{"id":"f79cbfc3-e504-4f56-9207-41b25519244a","arxiv_id":"2509.17590","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For planar random walks with finite second moments, the diameter and perimeter of their convex hull converge to Gaussian limits in generic cases and to explicit non-Gaussian limits in degenerate cases, completing the two-walk picture.","lead":"This paper classifies the large-time fluctuations of the diameter and perimeter of the convex hull generated by several independent planar random walks. It shows that generic drift configurations give Gaussian limits, while the previously open degenerate cases for two walks have explicit non-Gaussian limits involving Brownian functionals and an Itô integral.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The §7.2 fifth approximation relies on a g-weighted conditional Berry-Esseen bound that is not proved under the stated finite-second-moment assumption; Theorem 7.1's Itô-integral limit depends on it.","rationale":"The reader's weakest_assumption singles out the uniform Berry–Esseen approximation in §7.2, and this is indeed the most load-bearing premise for Theorem 7.1, the paper's most novel limit. The paper's hypotheses are the minimal finite-second-moment condition (M), and the cited theorem in [18] is, in the standard reference, a third-moment Berry–Esseen estimate; the g-weighted second-moment version is nonstandard and unproved in the text. If that estimate fails, the fifth approximation collapses and the Itô-integral representation has no support. This concern is precise and can be settled by checking the cited theorem and, if needed, supplying a proof. I do not claim the result is false; it is a proof gap that may be patchable. The centring error in Theorem 5.2 is definite but statement-level: the uncentred max of Gaussians has positive mean, so the displayed convergence to max(...) cannot hold for the centred statistic; the proof's L2 approximation actually yields max(...) - E[max(...)]. This should be corrected but does not alter the main framework or the Theorem 7.1 proof. Consequently, the appropriate verdict remains CONDITIONAL, matching the reader's stance: accept only after the Berry–Esseen step is justified and the diameter statements are recentred.","tokens_in":53490,"tokens_out":13383,"duration_ms":119893,"concrete_test":"Recover the exact statement of Theorem 6.6.3 from Gut, Probability: A Graduate Course, 2nd ed. (2013), and check whether it asserts the g-weighted bound under only E[φ(|ΔZ|²)]<∞, or whether it requires a finite third moment. If the theorem does not supply the bound, attempt an independent proof of the estimate in §7.2 (fifth approximation) from a known nonuniform Berry–Esseen theorem; if no such proof is available, the convergence n^{-1/2}Σ R5_{n,i}→0 in L2, and therefore Theorem 7.1, is not established. Separately, re-derive the continuous-mapping step in the proof of Theorem 5.2: since (M_n - E M_n)/√n converges to M - E[M], the displayed limit should be max(...) - E[max(...)], not max(...).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.1, the headline Itô-integral limit for the perimeter when the two drifts coincide, is reached in §7.2–§7.3 by an L2 approximation of (L_n - E L_n)/√n by n^{-1/2}Σ L5_{n,i}. The fifth approximation, displayed around (7.28)–(7.29), replaces the conditional probability in L4_{n,i} by Φ(A_{n,i}(θ)) with error bound\n|P(Σ_{j=i+1}^n e_θᵀΔZ_j / √((n-i)σ²(θ)) ≤ A | F_{i-1}) - Φ(A)| ≤ C E[|e_θᵀΔZ_1|² g(|e_θᵀΔZ_1|)] / g(√(n-i)),\nwhere g(x)=φ(x²)/x² and φ is chosen via [12, Thm 22] so that E[φ(|ΔZ|²)]<∞. This bound is attributed to [18, Thm 6.6.3]. The paper works throughout under (M), i.e., only finite second moments. The classical Berry–Esseen theorem quoted in [18] as Theorem 6.6.3 is normally stated under a finite third moment; the g-weighted, finite-second-moment version used here is not proved in the paper and is not a standard corollary of the classical statement. Since the estimate is used to show n^{-1}Σ E[(R5_{n,i})²]→0, a failure of this uniform approximation would break the martingale-difference approximation and hence the Itô-integral representation (7.3). This is the most delicate and least independent part of the proof: no alternative argument or verification is supplied at this step. The centring omission in Theorem 5.2 and Corollary 5.5 (the right-hand side should be max(...) - E[max(...)]) is a real but easily repairable statement error; it does not affect the central proof structure of Theorem 7.