Arcsine and Darling--Kac laws for piecewise linear random interval maps

TL;DR

This paper presents examples of piecewise linear random interval maps that exhibit arcsine and Darling--Kac laws, extending classical results to stochastic settings with maps having indifferent fixed points.

Contribution

It introduces new random interval maps satisfying these laws, demonstrating their behavior in stochastic systems similar to deterministic cases.

Findings

Maps satisfy arcsine law analogous to Thaler’s result

Maps satisfy Darling--Kac law similar to Aaronson’s result

Behavior mimics indifferent fixed points in deterministic maps

Abstract

We give examples of piecewise linear random interval maps satisfying arcsine and Darling--Kac laws, which are analogous to Thaler's arcsine and Aaronson's Darling--Kac laws for the Boole transform. They are constructed by random switch of two piecewise linear maps with attracting or repelling fixed points, which behave as if they were indifferent fixed points of a deterministic map.