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 normalized Cauchy distribution densities $K_a$, $a>0$, form a convolution semigroup: convolution adds their scale parameters. This follows from the convolution theorem, since their Fourier transforms are proportional to $e^{-a|k|}$. Consequently an integral equation with a Cauchy kernel can be solved by subtracting scale parameters when the resulting scale is positive.

If $s>0$ and $\Theta$ has the continuous uniform distribution on $(-\pi /2,\pi /2)$, then $Y=s\tan \Theta$ has the Cauchy distribution with location zero and scale $s$. The tangent function is strictly increasing on this interval, so its cumulative distribution function isDifferentiating gives the probability density function $s/[\pi (s^2+y^2)]$. The probability of either endpoint is zero, so including endpoints in the uniform interval does not alter this probability distribution. The slow quadratic decay makes the expected value undefined: its positive and negative parts both have infinite integrals. Its second moment is also infinite, so it has no finite standard deviation.

Independent standard Cauchy distribution variables sum to a Cauchy law of scale $n$, so their sample mean has its original distribution for every sample size. This follows by induction using convolution of probability densities. It does not contradict the usual weak law of large numbers, since the Cauchy law has no finite first absolute moment and its expectation is undefined. Symmetry does not turn a divergent expectation into zero.

The absolute value of a centered Cauchy distribution variable with scale $\gamma >0$ has this probability density function. Its positive support and doubled density distinguish it from the two-sided distribution. Its unit-scale case arises as the ratio of two independent, equally scaled half-normal distribution variables.

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 symmetric stable distribution has this characteristic function, with $c>0$ and location $\mu$. At location zero it is a strictly stable distribution: weighted sums of independent copies have scale factor $({\sum }_j|a_j{|}^{\alpha }{)}^{1/\alpha }$. The index $\alpha$ need not permit a finite mean or variance. The scale in this convention is $c^{1/\alpha }$, so the coefficient of $|t{|}^{\alpha }$ and the scale parameter should not be confused.

For $0<\alpha <2$, represent $u^{-1-\alpha }$ as $\Gamma (1+\alpha {)}^{-1}{\int }_0^{\infty }{v}^{\alpha }{e}^{-vu}dv$. The Tonelli theorem applies to the nonnegative integrand. The inner integral in $u$ is $1/[v(1+v^2)]$, leaving $\Gamma (1+\alpha {)}^{-1}{\int }_0^{\infty }{v}^{\alpha -1}/(1+v^2)dv$. Substitution $z=v^2$ and the Euler beta function together with the Gamma reflection formula give the result. The bounds at zero and infinity explain exactly the index range.

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.

Let $X$ have a strictly stable distribution of index $\alpha$, and let $X^{\prime }$ be an independent copy. For integers $1\le k<n$, split a sum of $n$ independent copies into blocks of sizes $k$ and $n-k$. Strict stability of each block and of the full sum givesFor any $0<c<1$, choose rationals $k_m/n_m\to c^{\alpha }$. On the fixed probability space carrying $X,X^{\prime }$, the left side converges almost surely to $cX+(1-c^{\alpha }{)}^{1/\alpha }{X}^{\prime }$. Thus it converges in distribution; since its probability distribution is constant, the limit also has the law of $X$. This proves a self-decomposable distribution with the displayed remainder. The proof does not require any finite moments.