Editing Cauchy distribution (section)

==Related distributions==
===General===
*<math>\operatorname{Cauchy}(0,1) \sim \textrm{t}(\mathrm{df}=1)\,</math> [[Student's t distribution|Student's ''t'' distribution]]
*<math>\operatorname{Cauchy}(\mu,\sigma) \sim \textrm{t}_{(\mathrm{df}=1)}(\mu,\sigma)\,</math> [[Student's t distribution#location-scale|non-standardized Student's ''t'' distribution]]
*If <math>X, Y \sim \textrm{N}(0,1)\, X, Y</math> independent, then <math> \tfrac X Y\sim \textrm{Cauchy}(0,1)\,</math>
*If <math>X \sim \textrm{U}(0,1)\,</math> then <math> \tan \left( \pi \left(X-\tfrac{1}{2}\right) \right) \sim \textrm{Cauchy}(0,1)\,</math>
*If <math>X \sim \operatorname{Log-Cauchy}(0, 1)</math> then <math>\ln(X) \sim \textrm{Cauchy}(0, 1)</math>
*If <math>X \sim \operatorname{Cauchy}(x_0,\gamma)</math>  then <math>\tfrac1X \sim \operatorname{Cauchy}\left(\tfrac{x_0}{x_0^2+\gamma^2},\tfrac{\gamma}{x_0^2+\gamma^2}\right)</math> 
*The Cauchy distribution is a limiting case of a [[Pearson distribution]] of type 4{{Citation needed|date=March 2011}}
*The Cauchy distribution is a special case of a [[Pearson distribution]] of type 7.<ref name=jkb1/>
*The Cauchy distribution is a [[stable distribution]]: if <math>X \sim \textrm{Stable}(1, 0, \gamma, \mu)</math>, then <math>X \sim \operatorname{Cauchy}(\mu, \gamma)</math>.
*The Cauchy distribution is a singular limit of a [[hyperbolic distribution]]{{Citation needed|date=April 2011}}
*The [[wrapped Cauchy distribution]], taking values on a circle, is derived from the Cauchy distribution by wrapping it around the circle.
*If <math>X \sim \textrm{N}(0,1)</math>, <math>Z \sim \operatorname{Inverse-Gamma}(1/2, s^2/2)</math>, then <math>Y = \mu + X \sqrt Z \sim \operatorname{Cauchy}(\mu,s)</math>. For half-Cauchy distributions, the relation holds by setting <math>X \sim \textrm{N}(0,1) I\{X\ge0\}</math>.

=== Lévy measure ===
The Cauchy distribution is the [[stable distribution]] of index 1. The [[Lévy process#L.C3.A9vy.E2.80.93Khintchine representation|Lévy–Khintchine representation]] of such a stable distribution of parameter <math> \gamma </math> is given, for <math> X \sim \operatorname{Stable}(\gamma, 0, 0)\,</math> by:

<math display="block">\operatorname{E}\left( e^{ixX} \right) = \exp\left( \int_{ \mathbb{R} } (e^{ixy} - 1) \Pi_\gamma(dy) \right)</math>

where

<math display="block">\Pi_\gamma(dy) = \left( c_{1, \gamma}  \frac{1}{y^{1 + \gamma}} 1_{ \left\{y > 0\right\} } + c_{2,\gamma}  \frac{1}{|y|^{1 + \gamma}} 1_{\left\{ y < 0 \right\}} \right) \, dy 
</math>

and <math> c_{1, \gamma}, c_{2, \gamma} </math> can be expressed explicitly.<ref>{{cite book |author=Kyprianou, Andreas |year=2009 |title=Lévy processes and continuous-state branching processes:part I |page=11 |url=http://www.maths.bath.ac.uk/~ak257/LCSB/part1.pdf |access-date=2016-05-04 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303235654/http://www.maths.bath.ac.uk/~ak257/LCSB/part1.pdf |url-status=live }}</ref> In the case <math> \gamma = 1 </math> of the Cauchy distribution, one has <math> c_{1, \gamma} = c_{2, \gamma}  </math>.

