Self-similarity of a Cauchy process (${\alpha }^{-1}{X}_{\alpha t}\stackrel{d}{=}{X}_t$)

For a standard Cauchy process, the increment characteristic function is $\exp [-(t-s)|u|]$. Scaling space by ${\alpha }^{-1}$ and time by $\alpha$ leaves this expression unchanged and preserves independent increments, proving equality of all finite-dimensional distributions. Equivalently, $\alpha X_{t/\alpha }$ has the original process law. Scaling both space and time by $\alpha$ instead changes the exponent by ${\alpha }^2$ and is not an invariance unless $\alpha =1$.