{"id":"e3f921d6-b9b3-4a07-b815-b670243cf651","arxiv_id":"2506.13167","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For random Young towers with ergodic driving, self-normalized Birkhoff sums converge to a standard Brownian motion in Wasserstein distance at rate O(n^{-1/4+1/(2q)}).","lead":"This paper proves a quenched invariance principle with an explicit Wasserstein convergence rate for random Young towers driven by an ergodic system. It introduces a secondary martingale-coboundary decomposition that controls the squared sums of the approximating martingale, yielding rates for self-normalized Birkhoff sums.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.4 proves the key annealed L1 decay by applying (P8) to the sign of P^n phi, a non-Lipschitz function; since (P8) requires test functions in F^K_beta and C_{phi,psi} depends on their Lipschitz norm, the estimate is not justified and the martingale-coboundary decompositions built on it…","rationale":"After reading the full manuscript, I agree with the Reader's conditional verdict. The central result is Theorem 3.2, which gives a quenched Wasserstein rate of order n^{-1/4+1/(2q)}. The proof route is: Proposition 2.4 gives annealed L1 decay of P^n applied to centered Lipschitz observables; Lemma 4.1 uses this to embed the observable as a reverse martingale plus L^q coboundary; Lemma 4.3 uses it again to control the secondary decomposition of ˘φ; the remaining steps (Lemmas 5.1–6.6 and Appendix A) are largely standard martingale/Skorokhod arguments. The weakest point is precisely Proposition 2.4. Its proof applies the quenched decay of correlations (P8) with ψ equal to the sign of P^n_ω(φ_ω−mean). This function is not in the admissible class F^K_β, and the constant C_{φ,ψ} in (P8) normally depends on the Lipschitz norm of ψ. Since the sign function has no finite Lipschitz norm, the estimate is not justified as written. This is not a stylistic objection: the same Proposition is used to prove χ∈L^q, and the secondary decomposition in Lemma 4.3 repeatedly invokes the same L1 decay. If the rate in Proposition 2.4 cannot be established, the martingale-coboundary machinery collapses and Theorem 3.2 is unproved. The concern is potentially repairable: using (P8) with a Lipschitz test function and then approximating the sign, or using the Lipschitz regularity of P^nφ (Proposition 2.5) to control the level set of its sign, might recover the required bound. But the paper does not provide such an argument. I also note a secondary issue not flagged by the Reader: Theorem 3.2 states the result without assuming the limiting variance Σ^2 is positive; when Σ^2=0, the self-normalized process W~^ω_n is not close to a standard Brownian motion, so the statement needs an explicit nondegeneracy hypothesis. This is a statement-level flaw that is easy to fix, but it does not affect the main proof gap. For these reasons the verdict remains CONDITIONAL, with the condition being a repair of Proposition 2.4 (and a clarification of Σ^2>0).","tokens_in":34470,"tokens_out":15292,"duration_ms":165334,"concrete_test":"Independently re-derive Proposition 2.4 without using a non-Lipschitz test function: for each n, bound E∫|P^n_ω(φ_ω−mean)| dμ_{σ^nω} by sup over Lipschitz ψ (with controlled Lipschitz constants) of the correlation expression, and check whether the optimal rate is n^{-(a−1−δ)}. If only n^{-(a−1−δ)/2} follows, recompute Lemma 4.1 and Lemma 4.3 with the weaker decay and verify whether the final exponent −1/4+1/(2q) in Theorem 3.2 remains valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.4 is the linchpin of the proof. It asserts E∫|P^n_ω(φ_ω−∫φ_ω dμ_ω)| dμ_{σ^nω} ≤ C n^{-(a−1−δ)}. The proof evaluates the L1 norm by duality against ψ = sgn(P^n_ω(φ_ω−∫φ_ω dμ_ω)) and invokes (P8). But (P8) is stated only for ψ∈F^K_β, a Lipschitz class with random constant K_ω, and the constant C_{φ,ψ} is expected to depend on the Lipschitz norm of ψ. A sign function is not Lipschitz and has no finite such norm; nor is it in F^K_β. The same defect appears whenever (P8) is used through Proposition 2.4, including the proof of χ,ψ∈L^q in Lemma 4.1 and the estimates in Lemma 4.3 that justify the secondary martingale-coboundary decomposition. Without Proposition 2.4, the L^q integrability of χ and the O(1) bound on the partial sums of P^k applied to ˘φ fail, so the Wasserstein rate in Theorem 3.2 has no proof. A repair might be possible by approximating the sign by Lipschitz functions, but the paper gives no such argument, and a naive approximation with Lipschitz constant N would only give a rate of order n^{-(a−1−δ)/2} after optimizing N, unless additional structure is exploited.