## 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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### References:

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