Moments of Maximum: Segment of AR(1)

Steven Finch

REVIEW 3 major objections 4 minor 25 references

Moments of Maximum: Segment of AR(1)

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For segments of length at least three from a stationary first-order autoregressive process, the expected maximum has an interior peak in the lag-one correlation, and the variance is strictly increasing.

desk verdict Useful decoding-and-correction note on Afonja's Gaussian-maximum moment formulas, with real benchmark value for short AR(1) segments; the unproved variance monotonicity and the omitted E(M5) need to be flagged before the numbers are used. read the letter →

arxiv 1908.04179 v1 pith:VVD2PK5G submitted 2019-08-12 math.HO math.STstat.TH

Steven Finch This is my paper · ORCID

classification math.HOmath.STstat.TH MSC 62E1560G7062M10

keywords AR(1)processexpectedmaximumvariancemonotonicitylag-onecorrelationmultivariatenormalpartialgoldenratioconjugateorderstatistics

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 studies five or fewer consecutive observations of a stationary first-order autoregressive Gaussian process and asks how the moments of their maximum $M_\ell$ depend on the lag-one correlation $\rho$. It shows that the expected maximum $\mathbb{E}(M_\ell)$ is not monotone in $\rho$: for $\ell \ge 3$ it has an interior maximum at a negative value, and for $\ell = 3$ the maximizer is exactly $\rho = (1-\sqrt{5})/2$, with the $\ell = 4$ and $\ell = 5$ maximizers given numerically. The second moment $\mathbb{E}(M_\ell^2)$ also attains an interior maximum, but the variance $\mathbb{V}(M_\ell)$ is strictly increasing for all $\rho \in (-1,1)$ when $\ell \ge 3$. These facts matter because they show that short blocks of autocorrelated normal data have their largest expected extremes at negative serial correlation, contrary to the usual worst-case assumption of independence or positive correlation.

What carries the argument

The load-bearing object is the partial-correlation machinery inside a 1972 formula for the first two moments of the maximum of $\ell$ jointly normal variables: the formulas express $\mathbb{E}(M)$ and $\mathbb{E}(M^2)$ as finite sums of terms involving the orthant probabilities $\Phi_{\ell-2}$ and $\Phi_{\ell-3}$ of difference variables with partial correlations, together with arcsin and arccos factors. The paper specifies the matrices $R_{i,j}$ and $R_{i,jk}$ whose entries are those partial correlations, notes that symmetry fails for $\ell \ge 4$, corrects two typographical errors in the source formula, and introduces the function $h(x,y,z)$ to organise the many terms. Substituting $\rho_{ij} = \rho^{|j-i|}$ reduces the $\ell = 4,5$ sums to one-dimensional functions of $\rho$, which are then maximized numerically.

What would settle it

For $\ell = 4$, evaluate $\mathbb{E}(\max\{X_1,\ldots,X_4\})$ for the AR(1) covariance by direct four-dimensional numerical integration on a fine grid of $\rho$, and check whether the maximum occurs at $\rho = -0.4973597615161907364022217\ldots$; repeat for $\ell = 5$ and check whether $\mathbb{V}(M_5)$ increases monotonically over the whole interval. A shift in the maximizing $\rho$ or any local decrease in the variance would refute the central claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the curves $\rho \mapsto \mathbb{E}(M_\ell)$ and $\rho \mapsto \mathbb{E}(M_\ell^2)$ each have an interior maximum for every segment length $\ell \ge 3$, while $\rho \mapsto \mathbb{V}(M_\ell)$ is strictly increasing on $(-1,1)$ for $\ell \ge 3$. The maximizing $\rho$ for $\mathbb{E}(M_3)$ is the golden-ratio conjugate $(1-\sqrt{5})/2 = -0.6180339887498948482045868\ldots$; the maximizing values for $\mathbb{E}(M_4)$ and $\mathbb{E}(M_5)$ are $-0.4973597615161907364022217\ldots$ and $-0.4336476843162656141275672\ldots$. At $\rho \to -1$ the mean maximum tends to $\sqrt{2/\pi}$, the mean of a standard half-normal variable, and at $\rho = 1$ it vanishes. The paper achieves this by decoding and correcting a 1972 formula for the moments of the maximum of correlated normal variables, specializing it to the Toeplitz covariance $\rho^{|j-i|}$ of an AR(1) process, and verifying the $\ell = 3$ expectation from first principles.

