On the rate of convergence for Takagi class functions

Masato Takei; Shoto Osaka

REVIEW 3 major objections 3 minor 13 references

On the rate of convergence for Takagi class functions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For Takagi class functions, the paper derives precise probabilistic rates of convergence: coefficient conditions give L2 and strong laws, a central limit theorem, and a law of the iterated logarithm, while geometric coefficients produce a…

desk verdict Nice results on Takagi-class tail sums, including a clean geometric-coefficient phase transition, but the LIL upper-bound proof has a load-bearing variance-proxy mismatch that must be fixed. read the letter →

arxiv 1908.06686 v1 pith:TPLOLJQP submitted 2019-08-19 math.PR math.CA

Shoto Osaka , Masato Takei This is my paper · ORCID

classification math.PRmath.CA MSC 60F0560F1526A2760G42

keywords TakagifunctionclassrateofconvergencereversemartingalelawlargenumberscentrallimittheoremtheiteratedlogarithmRademacherfunctions

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

The paper asks how quickly the partial sums $f_N(x)=\sum_{n=1}^{N-1} c_n \phi^{(n)}(x)$ of a generalized Takagi function converge to $f(x)=\sum_{n=1}^\infty c_n \phi^{(n)}(x)$, treating the tail $f-f_N$ as a random variable on the Lebesgue space $[0,1]$. The main theorem gives coefficient conditions under which the normalized tail $\bigl(f(x)-f_N(x)\bigr)/\bigl(\tfrac12\sum_{n=N}^\infty c_n\bigr)$ tends to $1$ in the usual probabilistic senses: in $L^2$ if and only if the square tail is small relative to the squared tail sum, almost surely under a stronger exponential condition, and with central-limit and law-of-the-iterated-logarithm fluctuations under further conditions. The paper then shows the opposite behavior for geometric coefficients $c_n=r^n$: the normalized tail converges in distribution to a nonconstant random variable $L_r=2(1-r)r^{-1}f_r$, so the classical Takagi function itself does not satisfy the usual law of large numbers. The upshot is a criterion for when partial-sum approximation is typical at the scale of half the coefficient tail, and when the error keeps random fluctuations of that same order.

What carries the argument

The central object is the centered tail sum $M_N=\sum_{n=N}^\infty c_n\phi_*^{(n)}$, where $\phi_*^{(n)}=\phi^{(n)}-\tfrac12$ is uniform on $[-1/2,1/2]$. Because the tent map preserves Lebesgue measure, $M_N$ is a reverse martingale—a martingale-like sum adapted to the decreasing tail $\sigma$-fields generated by the Rademacher functions—and its differences are orthogonal, with variance $\tfrac1{12}c_n^2$ and fourth moment $\tfrac1{80}c_n^4$. Variance computations turn the $L^2$ theorem into a ratio of tail sums, while an exponential tail inequality for the reverse martingale (Lemma 2.3) drives the strong law and the upper law of the iterated logarithm. For geometric coefficients the mechanism is different: the exact identity of Lemma 3.1 rewrites the normalized tail as $f_r(2^{N-1}x)/E[f_r]$, and the measure-preserving, ergodic doubling map $x\mapsto 2x\bmod 1$ converts this into a fixed nonconstant limiting distribution and, by the ergodic theorem, Cesàro convergence to the constant $1$.

