  A moisture process with dynamics that switch after hitting a threshold gives
rise to a rainfall process. This rainfall process is characterized by its
random holding times for dry and wet periods. On average, the holding times for
the wet periods are much shorter than the dry. Here convergence is shown for
the rain fall process to a point process that is a spike train. The underlying
moisture process for the point process is a threshold model with a teleporting
boundary condition. This approximation allows simplification of the model with
many exact formulas for statistics. The convergence is shown by a Fokker-Planck
derivation, convergence in mean-square with respect to continuous functions, of
the moisture process, and convergence in mean-square with respect to
generalized functions, of the rain process.
