Lévyfördelning

Lévyfördelning, often referred to as a Lévy distribution, is a family of probability distributions characterized by heavy tails. Unlike the normal distribution, which decays exponentially, Lévy distributions exhibit power-law decay, meaning extreme values occur more frequently. This property makes them suitable for modeling phenomena where sudden, large jumps or fluctuations are common.

The defining characteristic of a Lévy distribution is that it satisfies the generalized central limit theorem. This theorem states that the sum of independent and identically distributed random variables, when properly normalized, can converge to a Lévy distribution, even if the individual variables do not have finite variance. This is a significant departure from the classical central limit theorem, which requires finite variance for convergence to the normal distribution.

A key parameter of a Lévy distribution is its stability. Stable distributions are those for which a scaled and shifted version of the distribution is identical to the original distribution. Lévy distributions are a subset of stable distributions. The normal distribution is also a stable distribution, but it is a special case with an exponent of 2. Other Lévy distributions have exponents between 0 and 2.

The Cauchy distribution and the normal distribution are specific examples of Lévy distributions. The Cauchy distribution is a stable distribution with an exponent of 1. The normal distribution is a stable distribution with an exponent of 2. Lévy distributions with exponents between 0 and 2 are sometimes called strictly stable distributions.

Lévy distributions find applications in various fields, including finance, where they can model asset price movements exhibiting fat tails and volatility clustering. They are also used in physics to describe anomalous diffusion processes and in other areas where extreme events are prevalent.