Existence of an infinite particle limit of stochastic ranking process

TL;DR

This paper proves that in an infinite particle limit, the stochastic ranking process converges to a deterministic distribution, aligning with observed web ranking data, and establishes a law of large numbers for dependent variables.

Contribution

It demonstrates the convergence of a stochastic ranking model to a deterministic limit and introduces a law of large numbers for dependent random variables.

Findings

Empirical distribution converges to a deterministic space-time distribution.

Random particle trajectories become deterministic in the infinite limit.

Model aligns with real-world ranking data.

Abstract

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.