### 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 |
---|---|

Pages (from-to) | 359-396 |

Number of pages | 38 |

Journal | Tohoku Mathematical Journal |

Volume | 71 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 Sep |

### Fingerprint

### 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)

### Cite this

**Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities.** / Hattori, Tetsuya.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities

AU - Hattori, Tetsuya

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Complete convergence

KW - Gronwall inequality

KW - Hydrodynamic limit

KW - Last-arrival-time dependent intensity

KW - Law of large numbers

KW - Stochastic ranking process

UR - http://www.scopus.com/inward/record.url?scp=85072990447&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072990447&partnerID=8YFLogxK

U2 - 10.2748/tmj/1568772177

DO - 10.2748/tmj/1568772177

M3 - Article

AN - SCOPUS:85072990447

VL - 71

SP - 359

EP - 396

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

SN - 0040-8735

IS - 3

ER -