A Lévy flight, named after the French mathematician Paul Pierre Lévy, is a type of random walk in which the probability of each step being a particular length is distributed according to a "heavy tail" distribution.
A heavy tail distribution is a probability distribution which falls to zero as 1/|x|α+1 where 0 < α < 2 and therefore has an infinite variance. According to the central limit theorem, if the distribution were to have a finite variance, then after a large number of steps, the distance from the origin of the random walk would tend to a normal distribution. (This type of random walk is also known as Brownian motion). In contrast, if the distribution is heavy-tailed, then after a large number of steps, the distance from the origin of the random walk will tend to a Lévy distribution. Lévy flight is part of a class of Markov processes.
Two-dimensional Lévy flights were described by Benoît Mandelbrot in The Fractal Geometry of Nature. The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering.
This method of simulation stems heavily from the mathematics related to chaos theory and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomenon. Examples include earthquake data analysis, stock analysis, cryptography, signals analysis as well as many applications in astronomy and biology.
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