How to Define Mean, Variance, etc., for Heavy-Tailed Distributions: A Fractal-Motivated Approach

In many practical situations, we encounter heavy-tailed distributions for which the variance – and even sometimes the mean – are infinite. We propose a fractal-based approach that enables us to gauge the mean and variance of such distributions. 1 Formulation of the Problem There are many practical situations in which the probability distribution is drastically different from normal. In many such situations, the variance is infinite; such distributions are called heavy-tailed. These distributions surfaced in the 1960s, when Benoit Mandelbrot, the author of fractal theory, empirically studied the fluctuations and showed [11] that large-scale fluctuations follow the Pareto power-law distribution, with the probability density function ρ(x) = A ·x−α for x ≥ x0, for some constants α ≈ 2.7 and x0. For this distribution, variance is infinite. The above empirical result, together with similar empirical discovery of heavy-tailed laws in other application areas, has led to the formulation of fractal theory; see, e.g., [12, 13]. Since then, similar heavy-tailed distributions have been empirically found in other financial situations [2, 3, 4, 7, 14, 16, 17, 20, 21, 22, 23], and in many other application areas [1, 8, 12, 15, 19]. For heavy-tailed distributions, variance is infinite, so we cannot use variance to describe the deviation from the “average”. Thus, we need to come up with other characteristics for describing this deviation. This situation is typical in financial and economic applications, where this deviation is known as volatility. At first, economists followed a natural idea to use standard deviation as a quantitative measure of volatility. However, since the empirical distribution is heavy-tailed, its standard deviation is infinite, so