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a quenched invariance principle with an explicit Wasserstein convergence rate for random Young towers driven by an ergodic base. Under assumptions (P1)-(P8), with return-time tail exponent a>5, it proves for centered observables in a random Lipschitz class that the self-normalized process converges to a standard Brownian motion at rate O(n^{-1/4+1/(2q)}) in W_{q/4}, where q=(a-1-δ)/(δ+1)≥4. The proof hinges on a primary martingale-coboundary decomposition and a novel secondary decomposition controlling the conditional variance of the approximating martingale; the Skorokhod embedding method then yields the rate. Applications are given to i.i.d. translations of Viana maps, i.i.d. perturbations of intermittent interval maps, and small random perturbations of Anosov diffeomorphisms with ergodic driving.","tokens_in":34802,"tokens_out":8786,"duration_ms":90373,"significance":"If the proof gap identified below is repaired, this is a substantial contribution: it provides the first quenched Wasserstein convergence rate for random Young towers, introduces a secondary martingale-coboundary decomposition that controls conditional variance sums, and covers several important classes of random dynamical systems. The assumptions (P1)-(P8) are clearly stated as hypotheses and are verified in the cited literature, so there is no circularity. The paper is honest about the limitations of the Skorokhod embedding method (essentially optimal n^{-1/4} barrier) and gives a careful passage from the tower to the original system. The main issue is that a key annealed decay estimate is currently unjustified.","major_comments":[{"comment":"The proof of the annealed L^1 decay applies (P8) with ψ_{σ^nω} = sgn(P^n_ω(φ_ω - ∫φ_ω dμ_ω)). However, (P8) is stated only for test functions ψ ∈ F^K_β, a Lipschitz class with random constant K_ω. A sign function of a general L^∞ function is not Lipschitz and has no finite Lipschitz constant, so the hypothesis of (P8) is not satisfied, and the constant C_{φ,ψ} in (P8), which in standard formulations depends on the Lipschitz norm of ψ, cannot be taken finite for this choice. This estimate is used in Lemma 4.1 to prove χ,ψ ∈ L^q and in Lemma 4.3 to control (4.1)-(4.3), so the primary and secondary martingale-coboundary decompositions, and hence the rate in Theorem 3.2, are not justified as written. A repair would require either an extension of (P8) to L^∞ test functions with a constant independent of the Lipschitz norm, or a separate annealed L^1 decay argument via Lipschitz approximation with a quantitative error; neither appears in the manuscript, and a naive Lipschitz approximation with constant N would yield only n^{-(a-1-δ)/2} after optimizing N.","section":"Section 2.2, Proposition 2.4"},{"comment":"The final combination of estimates reads \"Cn^{-1/4+1/(2q)} + Cn^{-1/4+ε} ≤ Cn^{-1/4+1/(2q)} where the last inequality holds because ε>0 can be taken arbitrarily small.\" This is imprecise: for a fixed q, one must choose ε = 1/(2q) in Lemma 6.4 (which is allowed, since Lemma 6.4 holds for any ε>0 with a constant depending on ε). As written, the sentence suggests a uniform inequality in n that is false for arbitrary small ε. This is a presentation issue rather than a fatal gap, but it should be fixed by explicitly choosing ε=1/(2q) in the application of Lemma 6.4.","section":"Section 6.2, proof of Theorem 3.2"}],"minor_comments":[{"comment":"There are numerous typos, including \"Wassertein\" (title, abstract), \"eatimate\" and \"eati\" (Lemma 4.3), \"Gaussion\" (proof of Corollary 5.6), and \"reflexible\" (Lemma 4.2, should be \"reflexive\"). A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The notation K_{σ^{-n}ω}+C_{h,F} in Proposition 2.5 is used without explicitly defining the sum of a random variable K and a constant inside the function class; please clarify that the Lipschitz constant in the class is allowed to depend on this shifted random constant.","section":"Section 2.1, definition of F^K_β"},{"comment":"In the estimate (6.17), the inequality n^{-(γ-2γ/q)} ≤ n^{-γ/2} requires q≥4; this holds by assumption but should be stated explicitly at that point rather than as a side remark.","section":"Section 6.2, Lemma 6.