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Local Subexponentiality and Self-decomposability

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  • Toshiro Watanabe

    (The University of Aizu)

  • Kouji Yamamuro

    (Gifu University)

Abstract

The class of exponential tilts of convolution equivalent distributions is determined. As a corollary, the local subexponentiality of one-sided infinitely divisible distributions is characterized. It is applied to the subexponentiality of the densities of a self-decomposable distribution and its Lévy measure. Bondesson’s conjecture on the density of the Lévy measure of a lognormal distribution is solved as an example. Results of Denisov et al. on the distributions of random sums are extended to the two-sided case. Finally, the local subexponentiality of the distribution of the supremum of a random walk is characterized.

Suggested Citation

  • Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:4:d:10.1007_s10959-009-0240-8
    DOI: 10.1007/s10959-009-0240-8
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    References listed on IDEAS

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    1. 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.
    2. Lennart Bondesson, 2002. "On the Lévy Measure of the Lognormal and the LogCauchy Distributions," Methodology and Computing in Applied Probability, Springer, vol. 4(3), pages 243-256, September.
    3. 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.
    4. Ole E. Barndorff‐Nielsen & Neil Shephard, 2003. "Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 277-295, June.
    5. 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.
    6. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    7. 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. 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.
    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.

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