Symmetric Birkhoff sums in infinite ergodic theory

TL;DR

This paper proves that in infinite ergodic systems, symmetric Birkhoff sums of positive integrable functions do not converge pointwise, despite being almost surely bounded away from zero and infinity.

Contribution

It establishes a fundamental non-convergence result for symmetric Birkhoff sums in infinite ergodic theory, highlighting a key difference from finite systems.

Findings

Symmetric Birkhoff sums do not converge pointwise in infinite ergodic systems.

Such sums can be almost surely bounded away from zero and infinity.

The result clarifies limitations of normalized sums in infinite ergodic theory.

Abstract

We show that the absolutely normalized, symmetric Birkhoff sums of positive integrable functions in infinite, ergodic systems never converge pointwise even though they may be almost surely bounded away from zero and infinity.