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Cram\'er-Type Moderate Deviations for Engel's Series via a Martingale Approach

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Pith's one-line read A martingale decomposition gives Cramér-type moderate deviation expansions for Engel series without rate restrictions.

desk verdict They get the full-range moderate deviation principle for Engel series by martingale decomposition, removing the prior lower-rate restriction, but the conditions on moments and quadratic variation need explicit verification. read the letter →

arxiv 2606.18866 v1 pith:65U5Q6WN submitted 2026-06-17 math.PR math.NT

Shaochen Wang , Guangyu Yang This is my paper · ORCID
classification math.PRmath.NT
keywords EngelseriesmoderatedeviationsmartingaledecompositionCramér-typeexpansionBerry-Esseenbounddigitsprobabilitytheory
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 authors show that for a uniform random x in (0,1), the Engel digits q_n satisfy a moderate deviation expansion for (log q_n - n) scaled by sqrt n. This is proved via a martingale decomposition to which standard asymptotic results apply. As a result, the moderate deviation principle holds over the entire range of scales from the central limit theorem to the law of large numbers. The paper also obtains a uniform Berry-Esseen bound of order (log n)/sqrt n. A reader would care because this removes the lower rate restriction that limited earlier results on such expansions.

What carries the argument

Martingale decomposition of the sequence (log q_n - n) allowing direct application of martingale moderate-deviation and Berry-Esseen asymptotics.

What would settle it

Compute the empirical distribution of (log q_n - n)/sqrt n for large n and check if the tail probabilities align with the predicted Cramér-type expansion, such as the Gaussian tail adjusted by moderate deviation factors.

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Extended reading notes

Core claim

Let x be uniformly distributed on (0,1), and let (q_n) be the digits of its Engel series expansion. We establish a Cramér-type moderate deviation expansion for (log q_n - n)/sqrt n. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry--Esseen bound of order (log n)/sqrt n.

Load-bearing premise

The Engel digit process allows a martingale decomposition that meets the moment and dependence conditions needed for the standard martingale moderate deviation theorems to apply.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The paper claims to prove a Cramér-type moderate deviation expansion for (log q_n - n)/√n, where q_n denotes the nth Engel digit of a uniform random variable x on (0,1). The argument proceeds by constructing a martingale decomposition of the centered log-digit process and invoking standard martingale moderate-deviation and Berry–Esseen theorems. As corollaries the authors obtain a moderate deviation principle on the full scale range between the CLT and the LLN (removing a lower-rate restriction present in earlier works) together with a uniform Berry–Esseen bound of order (log n)/√n.

Significance. If the martingale conditions are verified, the result would strengthen the existing moderate-deviation theory for Engel expansions by extending the admissible range of deviation scales and improving the rate in the Berry–Esseen estimate. The martingale approach supplies a transparent way to exploit the underlying dependence structure and may be adaptable to other digit processes arising in ergodic number theory.

major comments (1)
  1. [Martingale decomposition and application of limit theorems (likely §3)] The central claim rests on the direct applicability of martingale moderate-deviation and Berry–Esseen theorems to the constructed decomposition of log q_n - n. The Engel digit process possesses a recursive dependence (each digit is a function of the fractional part remaining after the preceding step). Consequently, the paper must explicitly establish that the martingale differences possess moments of order greater than 2 that are uniformly bounded in n and that the normalized quadratic variation converges in probability to a positive deterministic limit. These verifications are load-bearing for the invocation of the cited limit theorems and are not automatic from the decomposition alone.
minor comments (1)
  1. [Introduction] Clarify in the introduction the precise improvement over the lower-rate restrictions appearing in the cited earlier works on Engel moderate deviations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and the positive assessment of the significance of the martingale approach. We address the single major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Martingale decomposition and application of limit theorems (likely §3)] The central claim rests on the direct applicability of martingale moderate-deviation and Berry–Esseen theorems to the constructed decomposition of log q_n - n. The Engel digit process possesses a recursive dependence (each digit is a function of the fractional part remaining after the preceding step). Consequently, the paper must explicitly establish that the martingale differences possess moments of order greater than 2 that are uniformly bounded in n and that the normalized quadratic variation converges in probability to a positive deterministic limit. These verifications are load-bearing for the invocation of the cited limit theorems and are not automatic from the decomposition alone.

    Authors: We agree that the recursive dependence requires explicit verification of the moment and quadratic-variation conditions before the standard martingale theorems can be invoked. While the decomposition itself appears in Section 3, the uniform (2+δ)-moment bounds and the in-probability convergence of the normalized quadratic variation are only indicated via references to the underlying ergodic properties rather than proved in full detail. In the revised manuscript we will insert a new subsection (3.2) that (i) derives the uniform moment bound from the explicit tail estimate P(q_n ≥ k) ≤ C/k (which follows directly from the uniform distribution of the remainder) and (ii) proves convergence of the quadratic variation by applying the ergodic theorem to the invariant measure of the Engel map, thereby justifying the application of the cited moderate-deviation and Berry–Esseen results for martingales. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; applies external martingale theorems to constructed decomposition

full rationale

The paper constructs a martingale decomposition for (log q_n - n) and directly invokes standard martingale moderate-deviation and Berry-Esseen theorems. These are external results independent of the present work. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation chain is self-contained against external benchmarks, consistent with a score of 0.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the existence of a suitable martingale decomposition for the Engel digit process and on the applicability of general martingale moderate-deviation theorems; no free parameters or new entities are introduced.

assumptions (1)
  • domain assumption The Engel digit process admits a martingale decomposition whose increments satisfy the conditions of standard martingale moderate-deviation and Berry-Esseen theorems.
    Invoked to justify applying asymptotic martingale results to obtain the claimed expansion and bound.

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Cite this review

Pith. "Pith review of Cram\'er-Type Moderate Deviations for Engel's Series via a Martingale Approach." pith.science (2026). https://pith.science/paper/65U5Q6WN

@misc{pith2026260618866,
  author       = {Pith},
  title        = {Pith review of: Cram\'er-Type Moderate Deviations for Engel's Series via a Martingale Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/65U5Q6WN}},
  note         = {Machine review of arXiv:2606.18866}
}
abstract

Let $x$ be uniformly distributed on $(0,1)$, and let $(q_n)_{n\geq1}$ be the digits of its Engel series expansion. We establish a Cram\'er-type moderate deviation expansion for $(\log q_n-n)/\sqrt n$. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry--Esseen bound of order $(\log n)/\sqrt n$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Functional Limit Theorems for Random Least Common Multiples

    math.PR 2026-07 accept novelty 6.5 of 10

    For fixed retention probability θ, the process log L_⌊nt⌋ obeys a functional LDP with geometric-mark entropy rate, an MDP with the CLT Gaussian RKHS rate, and a Strassen LIL with that unit ball as cluster set.

Reference graph

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