Taylor's law and heavy-tailed distributions - pnas.org

Lionel Roy Taylor’s law of the mean (TLM), also known as Taylor’s law (1⇓–3), can be described succinctly as

for a sample of positive elements. Careful consideration of TLM (2, 4, 5) raises two unanswered fundamental questions in natural and statistical science.

• 1) Is TLM a unique phenomenon in nature, or should alternative extensions and refinements of TLM, exploiting higher moments and various measures for dependence (association), be explored?

• 2) What are the proper measures for the location, spread, asymmetry, and dependence (association) for random samples with infinite mean?

Applying non-Gaussian (stable) limit theorems, Cohen et al. (4) show that the limiting distribution of exists even if the sample is drawn from a distribution with support on having infinite mean. A similar limiting result was obtained by Drton and Xiao (6) and Pillai and Meng (7) with the Cauchy distribution as the limiting distribution. To describe how truly bewildering these results are, consider the following scenario from finance. Take a long-only portfolio consisting of risky assets, with a one-period, random portfolio return given by

source: https://www.pnas.org/content/118/50/e2118893118

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