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The overshoot of a random walk with negative drift. (English) Zbl 1108.60042
Summary: Let \(\{S_n,n\geq 0\}\) be a random walk starting from 0 and drifting to \(-\infty\), and let \(\tau(x)\) be the first time when the random walk crosses a given level \(x\geq 0\). Some asymptotics for the tail probability of the overshoot \(S_{\tau(x)}-x\), associated with the event \((\tau(x)<\infty)\), are derived for the cases of heavy-tailed and light-tailed increments. In particular, the formulae obtained fulfill certain uniform requirements.

MSC:
60G50 Sums of independent random variables; random walks
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[1] Asmussen, S.; Klüppelberg, C., Large deviations results for subexponential tails, with applications to insurance risk, Stochastic process. appl., 64, 1, 103-125, (1996) · Zbl 0879.60020
[2] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge · Zbl 0617.26001
[3] Chang, J.T., Inequalities for the overshoot, Ann. appl. probab., 4, 4, 1223-1233, (1994) · Zbl 0812.60021
[4] Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II, second ed. Wiley, New York, London, Sydney. · Zbl 0219.60003
[5] Janson, S., Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift, Adv. in appl. probab., 18, 4, 865-879, (1986) · Zbl 0612.60060
[6] Klüppelberg, C., Subexponential distributions and characterizations of related classes, Probab. theory related fields, 82, 2, 259-269, (1989) · Zbl 0687.60017
[7] Klüppelberg, C.; Kyprianou, A.E.; Maller, R.A., Ruin probabilities and overshoots for general Lévy insurance risk processes, Ann. appl. probab., 14, 4, 1766-1801, (2004) · Zbl 1066.60049
[8] Pakes, A.G., Convolution equivalence and infinite divisibility, J. appl. probab., 41, 2, 407-424, (2004) · Zbl 1051.60019
[9] Rogozin, B.A., On the constant in the definition of subexponential distributions, Theory probab. appl., 44, 2, 409-412, (2000), Translated from Rogozin, B.A., 1999. Teor. Veroyatnost. i Primenen. 44(2), 455-458 (in Russian). · Zbl 0971.60009
[10] Rogozin, B.A.; Sgibnev, M.S., Banach algebras of measures on the line with given asymptotics of distributions at infinity, Siberian math. J., 40, 3, 565-576, (1999), Translated from Rogozin, B.A., Sgibnev, M.S., 1999. Sibirsk. Mat. Zh. 40(3), 660-672 (in Russian) · Zbl 0936.46021
[11] Shimura, T.; Watanabe, T., Infinite divisibility and generalized subexponentiality, Bernoulli, 11, 3, 445-469, (2005) · Zbl 1081.60016
[12] Su, C., Chen, J., Hu, Z., 2004. Some discussions on the class \(\mathcal{L}(\gamma)\). J. Math. Sci. 122 (4), 3416-3425. · Zbl 1066.60017
[13] Tang, Q., A note on the severity of ruin in the renewal model with claims of dominated variation, Bull. Korean math. soc., 40, 4, 663-669, (2003) · Zbl 1046.62115
[14] Tang, Q., On convolution equivalence with applications, Bernoulli, 12, 3, 535-549, (2006) · Zbl 1114.60015
[15] Veraverbeke, N., Asymptotic behaviour of wiener – hopf factors of a random walk, Stochastic processes appl., 5, 1, 27-37, (1977) · Zbl 0353.60073
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