A large deviations principle for the Maki-Thompson rumour model

TL;DR

This paper establishes a large deviations principle for the Maki-Thompson rumour model, providing an explicit rate function that describes the probabilities of rare events in the rumour propagation process.

Contribution

It introduces a large deviations principle for the model, complementing existing results on convergence and normality, with an explicit rate function.

Findings

Proves a large deviations principle for the rumour model.

Provides an explicit formula for the rate function.

Complements previous asymptotic normality results.

Abstract

We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson. Sudbury established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to $0.2032$. Watson later derived the asymptotic normality of a suitably scaled version of this proportion. We prove a corresponding large deviations principle, with an explicit formula for the rate function.