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Note on the tail behavior of random walk maxima with heavy tails and negative drift. (English) Zbl 1084.60515

Summary: This paper investigates the asymptotic tail behavior of maxima of a random walk with negative mean and heavy-tailed increment distribution. A simple proof is given to improve the related result in Ng et al. (2002). Hence, as announced, the quantity \(P(S^{(\tau)} > x)\) can indeed be viewed as a ruin probability in finite horizon. In what follows, all limit relationships are for \(x\to\infty\). For two positive infinitesimals \(A(x)\) and \(B(x)\), we write \(A(x)\sim B(x)\), as usual, if \(\lim A(x)/B(x) = 1\).

MSC:

60G50 Sums of independent random variables; random walks
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