Editing Stable distribution (section)

===The distribution===
A stable distribution is therefore specified by the above four parameters. It can be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function.<ref name=":0" /> If <math> f(x; \alpha, \beta, c, \mu) </math> denotes the density of ''X'' and ''Y'' is the sum of independent copies of ''X'':
<math display="block"> Y = \sum_{i = 1}^N k_i (X_i - \mu)</math>
then ''Y'' has the density <math> \tfrac{1}{s} f(y / s; \alpha, \beta, c, 0) </math> with
<math display="block"> s = \left(\sum_{i = 1}^N |k_i|^\alpha \right )^{\frac{1}{\alpha}} </math>

The asymptotic behavior is described, for <math>\alpha < 2</math>, by:<ref name=":0" />
<math display="block"> f(x) \sim \frac{1}{|x|^{1 + \alpha}} \left (c^\alpha (1 + \sgn(x) \beta) \sin \left (\frac{\pi \alpha}{2} \right ) \frac{\Gamma(\alpha + 1) }{\pi} \right ) </math>
where Γ is the [[Gamma function]] (except that when <math>\alpha \geq 1</math> and <math>\beta  = \pm 1</math>, the tail does not vanish to the left or right, resp., of ''μ'', although the above expression is 0). This "[[heavy tail]]" behavior causes the variance of stable distributions to be infinite for all <math>\alpha <2</math>. This property is illustrated in the log–log plots below.

When <math>\alpha = 2</math>, the distribution is Gaussian (see below), with tails asymptotic to exp(−''x''<sup>2</sup>/4''c''<sup>2</sup>)/(2''c''{{radic|{{pi}}}}).