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arxiv: 2505.04415 · v2 · submitted 2025-05-07 · 🧮 math.DS

Regularity of the variance in quenched CLT for random intermittent dynamical systems

Davor Dragi\v{c}evi\'c , Juho Lepp\"anen This is my paper

Pith reviewed 2026-05-22 17:05 UTC · model grok-4.3

classification 🧮 math.DS
keywords quenched central limit theoremrandom dynamical systemsintermittent mapsLSV mapsstatistical stabilitylinear responsevariance regularity
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The pith

For random systems of LSV maps, a quenched central limit theorem holds and its variance changes continuously and differentiably with perturbations.

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

The paper studies random dynamical systems built from LSV maps that have different parameters at each step. Without assuming any mixing on the underlying random process, it proves a quenched central limit theorem. It also specifies conditions making the limit variance continuous and differentiable when the random dynamics are perturbed slightly. The proof uses established results on statistical stability and linear response for these intermittent maps.

Core claim

We establish a quenched central limit theorem for random dynamical systems composed of LSV maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics.

What carries the argument

Quenched central limit theorem for random intermittent maps whose limiting variance is continuous and differentiable under perturbations, based on statistical stability and linear response.

If this is right

  • The variance in the quenched CLT is a continuous function of perturbations to the random dynamics.
  • Differentiability of the variance holds when the perturbations satisfy the stated conditions.
  • The central limit theorem remains valid without mixing assumptions on the base space.
  • Small changes to map parameters produce controlled, predictable shifts in the scale of fluctuations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same regularity might hold for other families of intermittent maps with indifferent points.
  • This continuity could support numerical approximation of the variance from observed trajectories.
  • Extensions to response of higher moments or to non-stationary random bases seem plausible.

Load-bearing premise

The arguments depend on prior results about statistical stability and linear response for random intermittent maps.

What would settle it

A concrete family of parameter perturbations in the LSV maps where the limiting variance jumps or fails to be differentiable would disprove the regularity claim.

read the original abstract

We study random dynamical systems composed of LSV maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025), e70150).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies random dynamical systems generated by LSV maps with varying intermittency parameters, without mixing assumptions on the base space. It establishes a quenched central limit theorem for almost every realization of the random sequence and identifies conditions ensuring that the associated asymptotic variance varies continuously and differentiably under perturbations of the random dynamics. The arguments rely on statistical stability and linear response results from Dragicevic et al. (J. Lond. Math. Soc. 111 (2025)). The variance is expressed via a Green-Kubo-type formula involving the stationary measure and an infinite sum of correlations along random orbits.

Significance. If the claims hold, the work would provide a useful extension of quenched limit theorems to random intermittent systems and establish regularity of the variance, which is relevant for parameter dependence in ergodic theory and statistical mechanics. The reliance on recent linear response results for random maps is a methodological strength, allowing the authors to focus on the variance regularity without re-deriving the base stability estimates.

major comments (1)
  1. The justification for differentiating the Green-Kubo expression for the quenched variance: the manuscript invokes first-order linear response from Dragicevic et al. to differentiate the infinite sum of correlations with respect to the perturbation parameter. However, LSV maps exhibit only polynomial decay of correlations of order O(k^{-β}) where β depends on the intermittency parameter; without an explicit uniform bound on the response of the k-step transfer operators for all k or a dominated remainder estimate, term-by-term differentiation of the sum is not automatically justified. This step is load-bearing for the C^1 regularity claim.
minor comments (2)
  1. Clarify the precise statement of the quenched CLT (e.g., the normalization and the almost-sure set of realizations) in the introduction to make the main result immediately readable.
  2. Add a short remark on how the conditions on the intermittency parameters ensure the required polynomial mixing rates remain compatible with the linear response estimates from the cited work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise observation regarding the justification of term-by-term differentiation in the Green-Kubo formula. We address the major comment below and outline the planned clarification.

read point-by-point responses

  1. Referee: The justification for differentiating the Green-Kubo expression for the quenched variance: the manuscript invokes first-order linear response from Dragicevic et al. to differentiate the infinite sum of correlations with respect to the perturbation parameter. However, LSV maps exhibit only polynomial decay of correlations of order O(k^{-β}) where β depends on the intermittency parameter; without an explicit uniform bound on the response of the k-step transfer operators for all k or a dominated remainder estimate, term-by-term differentiation of the sum is not automatically justified. This step is load-bearing for the C^1 regularity claim.

    Authors: We agree that the polynomial decay rate O(k^{-β}) requires care when interchanging differentiation and summation. The first-order linear response theorem in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025)) supplies precisely the needed uniform bound on the response of the k-step transfer operators together with a dominated remainder estimate that is uniform in the intermittency parameter within the admissible range. These estimates are stated in terms of the same Banach space norms used for the statistical stability results we invoke, and they directly permit differentiation under the sum for the quenched variance expression. Nevertheless, to make the dependence on these estimates fully transparent, we will insert a short clarifying paragraph (new Section 3.3 or an expanded remark after the statement of Theorem 1.2) that recalls the relevant bound from Dragicevic et al. and verifies that the remainder series remains dominated after differentiation. This addition does not alter the proof strategy but improves readability. revision: partial

Circularity Check

1 steps flagged

Quenched CLT and variance regularity rest on self-cited statistical stability and linear response

specific steps
  1. self citation load bearing [Abstract]
    "Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established in Dragicevic et al. (J. Lond. Math. Soc. 111 (2025), e70150)."

    The quenched central limit theorem and the continuous/differentiable dependence of the Green-Kubo variance on perturbations of the random LSV family are obtained by direct invocation of the cited stability and response theorems. Because those theorems originate in prior work by the lead author, the new claims inherit their justification from the self-citation rather than from derivations internal to the present manuscript.

full rationale

The manuscript states that its arguments for the quenched CLT and C^1 regularity of the limit variance rely on statistical stability and linear response results from a 2025 paper by the same lead author. This is a load-bearing self-citation for the central claims. While the cited work is a separate peer-reviewed publication and therefore supplies independent theorems, the present paper's derivation chain reduces to those external results without reproducing or independently verifying the key estimates inside this manuscript. No self-definitional, fitted-prediction, or renaming patterns appear; the circularity is limited to the dependence on overlapping-author prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper depends on external results for statistical stability and linear response rather than deriving them anew; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Statistical stability and linear response hold for the random intermittent maps as established in the cited prior work.
    Invoked to obtain the quenched CLT and the regularity of the variance.

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