Editing Cauchy distribution (section)

===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.