Central limit theorems of occupation times of high-dimensional normalized binary contact path processes

TL;DR

This paper proves a central limit theorem for the occupation times of a normalized binary contact process on high-dimensional lattices, showing convergence to Brownian motion under certain conditions.

Contribution

It establishes the convergence in distribution of occupation times of NBCPP to Brownian motion in high dimensions, extending understanding of epidemic spread models.

Findings

Convergence to Brownian motion in high dimensions

Requires large lattice dimension and infection rate

Initial distribution must be invariant

Abstract

The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with the central limit theorem of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We show that the centred occupation time process of the NBCPP converges in finite dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with a particular invariant distribution.