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On the supremum of an infinitely divisible process

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  • Willekens, Eric

Abstract

It was shown by Berman in a recent paper that, for any infinitely divisible process X = {Xt, t[greater-or-equal, slanted]0} with symmetric increments, P(sup0[less-than-or-equals, slant]s[less-than-or-equals, slant]t Xs[greater-or-equal, slanted]u) ~ P(Xt[greater-or-equal, slanted]u) (u --> [infinity]) if the right tail of the Lévy measure is regularly varying with index 0

Suggested Citation

  • Willekens, Eric, 1987. "On the supremum of an infinitely divisible process," Stochastic Processes and their Applications, Elsevier, vol. 26, pages 173-175.
    • Handle: RePEc:eee:spapps:v:26:y:1987:i::p:173-175
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    Cited by:

    1. Braverman, Michael, 2000. "Suprema of compound Poisson processes with light tails," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 145-156, November.
    2. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 362-370, April.
    3. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
    4. Braverman, Michael, 1999. "Remarks on suprema of Lévy processes with light tailes," Statistics & Probability Letters, Elsevier, vol. 43(1), pages 41-48, May.
    5. Albin, J. M. P., 1999. "Extremes of totally skewed [alpha]-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 185-212, February.
    6. Michna, Zbigniew, 2011. "Formula for the supremum distribution of a spectrally positive [alpha]-stable Lévy process," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 231-235, February.
    7. Krzysztof Dȩbicki & Peng Liu & Michel Mandjes & Iwona Sierpińska-Tułacz, 2017. "Lévy-driven GPS queues with heavy-tailed input," Queueing Systems: Theory and Applications, Springer, vol. 85(3), pages 249-267, April.
    8. Braverman, Michael, 1997. "Suprema and sojourn times of Lévy processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 265-283, June.
    9. Braverman, Michael, 2010. "On suprema of Lévy processes with light tails," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 541-573, April.
    10. Griffin, Philip S. & Roberts, Dale O., 2016. "Sample paths of a Lévy process leading to first passage over high levels in finite time," Stochastic Processes and their Applications, Elsevier, vol. 126(5), pages 1331-1352.
    11. Albin, J.M.P. & Sundén, Mattias, 2009. "On the asymptotic behaviour of Lévy processes, Part I: Subexponential and exponential processes," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 281-304, January.
    12. Cheng, Ming & Konstantinides, Dimitrios G. & Wang, Dingcheng, 2022. "Uniform asymptotic estimates in a time-dependent risk model with general investment returns and multivariate regularly varying claims," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    13. Maulik, Krishanu & Zwart, Bert, 2006. "Tail asymptotics for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 156-177, February.
    14. Engelke, Sebastian & Kabluchko, Zakhar, 2015. "Max-stable processes and stationary systems of Lévy particles," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4272-4299.
    15. 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.
    16. Braverman, Michael & Samorodnitsky, Gennady, 1995. "Functionals of infinitely divisible stochastic processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 207-231, April.

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