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an L2-approximation framework for the diameter and perimeter of the convex hull generated by N independent planar random walks whose increments have finite second moments. For the diameter it proves a general max-type limit theorem (Theorem 5.1 and Theorem 5.2) expressed in terms of Gaussian projections and Brownian infimum functionals. For the perimeter it completes the picture for N=2: Theorem 6.2 covers one zero-drift and one non-zero-drift walk, giving a Gaussian plus semi-perimeter Brownian limit; Theorem 7.1 covers two walks with the same non-zero drift, giving a limit described by an Itô integral. For N≥3, Corollary 4.3 and Corollary 4.4 give perimeter limits when the origin is in the interior of the drift polygon. The paper also provides variance asymptotics, including the explicit constant 6−π in the identity-covariance same-drift case, and claims optimality of the finite-second-moment condition.","tokens_in":53909,"tokens_out":19626,"duration_ms":174943,"significance":"If the technical issues identified below are resolved, this is a significant contribution. It completes the N=2 classification left open by earlier work, introduces a unifying L2/Wald-type framework that is likely to be useful beyond the present setting, and identifies genuinely non-Gaussian limits (max-type, Brownian semi-perimeter, and Itô-integral) in previously inaccessible degenerate cases. The paper is unusually explicit about limit variances and includes detailed proofs of the one-dimensional Wald ingredients rather than simply citing them. The explicit falsifiable predictions, such as the variance limit 6−π, and the systematic treatment of boundary cases are clear strengths.","major_comments":[{"comment":"The distributional statement in Theorem 5.2, second display, and in Corollary 5.5(i)-(iv), is not centred. The left-hand side is (D_n - E[D_n])/sqrt(n), which has mean zero, while the right-hand side max(...) has strictly positive mean: a maximum of non-degenerate zero-mean Gaussians has positive mean, and the variables xi_{i,j} are nonnegative with positive mean. The stated convergence is therefore false as written. The limit must be the centred version, for example max(...) - E[max(...)]. This also affects the variance-asymptotics claim in Remark 5.3(a).","section":"Theorem 5.2 and Corollary 5.5"},{"comment":"The specification of the Gaussian variables zeta is inconsistent with the L2 approximation proved in the same theorem. Under that approximation, the endpoint projections have asymptotic variances sigma^2_{i,j} and sigma^2_j, not 4 sigma^2_e. For N=1 this is an internal contradiction: Theorem 3.1 gives the limit N(0,sigma^2_1), while Theorem 5.2 combined with (5.8) would give N(0,4 sigma^2_1). There is also a sign issue in (5.9): for e1=(0,j) and e2=(j,k), both projections involve S(j) with coefficient +1, so the covariance should be + bmu_j^T Sigma_j bmu_{j,k}, whereas the rule e1 oplus e2 = -1 gives the negative value. These errors change the max-type limit laws in Corollary 5.5(ii)-(iii) and need to be corrected before the diameter results can be accepted.","section":"Theorem 5.2, equations (5.7)-(5.9)"},{"comment":"The fifth approximation uses a weighted Berry-Esseen bound of the form |P( sum_{j=i+1}^n e_theta^T Delta Z_j / sqrt((n-i) sigma^2(theta)) <= A | F_{i-1}) - Phi(A)| <= C E[ |e_theta^T Delta Z_1|^2 g(|e_theta^T Delta Z_1|) ] / g(sqrt(n-i)), attributed to [18, Theorem 6.6.3]. This is load-bearing: it is exactly the estimate that gives n^{-1} sum_i E[(R5_{n,i})^2] -> 0 and hence the Ito-integral representation (7.3). The cited textbook theorem appears to be the classical Berry-Esseen theorem under a finite third moment, and the g-weighted finite-second-moment version is not proved in the paper. The authors should either provide a self-contained proof of this weighted bound (for example by citing and verifying the hypotheses of a Bikelis-type theorem, including the claimed monotonicity of g and x/g(x)) or state and prove the additional moment assumption needed. As it stands, the proof of Theorem 7.1 has a gap at this step.","section":"Section 7.2, equations (7.28)-(7.29)"}],"minor_comments":[{"comment":"The notation in the display of Theorem 4.1 and in equation (4.5) is missing a parenthesis: it should be rho_H(H_n,G_n), not rho_H(H_n,G_n).","section":"Theorem 4.1 and equation (4.5)"},{"comment":"In the last paragraph of the proof, the same expression 'n^{-1/2}(D_n - diam A_n) -> 0' appears twice; the second occurrence should presumably refer to Gamma_mu(n,0) or to the corresponding approximation for diam A_n.","section":"Proof of Theorem 5.1"},{"comment":"There is a typo in 'Gaussian, as was was known already for N in {1,2} walks'; 'was was' should be 'was'.","section":"Introduction, Section 1.1"},{"comment":"There are minor typos such as 'to chcek' in Section 7.3, and the sentence in Section 7.2 about the random variables iN_k being identically distributed as N_k is grammatically awkward. These do not affect the mathematics.","section":"Section 7.3 and Appendix D"},{"comment":"The abstract says the diameter results cover N>=2, but Theorem 5.2 excludes the all-zero-drift case; that case is treated separately in Appendix C. It would be helpful to mention this qualification in the abstract or in the overview of results.","section":"Abstract and Appendix C"}],"recommendation":"major_revision","confidential_remarks":"This is a substantial paper with a coherent framework and several novel results. The main obstacles are the uncentred and incorrectly specified diameter limits in Theorem 5.2 and Corollary 5.5, and the missing justification of the weighted Berry-Esseen