What would settle it

Compute the empirical $L^2$ ratio for $c_n=n^{-\alpha}$ with $\alpha>1$: Theorem 1.1(i) predicts $E[((f-f_N)/(\tfrac12\sum_{n=N}^\infty c_n)-1)^2]=\tfrac13\sum_{n=N}^\infty c_n^2/(\sum_{n=N}^\infty c_n)^2$, which decays like $((\alpha-1)^2)/(3(2\alpha-1))\,N^{-1}$; a mismatch in the constant or the rate would refute the $L^2$ claim. Separately, for $r=1/2$, the histogram of the normalized tail over uniformly random points should approach the nonconstant distribution of $2T(x)$ rather than concentrating at $1$, testing Theorem 1.7.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1.1. Define the centered tail $M_N=f-f_N-\tfrac12\sum_{n=N}^\infty c_n=\sum_{n=N}^\infty c_n(\phi^{(n)}-\tfrac12)$. The functions $\phi^{(n)}-\tfrac12$ are centered and uniformly distributed on $[-1/2,1/2]$, and $M_N$ is a reverse martingale with orthogonal differences $d_n=c_n(\phi^{(n)}-\tfrac12)$, so $s_N^2=E[M_N^2]=\tfrac1{12}\sum_{n=N}^\infty c_n^2$. Theorem 1.1 says: (i) $M_N/\tfrac12\sum_{n=N}^\infty c_n\to 0$ in $L^2$ if and only if condition (1.4) holds; (ii) under (1.5) the same ratio tends to $0$ almost surely; (iii) under (1.6) $M_N/\sqrt{s_N^2}$ converges in distribution to the standard normal; and (iv) the upper law of the iterated logarithm holds under (1.6), while the full two-sided version with norming $\varphi(s_N^2)$ holds under (1.7). For geometric coefficients $c_n=r^n$, Theorem 1.7 proves the exact self-similarity identity $\bigl(f_r-f_{r,N}\bigr)/\bigl(\tfrac12\sum_{n=N}^\infty r^n\bigr)=f_r(2^{N-1}x)/E[f_r]$, which makes the normalized tail have the nonconstant distribution of $f_r/E[f_r]$ for every $N$; ergodicity of the doubling map then gives Cesàro convergence of these normalized tails to $1$ almost surely and in $L^1$.

Load-bearing premise

The load-bearing assumption is the quoted reverse-martingale representation of the centered tail together with the unproved exponential tail estimate of Lemma 2.3; if either needs extra hypotheses on the coefficients beyond $\ell^1$, the strong-law and law-of-the-iterated-logarithm conclusions are not established.

Editorial extensions

If this is right

For coefficients $c_n=n^{-\alpha}$ with $\alpha>1$, all hypotheses of Theorem 1.1 are satisfied: the normalized tail obeys both laws of large numbers, and after rescaling by $\sqrt{N}$ it has a standard normal limit with explicit constant $(\alpha-1)/\sqrt{3(2\alpha-1)}$.
For subexponential coefficients $c_n=e^{-Kn^\beta}$ with $K>0$ and $0<\beta<1$, the weak and strong laws and the central limit theorem hold, but the full law of the iterated logarithm is obtained only for $\beta<1/2$; the paper leaves open whether the gap $1/2\le\beta<1$ can be filled.
For geometric coefficients, the normalized tail converges in distribution to a nonconstant random variable, so no deterministic normalization makes it concentrate at $1$; however, the Cesàro average of the normalized tails tends to $1$, giving an ergodic sense in which the half-tail sum is the correct average scale.
The $L^2$ equivalence in Theorem 1.1(i) gives a practical criterion: partial sums approximate the full function at the scale $\tfrac12\sum_{n=N}^\infty c_n$ exactly when the square tail is dominated by the squared tail sum.

Reading between the lines

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

A natural next step is to determine whether the exponential condition (1.5) is necessary for the strong law, and whether Lemma 2.3 can be proved directly for arbitrary $\ell^1$ coefficient sequences rather than imported from the classical martingale argument.
The same two-case dichotomy—reverse-martingale limit theorems versus self-similar distributional limits—should apply to other continuous functions built from iterated maps, such as Takagi-type sums with slowly varying coefficient blocks.
Because $f_{1/4}(x)=x(1-x)$, the limiting variable $L_{1/4}=6x(1-x)$ has an explicit distribution; exploring other rational $r$ could yield explicit laws for $L_r$ and link the distributional limit to familiar beta-type distributions.
For plotting or numerical approximation of geometric-coefficient Takagi functions, the distributional limit implies an irreducible random-looking vertical error on the scale of the entire tail mean, so error bounds for graphs of $f_N$ should be quantiles of $L_r$, not pointwise intervals around $1$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's main contributions are real: a sharp L2-law for the tail sum of Takagi class functions, and the geometric-coefficient case where the normalized tail converges to the nondegenerate L_r = 2(1−r)/r · f_r. That part is elegant and solid, built on an exact identity and ergodic theory. The CLT via Hall–Heyde is standard but carefully checked, and the L2 iff condition is clean. No fitted parameters, no circularity. The citation pattern is fine — Kono's reverse martingale structure is used as a black box, but that is legitimate and acknowledged.

