A statistical analysis of mesoscale rainfall as a random cascade

Abstract

The current mathematical foundations of the theory of random cascades involve the study of their ensemble properties and their sample-average properties in space. Recent results have shown that these two are not the same because the spatial law of large numbers does not hold for random cascades due to strong spatial correlations. Two key results from the current literature on the scaling of the statistical moments and the tail probability of the random cascade measures are derived. These ensemble computations only involve the marginal probability distributions of the random measures. By contrast, the sample-average computations involve their spatial-dependence structure and so far have been carried out only under certain boundedness assumptions on the generators of the random measures. -from Authors