{"id":"8d63eb35-62b9-4b7d-a0d3-085c43c899b7","arxiv_id":"2507.04427","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit generating functions are derived for persistence probabilities of MA(1) processes with uniform innovations in every parameter region.","lead":"This paper works out exact formulas for the probability that a moving average process with random inputs stays positive for n steps. The formulas cover all choices of the random input distribution and the process coupling, dividing the parameter space into regions with different behavior.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's generating-function identities survive scrutiny; the reader's singularity worry is resolved by Pringsheim plus positivity. The concrete flaw is Corollary 4, whose stated domain makes it false and whose proof wrongly invokes the blue-region formula.","rationale":"I checked the main theorem region by region, including the blue, green, yellow, and orange formulas, and the dualities in Lemmas 1-3 and Corollary 1. The generating-function identities are mutually consistent and match small-n checks. The reader's weakest assumption, concerning numerator cancellation at the smallest positive zero of the denominator, does not materialize: in the blue and green regions the second E-term is evaluated at a positive argument and cannot vanish, and in the yellow region the numerator of the fraction is identically 1 at a pole; the orange region is protected by Pringsheim because the persistence probabilities are nonnegative. Thus the persistence-exponent corollaries are supported by standard arguments, though the paper would benefit from stating them. The substantive problem I found is Corollary 4: as written it asserts a false identity for θ∈[0,1]. The source is a domain error in applying the blue-region formula to a=-θ, which is only valid for θ∈[-1,0]. This is a concrete, checkable false statement in the paper, but it does not undermine Theorem 1. Since the central claim stands and the error is localized to a corollary, the appropriate verdict remains conditional acceptance rather than rejection.","tokens_in":17229,"tokens_out":43960,"duration_ms":425785,"concrete_test":"Check Corollary 4 at θ=1/2, n=1: compute p^{-1/2}_1(1/2)=P(X2 ≥ X1/2) with X uniform on [1/2,1]; since X1≥1/2 implies X1/2≥1/4 and X2≥1/2, the probability is exactly 1. Independently compute J_3(1/2) from the defining identity (17), which gives J_3(θ)=θ+2, so J_3(1/2)/2 = 1.25. If the computation is correct, Corollary 4 is false as stated. Also re-derive the first equality of Lemma 6 for θ=1/2, a=-1/2: the inclusion θx∈[-a,1] fails at x=1/2, so the blue-region integral representation cannot be invoked there.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central generating-function identities are internally sound. The reader's weakest assumption does not land: in the blue region, at a zero z0>0 of E(θ,-z/(1+a)), the numerator term E(θ,a z0/(1+a)) is evaluated at a positive argument, where E has strictly positive coefficients, so the pole is simple and no cancellation occurs. The same positivity works in the green region, where the other E-term has argument -z0/(θ(1+a)) > 0 because θ<0, and in the yellow region, where at a zero of the denominator the fraction 1/(R-μz)-1 has numerator 1. For the orange region, the generating function has nonnegative coefficients, so Pringsheim's theorem forces the radius of convergence to be a positive real pole of F/(1-zF). The genuine error is Corollary 4: for θ∈[0,1] and a=-θ, the support is [θ,1], so θXi ≤ θ ≤ Xi+1 and p^a_n(θ)=1 for every n. But J_{n+2}(θ)/(n+1)! is not identically 1; already for θ=1/2, n=1, J_3(1/2)/2 = 2.5/2 = 1.25. The proof applies the blue-region formula (16), whose hypotheses fail for θ∈[0,1], a=-θ; the intended statement is θ∈[-1,0], where a=-θ is the lower blue boundary.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":17524,"tokens_out":27159,"duration_ms":271667,"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":[{"comment":"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.","section":"§4.2, Corollary 4"},{"comment":"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.","section":"Remark 7; Corollaries 3, 8, 10, 11"}],"minor_comments":[{"comment":"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.","section":"Theorem 1(O), Eq. (5); Lemma 9, Eq. (22)"},{"comment":"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).","section":"Corollary 1 proof"},{"comment":"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.","section":"Lemma 3 proof"},{"comment":"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.","section":"Lemma 9"},{"comment":"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.","section":"Section 7 opening"},{"comment":"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).","section":"Remark 8"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main results hold up. Theorem 1 gives explicit generating functions for persistence probabilities of MA(1) processes with uniform innovations in all parameter regions, and the special case a=1 reproduces the known Majumdar–Dhar result. The derivations are careful, from first principles via integral representations, with no fitted parameters or circular reasoning. The dualities in Lemmas 1–3 and Corollary 1 are genuinely useful and of independent interest. This is a solid extension of a known result, not just a repackaging.