Levy skew alpha-stable Probability density function
Symmetric centered Lévy distributions with unit scale factor
Skewed centered Lévy distributions with unit scale factor | Cumulative distribution function
CDF's for symmetric centered Lévy distributions
CDF's for skewed centered Levy distributions | | Parameters | ![alphain (0,2],](http://upload.wikimedia.org/math/e/2/2/e229ce4c045c082a1eaa7b64458c5c20.png) exponent (real)
![betain [-1,1],](http://upload.wikimedia.org/math/b/2/e/b2ea5d1e50a2ee5b54c112c14c7f534e.png) skewness (real)
 scale (real)
 location (real) Download high resolution version (1300x975, 176 KB) Wikipedia does not have an article with this exact name. ...
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In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ...
| | Support |  (real) | | Probability density function (pdf) | usually not analytically expressible (see text) | | Cumulative distribution function (cdf) | usually not analytically expressible (see text) | | Mean | undefined when α≤1, otherwise μ | | Median | usually not analytically expressible (see text). Equal to μ when β=0 | | Mode | usually not analytically expressible. Equal to μ when β=0 | | Variance | infinite except when α=2, when it is 2c2 | | Skewness | undefined | | Kurtosis | undefined | | Entropy | not analytically expressible (see text) | | mgf | undefined | | Char. func. | ![expleft[~itmu - |c t|^alpha,(1-i beta,mbox{sgn}(t)Phi)~right]](http://upload.wikimedia.org/math/8/2/e/82e5e6afb2d7ac50585a42b92be26fad.png)
 for 
 for  In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...
In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
Entropy of a Bernoulli trial as a function of success probability. ...
In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ...
In probability theory, the characteristic function of any random variable completely defines its probability distribution. ...
| In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. If X1,X2 are stable and independently and identically distributed (iid) and if Y = aX1 + bX2 + c is a linear combination of the two, then Y is of the same type: Y = dX + e. If e = 0 for all a, b and c it is called strictly stable.(Nolan 2005) Probability theory is the mathematical study of probability. ...
Paul Pierre Lévy (September 15, 1886 - December 15, 1971) was a French mathematician who was active especially in probability theory, introduced martingales and Lévy flights. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...
A random variable is a term used in mathematics and statistics. ...
In probability theory and statistics, the stability of a family of probability distributions is an important property which basically states that if you have a number of random variates that are in the family, any linear combination of these variates will also be in the family. Here a family of...
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Levy distributions are found in analysis of critical behavior and financial data. (Voit 2003 § 5.4.3) Benoît Mandelbrot found that changes in cotton prices followed a Lévy distribution with α equal to 1.7. Lévy distributions are also found in spectroscopy as a general expression for a quasistatically pressure-broadened spectral line. (Peach 1981 § 4.5) Benoît Mandelbrot Benoît B. Mandelbrot (born November 20, 1924) is a Polish-born French mathematician and leading proponent of fractal geometry. ...
Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of spectra, that is, the dependence of physical quantities on frequency. ...
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ...
The distribution A Lévy skew stable distribution is specified by scale c, exponent α, shift μ and skewness parameter β. The skewness parameter must lie in the range [−1, 1] and when it is zero, the distribution is symmetric and is referred to as a Lévy symmetric alpha-stable distribution. The exponent α must lie in the range (0, 2]. In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
The Lévy skew stable probability distribution is defined by the Fourier transform of its characteristic function  (Voit 2003 § 5.4.3)(Nolan 2005) The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...
Some mathematicians use the phrase characteristic function synonymously with indicator function. ...
 where is given by: ![varphi(t) = expleft[~itmu!-!|c t|^alpha,(1!-!i beta,textrm{sgn}(t)Phi)~right]](http://upload.wikimedia.org/math/6/1/9/619b1fb223873ae01e24c1cefef87eee.png) where sgn(t) is just the sign of t and Φ is given by Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...
for all α except α = 1 in which case:  μ is a shift parameter, β is a measure of asymmetry, with β=0 yielding a distribution symmetric about μ. c is a scale factor which is a measure of the width of the distribution and α is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution for α < 2. Note that this is only one of the parameterizations in use for stable distributions; it is the most common but is not continuous in the parameters.
The asymptotic behavior is described, for α<2, by:(Nolan, Theorem 1.12)
where Γ is the Gamma function (except that when α<1 and β=1 or -1, the tail vanishes to the left or right, resp., of μ). This "heavy tail" behavior causes the variance of Lévy distributions to be infinite for all α < 2. This property is illustrated in the log-log plots below. The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...
It has been suggested that this article or section be merged with The long tail. ...
 Log-log plot of symmetric centered Levy distribution PDF's showing the power law behavior for large x. The power law behavior is evidenced by the straight-line appearance of the PDF for large x, with the slope equal to -(α+1). (The only exception is for α=2, in black, which is a normal distribution.)
