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

TL;DR

This paper studies a stochastic ranking process with space-time dependent jump rates, proving that fluctuations among particles cancel in the infinite limit, leading to a deterministic distribution described by non-Markovian processes.

Contribution

It introduces a novel approach to analyze stochastic ranking processes with unbounded, space-time dependent jump rates and establishes almost sure convergence to a deterministic distribution.

Findings

Empirical distribution converges almost surely in the infinite particle limit.

Fluctuations among particles with different jump rates cancel out in the limit.

The limiting distribution is represented by non-Markovian point processes.

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.