The soft spot is serious. The upper-bound half of the LIL (Theorem 1.1(iv), first half, and the Remark 1.2 corollary) rests on Lemma 2.7. That lemma is stated with variance proxy Σ c_n², but the application in Section 2.4 needs it with s²_N = (1/12)Σ c_n². Substituting the lemma as written into the Markov step gives a positive exponent, not the negative one needed for Borel–Cantelli. The one-line proof 'by a similar argument to Azuma' does not establish the stronger form; a direct Azuma/Hoeffding bound on the reversed martingale differences gives a variance proxy controlled by Σ c_n²/8 = (3/2)s²_N, which is still not enough. So the LIL upper bound is not proven as written. This is an internal inconsistency, not a missing detail, and it is load-bearing.

The omitted proof of Lemma 2.3 is minor by comparison — standard exponential martingale arguments give it up to constants, and the strong law conclusion would survive with a slightly different constant.

Who this is for: probabilists working on Takagi-type functions or reverse martingale limit theorems. The geometric case result is genuinely worth knowing. The paper deserves a serious referee, but the authors need to fix Lemma 2.7 — either prove a maximal inequality with the correct s² proxy, or restructure the LIL argument so it only needs what can be proved. I'd send it to peer review, and I'd expect major revision.

Referee Report

3 major / 3 minor

Summary. Let f(x) = \sum_{n=1}^\infty c_n \phi^{(n)}(x) be a generalized Takagi function with c_n \in \ell^1. The paper studies the tail remainder f - f_N as a random variable on [0,1] under Lebesgue measure. Theorem 1.1 characterizes, in terms of tail sums of c_n and c_n^2, when the normalized remainder (f - f_N)/((1/2)\sum_{n=N}^\infty c_n) converges to 1 in L^2 or almost surely, and when the centered remainder satisfies a central limit theorem and a law of the iterated logarithm. Conditions (1.4)--(1.7) are given as sufficient conditions, with (1.4) also necessary and sufficient for the L^2 result. Examples with polynomial and exponential coefficients are worked out. Theorem 1.7 treats geometric coefficients c_n = r^n, for which the normalized tail converges in distribution to a nonconstant limit, namely 2(1-r)r^{-1} f_r, while the Ces\`aro averages of the normalized tail converge to 1.

Significance. The probabilistic framework for Takagi class functions is elegant, and the characterization in Theorem 1.1(i) is an exact, parameter-free result. The paper applies standard martingale limit theorems --- Hall--Heyde, Scott--Huggins, Azuma, Birkhoff --- to a deterministic approximation problem and gives concrete, falsifiable predictions in Examples 1.3 and 1.4. The geometric-coefficient result in Theorem 1.7 is a useful contrast, showing that the usual law of large numbers fails for the Takagi function itself. However, the proof of the LIL upper bound rests on an inconsistent application of Lemma 2.7, and Remark 1.2 is false as stated. These issues are local but load-bearing; once repaired, the paper would be a solid contribution.

major comments (3)

[§2.4, Lemma 2.7] The application of Lemma 2.7 in the proof of the upper LIL is inconsistent with the lemma as stated. Lemma 2.7 gives E[exp(λ max_{a≤N≤b}|Σ_{n=N}∞ c_n φ^{(n)}_*|)] ≤ 8 exp((λ²/2)Σ_{n=a}∞ c_n²), whereas the proof uses the bound 8 exp(λ² s_{N_k}²/2 − λu). Since s_N² = (1/12)Σ_{n=N}∞ c_n², the lemma literally gives 8 exp(6λ² s_{N_k}² − λu). With λ = (1+ε)φ(s²_{N_{k+1}})/s²_{N_k} and u = (1+ε)φ(s²_{N_{k+1}}), the exponent becomes 5(1+ε)²φ²/s²_{N_k} > 0, so the bound does not decay and the Borel--Cantelli argument collapses. The needed inequality, with variance proxy s_a² instead of Σ_{n=a}∞ c_n², is not a consequence of the cited argument from Azuma [3]; a direct bounded-martingale-difference argument gives only an exponent controlled by (3/2)λ² s_a², still three times too large. The upper half of Theorem 1.1(iv) therefore needs either a proof of a stronger maximal inequality under condition (1.6) or a different argument.
[Remark 1.2] The claimed corollary in Remark 1.2 is false as stated. For c_n = n^{-α} with α > 1, conditions (1.6) and (1.7) hold, so the second half of Theorem 1.1(iv) gives limsup_N ± M_N/φ(s_N²) = 1 a.s. Hence along a subsequence |M_N| / Σ_{n=N}∞ c_n² = |M_N|/(12 s_N²) ∼ φ(s_N²)/(12 s_N²) → ∞, not 0. This directly contradicts the asserted limit in Remark 1.2. The remark should be removed or replaced with a statement using the correct normalization, such as a bound involving φ(s_N²).
[Lemma 2.3] Lemma 2.3 is a load-bearing exponential inequality for the strong law in Theorem 1.1(ii), but it is stated with the note 'whose proof is quite similar to the original one and is omitted.' Because condition (1.5) involves the exact exponential rate, the proof should be supplied or a precise reference given. A standard Azuma/Hoeffding argument does yield a bound of this type, so the issue is likely fixable, but as written the SLLN relies on an unproved statement.

