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Log-normal distribution has been reported in a wide range of economic, biological, and sociological systems 17, including income, species abundance, family size, etc. Economists have proposed different kinds of generative mechanisms for log-normal distributions (and power-law distributions as well). One particular interest for us is the multiplicative process 18, 19. Starting from an initial value \({X}_{0}\), random variables in a multiplicative process follow an iterative formula \({X}_{i+1}=\exp ({\nu }_{i}){X}_{i}\) or \(\mathrm{ln}\,{X}_{i+1}=\,\mathrm{ln}\,{X}_{i}+{\nu }_{i}\). If the v i has finite mean and variance, and is independent and identically distributed, then according to the central limit theorem, for large i, ln X i will follow a normal distribution, which means X i will follow a log-normal distribution.

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