### Abstract

We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson timesof jumps to the origin.

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

Pages (from-to) | 571-607 |

Number of pages | 37 |

Journal | Alea |

Volume | 9 |

Issue number | 2 |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

- Hydrodynamic limit
- Inviscid Burgers equation
- Move-to-front rules.
- Poisson process
- Stochastic ranking process

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Alea*,

*9*(2), 571-607.

**Stochastic ranking process with space-time dependent intensities.** / Hattori, Tetsuya; Kusuoka, Seiichiro.

Research output: Contribution to journal › Article

*Alea*, vol. 9, no. 2, pp. 571-607.

}

TY - JOUR

T1 - Stochastic ranking process with space-time dependent intensities

AU - Hattori, Tetsuya

AU - Kusuoka, Seiichiro

PY - 2012

Y1 - 2012

N2 - We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson timesof jumps to the origin.

AB - We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson timesof jumps to the origin.

KW - Hydrodynamic limit

KW - Inviscid Burgers equation

KW - Move-to-front rules.

KW - Poisson process

KW - Stochastic ranking process

UR - http://www.scopus.com/inward/record.url?scp=84877072809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877072809&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84877072809

VL - 9

SP - 571

EP - 607

JO - Alea

JF - Alea

SN - 1980-0436

IS - 2

ER -