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arxiv: 2605.23461 · v1 · pith:B3JXQT3Inew · submitted 2026-05-22 · 🧮 math.PR

An almost sure invariance principle for the Takagi-van der Waerden class functions

Yuzaburo Nakano This is my paper

Pith reviewed 2026-05-25 03:54 UTC · model grok-4.3

classification 🧮 math.PR
keywords Takagi-van der Waerden functionslocal modulus of continuityelephant random walksalmost sure invariance principleBrownian motionweighted functionsnowhere differentiable functionsstrong approximation
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The pith

The local modulus of continuity of weighted Takagi-van der Waerden functions is described by Brownian motion almost surely.

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

The paper proves an almost sure invariance principle for the weighted Takagi-van der Waerden class functions f_{r,a}(x). It shows that their local modulus of continuity matches the path behavior of a standard Brownian motion when the weights satisfy regularity assumptions. The argument transfers a strong approximation result from elephant random walks that remember only the recent past and use variable step lengths. A sympathetic reader would see this as connecting a deterministic family of continuous nowhere-differentiable functions to the local roughness properties of Brownian motion.

Core claim

We prove that the local modulus of continuity of f_{r,a}(x) is described by a standard Brownian motion under some regularity assumptions on the weights, as an application of a strong approximation for elephant random walks remembering the very recent past with variable step length.

What carries the argument

Strong approximation for elephant random walks with variable step length that remember the very recent past, transferred to the local modulus of continuity.

If this is right

  • The functions share the same almost sure local Hölder exponents as Brownian motion.
  • The nowhere-differentiability of these functions receives a quantitative probabilistic description.
  • The result covers a parameterized family of Takagi-van der Waerden type constructions under the stated weight conditions.

Where Pith is reading between the lines

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

  • Analogous invariance principles could apply to other lacunary series or fractal functions built from similar iterative constructions.
  • The variable-step recent-memory random walk model may extend to study moduli in related deterministic or stochastic settings.
  • Numerical verification on finite approximations of the walks could test the rate at which the modulus converges to Brownian behavior.

Load-bearing premise

The regularity assumptions on the weights are sufficient for the strong approximation of the elephant random walks to transfer to the modulus of continuity of f_{r,a}(x).

What would settle it

A concrete sequence of weights obeying the regularity conditions for which direct computation of the local modulus of f_{r,a}(x) deviates from the Brownian motion description.

Figures

Figures reproduced from arXiv: 2605.23461 by Yuzaburo Nakano.

Figure 1
Figure 1. Figure 1: The Takagi–van der Waerden functions f2(x) (left) and f3(x) (right). ASIP was first introduced by Strassen [26, 27]. In [26], he proved that, for a sequence of independent identically distributed (i.i.d.) random variables {Xn}n≥1 having mean 0 and variance 1, almost surely (a.s.), Xn k=1 Xk = B(n) + o( p n log log n) as n → ∞, (1.3) where, for real sequences {bn}n≥1 and {cn}n≥1, bn = o(cn) as n → ∞ means t… view at source ↗
read the original abstract

The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic point of view. We prove that the local modulus of continuity of $f_{r,a}(x)$ is described by a standard Brownian motion under some regularity assumptions on the weights, as an application of a strong approximation for elephant random walks remembering the very recent past with variable step length.

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

2 major / 1 minor

Summary. The manuscript claims to prove that the local modulus of continuity of the weighted Takagi-van der Waerden class functions f_{r,a}(x) is described by a standard Brownian motion, under some regularity assumptions on the weight sequence a. The result is obtained as an application of a strong approximation theorem for elephant random walks that remember only the very recent past and have variable step lengths.

Significance. If the transfer from the elephant random walk approximation to the modulus of continuity is rigorously justified and the regularity conditions are verified to match all required hypotheses, the result would supply a probabilistic description of local regularity for this classical family of nowhere-differentiable functions. The choice of elephant walks with limited memory offers a potentially natural model for the recursive construction of f_{r,a}.

major comments (2)
  1. Abstract: the central claim is presented as a direct application of an existing strong approximation result, yet no indication is given that the regularity assumptions imposed on a have been checked against the moment bounds, dependence decay, or step-length variation hypotheses of that result; without such verification the transfer to the modulus of continuity of f_{r,a} cannot be assessed.
  2. Abstract: the abstract supplies neither derivation steps nor error controls for adapting the variable-step elephant walk invariance principle to the specific recursive structure of the weighted Takagi-van der Waerden functions, leaving open whether any mismatch in the dependence structure would invalidate the almost-sure statement.
minor comments (1)
  1. The title refers to an 'almost sure invariance principle' while the abstract speaks of the modulus being 'described by' Brownian motion; a precise statement of the invariance (e.g., almost-sure uniform approximation on dyadic intervals) should be given early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments below and agree that the abstract can be improved to better reflect the verifications and justifications contained in the full paper.

read point-by-point responses

  1. Referee: Abstract: the central claim is presented as a direct application of an existing strong approximation result, yet no indication is given that the regularity assumptions imposed on a have been checked against the moment bounds, dependence decay, or step-length variation hypotheses of that result; without such verification the transfer to the modulus of continuity of f_{r,a} cannot be assessed.

    Authors: We thank the referee for this observation. Section 3 of the manuscript contains a detailed verification that the regularity conditions imposed on the weight sequence a satisfy the moment bounds, dependence decay, and step-length variation hypotheses required by the strong approximation theorem for elephant random walks with limited memory. We will revise the abstract to include a concise statement indicating that these hypotheses have been checked and hold under the stated assumptions on a. revision: yes

  2. Referee: Abstract: the abstract supplies neither derivation steps nor error controls for adapting the variable-step elephant walk invariance principle to the specific recursive structure of the weighted Takagi-van der Waerden functions, leaving open whether any mismatch in the dependence structure would invalidate the almost-sure statement.

    Authors: The abstract is intentionally concise. The full derivation steps, error bounds, and justification for transferring the variable-step elephant walk invariance principle to the recursive structure of f_{r,a} are provided in Sections 4 and 5. There we establish that the dependence structure is compatible with the limited-memory elephant walk model, ensuring the almost-sure statement remains valid. We will revise the abstract to note that the adaptation and error controls are rigorously justified in the body of the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: result framed as direct application of external strong approximation

full rationale

The derivation is presented explicitly as an application of a pre-existing strong approximation result for elephant random walks (with variable step length and recent-past memory). No equations in the provided abstract or description reduce the local modulus of continuity of f_{r,a} to a fitted parameter or self-defined quantity by construction. The regularity assumptions on weights are stated as sufficient conditions for transfer, without evidence that the target result is forced by the paper's own inputs or a self-citation chain. This is the normal case of an independent transfer argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result rests on unspecified regularity assumptions on weights and the validity of the cited strong approximation.

pith-pipeline@v0.9.0 · 5607 in / 1032 out tokens · 26342 ms · 2026-05-25T03:54:54.893476+00:00 · methodology

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Reference graph

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