Distributional limits of positive, ergodic stationary processes and infinite ergodic transformations

TL;DR

This paper characterizes the distributional limits of non-negative, ergodic stationary processes, demonstrating their full range of possibilities, and explores implications for infinite ergodic theory with new examples of stable and rationally ergodic transformations.

Contribution

It identifies the possible distributional limits of ergodic stationary processes and introduces new examples of distributionally stable and alpha-rationally ergodic transformations.

Findings

All distributional limits of non-negative ergodic stationary processes are possible.

New examples of distributionally stable transformations are provided.

Implications for infinite ergodic theory are discussed.

Abstract

In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable- and $\alpha$-rationally ergodic tranformations are presented.