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Which functions preserve Cauchy laws?

Author: Gérard Letac
Journal: Proc. Amer. Math. Soc. 67 (1977), 277-286
MSC: Primary 28A65; Secondary 60E05
MathSciNet review: 0584393
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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.

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Keywords: Measure preserving transformations, Poisson kernel, Cauchy laws
Article copyright: © Copyright 1977 American Mathematical Society

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