Almost sure invariance principle for displacement of hyperbolic-group random walks

Prove an almost sure invariance principle for the word-metric displacement of random walks on word-hyperbolic groups, including the stationary-increment random walks with dependent increments modeled by the skew product in Section 5.

Background

The paper develops an almost sure invariance principle for the drift of random walks with dependent increments on word-hyperbolic groups, where the relevant displacement is measured using a dynamically defined Green metric. It also applies this framework to fluctuations along geodesic flows on regular covers.

The authors explicitly distinguish this result from an almost sure invariance principle for the displacement itself. In particular, the unresolved issue concerns establishing such a strong approximation for the word-metric displacement of the random walk, rather than merely obtaining a central limit theorem or an almost sure invariance principle for the Green-metric drift.

References

However, to the best of the authors' knowledge, no almost sure invariance principle is known for the displacement of the random walk, nor are there analogous limit laws for the geodesic flow as in the second part of Theorem \ref{mainthm:vasip-for-the-flow}.

— On fluctuations of the drift in hyperbolic groups  (2609.18053 - Hernandez et al., 16 Sep 2026) in Section 1, Introduction, paragraph beginning “There are several parallel results for random walks with independent increments on word-hyperbolic groups”