### Abstract

We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon.co.jp). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time-dependent distribution. A core of the proof is the law of large numbers for dependent random variables.

Original language | English |
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Pages (from-to) | 966-979 |

Number of pages | 14 |

Journal | Stochastic Processes and their Applications |

Volume | 119 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 Mar 1 |

Externally published | Yes |

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### Keywords

- Dependent random variables
- Hydrodynamic limit
- Law of large numbers
- Stochastic ranking process

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics