> [!theorem] > > A [[Probability|probability]] measure $\mu$ is **Cauchy-distributed** if it has [[Density|density]] > $ > f(x) = \frac{1}{\pi(1 + x^2)} > $ > with respect to the Lebesgue measure, which has [[Characteristic Function|characteristic function]] > $ > \wh \mu(\xi) = e^{-\abs{\xi}} > $ > *Proof*. Suppose that $\xi > 0$, then over a contour integral, > $ > \int_C \frac{e^{i z\xi}}{\pi(1 + z^2)}dz = 2\pi i \cdot \frac{e^{-\xi}}{2\pi i} = e^{-\xi} > $ > where dropping the curve keeps the value. > [!theorem] > > Let $\mu$ be a Cauchy probability measure, then $\wh \mu(\xi) = e^{-\abs{\xi}} = (e^{-\abs{\xi/n}})^n$. Since $\wh \mu_{1/n}$ can be identified with $e^{-\abs{\xi/n}}$, $\mu$ is [[Infinitely Divisible Distribution|infinitely divisible]] and $1$-[[Alpha-Stable Distribution|stable]].