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On distribution tail of the maximum of a random walk

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  • Korshunov, D.

Abstract

Let Sn, n [greater-or-equal, slanted] 1, be the partial sums of i.i.d. random variables with negative mean value. Many papers (see, for example, [1,2,5,6,7,9,11]) give us different theorems on the tail behavior of the distribution of sup {Sn,n [greater-or-equal, slanted] 1}. In this paper the final versions of these theorems (with necessary and sufficient conditions) are presented. The main attention is paid to the necessity part of these theorems.

Suggested Citation

  • Korshunov, D., 1997. "On distribution tail of the maximum of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 97-103, December.
    • Handle: RePEc:eee:spapps:v:72:y:1997:i:1:p:97-103
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    References listed on IDEAS

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    1. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    2. Stadje, W., 1995. "A note on the maximum of a random walk," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 227-231, May.
    3. Veraverbeke, N., 1977. "Asymptotic behaviour of Wiener-Hopf factors of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 27-37, February.
    4. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
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    Cited by:

    1. Lin, Jianxi, 2012. "Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 422-429.
    2. Wang, Yuebao & Yang, Yang & Wang, Kaiyong & Cheng, Dongya, 2007. "Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 256-266, March.
    3. Yuebao Wang & Kaiyong Wang, 2009. "Equivalent Conditions of Asymptotics for the Density of the Supremum of a Random Walk in the Intermediate Case," Journal of Theoretical Probability, Springer, vol. 22(2), pages 281-293, June.
    4. Yuebao Wang & Hui Xu & Dongya Cheng & Changjun Yu, 2018. "The local asymptotic estimation for the supremum of a random walk with generalized strong subexponential summands," Statistical Papers, Springer, vol. 59(1), pages 99-126, March.
    5. Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    6. Gao, Qingwu & Wang, Yuebao, 2009. "Ruin probability and local ruin probability in the random multi-delayed renewal risk model," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 588-596, March.
    7. Nelly Litvak & Maria Vlasiou, 2010. "A survey on performance analysis of warehouse carousel systems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 401-447, November.
    8. Mandjes, M. & Ravner, L., 2021. "Hypothesis testing for a Lévy-driven storage system by Poisson sampling," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 41-73.
    9. Boxma, Onno & Kella, Offer & Mandjes, Michel, 2023. "On fluctuation-theoretic decompositions via Lindley-type recursions," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 316-336.
    10. Ilya Tkachev & Alessandro Abate, 2013. "Computation of ruin probabilities for general discrete-time Markov models," Papers 1308.5152, arXiv.org.
    11. Søren Asmussen & Serguei Foss & Dmitry Korshunov, 2003. "Asymptotics for Sums of Random Variables with Local Subexponential Behaviour," Journal of Theoretical Probability, Springer, vol. 16(2), pages 489-518, April.
    12. Korshunov, Dmitry, 2018. "On subexponential tails for the maxima of negatively driven compound renewal and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1316-1332.

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