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its conditional assumptions and does not overclaim. The sign-function gap in Proposition 2.4 is, however, load-bearing and not a trivial presentation issue; whether it can be repaired depends on whether (P8) can be extended or a new annealed estimate can be proved. I recommend major revision rather than rejection because the central framework and the secondary decomposition are potentially valuable, and the gap, while serious, is localized to one proposition and its downstream uses. The scope fits the journal, and the literature review is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline result is real: the paper gives the first Wasserstein convergence rate for the quenched invariance principle in random Young towers with an ergodic driving system, and a quenched almost sure invariance principle as a byproduct. The secondary martingale-coboundary decomposition is a genuine technical step, and the examples (i.i.d. Viana maps, intermittent maps, Anosov perturbations) are appropriate and non-trivial. The paper is clearly written and honestly positioned relative to Su, Liu-Wang, Paviato, and others. There's no circularity: (P1)-(P8) are external assumptions, not derived from the conclusions.\n\nThe soft spot is load-bearing. Proposition 2.4 is the linchpin: it proves an annealed L1 decay estimate on P^n(phi - ∫phi). The proof invokes the quenched decay of correlations (P8) with psi = sgn(P^n(phi - ∫phi)). But (P8) is stated for psi in the Lipschitz class F^K_beta, and the constant C_{phi,psi} depends on the Lipschitz norm of psi. A sign function is not Lipschitz. This is not a minor regularity quibble: the same move is used repeatedly through Lemma 4.1 (L^q integrability of chi), Lemma 4.3, and Corollary 5.5. Without Proposition 2.4, the martingale-coboundary decomposition lacks the needed control, and Theorem 3.2's rate has no proof as written.\n\nThe gap looks repairable: approximate the sign by Lipschitz functions with a carefully chosen Lipschitz constant and track the dependence of C_{phi,psi}. The paper gives no such argument. A naive approximation with Lipschitz constant N gives a rate of n^{-(a-1-delta)/2} after optimizing N, much worse than the claimed n^{-1/4+1/(2q)}. So the current text does not justify the main theorem.\n\nOther issues are minor. The self-normalized process W~ is unusual but a legitimate way to avoid regularity assumptions in omega. The examples rely on prior constructions for (P8); that's fine. The appendices are standard.\n\nWho should read this? Researchers working on statistical properties of random dynamical systems, especially those using random Young towers. If the gap is repaired, this is a solid paper worth citing. As it stands, it's a promising preprint with a specific, clearly identified problem.\n\nMy recommendation: send to peer review. The result is novel and the gap is specific enough that a good referee can determine whether Proposition 2.4 can be made rigorous. If yes, accept after substantial revision; if no, the main theorem falls. Either way, it deserves referee time.","headline":"New Wasserstein rate for quenched WIP in random Young towers, but the key decay estimate applies correlation bounds to a sign function; likely repairable but currently not justified.","tokens_in":35376,"tokens_out":3029,"would_cite":false,"duration_ms":27466,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37H30","60F17","37A50","60B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For random Young towers driven by an ergodic system, zero-mean Hölder observables have self-normalized Birkhoff sums converging to a standard Brownian motion with Wasserstein rate O(n^{-1/4+1/(2q)}), where q=(a-1-δ)/(δ+1)≥4 and a is the…","keywords":["quenched invariance principle","random Young towers","Wasserstein distance","martingale-coboundary decomposition","random dynamical systems","ergodic driving system","return-time tails","intermittent maps"],"falsifier":"Take the intermittent-map random Young tower of Section 5 in [12] with tail exponent a, fix a centered Hölder observable φ, and compute E∫|P^n_ω(φ_ω−∫φ_ω dμ_ω)|dμ_{σ^nω} numerically or analytically for increasing n; if the decay exponent is below a−1 (or no decay appears), the estimate in Proposition 2.4 fails and with it the rate in Theorem 3.2.","tokens_in":34236,"feed_emoji":"📈","tokens_out":9130,"duration_ms":90383,"temperature":0.7,"pith_summary":"This paper claims that quenched convergence to Brownian motion for random Young towers—random dynamical systems in which fiber maps are chosen by an ergodic driving system—comes with an explicit Wasserstein rate. The main theorem states that for return-time tails with exponent a>5, the distance between the self-normalized Birkhoff process and a standard Brownian motion is O($n^{{-1/4+1/(2q)}}$) with q=(a-1-δ)/(δ+1)≥4. The rate matters because qualitative invariance principles say nothing about how fast the limit is approached, and rates are what statistical applications need. The proof's engine is a new secondary martingale-coboundary decomposition that controls the fluctuations of the conditional variance of the approximating martingale, complementing