Functional central limit theorem for tagged particle dynamics in stochastic ranking process with space-time dependent intensities

Abstract

. In this paper, we consider a “parabolic” scaling limit of tagged particle dynamics and that of empirical measure of the position of particles for stochastic ranking process with space-time dependent intensities. A stochastic ranking process is driven according to an algorithm for a self-organizing linear list of a finite number of items. We regard this process as a particle system. We fasten a tag to a “particle” (item) and observe the (normalized) motion of the “tagged particle”. We obtain a sum of diffusion processes between each two successive jump time for a “parabolic” scaling limit of tagged particle dynamics. In order to obtain the diffusion process, we have to observe a “parabolic” scaling limit of empirical measure of the position of particles. We also obtain a generalized Ornstein-Uhlenbeck process for a “parabolic” scaling limit of empirical measure of the position of particles.