This last representation is a consequence of the formula

<math display="block">\pi |x| = \operatorname{PV }\int_{\mathbb{R} \smallsetminus\lbrace 0 \rbrace} (1 - e^{ixy}) \, \frac{dy}{y^2} </math>

===Multivariate Cauchy distribution===
A [[random vector]] <math>X=(X_1, \ldots, X_k)^T</math> is said to have the multivariate Cauchy distribution if every linear combination of its components <math>Y=a_1X_1+ \cdots + a_kX_k</math> has a Cauchy distribution. That is, for any constant vector <math>a\in \mathbb R^k</math>, the random variable <math>Y=a^TX</math> should have a univariate Cauchy distribution.<ref name=ferg2>{{cite journal|last1=Ferguson|first1=Thomas S.|title=A Representation of the Symmetric Bivariate Cauchy Distribution|journal=The Annals of Mathematical Statistics |volume= 33|issue= 4|pages=1256–1266|year=1962 |jstor=2237984|doi=10.1214/aoms/1177704357|url=http://projecteuclid.org/download/pdf_1/euclid.aoms/1177704357|access-date=2017-01-07 |doi-access=free}}</ref>  The characteristic function of a multivariate Cauchy distribution is given by:

<math display="block">\varphi_X(t) =  e^{ix_0(t)-\gamma(t)}, \!</math>

where <math>x_0(t)</math> and <math>\gamma(t)</math> are real functions with <math>x_0(t)</math> a [[homogeneous function]] of degree one and <math>\gamma(t)</math> a positive homogeneous function of degree one.<ref name=ferg2/>  More formally:<ref name=ferg2/>

<math display="block">\begin{align}
x_0(at) &= a x_0(t), \\
\gamma (at) &= |a| \gamma (t),
\end{align}</math>

for all <math>t</math>.

An example of a bivariate Cauchy distribution can be given by:<ref name=bivar>{{cite journal|title=Non-linear Integral Equations to Approximate Bivariate Densities with Given Marginals and Dependence Function|last1=Molenberghs|first1=Geert|last2=Lesaffre|first2=Emmanuel|journal=Statistica Sinica|volume=7|year=1997|pages=713&ndash;738|url=http://www3.stat.sinica.edu.tw/statistica/oldpdf/A7n310.pdf|url-status=dead|archive-url=https://web.archive.org/web/20090914055538/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A7n310.pdf|archive-date=2009-09-14}}</ref>
<math display="block">f(x, y; x_0,y_0,\gamma) = \frac{1}{2 \pi} \, \frac{\gamma}{{\left({\left(x - x_0\right)}^2 + {\left(y - y_0\right)}^2 + \gamma^2\right)}^{3/2}} .</math>
Note that in this example, even though the covariance between <math>x</math> and <math>y</math> is 0, <math>x</math> and <math>y</math> are not [[Independence (probability theory)|statistically independent]].<ref name=bivar/>

We also can write this formula for complex variable. Then the probability density function of complex Cauchy is :

<math display="block">f(z; z_0,\gamma) = \frac{1}{2\pi} \,\frac{\gamma}{{\left({\left|z - z_0\right|}^2 + \gamma^2\right)}^{3/2} }  .</math>

Like how the standard Cauchy distribution is the Student t-distribution with one degree of freedom, the multidimensional Cauchy density is the [[multivariate Student distribution]] with one degree of freedom. The density of a <math>k</math> dimension Student distribution with one degree of freedom is:

<math display="block">f(\mathbf{x}; \boldsymbol{\mu},\mathbf{\Sigma}, k)= \frac{\Gamma{\left(\frac{1+k}{2}\right)}}{\Gamma(\frac{1}{2}) \pi^{\frac{k}{2}} \left|\mathbf{\Sigma}\right|^{\frac{1}{2}} \left[1 + ({\mathbf x}-{\boldsymbol\mu})^\mathsf{T} {\mathbf\Sigma}^{-1} ({\mathbf x}-{\boldsymbol\mu})\right]^{\frac{1+k}{2}}} .</math>

The properties of multidimensional Cauchy distribution are then special cases of the multivariate Student distribution.