### 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.

Original language | English |
---|---|

Pages (from-to) | 99-118 |

Number of pages | 20 |

Journal | Probability and Mathematical Statistics |

Volume | 31 |

Issue number | 1 |

Publication status | Published - 2011 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Probability and Mathematical Statistics*,

*31*(1), 99-118.

**Several forms of stochastic integral representations of gamma random variables and related topics.** / Aoyama, Takahiro; Maejima, Makoto; Ueda, Yohei.

Research output: Contribution to journal › Article

*Probability and Mathematical Statistics*, vol. 31, no. 1, pp. 99-118.

}

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

Y1 - 2011

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

UR - http://www.scopus.com/inward/record.url?scp=79960752875&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960752875&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79960752875

VL - 31

SP - 99

EP - 118

JO - Probability and Mathematical Statistics

JF - Probability and Mathematical Statistics

SN - 0208-4147

IS - 1

ER -