TY - JOUR

T1 - Several forms of stochastic integral representations of gamma random variables and related topics

AU - Aoyama, Takahiro

AU - Maejima, Makoto

AU - Ueda, Yohei

PY - 2011/8/1

Y1 - 2011/8/1

N2 - 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.

AB - 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.

KW - Gamma distribution

KW - Infinitely divisible distribution

KW - Lévy process

KW - Stochastic integral representation

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M3 - Article

AN - SCOPUS:79960752875

SN - 0208-4147

VL - 31

SP - 99

EP - 118

JO - Probability and Mathematical Statistics

JF - Probability and Mathematical Statistics

IS - 1

ER -