A vector-valued almost sure invariance principle for random expanding on average cocycles

TL;DR

This paper establishes a vector-valued almost sure invariance principle for random expanding on average cocycles, improving error rates with martingale techniques and addressing a key scaling condition through an example.

Contribution

It introduces a quenched vector-valued ASIP for such cocycles by combining Gou"{e}zel's approach with adapted norms, and extends martingale approximation techniques to real-valued observables.

Findings

Established vector-valued ASIP for random expanding on average cocycles.

Improved error rates using martingale approximation for real-valued observables.

Demonstrated the necessity of a scaling condition with a specific example.

Abstract

We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gou\"{e}zel's approach for establishing ASIP and the recent construction of the so-called adapted norms, which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of a scaling condition, answering a question of the first and third authors.