### Abstract

Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.

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

Pages (from-to) | 33-50 |

Number of pages | 18 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 32 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- Characteristic function
- Infinitely divisible distribution
- Multinomial distribution
- Zeta function

### ASJC Scopus subject areas

- Applied Mathematics
- Engineering(all)

### Cite this

*Japan Journal of Industrial and Applied Mathematics*,

*32*(1), 33-50. https://doi.org/10.1007/s13160-015-0165-9

**Multinomial distributions in Shintani zeta class.** / Aoyama, Takahiro; Yoshikawa, Kazuhiro.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 32, no. 1, pp. 33-50. https://doi.org/10.1007/s13160-015-0165-9

}

TY - JOUR

T1 - Multinomial distributions in Shintani zeta class

AU - Aoyama, Takahiro

AU - Yoshikawa, Kazuhiro

PY - 2015

Y1 - 2015

N2 - Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.

AB - Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.

KW - Characteristic function

KW - Infinitely divisible distribution

KW - Multinomial distribution

KW - Zeta function

UR - http://www.scopus.com/inward/record.url?scp=84925467439&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925467439&partnerID=8YFLogxK

U2 - 10.1007/s13160-015-0165-9

DO - 10.1007/s13160-015-0165-9

M3 - Article

AN - SCOPUS:84925467439

VL - 32

SP - 33

EP - 50

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -