Tokyo Journal of Mathematics

Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties


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This paper is concerned with invariant densities for transformations on $\mathbb{R}$ which are the boundary restrictions of inner functions of the upper half plane. G. Letac [9] proved that if the corresponding inner function has a fixed point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ or a periodic point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ with period 2, then a Cauchy distribution $(1/\pi)\mathrm{Im}\left(1/(x-z_{0}) \right)$ is an invariant probability density for the transformation. Using Cauchy's integral formula, we give an easier proof of Letac's result. An easy sufficient condition for such transformations to be isomorphic to piecewise expanding transformations on an finite interval is given by the explicit form of the density. Transformations of the forms $\alpha x + \beta - \sum ^{n }_{k=1}b_{k}/(x-a_{k})$, \:$\alpha x-\sum ^{\infty }_{k=1}\left\{ b_{k}/(x-a_{k})+b_{k}/(x+a_{k}) \right\}$ and $\alpha x +\beta\tan x$ are studied as examples.

Article information

Tokyo J. Math., Volume 36, Number 1 (2013), 177-193.

First available in Project Euclid: 22 July 2013

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Zentralblatt MATH identifier

Primary: 37A05: Measure-preserving transformations
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60F05: Central limit and other weak theorems


ISHITANI, Hiroshi. Transformations Which Preserve Cauchy Distributions and Their Ergodic Properties. Tokyo J. Math. 36 (2013), no. 1, 177--193. doi:10.3836/tjm/1374497518.

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