Load-bearing premise

The reported maximizers for $\ell = 4$ and $\ell = 5$, and the monotonicity of the variance for $\ell \ge 4$, rest on the accuracy of the 1972 general moment formulas after the paper's two typo corrections, because the paper does not re-derive those formulas from first principles for those lengths.

Editorial extensions

If this is right

For any stationary AR(1) Gaussian series, a block of three to five consecutive observations is expected to be largest when the lag-one correlation is negative, because negative correlation spreads the values apart.
The variance of the block maximum is strictly increasing in $\rho$ for $\ell \ge 3$, so stronger positive serial correlation widens the spread of the maximum even as its mean decreases.
The length-three case yields the exact interior maximizer $\rho = (1-\sqrt{5})/2$, a rare closed-form extremum for an expectation over correlated normals.
With the corrected formulas, the first two moments of the block maximum can be computed for $\ell$ up to six without simulation, and the $\ell = 3$ expectation is verified from first principles.

Reading between the lines

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

A natural numerical extension is to compute the interior maximizer for $\ell \ge 6$; the large-segment extreme-value theorem cited in the paper does not put $\rho$ into its centering constant, so the location of the maximum may be a purely finite-length effect.
The strict monotonicity of the variance invites a coupling or convex-order proof: pairing AR(1) processes with different $\rho$ values might show that the block maximum for larger $\rho$ is larger in convex order, which would supply the intuitive explanation the paper leaves open.
For worst-case analyses that use short blocks of autocorrelated normal data, the riskiest serial correlation is negative rather than independence or positive correlation, so scanning $\rho < 0$ could change reported worst-case maxima in timing or finance applications.

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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 is a small but honest piece of cleanup work on Afonja's 1972 formulas for the moments of the maximum of correlated normals. Finch decodes the formulas, corrects several typos, applies them to short AR(1) segments, and records the specific ρ values that maximize E(M_ℓ) for ℓ = 3, 4, 5. It also notes — without proof — that Var(M_ℓ) increases in ρ for ℓ ≥ 3.

The genuinely useful parts are the corrections to Afonja (√(2π) in E(M), a missing factor in E(M) for ℓ = 4, a missing constant 1 and a misstated summation in E(M^2) for ℓ = 5/6), the algebraic equivalence check in Section 4, and the first-principles derivation of E(M_3) in Section 5 using the max-of-maxes identity. The ρ → 0 limits match known independence values, which is a good sanity check. The specific maximizer values and the variance monotonicity observation are new, even if they are direct evaluations of known formulas.

The soft spots are real, and they are mostly about what is not shown. Section 3's displayed formula for E(M_3) has a typo — the second term is written √(1−ρ^2)/(4π) where it should be √(1−ρ^2)/(2√π). For a note whose main service is correcting other people's typos, that is unfortunate. More substantively, the ℓ = 5 maximizer is reported to six decimals without ever writing E(M_5), which is dismissed as 'too lengthy.' That means the central numerical claim is not independently checkable from the manuscript. The same is true of E(M^2_5) and E(M^2_6). The variance monotonicity is drawn from a figure and explicitly left unproved; it should be labeled a numerical conjecture. The stress-test concern is warranted: limiting checks at ρ → 0 do not validate the ρ-dependence of the formulas, and since the paper does not derive the ℓ = 4, 5 formulas from first principles, an undetected error in Afonja's orthant probabilities would shift the reported maximizers.

None of this kills the paper. It is a modest note, clearly written, with an honest separation of what is proved and what is observed. The fix is simple: put the full ℓ = 5 and ℓ = 6 formulas in an appendix or a supplementary file, fix the E(M_3) typo, and mark the variance claim as a conjecture. After that, I would be comfortable with it as a published short note. I would send it to a referee, not because the result is revolutionary, but because someone should check the arithmetic and the transcription before these numbers get cited as benchmarks.

