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Asymptotic behaviour of Wiener-Hopf factors of a random walk

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  • Veraverbeke, N.

Abstract

For a random walk governed by a general distribution function F on (-[infinity], +[infinity]), we establish the exponential and subexponential asymptotic behaviour of the corresponding right Wiener-Hopf factor F+. The results apply to classes of distribution functions in recent publications: the subexponential class and a related (exponential) class [gamma]. Given the behaviour of F+, the Wiener-Hopf identity is used, to obtain the behaviour of F. To reverse the argument, we derive a new identity, similar in form to the first one. The results for F+ are then fruitfully applied to give a full description of the tail behaviour of the maximum of the randon walk. Also they provide new proofs for recent theorems on the tail of the waiting-time distribution in the GI/G/1 queue.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:5:y:1977:i:1:p:27-37
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    Cited by:

    1. Sgibnev, M. S., 2001. "Exact asymptotic behaviour of the distribution of the supremum," Statistics & Probability Letters, Elsevier, vol. 52(3), pages 301-311, April.
    2. 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.
    3. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Asmussen, Søren & Klüppelberg, Claudia, 1996. "Large deviations results for subexponential tails, with applications to insurance risk," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 103-125, November.
    9. Krzysztof Burdzy & Tvrtko Tadić, 2018. "Random Reflections in a High-Dimensional Tube," Journal of Theoretical Probability, Springer, vol. 31(1), pages 466-493, March.
    10. Hägele, Miriam, 2020. "Precise asymptotics of ruin probabilities for a class of multivariate heavy-tailed distributions," Statistics & Probability Letters, Elsevier, vol. 166(C).
    11. Youri Raaijmakers & Sem Borst & Onno Boxma, 2023. "Fork–join and redundancy systems with heavy-tailed job sizes," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 131-159, February.
    12. Bert Zwart, 2015. "Loss rates in the single-server queue with complete rejection," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 299-315, June.
    13. Serguei Foss & Takis Konstantopoulos & Stan Zachary, 2007. "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments," Journal of Theoretical Probability, Springer, vol. 20(3), pages 581-612, September.
    14. Cheng, Yebin & Tang, Qihe & Yang, Hailiang, 2002. "Approximations for moments of deficit at ruin with exponential and subexponential claims," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 367-378, October.
    15. 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.
    16. Tang, Qihe, 2007. "The overshoot of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 158-165, January.
    17. Wang, Kaiyong & Yang, Yang & Yu, Changjun, 2013. "Estimates for the overshoot of a random walk with negative drift and non-convolution equivalent increments," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1504-1512.
    18. Sgibnev, M. S., 1997. "Submultiplicative moments of the supremum of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 377-383, April.
    19. 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.
    20. Mihalis G. Markakis & Eytan Modiano & John N. Tsitsiklis, 2018. "Delay Analysis of the Max-Weight Policy Under Heavy-Tailed Traffic via Fluid Approximations," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 460-493, May.
    21. M. S. Sgibnev, 1998. "On the Asymptotic Behavior of the Harmonic Renewal Measure," Journal of Theoretical Probability, Springer, vol. 11(2), pages 371-382, April.
    22. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    23. Sgibnev, M. S., 2001. "On the exact asymptotic behaviour of the distribution of the supremum in the "critical" case," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 357-362, October.
    24. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 299-325, December.
    25. Barbe, Ph. & McCormick, W.P. & Zhang, C., 2007. "Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1835-1847, December.

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