A note on the tail behavior of randomly weighted and stopped dependent sums

Abstract

In this paper, we deal with the tail behavior of the maximum of randomly weighted and stopped sums. We assume that primary random variables (with a certain dependence structure) are identically distributed with heavy-tailed distribution function and random weights are nonnegative. In this note, we specify some conditions for the (weak) asymptotics of the tail of random maximum.

A note on the asymptotics for the randomly stopped weighted sums

“…The third lemma investigates the asymptotic tail behavior of the random sum P(Z τ > x), which extends Proposition 1(i) in [4] and plays an important role in the proof of our main result.…”

“…Based on Theorem A, [4] considered a more specific case, in which θ i 's are assumed to be UEND and identically distributed r.v.s with common distribution F θ ∈ D, and their result makes Theorem A more clear and verifiable. In the present paper, we aim to consider the case that {θ i , i…”

“…Proof. We follow the line of the proof of Proposition 1 in [4] to prove this lemma. For any positive integer M and any ε ∈ (0, 1), we divide the tail probability P(Z τ > x) into three parts as P(Z τ > x) =:…”

“…In this paper, we are interested in the asymptotic behavior for the tail probability of a random number of randomly weighted sums and their maximum defined in (1), which is motivated by two recent papers [15] and [4]. The former considered a random number of randomly weighted sums and their maximum and studied the tail behaviors of S τ and M τ under a restrictive dependence structure.…”

“…The former considered a random number of randomly weighted sums and their maximum and studied the tail behaviors of S τ and M τ under a restrictive dependence structure. [4] further specified some conditions in [15] to make the results more clear and verifiable by assuming {θ i , i 1} to be upper extended negatively dependent and identically distributed. Following the work [4], in the present paper, we aim to consider the case that {θ i , i 1} follow the more general widely upper orthant dependence structure.…”