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

The paper cleanly proves a quenched vector‑valued ASIP for random expanding-on-average cocycles under the “good cocycle” hypotheses, via adapted norms for the normalized transfer-operator cocycle and a Gouëzel-style spectral argument. The main theorem (Theorem 9) states the existence/characterization of the asymptotic covariance and an O(n^{1/4+δ}) strong invariance principle with L2 control, and the proof supplies all needed technical estimates (Definition 3, Theorem 12, Corollary 13, Lemma 14, completion in §2.6) . By contrast, the candidate solution sketches a martingale–coboundary approach and then simply invokes the Cuny–Merlevède reverse martingale SIP to obtain the same rate. It does not verify the reverse‑martingale hypotheses (conditional variance control, moment conditions, vector-valued extension) in this random, quenched setting, nor does it address the tempered/adapted‑norm machinery that the paper shows is necessary to absorb non‑uniformity; it also asserts uniform-in-ω BV bounds where the paper establishes only tempered bounds. The scalar martingale route with improved rates in the paper requires an extra assumption (54), underscoring the nontriviality of these checks . Hence the paper is correct and complete, while the model’s proof is incomplete as written.