  This paper analyzes the convergence of the finite population optimal stopping
problem towards the corresponding mean field limit. Building on the viscosity
solution characterization of the mean field optimal stopping problem of our
previous papers [Talbi, Touzi & Zhang 2021 & 2022], we prove the convergence of
the value functions by adapting the Barles-Souganidis [1991] monotone scheme
method to our context. We next characterize the optimal stopping policies of
the mean field problem by the accumulation points of the finite population
optimal stopping strategies. In particular, if the limiting problem has a
unique optimal stopping policy, then the finite population optimal stopping
strategies do converge towards this solution. As a by-product of our analysis,
we provide an extension of the standard propagation of chaos to the context of
stopped McKean-Vlasov diffusions.
