Lévy distribution

The Lévy distribution is a heavy-tailed probability distribution characterized by infinite variance and, for certain parameter values, infinite mean. Unlike the normal distribution, where the central limit theorem guarantees that sums of independent random variables converge to a Gaussian, the Lévy distribution is one of the stable distributions — it is its own attractor under summation. This means that sums of Lévy-distributed variables remain Lévy-distributed, never becoming normal regardless of sample size.

The distribution appears in systems governed by anomalous diffusion, where particles do not spread according to the standard Brownian model but exhibit long-range jumps whose sizes follow a power law. In finance, Lévy processes model price fluctuations with sudden large movements that Gaussian models systematically underestimate. In network science, Lévy flights — random walks with step lengths drawn from a Lévy distribution — have been proposed as search strategies on scale-free networks, where the heavy-tailed step distribution allows the walker to traverse the network's hubs efficiently.

The Lévy distribution is what happens when the central limit theorem fails because the underlying process has no characteristic scale. It is the mathematical voice of systems where outliers are not errors but structural features. Any model that assumes normality for a Lévy process is not just imprecise — it is systematically blind to the events that matter most.