The law of large numbers for stochastic rumor models

Abstract

We investigate the generalization of the Maki–Thompson model for the spreading of a rumor through a finite population in which each spreader stops transmitting the rumor right after being involved in $k$ unsuccessful telling interactions. We prove that the proportion of people unaware of the rumor at the end of the process converges in probability to a constant, as the population size goes to $\infty$. The proof relies on an application of the martingale stopping theorem and is based upon the case $k=1$ established by Sudbury (1985), but our approach for proving the convergence is simpler, reducing technicalities.