Journal of Theoretical Probability

, Volume 23, Issue 4, pp 1039–1067 | Cite as

Local Subexponentiality and Self-decomposability

  • Toshiro Watanabe
  • Kouji Yamamuro


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.


Self-decomposability Local subexponentiality Infinite divisibility Convolution equivalence Supremum of a random walk 

Mathematics Subject Classification (2000)

60E07 60G50 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Center for Mathematical SciencesThe University of AizuAizu-WakamatsuJapan
  2. 2.Faculty of EngineeringGifu UniversityGifuJapan

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