Editing Cauchy distribution (section)

=== 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>