This paper proposes new analytical results based on the theory of stochastic processes that establish a theoretical dependence of point rainfall variance on the scale of observation under somewhat general assumptions. It is shown that power law scaling, commonly assumed by analyses and models of point rainfall, cannot hold over all scales of observation and that an inner regime exists for small sampling intervals where the variance is a quadratic function of interval length. A power law scaling regime is shown to exist for larger aggregation intervals, whose characteristics depend on the memory of the continuous rainfall process. The presence of a transition regime between the inner and the scaling regime is also shown and may explain the deviations from a power law scaling behavior in observed rainfall reported in the literature. Furthermore, the application of the theory to rainfall data from a representative variety of climate types shows that fractal and multifractal analysis techniques and models may indeed fail to capture observed rainfall properties over wide ranges of aggregation scales. Finally, correlation structures with short- and long-term memory are considered and different theoretical relationships expressing rainfall variance as a function of aggregation interval are derived and tested against rainfall observed at different locations with aggregation timescales ranging from 15 min to 2 days. The theoretical and observational analyses show that both finite and infinite memory rainfall processes may be observed in nature.