Approximation of the tail probability of randomly weighted sums and applications

Consider the problem of approximating the tail probability of randomly weighted sums ∑ i = 1 n Θ i X i and their maxima, where { X i , i ≥ 1 } is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation class, and { Θ i , i ≥ 1 } is a sequence of nonnegative random variables, independent of { X i , i ≥ 1 } and satisfying certain moment conditions. Under the assumption that { X i , i ≥ 1 } has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations: Pr ( max 1 ≤ k ≤ n ∑ i = 1 k Θ i X i > x ) ∼ Pr ( ∑ i = 1 n Θ i X i > x ) ∼ ∑ i = 1 n Pr ( Θ i X i > x ) , and Pr ( max 1 ≤ k < ∞ ∑ i = 1 k Θ i X i > x ) ∼ Pr ( ∑ i = 1 ∞ Θ i X i + > x ) ∼ ∑ i = 1 ∞ Pr ( Θ i X i > x ) , as x → ∞ . In doing so, no assumption is made on the dependence structure of the sequence { Θ i , i ≥ 1 } .