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Persistence probabilities for MA(1) sequences with uniform innovations

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Pith's one-line read This paper gives explicit generating functions for the persistence probability of an MA(1) sequence with uniform innovations, in every parameter region, using deformed exponential functions and duality formulas.

desk verdict The generating-function results are real and mostly correct, but Corollary 4 is false as stated and the paper needs a round of typo fixes before it is publishable. read the letter →

arxiv 2507.04427 v1 pith:QVRBXELH submitted 2025-07-06 math.PR math.CO

Frank Aurzada , Kilian Raschel This is my paper
classification math.PRmath.CO MSC 60G1060J0505A15
keywords persistenceprobabilitymovingaverageprocessMA(1)uniformdistributiongeneratingfunctiondeformedexponentialMallows-Riordanpolynomialsduality
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 the persistence probability $p^a_n(\theta)=P(X_2\ge\theta X_1,\dots,X_{n+1}\ge\theta X_n)$ for i.i.d. innovations uniformly distributed on $[-a,1]$. It establishes that, in each of five regions of the $(a,\theta)$-phase diagram, the generating function $\sum_{n\ge0}p^a_n(\theta)z^n$ has an explicit closed form. In the blue region the formula is $(E(\theta, az/(1+a))-E(\theta, -z/(1+a)))/(zE(\theta, -z/(1+a)))$, where $E(r,z)=\sum_{n\ge0}r^{n(n-1)/2}z^n/n!$ is the deformed exponential function; the green, yellow, orange, and white regions have analogous formulas of their own. The paper also proves duality identities that carry the results to $|\theta|>1$, and it derives persistence exponents as reciprocals of the smallest positive zero of the denominator. A sympathetic reader cares because exact persistence probabilities are rare for nontrivial stochastic processes; here an entire two-parameter family is solved in closed form and tied to combinatorial quantities.

What carries the argument

The central object is the deformed exponential function $E(r,z)=\sum_{n\ge0}r^{n(n-1)/2}z^n/n!$. It arises from integrating the nested volume of the polytope $\{\theta x_i\le x_{i+1}\le1\}$, and it appears in the denominator of every nontrivial generating function; its reciprocal has a purely combinatorial expansion in terms of $\ell$-profiles that the paper works out in Lemma 5. The arguments $az/(1+a)$ and $-z/(1+a)$, together with their $\theta$-scaled variants, encode the asymmetry of the uniform interval $[-a,1]$. Three duality operations—reversing the index order, replacing $\theta$ by $1/\theta$, and swapping $a$ with $1/a$—carry the identities across parameter regions, and the generating-function relation $\hat P_\theta(z)=\hat P_{1/\theta}(-z)/(1-z\hat P_{1/\theta}(-z))$ is the engine that connects the regions.

What would settle it

Compute $p^a_n(\theta)$ exactly for small $n$ by numerical integration or exact polytope volume at a parameter point inside the blue region, say $a=1$, $\theta=1/2$, and compare the coefficients of the power series of the right-hand side of the blue-region identity; a mismatch would refute the identity. To test the exponent claim, locate all zeros of $E(\theta,-z/(1+a))$ in a disk around the origin and check whether the smallest-modulus zero is real and positive; a counterexample would invalidate Corollaries 3, 8, 10, and 11 without touching Theorem 1.

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Extended reading notes

Core claim

The central discovery is that the persistence probability of an MA(1) process with uniform innovations is governed by the deformed exponential function $E(r,z)=\sum_{n\ge0}r^{n(n-1)/2}z^n/n!$. For parameters in the blue region, the generating function identity $\sum_{n\ge0}p^a_n(\theta)z^n=(E(\theta, az/(1+a))-E(\theta, -z/(1+a)))/(zE(\theta, -z/(1+a)))$ holds, and analogous identities hold in the green, yellow, orange, and white regions, with the white region giving $p^a_n(\theta)\equiv1$. Duality relations, including $p^a_n(\theta)=p^{1/a}_n(1/\theta)$ for $\theta>0$ and the generating-function identity $\hat P_\theta(z)=\hat P_{1/\theta}(-z)/(1-z\hat P_{1/\theta}(-z))$, extend these formulas to $|\theta|>1$. In the blue and green regions the coefficients can be rewritten through the combinatorial quantities $\phi_\ell(\theta)$ defined by the series expansion of $1/E(\theta,z)$, and in the special case $a=-\theta$ they are Mallows-Riordan polynomials, the polynomials defined by the logarithmic generating function of the deformed exponential.

Load-bearing premise

The corollaries that turn the generating functions into exponential decay rates assume that the decay rate is controlled by the smallest positive zero of the denominator; this singularity-analysis step is stated but not proved in the paper.