Referee Report

3 major / 4 minor

Summary. This note studies the maximum M_ℓ of ℓ consecutive observations of a stationary AR(1) process with standard normal white noise. It specializes Afonja's (1972) formulas for the moments of the maximum of correlated normal variables to the Toeplitz correlation structure ρ_{ij}=ρ^{|j−i|}, obtaining explicit expressions for E(M_ℓ) and E(M_ℓ^2) for ℓ≤5 (with ℓ=6 for the second moment). The paper reports that E(M_ℓ), for ℓ≥3, is maximized at an interior negative value of ρ, with closed form ρ=(1−√5)/2 for ℓ=3 and numerical values for ℓ=4,5; that E(M_ℓ^2) also has interior maxima; and that Var(M_ℓ) is strictly increasing in ρ∈(−1,1) for ℓ≥3. It includes an algebraic proof of the equivalence of its E(M^2) formula to Afonja's, a first-principles derivation of E(M_3), and limiting checks at ρ=0.

Significance. If the reported results are correct, the paper gives a complete answer to a natural question about maxima of short AR(1) segments and highlights an interesting contrast: the mean has an interior maximum at a negative serial correlation, while the variance is monotone. The independent derivation of E(M_3) in Section 5 and the algebraic equivalence proof in Section 4 are valuable, as are the checks against known independent-normal limits. However, the results for ℓ=4 and ℓ=5 depend on external formulas that are only partially reproduced, and the variance-monotonicity claim is unsupported. The paper is a useful note but currently falls short of being self-contained.

major comments (3)

[Section 3, displayed formula for E(M_3)] The displayed formula for E(M_3) is incorrect as printed: it reads √(1−ρ)/π + √(1−ρ^2)/(4π), but the correct value, obtained from the Section 2 formula by substituting ρ12=ρ23=ρ and ρ13=ρ^2, is √((1−ρ)/π) + (1/2)√((1−ρ^2)/π). This is a load-bearing typo in a central formula, even though the limiting value 3/(2√π) at ρ=0 is consistent with the corrected expression.
[Section 3, paragraph 'E(M5) is too lengthy to record here'] The paper reports the ℓ=5 maximizer of E(M_5) to 25 decimal places but never records E(M_5) or E(M_5^2), stating only that they are 'too lengthy.' Since the claimed maximizer is computed from Afonja's formulas, and since the paper itself corrects two typographical errors in Afonja's paper, the reader cannot verify the ℓ=5 result or rule out further undetected errors. To make the central numerical claim reproducible, the full formulas for ℓ=5 (or an evaluation script with clear definitions of the partial correlations) must be included.
[Section 3, Figure 2 and the paragraph following it] The claim that Var(M_ℓ) is strictly increasing in ρ for all ℓ≥3 is one of the two central questions in the abstract, but it is supported only by visual inspection of Figure 2. The sentence 'An intuitive reason for such behavior would be good to establish someday' acknowledges that no proof is offered. Since closed-form formulas for ℓ=3 and ℓ=4 are available in Section 3, the authors should either provide a proof (at least for these cases) or clearly label the monotonicity as a numerical conjecture.

minor comments (4)

[Section 5, first displayed identity] The identity 'max {X1, X2, X2} = max {max {X1, X2}, max {X2, X3}}' contains an obvious typo: the left-hand side should be max {X1, X2, X3}.
[Section 1, definitions of sets] Several set-builder notations use mismatched parentheses, e.g., '( i, j, m, n} = {1, 2, 3, 4}' and '( i, j, m, n, o} = {1, 2, 3, 4, 5}'. These should be written with matching braces.
[Section 2, general formulas] The notation 'Φ ℓ−3(Ri,,jk)' contains a doubled comma and should be 'Φ ℓ−3(R_{i,jk})'.
[Section 4, displayed identity] The expression '1/(2π) ri,ji · ri,ki − ri,jk ri,ji/√(1−r_{i,jk}^2)' is ambiguous because a bracket is missing; the intended form, as used in the following line, is '1/(2π) [r_{i,ji}(r_{i,ki} − r_{i,jk} r_{i,ji}) / √(1−r_{i,jk}^2)]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central formulas come from external Afonja source with independent checks; self-citations are auxiliary.