minor comments (3)

[§2.4, equation (2.8)] The asymptotic relations 1/s²_{N_k} ∼ p^k and s²_{N_{k+1}}/s²_{N_k} → 1/p are stated without derivation; they follow from (1.6), (2.7), and the definition of s_N², but a short explanatory sentence would improve readability.
[§2.4, paragraph before Lemma 2.7] The indexing in the definition of the forward martingale (Y_n, G_n) should be made explicit, since n ranges from 0 to b-a and the indices of M_N and T_N decrease as n increases; the current one-sentence description is correct but terse.
[Example 1.4] The statement that condition (1.7) holds only when 0 < β < 1/2 is correct, but the summability threshold could be stated more explicitly by noting that the relevant series behaves like Σ_N N^{-2+2β}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all main results follow from exact identities and external martingale/ergodic theorems; no fitted parameters or self-citations are load-bearing.

full rationale

The derivation chain is self-contained. Theorem 1.1(i) follows from an explicit identity: E[(MN/mN)^2] = (1/12)sum c_n^2 / ((1/2)sum c_n)^2, so convergence is literally equivalent to condition (1.4). Theorem 1.1(ii) uses the Azuma-type exponential inequality stated as Lemma 2.3; Theorem 1.1(iii) is obtained by verifying the hypotheses of Hall and Heyde's reverse-martingale CLT with explicit variance computations; Theorem 1.1(iv) is derived from Azuma's maximal inequality (Lemma 2.7) and the Scott-Huggins reverse-martingale LIL (Lemma 2.8). Theorem 1.7 is an exact algebraic identity, Lemma 3.1, combined with the measure-preserving property of the dyadic map and Birkhoff's ergodic theorem. No parameter is fitted and no claimed conclusion is assumed in its own proof. The paper reuses Kono's reverse-martingale representation and multiplicative-system facts, but these are independently published results, not author self-citations, and they do not smuggle in the target rates. The manuscript does contain two omitted proofs: Lemma 2.3 is said to follow by a proof 'quite similar to the original one', and Lemma 2.7 is said to follow 'by a similar argument to Lemma 2 in [3]'. These omissions are completeness/correctness concerns, not circularity; they do not make the derived statements equivalent to their inputs by construction. In particular, the skeptical observation that Lemma 2.7's stated exponent uses sum c_n^2 while its application uses s_N^2 would point to a possible proof gap, but such a gap is not a circular reduction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper uses no fitted constants or invented entities. It does rely on several quoted or unproved martingale and ergodic theorems; the most fragile is Lemma 2.3, an omitted proof of the exponential inequality that directly supports the strong law.

assumptions (7)

standard math Reverse martingale structure of (M_N, T_N) (Kono, [9], Theorem 1(vi))
The paper's probabilistic limit theorems are applied to M_N as a reverse martingale; this structural fact is quoted from Kono [9] and not re-proved.
ad hoc to paper Azuma-type exponential inequality for reverse martingales with bounded differences (Lemma 2.3)
Stated without proof in Section 2.2; used to derive the exponential bound that yields the SLLN via Borel-Cantelli. The paper says the proof is similar to Azuma [3] and omits it.
ad hoc to paper Maximal exponential inequality for the partial maxima of the tail sum (Lemma 2.7)
Adapted from Azuma [3], Lemma 2; proof is only sketched. Used to get the limsup part of the LIL.
standard math Reverse martingale CLT of Hall and Heyde (Corollary 3.4)
Used as a black box in Section 2.3 to derive the CLT; conditions (2.1) and (2.2) are verified.
standard math Reverse martingale LIL of Scott and Huggins (Theorem 6), specialized to deterministic norming in Lemma 2.8
Quoted and adapted with a deterministic norming sequence; Remark 2.9 explains the simplification. The conditions (2.9)-(2.12) are verified.
standard math Birkhoff and von Neumann ergodic theorems for the dyadic transformation
Used in Section 3 to establish Cesaro convergence of the normalized tails for geometric coefficients.
standard math Tail version of Kronecker's lemma (Heyde [8], Lemma 1(ii))
Used at the end of Section 2.4 to show the almost-sure convergence of Q_N^2/s_N^2 to 1.

