Probability density function

The general form of the centered Kaniadakis κ-Gaussian probability density function is:^[3]

${\displaystyle f_{_{\kappa }}(x)=Z_{\kappa }\exp _{\kappa }(-\beta x^{2})}$

where ${\displaystyle |\kappa |<1}$ is the entropic index associated with the Kaniadakis entropy, ${\displaystyle \beta >0}$ is the scale parameter, and

${\displaystyle Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi }}}{\Bigg (}1+{\frac {1}{2}}\kappa {\Bigg )}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}}$

is the normalization constant.

The standard Normal distribution is recovered in the limit ${\displaystyle \kappa \rightarrow 0.}$

Cumulative distribution function

The cumulative distribution function of κ-Gaussian distribution is given by

${\displaystyle F_{\kappa }(x)={\frac {1}{2}}+{\frac {1}{2}}{\textrm {erf}}_{\kappa }{\big (}{\sqrt {\beta }}x{\big )}}$

where

${\displaystyle {\textrm {erf}}_{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt}$

is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function ${\displaystyle {\textrm {erf}}(x)}$ as ${\displaystyle \kappa \rightarrow 0}$.

Moments, mean and variance

The centered κ-Gaussian distribution has a moment of odd order equal to zero, including the mean.

The variance is finite for ${\displaystyle \kappa <2/3}$ and is given by:

${\displaystyle \operatorname {Var} [X]=\sigma _{\kappa }^{2}={\frac {1}{\beta }}{\frac {2+\kappa }{2-\kappa }}{\frac {4\kappa }{4-9\kappa ^{2}}}\left[{\frac {\Gamma \left({\frac {1}{2\kappa }}+{\frac {1}{4}}\right)}{\Gamma \left({\frac {1}{2\kappa }}-{\frac {1}{4}}\right)}}\right]^{2}}$

Kurtosis

The kurtosis of the centered κ-Gaussian distribution may be computed thought:

${\displaystyle \operatorname {Kurt} [X]=\operatorname {E} \left[{\frac {X^{4}}{\sigma _{\kappa }^{4}}}\right]}$

which can be written as

${\displaystyle \operatorname {Kurt} [X]={\frac {2Z_{\kappa }}{\sigma _{\kappa }^{4}}}\int _{0}^{\infty }x^{4}\,\exp _{\kappa }\left(-\beta x^{2}\right)dx}$

Thus, the kurtosis of the centered κ-Gaussian distribution is given by:

${\displaystyle \operatorname {Kurt} [X]={\frac {3{\sqrt {\pi }}Z_{\kappa }}{2\beta ^{2/3}\sigma _{\kappa }^{4}}}{\frac {|2\kappa |^{-5/2}}{1+{\frac {5}{2}}|\kappa |}}{\frac {\Gamma \left({\frac {1}{|2\kappa |}}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{|2\kappa |}}+{\frac {5}{4}}\right)}}}$

or

${\displaystyle \operatorname {Kurt} [X]={\frac {3\beta ^{11/6}{\sqrt {2\kappa }}}{2}}{\frac {|2\kappa |^{-5/2}}{1+{\frac {5}{2}}|\kappa |}}{\Bigg (}1+{\frac {1}{2}}\kappa {\Bigg )}\left({\frac {2-\kappa }{2+\kappa }}\right)^{2}\left({\frac {4-9\kappa ^{2}}{4\kappa }}\right)^{2}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}}\right]^{3}{\frac {\Gamma \left({\frac {1}{|2\kappa |}}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{|2\kappa |}}+{\frac {5}{4}}\right)}}}$