A note on scaling limits for truncated birth-and-death processes with interaction

TL;DR

This paper investigates the asymptotic behavior of a finite system of interacting birth-and-death processes, demonstrating convergence to a diffusion process as the state space grows large and analyzing conditions for its stationary distribution.

Contribution

It introduces a scaling limit for interacting birth-and-death processes, establishing convergence to a diffusion and providing conditions for stationary distribution existence.

Findings

Scaled Markov chain converges to a diffusion process

Conditions for stationary distribution are derived in special cases

Asymptotic behavior analyzed as state space size increases

Abstract

In this note we consider a Markov chain formed by a finite system of interacting birth-and-death processes on a finite state space. We study an asymptotic behaviour of the Markov chain as its state space becomes large. In particular, we show that the appropriately scaled Markov chain converges to a diffusion process, and derive conditions for existence of the stationary distribution of the limit diffusion process in special cases.