full rationale

The paper derives its numerical answers from Afonja's 1972 moment formulas, an external source, after correcting typographical errors. The chain is: Afonja formulas -> specialization to AR(1) correlations -> numerical maximizers and variance plots. No step fits a parameter to the claimed output and then renames it a prediction; no definition smuggles the target result into an assumption. The paper also provides independent validation: Section 4 proves algebraic equivalence between its displayed E(M^2) formula and Afonja's, and Section 5 gives a first-principles derivation of E(M3) from max identities and bivariate normal integrals. The limiting checks at rho -> 0 match known independent values, providing further confirmation. Finch's self-citations ([4], [5], [7], [10], [11], [12]) are used only for contextual constants, exercises, or related results and are not load-bearing for the central claims. The reported typo in the Section 3 display of E(M3) and the omission of the lengthy E(M5) expression are reproducibility or correctness concerns, not circularity: they do not make the derivation equivalent to its inputs. Accordingly, no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; all numbers are computed from the stated formulas. The main external input is Afonja's formulas, which the paper corrects but does not re-derive for ℓ ≥ 4. No new entities are introduced.

assumptions (4)

domain assumption Afonja's moment formulas for the maximum of correlated normal variates are correct after the typo corrections in Section 2.
The paper relies on Afonja (1972) as the source of E(M) and E(M^2) formulas; the corrections indicated are not independently derived for ℓ = 4, 5.
domain assumption The AR(1) process is stationary with Gaussian innovations, so the ℓ-tuple has mean zero and covariance matrix with entries ρ^{|j-i|}.
This is the standard definition of the process under study, stated in Section 3.
standard math Berman's extreme value theorem applies to the AR(1) maximum as ℓ tends to infinity.
Used in Section 6 to describe the asymptotic distribution; the conditions hold for AR(1) correlations.
standard math The identity max{X1,X2,X3} = max{max{X1,X2}, max{X2,X3}} holds.
Used in Section 5 for the first-principles derivation of E(M_3).

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Cite this review

Pith. "Pith review of Moments of Maximum: Segment of AR(1)." pith.science (2026). https://pith.science/paper/VVD2PK5G

@misc{pith2026190804179,
  author       = {Pith},
  title        = {Pith review of: Moments of Maximum: Segment of AR(1)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVD2PK5G}},
  note         = {Machine review of arXiv:1908.04179}
}

abstract

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider five contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{5}$). Let $M$ denote the maximum of these. Let $\rho$ be the lag-one serial correlation, which satisfies $|\rho| < 1$. For what value of $\rho$ is $\mathbb{E}(M)$ maximized? How does $\mathbb{V}(M)$ behave for increasing $\rho$? Answers to these questions lie in Afonja (1972), suitably decoded.

Figures

Figures reproduced from arXiv: 1908.04179 by the authors.

**Figure 2.** Linear (with slope 1/π and vertical intercept 1 − 1/π) for ℓ = 2; strictly increasing for ℓ ≥ 3 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages

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Pith tools

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

[1] [1]

Afonja, The moments of the maximum of correlated normal an d t-variates, J

B. Afonja, The moments of the maximum of correlated normal an d t-variates, J. Royal Statist. Soc. Ser. B 34 (1972) 251–262; MR0331637

work page 1972

[2] [2]

Prokhorov, Partial correlation coeﬃcient, Encyclopaedia of Mathematics , Kluwer, 1988; https://www.encyclopediaofmath.org/index.php/Partial correlation coeﬃcient; MR1263687

A.V. Prokhorov, Partial correlation coeﬃcient, Encyclopaedia of Mathematics , Kluwer, 1988; https://www.encyclopediaofmath.org/index.php/Partial correlation coeﬃcient; MR1263687

work page 1988

[3] [3]

K. Baba, R. Shibata and M. Sibuya, Partial correlation and condit ional correla- tion as measures of conditional independence, Austral. New Zealand J. Statist. 46 (2004) 657–664; MR2115961

work page 2004

[4] [4]

Expected n-Step Product for Gaussian Tours

S. Finch, Expected n-step product for Gaussian tours, arXiv:1512.05592

work page Pith review arXiv

[5] [5]

S. R. Finch, Extreme value constants, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 363–367; MR2003519

work page 2003

[6] [6]

H. A. David and H. N. Nagaraja, Order Statistics, 3 th ed., Wiley, 2003, pp. 7 & 54; MR1994955

work page 2003

[7] [7]

Finch, Capturing, ordering and Gaussianity in 2D, arXiv:1601.04 937

S. Finch, Capturing, ordering and Gaussianity in 2D, arXiv:1601.04 937

work page

[8] [8]

J. J. Hunter, Renewal theory in two dimensions: asymptotic res ults, Adv. Appl. Probab. 6 (1974) 546–562; MR0346929

work page 1974