Editorial extensions

If this is right

  • The value $p^a_n(1)=1/(n+1)!$ follows for every $a>-1$, recovering the previously known formula for $\theta=1$.
  • In the blue and green regions the persistence exponent is the reciprocal of the smallest positive zero of $E(\theta,-z/(1+a))$, with the asymptotic form $p^a_n(\theta)\sim (1+a)E(\theta,a/(\lambda(1+a)))/E(\theta,-\theta/(\lambda(1+a)))\cdot\lambda^{n+2}$.
  • In the orange region the generating function is piecewise rational; for $\theta>1$ and $-1/\theta\le a<0$ it is piecewise polynomial, and the probabilities vanish once $\theta^n(-a)\ge1$.
  • For $a=-\theta$ in the blue region, one obtains $p^{-\theta}_n(\theta)=J_{n+2}(\theta)/(n+1)!$, connecting MA(1) persistence to the Mallows-Riordan polynomials studied for autoregressive processes.
  • The duality identities extend all five formulas to $|\theta|>1$, so the phase diagram covers the entire $(a,\theta)$-plane.

Reading between the lines

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

  • The volume-of-polytope reading suggests a direct numerical check that the paper does not report: for small $n$, exact polytope-volume algorithms could verify the generating-function coefficients at specific parameters such as $a=1$, $\theta=1/2$.
  • Because the duality in Lemma 2 is proved for general continuous distributions, the relation $\hat P_\theta(z)=\hat P_{1/\theta}(-z)/(1-z\hat P_{1/\theta}(-z))$ is a transfer principle that likely applies beyond uniform innovations; applying it to other innovation laws is not attempted here.
  • The persistence exponents in the blue and green regions reduce to zeros of the deformed exponential, so any future information about those zeros would immediately sharpen the asymptotic constants, not just the decay rates.
  • The rational-versus-transcendental classification of the generating functions mirrors the quarter-plane-walks dichotomy, and the same classification question could reasonably be asked for moving average processes of higher order.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper computes persistence probabilities p^a_n(θ) = P(X_2 ≥ θ X_1, …, X_{n+1} ≥ θ X_n) for an MA(1)-type condition when the i.i.d. innovations are uniform on [−a, 1]. The main theorem gives explicit generating functions in five parameter regions (blue, green, yellow, orange, white), expressed through the deformed exponential function E(θ, z), and also states dualities that extend the results to |θ| > 1. The proofs use integral recursions, conditioning arguments, dualities valid for general continuous distributions, and a combinatorial expansion for the reciprocal 1/E(θ, z). Several corollaries give persistence exponents and representations via Mallows–Riordan polynomials or the combinatorial coefficients φ_ℓ(θ).

Significance. If the central identities hold, the paper is a valuable contribution: explicit formulas for persistence probabilities of this MA(1) model were previously known only in special cases, and the five-region phase diagram with rational versus transcendental generating functions is elegant and potentially useful. The main derivations are careful, self-contained, and checkable, and the dualities in Lemmas 1–3 are of independent interest. However, the paper currently contains a false corollary (Corollary 4) and the persistence-exponent statements are asserted without a complete proof in the case where the relevant deformed exponentials have sign-changing coefficients. The core generating-function identities of Theorem 1 appear sound, but these secondary claims need correction before the paper can be accepted.

major comments (2)
  1. [§4.2, Corollary 4] Corollary 4 is false as stated. It claims p^{−θ}_n(θ) = J_{n+2}(θ)/(n+1)! for θ ∈ [0, 1]. For θ ∈ [0, 1] and a = −θ, the innovation distribution is uniform on [θ, 1], so θX_i ≤ θ ≤ X_{i+1} for all i and the persistence probability is identically 1. This is not equal to J_{n+2}(θ)/(n+1)! in general; for example θ = 1/2 and n = 1 gives J_3(1/2)/2! = (5/2)/2 = 5/4. The proof applies the blue-region formula (16), but its hypotheses are not satisfied for θ ∈ [0, 1], a = −θ. The intended domain is evidently θ ∈ [−1, 0], for which a = −θ is the lower boundary of the blue region and the identity follows from (16). The statement and proof must be corrected accordingly.
  2. [Remark 7; Corollaries 3, 8, 10, 11] The persistence-exponent corollaries are asserted without a complete proof. In the blue and green regions with θ ∈ [−1, 0), the coefficients of E(θ, ·) are not nonnegative, so the numerator E(θ, a z/(1+a)) (blue) or E(θ, −z/(θ(1+a))) (green) evaluated at a positive zero of E(θ, −z/(1+a)) is not automatically nonzero. Pringsheim's theorem applied to the full generating function ensures that the radius of convergence is a positive real singularity, but it does not by itself identify that singularity with the smallest positive zero of the denominator when numerator cancellation is possible. The paper needs either a short proof of the non-cancellation and dominant-zero property, or a supporting reference from the theory of deformed exponential zeros. This is needed to justify the stated formulas for the persistence exponent in these regimes.
minor comments (6)
  1. [Theorem 1(O), Eq. (5); Lemma 9, Eq. (22)] The definition x_+ := min(x, 0) is a typo; it should be x_+ := max(x, 0). The displayed formulas use positive parts of quantities such as (1/θ^n − b), which are nonnegative under the stated hypotheses, so min would make the expressions identically zero in some cases.
  2. [Corollary 1 proof] In the proof of Corollary 1, the displayed chain `¯θ(−X1/a) ⩽ (−X2/a), …, ¯θ(−Xn/a) ⩾ (−Xn+1/a)` mixes ≤ and ≥ signs. All inequalities should be ≤ (or the notation should be made consistent) so that the expression matches the definition of p^{1/a}_n(θ).
  3. [Lemma 3 proof] The sentence 'We proceed exactly as in the proof of Lemma 8' refers to a lemma that is defined later in the paper; it should refer to Lemma 2, whose proof is the one being adapted.
  4. [Lemma 9] The condition `1 ⩽ θ ⩽ −1/a` is not well-defined when a = 0. The case b = 0 should be stated separately, or the right endpoint should be interpreted as +∞ when b = 0.
  5. [Section 7 opening] In the first paragraph of Section 7, the phrase 'for b < 0' should read 'for b > 0', since b := −a and a ∈ (−1, 0] implies b ≥ 0.
  6. [Remark 8] The claim that the generating function in the orange region (O) is piecewise rational should be qualified. For a = 0, which is included in the stated region of Theorem 1(O), formula (5) reduces to (1 − E(θ, −z))/(z E(θ, −z)), which is not a rational function of z for fixed θ ∈ (0, 1).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1's generating functions are derived from first principles by polytope integration, conditioning, and general distributional dualities, with no fitted inputs and no load-bearing self-citations.