estimate in Section 7.2. The first set of issues is concrete and repairable; the second may require either a proof of a Bikelis-type result or a strengthening of the moment assumption. I would not reject the manuscript, but it should not be accepted until these points are resolved and the authors verify the diameter statements against the N=1 case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know before reading. The paper is a genuine advance: for two planar random walks with finite second moments, it gives the complete fluctuation picture for diameter and perimeter of the convex hull, including the non-Gaussian exceptional cases (equal non-zero drift: Itô-integral limit; one zero drift: Brownian semi-perimeter). The method, an L2 approximation framework built on Wald-type maximal CLTs, is clean and unifies earlier special cases. The general max-type diameter theorem for N walks and the Hausdorff approximation result are also new.\n\nThe soft spot is in the statement of Theorem 5.2 and Corollary 5.5. The centred diameter limit is written as max(...) but the right-hand side has positive mean (because the ξ terms are positive), while (D_n - E D_n)/√n has mean zero and is uniformly square-integrable. The limit should be max(...) - E[max(...)]. This is a statement error, not a proof error: the L2 approximation in the same theorem is correct and the centering is easily fixed.\n\nThe second, more delicate issue is in §7.2, the fifth approximation. The paper uses a g-weighted Berry-Esseen bound under only finite second moments, citing [18, Thm 6.6.3]. In standard references that theorem is stated under a third moment. The bound is plausible—it's the kind of weighted Berry-Esseen that exists in the literature—but the authors don't prove it and the citation may be wrong. Since this bound is what lets them replace the conditional probability by Φ(A_{n,i}) and then get the Itô-integral limit, it is load-bearing. A referee should ask for a proof or an exact citation.\n\nThe variance computations, the L2 approximations for Hausdorff distance, and the treatment of degenerate cases all look careful. The paper is long and notation-heavy, but the structure is transparent. I'd say it deserves a serious referee: the main results are new, the proofs are detailed, and the flaws are repairable.","headline":"Substantial paper completing the two-walk fluctuation picture for convex hulls with genuinely new non-Gaussian limits; the diameter limit statement has a missing centering constant, and the §7.2 Berry-Esseen step needs a verified citation.","tokens_in":54483,"tokens_out":5442,"would_cite":true,"duration_ms":47413,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G50","60D05","60F15","60J65","52A22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For random walks with finite second moments, hull fluctuations are governed by the drift polygon, with explicit non-Gaussian limits in all exceptional two-walk cases.","keywords":["convex hull","random walk","perimeter","diameter","central limit theorem","non-Gaussian limits","Itô integral","drift polygon"],"falsifier":"Simulate two independent planar random walks with identical non-zero drift and identity increment covariance, measure $n^{{-1/2}}$(L_n - E[L_n]) at large n, and check that the limiting sample variance equals 6 - pi (about 2.8584); a different value would falsify Theorem 7.1(ii).","tokens_in":53314,"feed_emoji":"📐","tokens_out":6150,"duration_ms":52373,"temperature":0.7,"pith_summary":"This paper tries to establish that the large-time fluctuations of the diameter and perimeter of the convex hull generated by N independent planar random walks are controlled by the geometry of the polygon spanned by their mean drifts. Under only finite second moments, after centring and dividing by sqrt(n), both statistics converge in distribution to explicit limit laws: Gaussian in generic configurations and non-Gaussian whenever several diametrical drift pairs compete or a drift degeneracy occurs. For two walks the classification is complete, with the previously open exceptional cases producing a maximum of correlated Gaussians, a Gaussian plus a Brownian semi-perimeter, or an Itô-integral limit when the two walks share the same non-zero drift. The paper matters because it turns a collection of isolated two-walk results into a single L2-approximation framework, and it supplies explicit variance asymptotics alongside every distributional limit.","feed_headline":"Non-Gaussian limits found for all exceptional two-walk hulls","feed_subtitle":"The drift polygon decides Gaussian versus non-Gaussian fluctuations; two-walk exceptions are now explicit.","key_machinery":"The load-bearing object is an L2-approximation framework: the hull H_n is approximated, in mean square and at sqrt(n) scale, by a simpler set G_n built from the trajectories and endpoints of only the boundary-relevant walks, together with explicit one-dimensional projection approximations for distances. For perimeter, the machinery also uses Cauchy's formula to write perimeter as the integral of the support function over angles, and a