Approximation of the tail probability of randomly weighted sums and applications

Consider the problem of approximating the tail probability of randomly weighted sums and their maxima, where {Xi,i>=1} is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation class, and {[Theta]i,i>=1} is a sequence of nonnegative random variables, independent of {Xi,i>=1} and satisfying certain moment conditions. Under the assumption that {Xi,i>=1} has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations: and as x-->[infinity]. In doing so, no assumption is made on the dependence structure of the sequence {[Theta]i,i>=1}.