 Log-log plot of skewed centered Levy distribution PDF's showing the power law behavior for large x. Again the slope of the linear portions is equal to -(α+1)
Download high resolution version (1300x975, 148 KB) Wikipedia does not have an article with this exact name. ...
Download high resolution version (1300x975, 148 KB) Wikipedia does not have an article with this exact name. ...
Download high resolution version (1300x975, 204 KB) Wikipedia does not have an article with this exact name. ...
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When α=2, the distribution is Gaussian (see below), with tails asymptotic to exp(-x2/4c2)/(2c√π).
Special cases There is no general analytic solution for the form of p(x). There are, however three special cases which can be analytically expressed as can be seen by inspection of the characteristic function. Some mathematicians use the phrase characteristic function synonymously with indicator function. ...
Other special cases are: Probability density function of Gaussian distribution (bell curve). ...
The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...
In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ...
- In the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function δ(x − μ).
The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...
Stability property (See (Voit 2003 § 5.4.3) and (Nolan 2005)
The Lévy alpha-stable distributions have the "stability" property that if N alpha-stable variates Xi are drawn from the distribution
 then the sum  will also be distributed as an alpha-stable variate,  where  This can be easily proven using the properties of characteristic functions. Some mathematicians use the phrase characteristic function synonymously with indicator function. ...
The generalized central limit theorem Another important property of Lévy distributions is the role that they play in a generalized central limit theorem. The central limit theorem states that the sum of a number of random variables with finite variances will tend to a normal distribution as the number of variables grows. A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 (and therefore having infinite variance) will tend to a stable Levy distribution f(x;α,0,c,0) as the number of variables grows. (Voit 2003 § 5.4.3) Central limit theorems are a set of weak-convergence results in probability theory. ...
The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...
Boris Vladimirovich Gnedenko was a Russian mathematician and a student of Andrey Nikolaevich Kolmogorov. ...
Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ...
Series representation The stable distribution can be restated as the real part of simpler integral:(Peach 1981 § 4.5) ![f(x;alpha,beta,c,mu)=frac{1}{pi}Releft[ int_0^infty e^{it(x-mu)}e^{-(ct)^alpha(1-ibetaPhi)},dtright]](http://upload.wikimedia.org/math/0/e/c/0ecf4bda24d07111a640f696123ac313.png) Expressing the second exponential as a Taylor series, we have: As the degree of the Taylor series rises, it approaches the correct function. ...
![f(x;alpha,beta,c,mu)=frac{1}{pi}Releft[ int_0^infty e^{it(x-mu)}sum_{n=0}^inftyfrac{(-qt^alpha)^n}{n!}right]](http://upload.wikimedia.org/math/f/3/2/f32433d88056521a0e340e7cca010fc3.png) where q = cα(1 − iβΦ). Reversing the order of integration and summation, and carrying out the integration yields: ![f(x;alpha,beta,c,mu)=frac{1}{pi}Releft[ sum_{n=1}^inftyfrac{(-q)^n}{n!}left(frac{1}{i(x-mu)}right)^{alpha n+1}Gamma(alpha n+1)right]](http://upload.wikimedia.org/math/1/c/3/1c3a68f2ecfd2d07213db3f2bbc8e250.png) which will be valid for  and will converge for appropriate values of the parameters. (Note that the n=0 term which yields a delta function in x − μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x − μ which is generally less useful. The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...
See also A Lévy flight, named after the French mathematician Paul Pierre Lévy, is a type of random walk in which the increments are distributed according to a heavy tail distribution. ...
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has stationary independent increments -- this phrase will be explained below. ...
The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...
In probability theory and statistics, the zeta distribution is a discrete probability distribution. ...
Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ...
In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ...
References - GNU Scientific Library - Reference Manual Edition 1.6, for GSL Version 1.6, 27 December 2004
- "The Levy alpha-Stable Distributions." GNU Scientific Library - Reference Manual. URL accessed on December 22, 2005.
- "The Levy skew alpha-Stable Distribution." GNU Scientific Library - Reference Manual. URL accessed on December 22, 2005.
- B. V. Gnedenko and A. N. Kolmogorov (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley.
- Johannes Voit (2003). The Statistical Mechanics of Financial Markets (Texts and Monographs in Physics). Springer-Verlag. ISBN 3540009787.
- "Some improvements in numerical evaluation of symmetric stable density and its derivatives." CIRGE Discussion paper. URL accessed on July 13, 2005.
- John P. Nolan. "Information on stable distributions." URL accessed on December 22, 2005.
- John P. Nolan (January 11, 2005). "Stable Distributions Models for Heavy Tailed Data." (PDF) URL accessed on December 22, 2005.
- John P. Nolan (November 29, 2005). "Bibliography on stable distributions, processes and related topics." (PDF) URL accessed on December 22, 2005.
- I. Ibragimov, Yu. Linnik (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing Groningen, The Netherlands.
- Peach, G. (1981). "Theory of the pressure broadening and shift of spectral lines". Advances in Physics 30 (3): 367-474.
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