Lévy distribution

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Lévy distribution is a continuous probability distribution named after the French mathematician Paul Lévy, who was instrumental in its development. This distribution is a type of stable distribution that is important in various fields such as physics, finance, and economics for modeling data that exhibit heavy tails and skewness. The Lévy distribution is particularly known for its application in the study of random walks and Brownian motion, providing insights into phenomena that exhibit anomalous diffusion.

Definition

The Lévy distribution is defined by two parameters: the location parameter \( \mu \) and the scale parameter \( c \). The scale parameter \( c > 0 \) controls the spread of the distribution, while the location parameter \( \mu \) shifts the distribution along the real line. The probability density function (PDF) of the Lévy distribution for \( x > \mu \) is given by:

\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}} \frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}} \]

The distribution does not have a defined mean or variance for any value of \( c \), which is a characteristic feature of heavy-tailed distributions.

Properties

The Lévy distribution is a special case of the stable distributions with stability parameter \( \alpha = 1/2 \) and skewness parameter \( \beta = 1 \), making it a skewed distribution with heavy tails. The lack of a defined mean or variance means that traditional statistical methods that rely on these moments are not applicable to data modeled by the Lévy distribution.

Applications

The Lévy distribution has been applied in various domains:

- In finance, it is used to model the distribution of returns for certain types of assets, capturing the heavy tails observed in empirical data. - In physics, the distribution describes the step lengths of particles undergoing Lévy flights, which are a form of random walk that allows for long jumps, in contrast to the normal diffusion modeled by Gaussian distributions. - In environmental science, it models the distribution of resources such as food patches or nesting sites, where the Lévy flight behavior of animals searching for these resources can be observed.

See also

• Stable distribution
• Paul Lévy
• Random walk
• Brownian motion
• Anomalous diffusion

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