Heavy-Tailed Probability Distributions: Some Examples of Their Appearance

Abstract

Heavy-tailed distributions, such as Pareto's law and Lotka's law, are characterized by their propensity to produce extreme values more frequently than distributions with lighter tails, like the normal distribution. In social sciences, these distributions often describe phenomena where a small number of occurrences account for a large proportion of the effect, such as wealth distribution or scientific productivity. The authors provide two illustrative examples using toy models to demonstrate the emergence of these distributions: Wealth Distribution: The study revisits Pareto's observation from 1896, which noted that a small fraction of the population controls a large portion of total wealth. The authors show that the Pareto distribution can naturally arise as a limit distribution for the product of a random number of independent, identically distributed positive random variables. This finding underscores that heavy-tailed distributions can result from purely random processes without the need for specific dependencies among variables. Scientific Productivity: The authors examine the distribution of scientific output, specifically the number of publications per scientist. They highlight Lotka's law, which suggests that the number of scientists producing n papers is inversely proportional to n². This heavy-tailed behavior indicates that a small number of scientists contribute a large portion of total publications. These examples emphasize that heavy-tailed distributions can emerge from random processes in social phenomena, leading to significant disparities such as wealth concentration and unequal scientific output. Understanding these distributions is crucial for accurately modeling and analyzing such phenomena. We provide two examples of the appearance of heavy-tailed distributions in social sciences applications. Among these distributions are the laws of Pareto and Lotka and some new ones. The examples are illustrated through the construction of suitable toy models.

Key Questions

What are heavy-tailed probability distributions?

Heavy-tailed probability distributions are statistical distributions with tails that are not exponentially bounded. This means they have a higher likelihood of producing extreme values compared to light-tailed distributions.

Which laws are examples of heavy-tailed distributions?

Pareto's law and Lotka's law are classic examples of heavy-tailed distributions. These laws describe phenomena where a small number of occurrences account for a large proportion of the effect.

How do heavy-tailed distributions appear in social sciences?

The study provides examples illustrating the appearance of heavy-tailed distributions in social sciences through the construction of suitable toy models, demonstrating how such distributions can emerge from random processes.

Why is understanding heavy-tailed distributions important?

Understanding heavy-tailed distributions is crucial because they often model real-world phenomena more accurately than light-tailed distributions, especially in fields like finance, insurance, and social sciences where extreme events have significant impacts.