Cauchy distribution ($\mathrm{Cauchy}(x_0,\gamma )$)

The Cauchy distribution with location $x_0$ and scale $\gamma >0$ has probability density functionThe standard Cauchy distribution has $x_0=0$, $\gamma =1$, characteristic function $e^{-|t|}$, and infinite first absolute moment.

The standard Cauchy distribution is $\mathrm{Cauchy}(0,1)$, with density $1/[\pi (1+x^2)]$ and characteristic function $e^{-|t|}$.

For independent Brownian motions $W,B$, the strong law for Brownian motion makes ${\int }_0^{\infty }{e}^{2W_s-s}ds$ finite almost surely, so bracket localization gives an almost sure limit of the stochastic integral. The arctangent transform of a two-noise affine diffusion and endpoint convergence of a bounded angle diffusion identify its distribution function as $1/2+\arctan (x)/\pi$. Conditional Gaussianity rules out atoms.

For $X\sim \mathrm{Cauchy}(x_0,\gamma )$,In particular, the Standard Cauchy distribution has characteristic function $e^{-|t|}$.

A Cauchy random variable is a random variable having a Cauchy distribution. For a standard Cauchy random variable $V$,

If $X_1,\dots ,X_n$ are independent standard Cauchy random variables, thenis again standard Cauchy. Indeed, its characteristic function is $(e^{-|t|/n}{)}^n=e^{-|t|}$.

A probability distribution is stable when every linear combination of independent copies has the same distribution up to a location and scale change. The Cauchy distribution is stable with stability parameter one.

A probability distribution is strictly stable with index $\alpha$ if $n^{-1/\alpha }(T^{(1)}+\cdots +T^{(n)})\stackrel{d}{=}T$ for every positive integer $n$, for independent copies. Here no centering term is allowed, unlike in the general definition of a stable distribution.