= Self-similarity of a Cauchy process
{title2=$\alpha^{-1}X_{\alpha t}\stackrel{d}=X_t$}
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 $\alpha=1$.
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