymptotic behaviors of randomly weighted sums: there exist limiting constants qFθ, qWθ and qGθ such that for large t, P(Σdi=1 ΘiXi>t) ~ E qFΘ P(X1 > t), P(Σdi=1 Θi(ωF-Xi) < 1/t) ~ E qWΘ·P(X1 > ωF -1/t), and for Σdi=1 Θi = 1 and t approaching to ωF , P(Σdi=1 ΘiXi > t) ~ E qGΘ P(X1 > t) according to F belonging to the maximum domains of attraction of the Fr´echet, Weibull and Gumbel distributions, respectively. Moreover, some basic properties of the proportionality factor E qFΘ are presented.