Rain process models and convergence to point processes

Abstract. A variety of stochastic models have been used to describe time series of precipitation or rainfall. Since many of these stochastic models are simplistic, it is desirable to develop connections between the stochastic models and the underlying physics of rain. Here, convergence results are presented for such a connection between two stochastic models: (i) a stochastic moisture process as a physics-based description of atmospheric moisture evolution, and (ii) a point process for rainfall time series as spike trains. The moisture process has dynamics that switch after the moisture hits a threshold, which represents the onset of rainfall and thereby gives rise to an associated 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, and, in the limit of short wet periods, the rainfall process converges to a point process that is a spike train. Also, in the limit, the underlying moisture process becomes a threshold model with a teleporting boundary condition. To establish these limits and connections, formal asymptotic convergence is shown using the Fokker-Planck equation, which provides some intuitive understanding. Also, rigorous convergence is proved 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.