Several forms of stochastic integral representations of gamma random variables and related topics

Takahiro Aoyama, Makoto Maejima, Yohei Ueda

Research output: Contribution to journalArticle

4 Citations (Scopus)


Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.

Original languageEnglish
Pages (from-to)99-118
Number of pages20
JournalProbability and Mathematical Statistics
Issue number1
Publication statusPublished - 2011
Externally publishedYes



  • Gamma distribution
  • Infinitely divisible distribution
  • Lévy process
  • Stochastic integral representation

ASJC Scopus subject areas

  • Statistics and Probability

Cite this