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The standard symmetric Cauchy process is the Lévy process with Lévy characteristic exponent $\Psi (u)=|u|$. Its time-$t$ distribution for $t>0$ is the Cauchy distribution of location zero and scale $t$. It can be constructed by subordination of a Lévy process: evaluate an independent standard Brownian motion at the Brownian first-passage subordinator, whose Laplace exponent is $\sqrt{2\lambda }$.

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

Specify the sign convention for the Lévy characteristic exponent byConditioning on $T_a$ and using the Brownian first-passage Laplace transform givesThereforeThe absolute value is essential for negative $u$. The process is the standard symmetric Cauchy process; for $a>0$, $X_a$ has the Cauchy distribution with location zero and scale $a$. If the exponent convention instead uses $\mathbb{E}{e}^{iu{X}_a}=e^{a\psi (u)}$, the answer is $\psi (u)=-|u|$.

Let $(Y_n)$ be independent and identically distributed random variables with the standard Cauchy distribution, whose probability density function isIts first absolute moment diverges sinceOn the other hand, its characteristic function is $\phi (t)=e^{-|t|}$. The characteristic function of a sum of independent variables therefore givesThus the stability of the Cauchy distribution implies that $S_n/n$ has the standard Cauchy distribution for every $n$. In particular,where $Z$ is standard Cauchy and $\stackrel{d}{\to }$ denotes convergence in distribution.