Holtsmark distribution

The characteristic function of a symmetric stable distribution is:

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

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

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

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

Since the Holtsmark distribution is a stable distribution with α > 1, ${\displaystyle \mu }$ represents the mean of the distribution.[6][7] Since β = 0, ${\displaystyle \mu }$ 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]

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

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

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 ${\displaystyle \mu }$ is equal to 0 and the scale parameter is equal to 1, the Holtsmark distribution has the probability density function:

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

where ${\displaystyle {\Gamma (x)}}$ is the gamma function and ${\displaystyle \;_{m}F_{n}()}$ is a hypergeometric function.[1] One has also[8]

${\displaystyle {\begin{aligned}f(x;0,1)&={\frac {-x^{2}}{6\pi }}\left[~_{2}F_{2}\left(1,{\frac {3}{2}};{\frac {4}{3}},{\frac {5}{3}};-{\frac {4ix^{3}}{27}}\right)+~_{2}F_{2}\left(1,{\frac {3}{2}};{\frac {4}{3}},{\frac {5}{3}};{\frac {4ix^{3}}{27}}\right)\right]\\&+{\frac {4}{3\times 3^{2/3}}}\left[\mathrm {Bi} '\left(-{\frac {x^{2}}{3\times 3^{1/3}}}\right)\cos \left({\frac {2x^{3}}{27}}\right)+{\frac {x}{3^{2/3}}}~\mathrm {Bi} \left(-{\frac {x^{2}}{3\times 3^{1/3}}}\right)\sin \left({\frac {2x^{3}}{27}}\right)\right],\end{aligned}}}$

where ${\displaystyle \mathrm {Bi} }$ is the Airy function of the second kind and ${\displaystyle \mathrm {Bi} '}$ its derivative. The arguments of the ${\displaystyle _{2}F_{2}}$ functions are pure imaginary complex numbers, but the sum of the two functions is real. For ${\displaystyle x}$ positive, the function ${\displaystyle \mathrm {Bi} (-x)}$ is related to the Bessel functions of fractional order ${\displaystyle J_{-1/3}}$ and ${\displaystyle J_{1/3}}$ and its derivative to the Bessel functions of fractional order ${\displaystyle J_{-2/3}}$ and ${\displaystyle J_{2/3}}$. Therefore, one can write[8]

${\displaystyle {\begin{aligned}f(x;0,1)&={\frac {4x^{2}}{27{\sqrt {3}}}}\left\{\cos \left({\frac {2x^{3}}{27}}\right)\left[J_{-2/3}\left({\frac {2x^{3}}{27}}\right)+J_{2/3}\left({\frac {2x^{3}}{27}}\right)\right]+\sin \left({\frac {2x^{3}}{27}}\right)\left[J_{-1/3}\left({\frac {2x^{3}}{27}}\right)-J_{1/3}\left({\frac {2x^{3}}{27}}\right)\right]\right\}\\&-{\frac {x^{2}}{6\pi }}\left[~_{2}F_{2}\left(1,{\frac {3}{2}};{\frac {4}{3}},{\frac {5}{3}};-{\frac {4ix^{3}}{27}}\right)+~_{2}F_{2}\left(1,{\frac {3}{2}};{\frac {4}{3}},{\frac {5}{3}};{\frac {4ix^{3}}{27}}\right)\right].\end{aligned}}}$

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