### Abstract

We consider the stochastic ranking process with space-time dependent unbounded jump rates for the particles. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution in the infinite particle limit. We assume topology of weak convergence for the space of distributions, which implies that the fluctuations among particles with different jump rates cancel in the limit. The results are proved by first finding an auxiliary stochastic ranking process, for which a strong law of large numbers is applied, and then applying a multi time recursive Gronwall's inequality. The limit has a representation in terms of non-Markovian processes which we call point processes with last-arrival-time dependent intensities. We also prove the propagation of chaos, i.e., the tagged particle processes also converge almost surely.

Original language | English |
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Pages (from-to) | 359-396 |

Number of pages | 38 |

Journal | Tohoku Mathematical Journal |

Volume | 71 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Sep |

### Keywords

- Complete convergence
- Gronwall inequality
- Hydrodynamic limit
- Last-arrival-time dependent intensity
- Law of large numbers
- Stochastic ranking process

### ASJC Scopus subject areas

- Mathematics(all)