The limit distributions of growth rate fluctuation of complex systems: An application to business firms

We consider statistical properties of aggregated systems consisting of randomly growing subunits which are governed by random multiplicative growth. The system exhibits typical features of complex systems such as power law system size distributions, fat-tailed growth rate distributions and nontrivial slow shrink of variance of growth rates as a function of the number of subunits. There are extraordinary cases in which variances of growth rates remain finite even in the limit of infinite numbers of sum of independent subunits having identical statistics. This result can be viewed as a generalization of the central limit theorem to the case of growth rates. As an example of real-world phenomena we analyze the business firm data of about 1 million Japanese firms and show that all features are consistent with our theory.