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Pith. "Pith review of On the rate of convergence for Takagi class functions." pith.science (2026). https://pith.science/paper/TPLOLJQP

@misc{pith2026190806686,
  author       = {Pith},
  title        = {Pith review of: On the rate of convergence for Takagi class functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPLOLJQP}},
  note         = {Machine review of arXiv:1908.06686}
}

read the original abstract

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[3]

Azuma, K. (1967). Weighted sums of certain dependent ran dom variables, Tˆ ohoku Math. J. (2) , 19, 357–367

work page 1967
[1]

Allaart, P. C. (2009). On a ﬂexible class of continuous fu nctions with uniform local structure, J. Math. Soc. Japan , 61, 237–262

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[2]

Allaart, P. C. and Kawamura, K. (2011/2012). The Takagi f unction: a survey, Real Analysis Exchange , 37, 1–54

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Billingsley, P. (1965). Ergodic theory and information , John Wiley & Sons, Inc

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Gamkrelidze, N. G. (1990). On a probabilistic propertie s of Takagi’s function, J. Math. Kyoto Univ. , 30, 227–229

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Hall, P. and Heyde, C. C. (1980). Martingale limit theory and its application, Prob- ability and Mathematical Statistics , Academic Press

work page 1980
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and Yamaguti, M

Hata, M. and Yamaguti, M. (1984). The Takagi function and its generalization, Japan J. Appl. Math. , 1, 183–199

work page 1984
[8]

Heyde, C. C. (1977). On central limit and iterated logari thm supplements to the martingale convergence theorem, J. Appl. Probab. , 14, 758–775

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[9]

Kˆ ono, N. (1987). On generalized Takagi functions, Acta Math. Hungar. , 49, 315–324

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Scott, D. J. and Huggins, R. M. (1983). On the embedding o f processes in Brownian motion and the law of the iterated logarithm for reverse mart ingales, Bull. Austral. Math. Soc. , 27, 443–459

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Yamaguti, M. and Hata, M. (1983). Weierstrass’s functi on and chaos, Hokkaido Math. J., 12, 333–342. 16 SHOTO OSAKA AND MASATO TAKEI Graduate School of Engineering Science, Yokohama National University E-mail address : osaka-shoto-jc@ynu.jp Department of Applied Mathematics, F...

1983

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

Azuma, K. (1967). Weighted sums of certain dependent ran dom variables, Tˆ ohoku Math. J. (2) , 19, 357–367

work page 1967

[2] [1]

Allaart, P. C. (2009). On a ﬂexible class of continuous fu nctions with uniform local structure, J. Math. Soc. Japan , 61, 237–262

work page 2009

[3] [2]

Allaart, P. C. and Kawamura, K. (2011/2012). The Takagi f unction: a survey, Real Analysis Exchange , 37, 1–54

work page 2011

[4] [4]

Billingsley, P. (1965). Ergodic theory and information , John Wiley & Sons, Inc

work page 1965

[5] [5]

Gamkrelidze, N. G. (1990). On a probabilistic propertie s of Takagi’s function, J. Math. Kyoto Univ. , 30, 227–229

work page 1990

[6] [6]

and Heyde, C

Hall, P. and Heyde, C. C. (1980). Martingale limit theory and its application, Prob- ability and Mathematical Statistics , Academic Press

work page 1980

[7] [7]

and Yamaguti, M

Hata, M. and Yamaguti, M. (1984). The Takagi function and its generalization, Japan J. Appl. Math. , 1, 183–199

work page 1984

[8] [8]

Heyde, C. C. (1977). On central limit and iterated logari thm supplements to the martingale convergence theorem, J. Appl. Probab. , 14, 758–775

work page 1977