On the generation of log-Lévy distributions and extreme randomness
The log-normal distribution is prevalent across the sciences, as it emerges from the combination of multiplicative processes and the central limit theorem (CLT). The CLT, beyond yielding the normal distribution, also yields the class of L?vy distributions. The log-L?vy distributions are the L?vy counterparts of the log-normal distribution, they appear in the context of ultraslow diffusion processes, and they are categorized by Mandelbrot as belonging to the class of extreme randomness. In this paper, we present a natural stochastic growth model from which both the log-normal distribution and the log-L?vy distributions emerge universally?the former in the case of deterministic underlying setting, and the latter in the case of stochastic underlying setting. In particular, we establish a stochastic growth model which universally generates Mandelbrot?s extreme randomness.