Holtsmark distribution

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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 equal to 3/2 and skewness parameter of zero. Since 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.

Probability density function
Symmetric stable distributions
Symmetric α-stable distributions with unit scale factor; α=1.5 (blue line) represents the Holtsmark distribution
Cumulative distribution function
CDF's for symmetric α-stable distributions; α=3/2 represents the Holtsmark distribution

c ∈ (0, ∞) — scale parameter

μ ∈ (−∞, ∞) — location parameter
Support xR
PDF expressible in terms of hypergeometric functions; see text
Mean μ
Median μ
Mode μ
Variance infinite
Skewness undefined
Ex. kurtosis undefined
MGF undefined

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 functionEdit

The characteristic function of a symmetric stable distribution is:


where   is the shape parameter, or index of stability,   is the location parameter, and c is the scale parameter.

Since the Holtsmark distribution has   its characteristic function is:[5]


Since the Holtsmark distribution is a stable distribution with α > 1,   represents the mean of the distribution.[6][7] Since β = 0,   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 functionEdit

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


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   is equal to 0 and the scale parameter is equal to 1, the Holtsmark distribution has the probability density function:


where   is the gamma function and   is a hypergeometric function.[1] One has also[8]


where   is the Airy function of the second kind and   its derivative. The arguments of the   functions are pure imaginary complex numbers, but the sum of the two functions is real. For   positive, the function   is related to the Bessel functions of fractional order   and   and its derivative to the Bessel functions of fractional order   and  . Therefore, one can write[8]



  1. ^ a b c d Lee, W. H. (2010). Continuous and Discrete Properties of Stochastic Processes (PDF) (PhD thesis). University of Nottingham. pp. 37–39.
  2. ^ Holtsmark, J. (1919). "Uber die Verbreiterung von Spektrallinien". Annalen der Physik. 363 (7): 577–630. Bibcode:1919AnP...363..577H. doi:10.1002/andp.19193630702.
  3. ^ 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.
  4. ^ Chandrasekhar, S. (1943-01-01). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 1–89. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1.
  5. ^ Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Providence, RI: American Mathematical Society. pp. 1, 41. ISBN 978-0-8218-4519-6. holtsmark.
  6. ^ a b c d Nolan, J. P. (2008). "Basic Properties of Univariate Stable Distributions" (PDF). Stable Distributions: Models for Heavy Tailed Data. pp. 3, 15–16. Retrieved 2011-02-06.
  7. ^ Nolan, J. P. (2003). "Modeling Financial Data". In Rachev, S. T. (ed.). Handbook of Heavy Tailed Distributions in Finance. Amsterdam: Elsevier. pp. 111–112. ISBN 978-0-444-50896-6.
  8. ^ a b Pain, Jean-Christophe (2020). "Expression of the Holtsmark function in terms of hypergeometric   and Airy   functions". Eur. Phys. J. Plus. 135: 236. doi:10.1140/epjp/s13360-020-00248-4.