A simple consensus model for an increasing population of agents with i.i.d incoming opinions

In this short note we study what happens in a symmetric opinion model when we send the total interacting population $N(t)$ to infinity as $t\to \mathrm{\infty }$. We assume that new population enters the system with opinions that are i.i.d random vectors with finite mean and variance. We give sharp conditions on the rate of population growth that is required for convergence to a global consensus in opinions. More particularly, we show that if the total population increases at a rate $N(t)\sim e^{t^{\alpha }}$, then $\alpha <1$ is necessary and sufficient condition for convergence to the mean of incoming opinions, and the convergence is achieved at an algebraic rate.