Approximation of the tail probability of randomly weighted sums and applications

Consider the problem of approximating the tail probability of randomly weighted sums ∑n i=1 ΘiXi 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 {Θi, 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