Existence of an infinite particle limit of stochastic ranking process

Kumiko Hattori, Tetsuya Hattori

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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 languageEnglish
Pages (from-to)966-979
Number of pages14
JournalStochastic Processes and their Applications
Issue number3
Publication statusPublished - 2009 Mar
Externally publishedYes


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


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