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arxiv: 2606.04179 · v1 · pith:AMTM25A6new · submitted 2026-06-02 · 🧮 math.PR

Functional Scaling Limits of Interpolated Correlated Random Walks in H\"older Topology

Felix Benning , Ivan Nourdin This is my paper

Pith reviewed 2026-06-28 08:20 UTC · model grok-4.3

classification 🧮 math.PR
keywords functional convergenceHölder topologyrandom walksGaussian sequencesscaling limitsBreuer-Major theoremDobrushin-Major-Taqqu theoremrough paths
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The pith

Interpolated random walks with Gaussian-sequence increments converge in Hölder topology to their classical scaling limits.

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

The paper strengthens the Dobrushin-Major-Taqqu and Breuer-Major theorems by replacing Skorokhod convergence with convergence in Hölder topology for the interpolated paths. In the summable-covariance regime the convergence holds even in rough Hölder topology. The stronger topology is chosen because Young integration and solution maps of differential equations remain continuous under it. A reader would care if these pathwise operations can now be applied directly to the limiting objects without further approximation arguments.

Core claim

When the increments are measurable functions of a stationary Gaussian sequence whose covariance satisfies either a regularly varying non-summable tail or a summable condition, the suitably scaled and linearly interpolated random walks converge in the Hölder topology (and, in the summable regime, in rough Hölder topology) to the same limiting processes given by the classical theorems.

What carries the argument

Functional convergence in (rough) Hölder topology of the interpolated processes, which upgrades the classical weak convergence in Skorokhod space.

If this is right

  • Young integrals against the limiting paths are well-defined as continuous functionals of the path.
  • Solutions to differential equations driven by the limiting paths exist as continuous images under the solution map.
  • Rough-path lifts of the limits can be obtained directly from the Hölder convergence in the summable regime.
  • Pathwise operations that are discontinuous in Skorokhod topology become continuous once the topology is strengthened to Hölder.

Where Pith is reading between the lines

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

  • The same Hölder upgrade might apply to scaling limits of other functionals of Gaussian sequences beyond the random-walk setting.
  • Numerical schemes that rely on pathwise integration could use these limits without additional regularization.
  • The result supplies a direct route from the covariance tail condition to the regularity needed for rough-path theory.

Load-bearing premise

The increments must be deterministic functions of a stationary Gaussian sequence whose covariance obeys either the regularly varying non-summable tail condition or the summable condition required by the classical theorems.

What would settle it

A covariance function satisfying the regularly varying non-summable tail condition for which the corresponding interpolated walks fail to converge in Hölder norm to the Dobrushin-Major-Taqqu limit.

read the original abstract

We prove functional scaling limits for interpolated random walks whose increments are functions of a stationary Gaussian sequence. In this setting, the classical Dobrushin-Major-Taqqu theorem describes the scaling limit when the covariance has a regularly varying, non-summable tail, while the Breuer-Major theorem describes the limit in the summable regime. We strengthen these convergence results to functional convergence in H\"older topology and, in the summable regime, in rough H\"older topology. These stronger topologies are useful because many operations on paths, such as Young integration and solution maps of differential equations, are continuous in (rough) H\"older topology, but not in Skorokhod topology.

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

0 major / 3 minor

Summary. The manuscript proves functional scaling limits in Hölder topology (and rough Hölder topology in the summable regime) for linearly interpolated random walks whose increments are functions of a stationary Gaussian sequence. It strengthens the classical Dobrushin-Major-Taqqu theorem (non-summable, regularly varying covariance tails) and Breuer-Major theorem (summable covariance) by establishing convergence in these stronger topologies rather than Skorokhod space, under the same covariance assumptions.

Significance. If the results hold, the work is significant because Hölder and rough Hölder topologies make maps such as Young integration and rough differential equation solution operators continuous, enabling direct transfer of convergence to pathwise operations that fail in Skorokhod topology. The approach supplies the necessary uniform moment bounds for Kolmogorov-type tightness criteria adapted to the Gaussian subordination setting, building directly on the classical finite-dimensional convergence results.

minor comments (3)
  1. [§2.2] §2.2, Definition 2.3: the precise range of the Hölder exponent α (in terms of the covariance decay parameter) is stated only implicitly via the moment bounds; an explicit interval for α would improve readability of the main theorems.
  2. [Theorem 3.1] Theorem 3.1 (DMT regime): the statement of convergence in C^α does not explicitly record the dependence of α on the regularly varying index; adding this dependence would make the result easier to apply.
  3. [§4.3] §4.3, proof of tightness: the application of the Kolmogorov-Chentsov criterion is correct but would benefit from a one-sentence reference to the precise version (e.g., the moment condition used for the interpolated process).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the Hölder and rough Hölder topologies, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained extension of classical theorems

full rationale

The paper extends the classical Dobrushin-Major-Taqqu and Breuer-Major theorems to Hölder (and rough Hölder) topologies by supplying additional uniform moment bounds for tightness via adapted Kolmogorov criteria. The covariance assumptions are exactly those of the input theorems, with no reduction of any limit statement to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The argument proceeds from finite-dimensional convergence (inherited from the cited classical results) plus tightness in the stronger metric; no step equates a derived object to its own input by construction. External benchmarks (standard tightness arguments, continuity of Young/rough integration in Hölder spaces) remain independent of the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and properties of stationary Gaussian sequences with prescribed covariance tails; these are standard background assumptions in the field rather than new postulates introduced by the paper.

axioms (1)
  • domain assumption Stationary Gaussian sequence with covariance satisfying regularly varying non-summable tail or summable condition
    Invoked to apply the classical Dobrushin-Major-Taqqu and Breuer-Major theorems before strengthening the topology.

pith-pipeline@v0.9.1-grok · 5641 in / 1134 out tokens · 27411 ms · 2026-06-28T08:20:25.463350+00:00 · methodology

discussion (0)

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Reference graph

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