Existence of an infinite particle limit of stochastic ranking process

Details Existence of an infinite particle limit of stochastic ranking process Existence of an infinite particle limit of stochastic ranking process

2008-04-02

arxiv.org/abs/0804.0321v2

Stochastic Processes and their Applications 119 (2009) 966-979

Mathematics

Probability

Scientific paper

10.1016/j.spa.2008.05.006

We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon). 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 {\it dependent} random variables.

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