Some properties of the arc-sine law related to its invariance under a family of rational maps

Jim Pitman, Marc Yor

Source: Anirban DasGupta, ed., A Festschrift for Herman Rubin (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004), 126-137.

Abstract

This paper shows how the invariance of the arc-sine distribution on (0, 1) under a family of rational maps is related on the one hand to various integral identities with probabilistic interpretations involving random variables derived from Brownian motion with arc-sine, Gaussian, Cauchy and other distributions, and on the other hand to results in the analytic theory of iterated rational maps.