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Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

Author

Listed:
  • Søren Asmussen

    (Lund University)

  • Serguei Foss

    (Sobolev Institute of Mathematics
    Heriot-Watt University)

  • Dmitry Korshunov

    (Sobolev Institute of Mathematics
    Heriot-Watt University)

Abstract

We study distributions F on [0,∞) such that for some T ≤ ∞, F *2(x, x+T] ∼ 2F(x, x+T]. The case T = ∞ corresponds to F being subexponential, and our analysis shows that the properties for T

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:2:d:10.1023_a:1023535030388
    DOI: 10.1023/A:1023535030388
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Asmussen, Søren & Kalashnikov, Vladimir & Konstantinides, Dimitrios & Klüppelberg, Claudia & Tsitsiashvili, Gurami, 2002. "A local limit theorem for random walk maxima with heavy tails," Statistics & Probability Letters, Elsevier, vol. 56(4), pages 399-404, February.
    4. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    5. 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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    Citations

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    Cited by:

    1. Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    2. Toshiro Watanabe & Kouji Yamamuro, 2017. "Two Non-closure Properties on the Class of Subexponential Densities," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1059-1075, September.
    3. Toshiro Watanabe, 2022. "Embrechts–Goldie’s Problem on the Class of Lattice Convolution Equivalent Distributions," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2622-2642, December.
    4. Jaap Geluk & Qihe Tang, 2009. "Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables," Journal of Theoretical Probability, Springer, vol. 22(4), pages 871-882, December.
    5. 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.
    6. Toshiro Watanabe, 2022. "Second-Order Behaviour for Self-Decomposable Distributions with Two-Sided Regularly Varying Densities," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1343-1366, June.
    7. Jianxi Lin, 2012. "Second order Subexponential Distributions with Finite Mean and Their Applications to Subordinated Distributions," Journal of Theoretical Probability, Springer, vol. 25(3), pages 834-853, September.
    8. Yang Yang & Shuang Liu & Kam Chuen Yuen, 2022. "Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2600-2621, December.
    9. Zhaolei Cui & Yuebao Wang & Hui Xu, 2022. "Local Closure under Infinitely Divisible Distribution Roots and Esscher Transform," Mathematics, MDPI, vol. 10(21), pages 1-24, November.
    10. Xu, Wei, 2023. "Asymptotics for exponential functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 1-42.
    11. Yang Yang & Xinzhi Wang & Shaoying Chen, 2022. "Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1221-1236, June.

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