Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

TL;DR

This paper extends classical arcsine laws to multidimensional occupation time ratios in infinite measure ergodic transformations, providing new convergence results applicable to interval maps and Markov chains.

Contribution

It introduces a multidimensional generalization of Lamperti's arcsine distribution for occupation times in infinite ergodic systems using the double Laplace transform method.

Findings

Joint distribution converges to a multidimensional arcsine law

Results apply to interval maps and Markov chains

Method utilizes double Laplace transform

Abstract

We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.