Tied-down occupation times of infinite ergodic transformations

TL;DR

This paper establishes limit theorems for occupation times in certain ergodic transformations, revealing connections to stable Lévy-bridges and refining mixing properties, with applications to Gibbs-Markov and AFU maps.

Contribution

It introduces new distributional limit theorems for occupation times of weakly mixing ergodic transformations, including tied-down renewal mixing and local limit theorems.

Findings

Limit theorems for occupation times after excursions

Identification of local times of p-stable Lévy-bridges as limits

Periodic local limit theorems for Gibbs-Markov and AFU maps

Abstract

We prove distributional limit theorems (conditional and integrated) for the occupation times of certain weakly mixing, pointwise dual ergodic transformations at "tied-down" times immediately after "excursions". The limiting random variables include the local times of $p$-stable L\'evy-bridges ($1<p\le 2$) and the transformations involved exhibit "tied-down renewal mixing" properties which refine rational weak mixing. Periodic local limit theorems for Gibbs-Markov and AFU maps are also established.