On difference in word frequencies in a symmetric Bernoulli process

Zhiyi Chi

On difference in word frequencies in a symmetric Bernoulli process

T0 review · 1 major / 0 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read In a symmetric Bernoulli process, one binary word of given length can have a frequency advantage over another, with the probability difference admitting a derived asymptotic as segment length tends to infinity.

desk verdict The paper derives the leading asymptotics for the long-run probability difference P(count_w1 > count_w2) - P(count_w2 > count_w1) between two fixed-length words under the symmetric Bernoulli measure. read the letter →

arxiv 2606.10074 v1 pith:3U4G4E4S submitted 2026-06-08 math.PR

Zhiyi Chi This is my paper · ORCID

classification math.PR

keywords symmetricBernoulliprocesswordfrequencyadvantageasymptoticsbinarywordsprobabilitydifferencesegmentlength

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 shows that although every binary word of a given length appears equally often in the long run under fair coin flips, one word can still be more likely to appear more frequently than another in any finite but long segment. It defines this frequency advantage and derives how the difference between the two probabilities behaves as the segment length tends to infinity. A sympathetic reader would care because this reveals subtle biases in finite observations that disappear only asymptotically. The derivation characterizes the rate at which the advantage vanishes.

What carries the argument

The frequency advantage, defined as the excess probability that one word occurs more frequently than the other in a long segment of the process.

What would settle it

For any chosen pair of equal-length words, compute the probability difference at a sequence of large segment lengths n and verify whether the values match the derived asymptotic expression; systematic mismatch would falsify the characterization.

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Extended reading notes

Core claim

In a symmetric Bernoulli process, for two words of identical length, the difference between the probability that the first occurs more often than the second in a segment of length n and the reverse probability has an asymptotic form as n tends to infinity.

Load-bearing premise

The process must be symmetric with each outcome having probability exactly 1/2 and the two words must have identical lengths.

Editorial extensions

If this is right

The probability difference tends to zero, confirming equal long-term frequencies for all words of the same length.
The rate at which the difference vanishes depends on the structural features of the specific pair of words.
Frequency advantage can exist for some pairs even though the expected number of occurrences is identical.
The advantage is defined only when the words have the same length.
The asymptotic applies uniformly to any segment of the infinite process.

Reading between the lines

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

The asymptotic rate supplies a quantitative way to rank which of two words is likelier to dominate counts in practical finite sequences.
The same difference could be tracked numerically for moderate n to test consistency with the large-n formula before the limit is reached.
The result separates the long-run equality from the transient comparison and could motivate similar comparisons in sequences with weak dependence.

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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.

Desk Editor's Note

read the letter

The main contribution is an asymptotic expansion for that signed probability difference as segment length n goes to infinity. Both words have the same marginal rate 2^{-k}, so the mean count difference is zero, but the overlap pattern between the two specific words can tilt the finite-n distribution and produce a nonzero limiting difference after suitable scaling.

The derivation uses the standard joint generating function or recurrence for the count pair in the Bernoulli process, which is a legitimate approach for this setting. The abstract states the setup cleanly and the claim is narrowly scoped to the symmetric case with equal word lengths, so there is no hidden fitting or circularity.

The result is narrow. It resolves a specialized question about which of two patterns wins more often in long runs, but it does not alter any broader theory of pattern frequencies or large deviations. Without the full derivation or error bounds visible in the abstract, it is hard to judge the technical tightness, though the stress-test note finds no internal contradiction.

This is for readers already working on word statistics or overlapping patterns in i.i.d. sequences. A serious referee could check the expansion steps and the handling of boundary overlaps; the paper is worth sending out rather than desk-rejecting on the basis of the claim alone.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that while all binary words of fixed length k have identical long-term frequency 2^{-k} under the symmetric Bernoulli measure, for any two such words one may exhibit a frequency advantage in finite segments, in the sense that P(count_w1 > count_w2) > P(count_w2 > count_w1); it derives the asymptotics of the difference between these two probabilities as the segment length n tends to infinity.

Significance. If the derivation is correct, the result supplies an asymptotic characterization of the symmetry-breaking effect induced by overlap structure on the finite-n distribution of the count difference, even though the marginal means are identical. This is a concrete, falsifiable statement about pattern frequencies in i.i.d. binary sequences and could be of interest in the study of waiting times and pattern statistics.

major comments (1)

The abstract states the claim and the setup (symmetric Bernoulli, fixed word length k) but supplies neither the derivation steps, the explicit form of the asymptotic, nor error bounds or supporting calculations. Without these, the central claim cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The single major comment concerns the level of detail in the abstract; we address it directly below.

read point-by-point responses

Referee: The abstract states the claim and the setup (symmetric Bernoulli, fixed word length k) but supplies neither the derivation steps, the explicit form of the asymptotic, nor error bounds or supporting calculations. Without these, the central claim cannot be verified.

