A controllable model of a random multiplicative process for the entire distribution of population

A random multiplicative process (RMP) is one of the basic models which can generate a power law distribution. Actually, the distribution generated by RMP has two parts, which are closely matched to the head of a log-normal distribution and the tail of a power law distribution. We investigated the relation between shapes of distributions and model variables. By changing input variables, we explained the origin of the cumulative population distributions of municipalities and prefectures in Japan from 1980 to 2006. This controllability of RMP can be applied to a power law distribution in various other fields.