SEVERAL FORMS OF STOCHASTIC INTEGRAL REPRESENTATIONS OF
GAMMA RANDOM VARIABLES AND RELATED TOPICS
TAKAHIRO AOYAMA
MAKOTO MAEJIMA
YOHEI UEDA
Abstract: 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.
2000 AMS Mathematics Subject Classification: Primary: 60E07; Secondary:
62E10.
Keywords and phrases: Infinitely divisible distribution, gamma distribution, stochastic
integral representation, Lévy process.