Editing Stable distribution (section)

== One-sided stable distribution and stable count distribution ==
When <math>\alpha < 1</math> and <math>\beta = 1</math>, the distribution is supported on [''μ'', ∞). This family is called '''one-sided stable distribution'''.<ref name="PhysRevLett 1007">{{Cite journal| last1=Penson|first1=K. A.| last2=Górska|first2=K.| date=2010-11-17| title=Exact and Explicit Probability Densities for One-Sided Lévy Stable Distributions| journal=Physical Review Letters| volume=105 | issue=21 | pages=210604 | doi=10.1103/PhysRevLett.105.210604| pmid=21231282| arxiv=1007.0193| bibcode=2010PhRvL.105u0604P | s2cid=27497684}}</ref> Its standard distribution (''μ''&nbsp;=&nbsp;0) is defined as

:<math>L_\alpha(x) = f\left(x;\alpha,1,\cos\left(\frac{\alpha\pi}{2}\right)^{1/\alpha},0\right)</math>, where <math>\alpha < 1.</math>

Let <math>q = \exp(-i\alpha\pi/2)</math>,  its characteristic function is <math> \varphi(t;\alpha) =  \exp\left (- q|t|^\alpha \right ) </math>. Thus the integral form of its PDF is (note: <math>\operatorname{Im}(q)<0</math>)
<math display="block"> \begin{align}
L_\alpha(x)
& = \frac{1}{\pi}\Re\left[ \int_{-\infty}^\infty e^{itx}e^{-q|t|^\alpha}\,dt\right]
\\ & = \frac{2}{\pi} \int_0^\infty e^{-\operatorname{Re}(q)\,t^\alpha}
       \sin(tx)\sin(-\operatorname{Im}(q)\,t^\alpha) \,dt, \text{ or }
\\ & = \frac{2}{\pi} \int_0^\infty e^{-\text{Re}(q)\,t^\alpha}
       \cos(tx)\cos(\operatorname{Im}(q)\,t^\alpha) \,dt .
\end{align}</math>

The double-sine integral is more effective for very small <math> x</math>.

Consider the Lévy sum <math display="inline">Y = \sum_{i=1}^N X_i</math> where <math display="inline">X_i \sim L_\alpha(x)</math>, then ''Y'' has the density <math display="inline">\frac{1}{\nu} L_\alpha \left(\frac{x}{\nu}\right)</math>  where <math display="inline">\nu = N^{1/\alpha}</math>. Set <math display="inline">x = 1</math> to arrive at the '''[[stable count distribution]]'''.<ref name=":4" /> Its standard distribution is defined as

:<math>\mathfrak{N}_\alpha(\nu)=\frac \alpha {\Gamma\left(\frac{1}{\alpha}\right)} \frac1\nu L_\alpha \left(\frac{1}{\nu} \right), \text{ where } \nu > 0 \text{ and } \alpha < 1.</math>

The stable count distribution is the [[conjugate prior]] of the one-sided stable distribution. Its location-scale family is defined as

:<math>\mathfrak{N}_\alpha(\nu;\nu_0,\theta) = \frac \alpha {\Gamma(\frac{1}{\alpha})} \frac{1}{\nu-\nu_0} L_\alpha \left(\frac{\theta}{\nu-\nu_0}\right), \text{ where } \nu > \nu_0</math>, <math>\theta > 0, \text{ and } \alpha < 1.</math>

It is also a one-sided distribution supported on <math>[\nu_0,\infty)</math>. The location parameter <math>\nu_0</math> is the cut-off location, while <math>\theta</math> defines its scale.

When <math display="inline">\alpha = \frac{1}{2}</math>, <math display="inline">L_{\frac{1}{2}}(x)</math> is the [[Lévy distribution]] which is an inverse gamma distribution. Thus <math>\mathfrak{N}_{\frac{1}{2}}(\nu; \nu_0, \theta)</math> is a shifted [[gamma distribution]] of shape 3/2 and scale <math>4\theta</math>,

:<math>\mathfrak{N}_{\frac{1}{2}}(\nu;\nu_0,\theta) = \frac{1}{4\sqrt{\pi}\theta^{3/2}}
(\nu-\nu_0)^{1/2} e^{-\frac{\nu-\nu_0}{4\theta}}, \text{ where } \nu > \nu_0, \qquad \theta > 0.</math>

Its mean is <math>\nu_0 + 6\theta</math> and its standard deviation is <math>\sqrt{24}\theta</math>. It is hypothesized that [[VIX]] is distributed like <math display="inline">\mathfrak{N}_{\frac{1}{2}}(\nu;\nu_0,\theta)</math> with <math>\nu_0 = 10.4</math> and <math>\theta = 1.6</math> (See Section 7 of <ref name=":4" />). Thus the [[stable count distribution]] is the first-order marginal distribution of a volatility process. In this context, <math>\nu_0</math> is called the "floor volatility".

Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of <ref name=":4" />)

:<math>\int_0^\infty e^{-z x} L_\alpha(x) dx = e^{-z^\alpha}, \text{ where } > \alpha<1. </math>

Let <math>x = 1 / \nu</math>, and one can decompose the integral on the left hand side as a [[product distribution]] of a standard [[Laplace distribution]] and a standard stable count distribution,

:<math>\int_0^\infty \frac{1}{\nu} \left ( \frac{1}{2} e^{-\frac{|z|}{\nu} }\right )
\left (\frac{\alpha}{\Gamma(\frac{1}{\alpha})}  \frac{1}{\nu} L_\alpha \left(\frac{1}{\nu}\right) \right ) \, d\nu
= \frac{1}{2} \frac{\alpha}{\Gamma(\frac{1}{\alpha})} e^{-|z|^\alpha}, \text{ where } \alpha<1. </math>

This is called the "lambda decomposition" (See Section 4 of <ref name=":4" />) since the right hand side was named as "symmetric lambda distribution" in Lihn's former works. However, it has several more popular names such as "[[exponential power distribution]]", or the "generalized error/normal distribution", often referred to when <math>\alpha > 1</math>.

The n-th moment of <math>\mathfrak{N}_\alpha(\nu)</math> is the <math>-(n + 1)</math>-th moment of <math>L_\alpha(x)</math>, and all positive moments are finite.