The limiting behavior of some infinitely divisible exponential dispersion models

Shaul K. Bar-Lev, Gérard Letac

Research output: Contribution to journalArticlepeer-review


Consider an exponential dispersion model (EDM) generated by a probability μ on [0,∞) which is infinitely divisible with an unbounded Lévy measure v. The Jørgensen set (i.e., the dispersion parameter space) is then R+, in which case the EDM is characterized by two parameters: θ0, the natural parameter of the associated natural exponential family, and the Jørgensen (or dispersion) parameter, t. Denote the corresponding distribution by EDM(θ0,t) and let Yt be a r.v. with distribution EDM(θ0,t). Then for v((x,∞))~-ℓlogx around zero, we prove that the limiting law F0 of Yt-t as t→0 is a Pareto type law (not depending on θ0) with the form F0(u)=0 for u<1 and the form 1-u-ℓ for u≥1. This result enables an approximation of the distribution of Yt to be found for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.

Original languageEnglish
Pages (from-to)1870-1874
Number of pages5
JournalStatistics and Probability Letters
Issue number23-24
StatePublished - Dec 2010


  • Exponential dispersion model
  • Infinitely divisible distributions
  • Limiting distributions
  • Natural exponential family

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'The limiting behavior of some infinitely divisible exponential dispersion models'. Together they form a unique fingerprint.

Cite this