full rationale

The derivation chain is self-contained. Lemma 6 computes the blue-region generating function by direct integration of the volume of the polytope defined by the inequalities theta x_i <= x_{i+1}, converting the integrated recursion into an identity involving the deformed exponential E(theta,z); no parameter is fit and no target generating function is used to set any constant. Lemma 7 reduces the green region to the blue boundary case by conditioning on the event X_i >= 1/theta, Lemma 8 obtains the yellow formula from the duality of Corollary 1, which itself is proved from Lemma 3 by an inclusion-exclusion argument valid for general continuous distributions, and Lemma 9 directly integrates the orange-region probabilities recursively before Lemma 10 applies the general duality (9). The white region is proved by pointwise inequalities in Lemma 4. The paper's citations to the authors' own prior work are not load-bearing: the symmetric case a = 1 from [15] is used as a check, and the theta = 1 result is re-proved in Lemma 4(b) to make the paper self-contained. The Mallows-Riordan formula (18) cited from [1] is re-derived in the text from the defining log-series (17) and is used only in the peripheral Corollary 4, not as an input to Theorem 1. The reader's singularity worry about cancellation at the smallest zero of the denominator is a technical verification question, not a circularity: the paper explicitly states the identities as formal power series or up to the smallest zero of the denominator in Remark 3, and positivity prevents numerator cancellation in the blue and green cases, as the skeptic's own analysis confirms. The genuine domain error in Corollary 4 is a correctness defect, not a circular step, because the incorrect application of formula (16) does not make any central claim equivalent to its own assumptions. Overall, there is no significant circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters are fitted; a and theta are the model parameters of the MA(1) process. The only external inputs are standard singularity analysis and cited results on zeros of the deformed exponential function. There are no invented entities.

assumptions (2)
  • domain assumption The persistence exponent is given by the inverse of the smallest positive zero of the denominator of the generating function.
    Invoked in Remark 7 and Corollaries 3, 8, 10, 11 as the basis for the exponential decay rate. The paper does not prove that the numerator is nonzero at this zero or that this singularity dominates; it is a standard singularity-analysis fact for generating functions with nonnegative coefficients.
  • standard math The deformed exponential function E(theta,z) admits a smallest positive real zero in the relevant parameter ranges.
    Used to define persistence exponents in the blue and green regions. Existence and properties of the zeros are cited from the literature (references [13], [21], [23]) rather than proved in this paper.

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Cite this review

Pith. "Pith review of Persistence probabilities for MA(1) sequences with uniform innovations." pith.science (2026). https://pith.science/paper/QVRBXELH

@misc{pith2026250704427,
  author       = {Pith},
  title        = {Pith review of: Persistence probabilities for MA(1) sequences with uniform innovations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVRBXELH}},
  note         = {Machine review of arXiv:2507.04427}
}
read the original abstract

We study the persistence probabilities of a moving average process of order one with uniform innovations. We identify a number of regions, characterized by the location of the uniform distribution and the coupling parameter of the process, where the persistence probabilities have qualitatively different behaviour. We obtain the generating functions of the persistence probabilities explicitly in all possible regions. In some of the regions, the persistence probabilities can be expressed explicitly in terms of various combinatorial quantities.

Figures

Figures reproduced from arXiv: 2507.04427 by the authors.

Figure 1
Figure 1. Phase diagram of the main results It is worth noting that, in the study of persistence problems, explicit and comprehensive results like Theorem 1 are rather rare; most works instead concentrate on asymptotic behavior, often through the persistence exponent. A few comments on these results are due: 1. We marked the different regions with colours for a better overview. Correspondingly, the letters in Theorem 1 stand … view at source ↗

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