Approximation of the tail probability of randomly weighted sums and applications

Consider the problem of approximating the tail probability of randomly weighted sums ∑i=1nΘiXi and their maxima, where {Xi,i≥1}{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}{Θi,i≥1} is a sequence of nonnegative random variables, independent of {Xi,i≥1}{Xi,i≥1} and satisfying certain moment conditions. Under the assumption that {Xi,i≥1}{Xi,i≥1} has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations: Pr(max1≤k≤n∑i=1kΘiXi>x)∼Pr(∑i=1nΘiXi>x)∼∑i=1nPr(ΘiXi>x), and Pr(max1≤k<∞∑i=1kΘiXi>x)∼Pr(∑i=1∞ΘiXi+>x)∼∑i=1∞Pr(ΘiXi>x), as x→∞x→∞. In doing so, no assumption is made on the dependence structure of the sequence {Θi,i≥1}{Θi,i≥1}.