Which functions preserve Cauchy laws?

Abstract:

A real random variable X has a Cauchy law $C(a,b)$ if its density is $b{\pi ^{ - 1}}{[{(x - a)^2} + {b^2}]^{ - 1}}$, with $b > 0$ and a real. Let f be a measurable function such that $f(X)$ also has a Cauchy law for any a and b. We prove that there exist $\alpha$ real, $k \geqslant 0,\varepsilon = \pm 1$ and a singular positive bounded measure $\mu$ on R such that for almost all x of R $f(X)$ has a Cauchy law when X has a Cauchy law. Furthermore, we prove that such a function preserves Lebesgue measure when $k = 1$, generalising a well-known Pólya and Szegö result.