Holtsmark distribution

Holtsmark
Probability density function Symmetric α-stable distributions with unit scale factor; α=1.5 (blue line) represents the Holtsmark distribution
Cumulative distribution function
Parameters	c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R
PDF	expressible in terms of hypergeometric functions; see text
Mean	μ
Median	μ
Mode	μ
Variance	infinite
Skewness	undefined
Ex. kurtosis	undefined
MGF	undefined
CF	$\exp \left[~it\mu \!-\!\|ct\|^{{3/2}}~\right]$

The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter $\alpha$ equal to 3/2 and skewness parameter $\beta$ of zero. Since $\beta$ equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the probability density function is known. However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions.

The Holtsmark distribution has applications in plasma physics and astrophysics.^[1] In 1919, Norwegian physicist J. Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to chaotic motion of charged particles.^[2] It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.^[3]^[4]

Characteristic function

The characteristic function of a symmetric stable distribution is:

\varphi (t;\mu ,c)=\exp \left[~it\mu \!-\!|ct|^{\alpha }~\right],

where $\alpha$ is the shape parameter, or index of stability, $\mu$ is the location parameter, and c is the scale parameter.

Since the Holtsmark distribution has $\alpha =3/2,$ its characteristic function is:^[5]

\varphi (t;\mu ,c)=\exp \left[~it\mu \!-\!|ct|^{{3/2}}~\right].

Since the Holtsmark distribution is a stable distribution with α > 1, $\mu$ represents the mean of the distribution.^[6]^[7] Since β = 0, $\mu$ also represents the median and mode of the distribution. And since α < 2, the variance of the Holtsmark distribution is infinite.^[6] All higher moments of the distribution are also infinite.^[6] Like other stable distributions (other than the normal distribution), since the variance is infinite the dispersion in the distribution is reflected by the scale parameter, c. An alternate approach to describing the dispersion of the distribution is through fractional moments.^[6]

Probability density function

In general, the probability density function, f(x), of a continuous probability distribution can be derived from its characteristic function by:

f(x)={\frac {1}{2\pi }}\int _{{-\infty }}^{\infty }\varphi (t)e^{{-ixt}}\,dt.

Most stable distributions do not have a known closed form expression for their probability density functions. Only the normal, Cauchy and Lévy distributions have known closed form expressions in terms of elementary functions.^[1] The Holtsmark distribution is one of two symmetric stable distributions to have a known closed form expression in terms of hypergeometric functions.^[1] When $\mu$ is equal to 0 and the scale parameter is equal to 1, the Holtsmark distribution has the probability density function:

{\begin{aligned}f(x;0,1)&={1 \over \pi }{\Gamma (5/3)}{}_{2}F_{3}(5/12,11/12;1/3,1/2,5/6;-4x^{6}/729)\\&{}\quad {}-{x^{2} \over 3\pi }{}_{3}F_{4}(3/4,1,5/4;2/3,5/6,7/6,4/3;-4x^{6}/729)\\&{}\quad {}+{7x^{4} \over 81\pi }{\Gamma (4/3)}{}_{2}F_{3}(13/12,19/12;7/6,3/2,5/3;-4x^{6}/729),\end{aligned}}

where ${\Gamma (x)}$ is the gamma function and $\;_{m}F_{n}()$ is a hypergeometric function.^[1]

References

1 2 3 4 Lee, W. H. (2010). Continuous and Discrete Properties of Stochastic Processes (PDF) (PhD thesis). University of Nottingham. pp. 37–39.
↑ Holtsmark, J. (1919). "Uber die Verbreiterung von Spektrallinien". Annalen der Physik. 363 (7): 577–630. Bibcode:1919AnP...363..577H. doi:10.1002/andp.19193630702. Retrieved 2009-01-06.
↑ Chandrasekhar, S.; J. von Neumann (1942). "The Statistics of the Gravitational Field Arising from a Random Distribution of Stars. I. The Speed of Fluctuations". The Astrophysical Journal. 95: 489. Bibcode:1942ApJ....95..489C. doi:10.1086/144420. ISSN 0004-637X. Retrieved 2011-03-01.
↑ Chandrasekhar, S. (1943-01-01). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 1. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1. Retrieved 2011-03-01.
↑ Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Providence, RI: American Mathematical Society. pp. 1, 41. ISBN 978-0-8218-4519-6.
1 2 3 4 Nolan, J. P. (2008). "Basic Properties of Univariate Stable Distributions". Stable Distributions: Models for Heavy Tailed Data (PDF). pp. 3, 15–16. Retrieved 2011-02-06.
↑ Nolan, J. P. (2003). "Modeling Financial Data". In Rachev, S. T. Handbook of Heavy Tailed Distributions in Finance. Amsterdam: Elsevier. pp. 111–112. ISBN 978-0-444-50896-6.

Hummer, D. G. (1986). "Rational approximations for the holtsmark distribution, its cumulative and derivative". Journal of Quantitative Spectroscopy and Radiative Transfer. 36: 1–5. doi:10.1016/0022-4073(86)90011-7.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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