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Embrechts–Goldie’s Problem on the Class of Lattice Convolution Equivalent Distributions

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  • Toshiro Watanabe

    (The University of Aizu)

Abstract

We show that the class of lattice convolution equivalent distributions is not closed under convolution roots. We prove that the class of lattice convolution equivalent distributions is closed under convolution roots under the assumption of the exponentially asymptotic decreasing condition. This result is extended to the class $$\mathcal {S}_{\Delta }$$ S Δ of $$\Delta $$ Δ -subexponential distributions. As a corollary, we show that the class $$\mathcal {S}_{\Delta }$$ S Δ is closed under convolution roots in the class $$\mathcal {L}_{\Delta }$$ L Δ . Moreover, we prove that the class of lattice convolution equivalent distributions is not closed under convolutions. Finally, we give a survey on the closure under convolution roots of the other distribution classes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01130-4
    DOI: 10.1007/s10959-021-01130-4
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    References listed on IDEAS

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    1. 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.
    2. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    3. Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    4. 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.
    5. Yu, Changjun & Wang, Yuebao & Cui, Zhaolei, 2010. "Lower limits and upper limits for tails of random sums supported on," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1111-1120, July.
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