\n\nThe soft spots are real but localized. The biggest problem is Corollary 4: as stated, for θ ∈ [0,1] and a = −θ, the support is [θ,1], so θXi ≤ θ ≤ Xi+1 always, and the persistence probability is identically 1. The claimed formula J_{n+2}(θ)/(n+1)! is not 1 (e.g. θ=1/2, n=1 gives 1.25), and the proof invokes the blue-region formula (16) outside its hypotheses. The intended statement is presumably θ ∈ [−1,0], where a = −θ lies on the blue boundary. This needs a fix, but it is a corollary, not load-bearing for Theorem 1.\n\nThere are also several typos that should be corrected: x+ should be max(x,0) in Lemma 9 and Theorem 1(O); the proof of Corollary 1 has reversed inequality signs in the last step; and Lemma 3's proof refers to Lemma 8, which does not exist (it should be Lemma 2). These are minor and easily fixed.\n\nThe reviewer's worry about the persistence-exponent corollaries relying on an unproved singularity assumption is not a real problem. The generating functions have nonnegative coefficients, so Pringsheim's theorem implies the radius of convergence is a positive real pole, and in the blue/green regions the numerator does not vanish at that pole. So those corollaries stand.\n\nOverall: this paper deserves a serious referee. The central contribution is correct, original, and clearly presented, and the flaws are correctable. I would send it to peer review, with instructions to fix Corollary 4 and the typos. The target audience is the persistence-probability community and people working on deformed exponentials and Mallows–Riordan polynomials.","headline":"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.","tokens_in":18041,"tokens_out":2822,"would_cite":true,"duration_ms":30743,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G10","60J05","05A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["persistence probability","moving average process","MA(1)","uniform distribution","generating function","deformed exponential function","Mallows-Riordan polynomials","duality"],"falsifier":"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.","tokens_in":17014,"feed_emoji":"🎲","tokens_out":6805,"duration_ms":68960,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact persistence odds found for MA(1) with uniform noise","feed_subtitle":"Generating functions in five parameter regions, built from a deformed exponential, cover every coupling strength and every uniform interval.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Solves the symmetric case $a=1$; the present paper extends that result to all uniform intervals and all coupling parameters.","marker":"[15]"},{"why":"Contains the special-case result $p^a_n(1)=1/(n+1)!$ that the paper rederives in Lemma 4.","marker":"[12]"},{"why":"Supplies the AR(1) persistence formula with uniform innovations and the Mallows-Riordan polynomials used in Corollary 4.","marker":"[1]"},{"why":"Defines the Mallows-Riordan polynomials, which appear in the closed form for $p^{-\\theta}_n(\\theta)$.","marker":"[16]"},{"why":"Studies zeros of the deformed exponential function, which determine the persistence exponents in the blue and green regions.","marker":"[23]"},{"why":"Provides the general eigenvalue-equation framework for persistence exponents in Markov chains that motivates the MA(1) analysis.","marker":"[4]"}],"fun_headline_variants":["MA(1) persistence solved via deformed exponential","Exact MA(1) persistence formulas via deformed exponential","MA(1) with uniform noise: persistence odds explicit","Persistence probabilities for MA(1) explicit in five regions","Deformed exponential yields MA(1) persistence probabilities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["MA(1) persistence solved via deformed exponential","Exact MA(1) persistence formulas via deformed exponential","MA(1) with uniform noise: persistence odds explicit","Persistence probabilities for MA(1) explicit in five regions","Deformed exponential yields MA(1) persistence probabilities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000966,"raw_usage":{"total_tokens":4081,"prompt_tokens":883,"completion_tokens":3198,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":3119}},"tokens_in":499,"tokens_out":3198,"duration_ms":21606,"temperature":1.0,"reasoning_tokens":3119,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:49:00.351124+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Majumdar and Deepak Dhar","cited_arxiv_id":null,"evidence_quote":"Solves the symmetric case $a=1$; the present paper extends that result to all uniform intervals and all coupling parameters."},{"cited_title":"Krishna and Manjunath Krishnapur","cited_arxiv_id":null,"evidence_quote":"Contains the special-case result $p^a_n(1)=1/(n+1)!$ that the paper rederives in Lemma 4."},{"cited_title":"Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials","cited_arxiv_id":null,"evidence_quote":"Supplies the AR(1) persistence formula with uniform innovations and the Mallows-Riordan polynomials used in Corollary 4."},{"cited_title":"Mallows and John Riordan","cited_arxiv_id":null,"evidence_quote":"Defines the Mallows-Riordan polynomials, which appear in the closed form for $p^{-\\theta}_n(\\theta)$."},{"cited_title":"Zeros of the deformed exponential function","cited_arxiv_id":null,"evidence_quote":"Studies zeros of the deformed exponential function, which determine the persistence exponents in the blue and green regions."},{"cited_title":"Persistence exponents in Markov chains","cited_arxiv_id":null,"evidence_quote":"Provides the general eigenvalue-equation framework for persistence exponents in Markov chains that motivates the MA(1) analysis."}],"review_version":1}