Asymptotics for the tail probability of random sums with a heavy-tailed random number and extended negatively dependent summands

Abstract

Let {X,X _k: k ≥ 1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX > 0. Let τ be a nonnegative integer-valued random variable, independent of {X,X _k: k ≥ 1}. In this paper, the authors obtain the necessary and sufficient conditions for the random sums S_\tau = \sum\limits_{n = 1}^\tau {X_n } to have a consistently varying tail when the random number τ has a heavier tail than the summands, i.e., \frac{{P(X > x)}}{{P(\tau > x)}} \to 0 as x→∞.

Keywords

Asymptotic behavior / Random sums / Heavy-Tailed distribution

References

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