Relative complexity of random walks in random sceneries

TL;DR

This paper investigates the relative complexity of certain zero entropy extensions called RWRSs, establishing distributional limit theorems that serve as invariants for their classification.

Contribution

It proves distributional limit theorems for the relative complexity of RWRSs with -stable CLT, providing new invariants for their relative isomorphism.

Findings

Distributional limit theorems for relative complexity of RWRSs.

Invariants for classifying relative isomorphism of these systems.

Extension of complexity theory to -stable CLT cases.

Abstract

Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the \alpha-stable CLT ($1<\alpha\le2$). The results give invariants for relative isomorphism of these.