the primary decomposition that already linearizes the observable. Applications cover i.i.d. translations of Viana maps, intermittent interval maps, and small random perturbations of Anosov maps with ergodic driving.","feed_headline":"Random tower sums approach Brownian motion at n^{-1/4+ε}","feed_subtitle":"Wasserstein rate for quenched invariance in random Young towers, with applications to intermittent and hyperbolic random maps.","key_machinery":"The central object is the random Young tower (RYT): a skew product over an ergodic base σ with fiber maps F_ω on levels of a tower, a random return-time function R_ω, and equivariant probability measures μ_ω, satisfying return-time tail bounds (P5) and a quenched decay of correlation (P8) on bounded random Lipschitz functions F^K_β (functions whose oscillations along tower partitions decay like $β^{{s_ω}}$ with a random Lipschitz norm K_ω). The argument is carried by two decompositions: the primary martingale-coboundary decomposition φ_ω = ψ_ω + χ_{σω}∘F_ω − χ_ω, where ψ is a reverse martingale difference and χ∈L^q is built from transfer operators, and the secondary decomposition for the conditional-variance observable φ̆_ω = [P_ω($ψ_ω^{2}$ − ∫$ψ_ω^{2}$ dμ_ω)]∘F_ω, obtained via a Cesàro/mean-ergodic argument, which controls the sum of squared martingale increments. The rate then follows by comparing the self-normalized process W̃^ω_n with a martingale process, applying the martingale Skorokhod embedding, and estimating the time-change fluctuations via Kolmogorov continuity. The quantity q=(a-1-δ)/(δ+1) records how many moments are available from the a>5 tail exponent.","core_discovery":"Under assumptions (P1)–(P8) on a random Young tower driven by an ergodic base, with return-time tail exponent a>5 and an observable φ in the random Lipschitz class F^K_β with zero fiberwise mean, the paper proves that for almost every environment ω the self-normalized continuous process W̃^ω_n satisfies W_{q/4}(W̃^ω_n, B) ≤ C $n^{{-1/4+1/(2q)}}$ for all n≥1, where B is a standard Brownian motion and q=(a-1-δ)/(δ+1)≥4. This is the quenched Wasserstein convergence rate: it upgrades the quenched invariance principle, proved here as well, to a quantitative statement. A key auxiliary result is the construction of two martingale-coboundary decompositions—one expressing the observable as a reverse martingale difference plus a coboundary, and a secondary one controlling sums of squares of the approximating martingale—which together yield the rate through the martingale version of Skorokhod embedding. The paper notes the rate's $n^{{-1/4}}$ floor is essentially optimal for this method, and the self-normalized formulation avoids requiring a fiberwise converging variance.","pith_inferences":["The rate's dependence on q is a moment-counting artifact of the martingale-coboundary method; a version using stronger L^p estimates for the coboundary could lower the 1/(2q) exponent for large a, approaching the n^{-1/4} floor the authors flag as optimal.","The secondary decomposition controls the conditional variance process in L^{q/2}; this same object is exactly what is needed for quenched Berry–Esseen bounds, so the machinery here is a plausible route to rates in the quenched CLT for these systems.","The self-normalized process W̃^ω_n avoids assuming fiberwise variance convergence; this construction could transfer the rate result to non-stationary environments or slowly driven systems where Σ^2_n(ω)/n oscillates, as long as the annealed decay estimates hold."],"forward_implications":["For i.i.d. translations of Viana maps and small random perturbations of Anosov maps with ergodic driving—cases with (stretched) exponential return-time tails—the Wasserstein rate becomes O(n^{-1/4+δ}) for arbitrarily small δ>0.","For i.i.d. perturbations of intermittent interval maps with a neutral fixed point, the rate O(n^{-1/4+1/(2q)}) is explicit in terms of the tail exponent α_0.","The distance W_p(W̃^ω_n, B) is bounded by the same rate for all 1≤p≤q/4, since Wasserstein distances are monotone in p.","The Lévy-Prokhorov distance between the self-normalized process and Brownian motion is O(n^{-(q-2)/(4(q+4))}), following from the general inequality relating these metrics.","When a>9 (so q>8), the secondary decomposition yields a quenched almost sure invariance principle with rate O(n^{1/4}(log n)^{1/2}(log log n)^{1/4})."],"supporting_citations":[{"why":"Establishes quenched exponential decay of correlations for RYTs with exponential tails under ergodic driving; supplies (P8) for the Anosov example and the general framework.","marker":"[4]"},{"why":"Constructs a random Young tower for i.i.d. intermittent maps and verifies conditions (P4),(P6)–(P7); by