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Quenched invariance principle with a rate for random dynamical systems

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.13167 v1 pith:WU5RFGR2 submitted 2025-06-16 math.DS math.PR

Zhenxin Liu , Benoit Saussol , Sandro Vaienti , Zhe Wang This is my paper
classification math.DSmath.PR MSC 37H3060F1737A5060B10
keywords quenchedinvarianceprinciplerandomYoungtowersWassersteindistancemartingale-coboundarydecompositiondynamicalsystemsergodicdrivingsystemreturn-timetailsintermittentmaps
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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}).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

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.

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 (2)
  1. [Section 2.2, Proposition 2.4] 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.
  2. [Section 6.2, proof of Theorem 3.2] 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.
minor comments (3)
  1. [Throughout] 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.
  2. [Section 2.1, definition of F^K_β] 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.
  3. [Section 6.2, Lemma 6.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a conditional result built on external assumptions (P1)-(P8), and no prediction reduces to a fitted input or to a self-citation by construction.

full rationale

I walked the derivation chain and found no circular step. Theorem 3.2 is a conditional Wasserstein-rate bound for the self-normalized process W~_n defined in (3.2); the normalization by Sigma_n^2(omega) is an explicit redefinition of the process, not a hidden use of the conclusion. The proof reduces to the primary martingale-coboundary decomposition (Lemma 4.1), the secondary decomposition (Lemma 4.3), and auxiliary estimates (Lemmas 5.1-5.4, 6.2, Proposition 6.3, Lemmas 6.4, 6.6). These lemmas use Proposition 2.4, which in turn invokes the externally assumed quenched decay of correlation (P8), and Proposition 2.1, cited to external works. No parameter is fitted to a subset of data and then renamed a prediction. The paper's self-citations, e.g. Liu-Wang [31,32], appear in the introduction for deterministic or sequential systems and are not load-bearing for the quenched random-tower argument. The most serious issue is a proof gap, not circularity: in Proposition 2.4, the proof applies (P8) with psi = sgn P^n_omega(phi_omega - integral phi_omega d mu_omega), while (P8) is stated for psi in the Lipschitz class F^K_beta; a sign function is generally not in that class, and the constant C_{phi,psi} in (P8) is allowed to depend on the Lipschitz data of psi. This is a correctness risk in the written proof, and if unrepairable it would invalidate the estimates in Lemma 4.1 and Lemma 4.3. But it is not a circularity: (P8) is an external hypothesis, not the theorem being proved, and the gap does not make the conclusion equivalent to an input by construction. The paper is honestly conditional on (P1)-(P8) and on the prior constructions cited for examples, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or dimensions. The only postulates are the structural assumptions (P1)-(P8) on the random Young tower and the ergodic base, all of which are standard in the field.

assumptions (4)
  • domain assumption The random Young tower satisfies (P1)-(P7): Markov, bounded distortion, weak expansion, aperiodicity, return-time asymptotics, finiteness, and annealed return-time asymptotics.
    These properties define the class of random towers under study. They are assumed throughout and cited from prior work such as [12,4,24].
  • domain assumption The quenched decay of correlations (P8) holds with rate n^{-(a-1-δ)}.
    This is the key mixing assumption, used in Proposition 2.4 and all subsequent estimates. It is not proved in this paper; it is known for Bernoulli driving from [12, Theorem 4.2] and [24, Theorem 1.2.6], and for ergodic automorphisms with exponential tails from [4].
  • domain assumption The driving system σ is an invertible ergodic measure-preserving transformation on (Ω,F,P).
    All quenched statements hold for a.e. ω with respect to this ergodic base. This is the standard setup for random dynamical systems.
  • standard math Standard probability tools: Birkhoff ergodic theorem, Burkholder-Davis-Gundy inequality, Skorokhod embedding theorem (Hall and Heyde), martingale CLT (Billingsley).
    These are invoked in Lemmas 5.1-5.3, 6.4, and Appendix A without proof, as is customary.

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Pith. "Pith review of Quenched invariance principle with a rate for random dynamical systems." pith.science (2026). https://pith.science/paper/WU5RFGR2

@misc{pith2026250613167,
  author       = {Pith},
  title        = {Pith review of: Quenched invariance principle with a rate for random dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WU5RFGR2}},
  note         = {Machine review of arXiv:2506.13167}
}
read the original abstract

In this paper, we consider the quenched invariance principle for random Young towers driven by an ergodic system. In particular, we obtain the Wassertein convergence rate in the quenched invariance principle. As a key ingredient, we derive a new martingale-coboundary decomposition for the random tower map, which provides a good control over sums of squares of the approximating martingale. We apply our results to a class of random dynamical systems that admit a random Young tower, such as independent and identically distributed (i.i.d.) translations of Viana maps, intermittent maps of the interval and small random perturbations of Anosov maps with an ergodic driving system.

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