Lévy distribution

Lévy distributions (named after the French mathematician Paul Lévy ) are a family of probability distributions with the special property of an infinite expected value .

definition

Lévy density functions of various scales and μ = 0

The density function of the Lévy distributions is

${\displaystyle f(x)={\sqrt {\frac {\gamma }{2\pi }}}\cdot {\frac {1}{(x-\mu )^{3/2}}}\cdot \exp \left(-{\frac {\gamma }{2(x-\mu )}}\right),\quad x>\mu }$., with the two parameters . ${\displaystyle \gamma >0,\,\mu \in \mathbb {R} }$

• ${\displaystyle \mu }$is a position parameter and defines the position on the -axis; ${\displaystyle x}$
• ${\displaystyle \gamma }$is a scale parameter (compression for ; stretch for ). ${\displaystyle \gamma <1}$ ${\displaystyle \gamma >1}$

Standard Lévy distribution

The standard Lévy distribution is the Lévy distribution with the parameter values ; its density function is thus: ${\displaystyle \gamma =1,\mu =0}$

${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \cdot x^{3}}}}\cdot e^{-{\frac {1}{2x}}},\quad x>0}$.

properties

The standard Lévy distribution (like the normal distribution and the Cauchy distribution ) belongs to the superordinate family of alpha-stable distributions , i.e. i.e. it fulfills the condition:

${\displaystyle (X_{1}+X_{2}+\dotsb +X_{n})\sim n^{1/\alpha }X}$

(here with ) for all independent standard Lévy-distributed random variables . Since the theory of stable distributions was significantly shaped by Lévy, one often speaks of the actual Lévy distribution in order to avoid confusion . ${\displaystyle \alpha =1/2}$ ${\displaystyle X_{1},X_{2},\ldots ,X_{n},X}$ ${\displaystyle \alpha }$

Moments

The Lévy distribution has no finite expectation, because it holds . The Lévy distribution thus belongs to the distributions with heavy margins , which are mainly used to model extreme events (e.g. a stock market crash in financial mathematics). ${\displaystyle \operatorname {E} (|X|)=\infty }$

application

The Lévy distribution can be used to describe various phenomena, particularly in nature:

• Brownian motion
• Course of stock exchange prices
• Polarity reversal of the earth's magnetic field

Individual evidence

1. ↑ ^{a b} Applebaum, D .: Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes (PDF; 282 kB) University of Sheffield. Pp. 37-53. July 22, 2010. Retrieved June 13, 2014.
2. ↑ Belle Dumé: Geomagnetic flip may not be random after all . In: physicsworld.com . March 21, 2006. Retrieved June 13, 2014.