Remark 2.2 this yields (P8) and is the basis of Example 7.3.","marker":"[12]"},{"why":"Provides the RYT with exponential return-time tails under which (P8) holds; used in Remark 2.2 and for the Viana example.","marker":"[24]"},{"why":"Proves the quenched almost sure invariance principle for i.i.d. random Young towers and supplies the regularity lemma (Proposition 2.5) for transfer operators used in Lemma 4.3.","marker":"[38]"},{"why":"Gives the Wasserstein-rate strategy via Skorokhod embedding for deterministic systems; Lemma 6.4 adapts its method to the quenched setting.","marker":"[31]"},{"why":"Provides the strong invariance principle with rate for reverse martingale differences; used in Corollary 5.6 to get the QASIP rate.","marker":"[19]"},{"why":"The martingale Skorokhod embedding theorem (Hall–Heyde) used to couple the martingale process with Brownian motion in Lemma 6.4.","marker":"[27]"},{"why":"Introduces the martingale-coboundary decomposition for families of dynamical systems that the secondary decomposition in Section 4.2 builds on.","marker":"[29]"}],"fun_headline_variants":["Rate for quenched random tower invariance: n^{-1/4+ε}","Wasserstein rate for quenched random dynamical systems","Random Young towers: quenched invariance at n^{-1/4+ε}","Quenched rate for random towers: Brownian motion at n^{-1/4}","n^{-1/4} Wasserstein rate for quenched random tower sums"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate proof depends on an annealed $L^{1}$ decay bound that is obtained by feeding the sign of the transfer operator's output into a correlation decay assumption stated only for Lipschitz test functions, and the associated constant is not known to be controlled for such a discontinuous function.","fun_headline_variants_meta":{"raw":{"variants":["Rate for quenched random tower invariance: n^{-1/4+ε}","Wasserstein rate for quenched random dynamical systems","Random Young towers: quenched invariance at n^{-1/4+ε}","Quenched rate for random towers: Brownian motion at n^{-1/4}","n^{-1/4} Wasserstein rate for quenched random tower sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000215,"raw_usage":{"total_tokens":1408,"prompt_tokens":904,"completion_tokens":504,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":404}},"tokens_in":520,"tokens_out":504,"duration_ms":4602,"temperature":1.0,"reasoning_tokens":404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:37:09.167854+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the intermittent-map random Young tower of Section 5 in [12] with tail exponent a, fix a centered Hölder observable φ, and compute E∫|P^n_ω(φ_ω−∫φ_ω dμ_ω)|dμ_{σ^nω} numerically or analytically for increasing n; if the decay exponent is below a−1 (or no decay appears), the estimate in Proposition 2.4 fails and with it the rate in Theorem 3.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes quenched exponential decay of correlations for RYTs with exponential tails under ergodic driving; supplies (P8) for the Anosov example and the general framework."},{"cited_title":"Bahsoun, C","cited_arxiv_id":null,"evidence_quote":"Constructs a random Young tower for i.i.d. intermittent maps and verifies conditions (P4),(P6)–(P7); by Remark 2.2 this yields (P8) and is the basis of Example 7.3."},{"cited_title":"Du, On mixing rates for random perturbations,PhD Thesis,National University of Singapore, 2015","cited_arxiv_id":null,"evidence_quote":"Provides the RYT with exponential return-time tails under which (P8) holds; used in Remark 2.2 and for the Viana example."},{"cited_title":"Su, Random Young towers and quenched limit laws,Ergodic Theory Dynam","cited_arxiv_id":null,"evidence_quote":"Proves the quenched almost sure invariance principle for i.i.d. random Young towers and supplies the regularity lemma (Proposition 2.5) for transfer operators used in Lemma 4.3."},{"cited_title":"Liu and Z","cited_arxiv_id":null,"evidence_quote":"Gives the Wasserstein-rate strategy via Skorokhod embedding for deterministic systems; Lemma 6.4 adapts its method to the quenched setting."},{"cited_title":"Cuny and F","cited_arxiv_id":null,"evidence_quote":"Provides the strong invariance principle with rate for reverse martingale differences; used in Corollary 5.6 to get the QASIP rate."},{"cited_title":"Hall and C","cited_arxiv_id":null,"evidence_quote":"The martingale Skorokhod embedding theorem (Hall–Heyde) used to couple the martingale process with Brownian motion in Lemma 6.4."},{"cited_title":"Korepanov, Z","cited_arxiv_id":null,"evidence_quote":"Introduces the martingale-coboundary decomposition for families of dynamical systems that the secondary decomposition in Section 4.2 builds on."}],"review_version":1}