Functional limits for "tied down" occupation time processes of infinite ergodic transformations

TL;DR

This paper establishes functional limit theorems for occupation times in infinite ergodic transformations, revealing tied-down Mittag-Leffler processes as limits and demonstrating strengthened renewal mixing properties.

Contribution

It introduces new functional limit theorems for occupation times in dual ergodic systems, with tied-down Mittag-Leffler processes as limits, and enhances understanding of renewal mixing.

Findings

Limit theorems for occupation times are proven.

Limiting processes are tied-down Mittag-Leffler processes.

Transformations exhibit strengthened renewal mixing properties.

Abstract

We prove functional, distributional limit theorems for the occupation times of pointwise dual ergodic transformations at "tied-down" times immediately after "excursions". The limiting processes are tied down Mittag-Leffler processes and the transformations involved exhibit functional tied-down renewal mixing properties strengthening those of [AS19].