The Difference-of-Log-Normals Distribution is Fundamental in Nature

Abstract The growth of many natural and social phenomena including pandemics1, firms,2 cities,3 and various economic indices,4,5 is known to be heavy-tailed. Most growth is modest, but we often observe explosive growth rates, such as firms doubling or halving in size within a short period. Neither a simple explanation nor a well-fitting distri10 butional form for these growth phenomena is known. Here we show that a hitherto obscure statistical distribution — the Difference-of-Log-Normals (DLN) — describes a plethora of growth phenomena remarkably well, and discuss why it arises as a natural consequence of the Central Limit Theorem (CLT). Our results demonstrate how growth phenomena subject to opposing random exponential forces are likely to distribute DLN. This provides both a framework for scientifically modeling these phenomena and a simple distributional form to be used when empirically modeling observed heavy-tailed growth. We hence posit that the DLN is a fundamental distribution in nature, in the sense that it emerges in many disparate natural phenomena, especially growth phe19 nomena, similar to the repeated disparate emergence of the Normal and log-Normal distributions.