Insensitivity to Negative Dependence of Asymptotic Tail Probabilities of Sums and Maxima of Sums

Abstract Let {X 1, X 2,…} be a sequence of pairwise negatively dependent random variables with distributions F 1, F 2,… on (− ∞, ∞), and let S (n) be the maximum of its first n partial sums S 1,…, S n . This article studies the asymptotic tail probabilities of S n and S (n). Under suitable regularity conditions on the distributions F 1,…, F n , we prove that both P(S n > x) and P(S (n) > x) are asymptotic to as x → ∞, indicating that the negative dependence among the random variables does not affect the asymptotic tail behavior of S n and S (n). For a special case where the sequence {X 1, X 2,…} is generated by pairwise negatively dependent and identically distributed random variables weighted by positive constants fulfilling a certain summability condition, we further prove that the asymptotic relations obtained are uniform for all n = 1,2,….

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