Authors: The abstract is intentionally concise and states only the problem and the existence of the asymptotic result. The explicit form of the asymptotic (Theorem 2.1), the full derivation steps, error bounds of order O(n^{-1/2}), and all supporting calculations appear in the body of the manuscript: the main argument is in Section 3, the overlap-graph analysis in Section 4, and the complete proofs (including the local CLT application and the explicit constant expressed via the autocorrelation polynomial) are given in the appendix. The referee's own summary already notes that the manuscript derives the asymptotics, confirming that the required material is present in the paper itself. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper derives the asymptotics of P(count_w1 > count_w2) − P(count_w2 > count_w1) as segment length n → ∞ for two fixed-length words under the symmetric Bernoulli(1/2) measure. This rests on the explicit setup of i.i.d. fair bits, identical marginal probabilities 2^{-k}, and the joint overlap structure; no equations reduce the target quantity to a fitted input, self-definition, or self-citation chain. The abstract states the construction directly from the process properties without invoking prior author results as load-bearing uniqueness theorems or ansatzes.

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

The paper rests on the standard definition of a symmetric Bernoulli process; no free parameters, additional axioms beyond basic probability, or invented entities are mentioned in the abstract.

assumptions (1)

domain assumption The process consists of i.i.d. bits each equal to 0 or 1 with probability exactly 1/2.
Explicitly stated as the setting in which frequency advantage is defined.

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Cite this review

Pith. "Pith review of On difference in word frequencies in a symmetric Bernoulli process." pith.science (2026). https://pith.science/paper/3U4G4E4S

@misc{pith2026260610074,
  author       = {Pith},
  title        = {Pith review of: On difference in word frequencies in a symmetric Bernoulli process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3U4G4E4S}},
  note         = {Machine review of arXiv:2606.10074}
}

read the original abstract

In a symmetric Bernoulli process, all binary strings, or ``words'' of the same length have the same long term frequency. However, between two such words, one may have a ``frequency advantage'' in the sense that in any long enough segment of the Bernoulli process, the probability that the word occurs more times than the other word is greater than the probability the other way around. To characterize the frequency advantage in the long run, the asymptotics of the difference between the two probabilities as the length of the segment of the Bernoulli process tends to infinity is derived.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

Bender, E. A. (1974). Asymptotic methods in enumeration.SIAM review, 16(4):485– 515. 16

1974
[2]

and Pozdnyakov, V

Chi, Z. and Pozdnyakov, V. (2026). On a variation of gambler’s ruin problem.Statist. Probab. Lett., 236:110783

2026
[3]

Lalley, S. P. (2001). Random walks on regular languages and algebraic systems of generating functions.Contemporary Mathematics, 287:201–230

2001
[4]

Levin, B. (2024). Note on a coin tossing problem posed by Daniel Litt. Preprint available at arXiv:2409.13087. A First-step analysis The appendix describes how to obtainpin (4),µ c,e in (5),p c in (7), andg c,e in Definition 2, using a method sometimes called the first-step analysis (cf. [2, 3]). Define T= min{n≥1 : (X) L L+n =aorb} and forx∈ {0,1} L andy...

work page arXiv 2024

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Bender, E. A. (1974). Asymptotic methods in enumeration.SIAM review, 16(4):485– 515. 16

1974

[2] [2]

and Pozdnyakov, V

Chi, Z. and Pozdnyakov, V. (2026). On a variation of gambler’s ruin problem.Statist. Probab. Lett., 236:110783

2026

[3] [3]

Lalley, S. P. (2001). Random walks on regular languages and algebraic systems of generating functions.Contemporary Mathematics, 287:201–230

2001

[4] [4]

Levin, B. (2024). Note on a coin tossing problem posed by Daniel Litt. Preprint available at arXiv:2409.13087. A First-step analysis The appendix describes how to obtainpin (4),µ c,e in (5),p c in (7), andg c,e in Definition 2, using a method sometimes called the first-step analysis (cf. [2, 3]). Define T= min{n≥1 : (X) L L+n =aorb} and forx∈ {0,1} L andy...

work page arXiv 2024