The power law distribution in the geometrically progressing system

Chol-Jun, Kim

The power law distribution is ubiquitous and its mechanism is various. We find a general mechanism for the distribution. If we suppose the normal distributions for a growth coefficient and a number of growth (or the duration of growth) and a correlation between them in a geometrically progressing system, the distribution of the system can reach approximately a log - completely quadratic of chi distribution with 1 degree of freedom (log-CQ$\chi_1$), which is asymptotically a power law distribution. Furthermore, the asymptotic exponent of the power law is inversely proportional to the product of standard deviations of both the number of growth and the growth coefficient and their correlation but independent on an initial distribution of the system. The mechanism shows a comprehensiveness to be involved in wide practice.