martingale-difference resampling device (resampling the i-th increment of each walk) that reduces the centred perimeter to a sum of conditional expectations, ultimately converging to a stochastic integral via semimartingale convergence criteria. The one-dimensional Wald maximal central limit theorem serves as the prototype for why running maxima can replace endpoints in these approximations.","core_discovery":"On the paper's own terms, the convex hull H_n of N walks with finite second moments is, after division by n, asymptotically the drift polygon C_mu = hull{0, mu_1,..., mu_N}. The paper's central discovery is that the next-order fluctuations are carried by a small, explicit list of random ingredients: endpoints of walks whose drifts are extremal vertices of C_mu, running maxima of one-dimensional projections of walks with non-zero drift, and Brownian support-function or semi-perimeter terms coming from zero-drift walks. For two walks this yields a complete dichotomy: the generic Gaussian limits previously known are complemented by non-Gaussian limits in every exceptional case, including an Itô-integral characterization of the perimeter when both drifts coincide.","pith_inferences":["A natural extension is to export the same L2-approximation scheme to higher-dimensional walks; the main obstruction there is the richer face structure of the drift polytope, not the martingale approximations.","The leader-probability asymptotics developed for the equal-drift case suggest quantitative answers to persistence-type questions about which of two identically drifted walks holds the running maximum over long time intervals.","Conjecture 8.1 indicates the perimeter for N >= 3 should depend only on a thin boundary layer of each relevant trajectory; if true, the limiting perimeter laws for generic large N would be built from the same Gaussian and Brownian-support-function ingredients identified here.","The explicit Itô-integral limit could be tested empirically by computing skewness or by comparing the empirical variance ratio to 6 - pi, which would distinguish the non-Gaussian law from a normal approximation in simulations."],"forward_implications":["For two walks, every open case in the earlier Gaussian theory now has an explicit limit: equal non-zero drifts give an Itô-integral perimeter limit, one zero drift gives a Gaussian plus a Brownian semi-perimeter, and isosceles or equilateral drift triangles give maxima of (possibly correlated) Gaussians for diameter.","The variance of the centered perimeter converges to explicit constants; for two identical walks with identity covariance and equal non-zero drift, n^{-1} Var(L_n) tends to 6 - pi, about 2.8584.","When the origin lies in the interior of the drift polygon and the extremal drift vertices are distinct, the perimeter of N walks obeys a Gaussian CLT with a closed-form variance.","For general N, the diameter has a max-type limit built from independent Gaussian and Brownian-extreme components, indexed by the diametrical pairs of the drift polygon.","The L2 approximation is strong enough to give convergence of variance and, with minor modifications, extends to walks whose increments are dependent across walks."],"supporting_citations":[{"why":"Supplies the earlier two-walk Gaussian-limit results whose exceptional cases this paper completes.","marker":"[20]"},{"why":"Provides the single-walk diameter and perimeter limit theorems and the line-segment L2 approximation that the multi-walk framework extends.","marker":"[31]"},{"why":"Gives the martingale-difference method and perimeter CLT for one planar random walk with drift.","marker":"[48]"},{"why":"Establishes the Brownian-hull scaling limits used for the zero-drift components of the perimeter and diameter limits.","marker":"[49]"},{"why":"Supplies Wald's maximal central limit theorem, the one-dimensional prototype for replacing maxima by endpoints in the L2 approximations.","marker":"[50]"},{"why":"Provides the semimartingale convergence criteria used to prove convergence to the Itô-integral limit in Theorem 7.1.","marker":"[21]"},{"why":"Supplies the moment-function bound used in the uniform Berry-Esseen approximation that carries the equal-drift perimeter proof.","marker":"[12]"},{"why":"Gives the correlation formula for maxima of correlated Brownian motions used in the explicit variance computations.","marker":"[40]"}],"fun_headline_variants":["Two-walk convex hull limits turn non-Gaussian in all exceptional cases","Drift polygon decides Gaussian vs non-Gaussian hull fluctuations","Non-Gaussian limits for every exceptional two-walk hull","Explicit non-Gaussian limits complete two-walk hull picture","Two-walk convex hulls: non-Gaussian limits in all edge cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The most delicate premise is that, for two walks with the same non-zero drift, the conditional probability that one walk's running maximum exceeds the other's is uniformly approximated by Phi(A_{n,i}(theta)) with a Berry-Esseen rate; this uniform approximation is what carries the Itô-integral representation.","fun_headline_variants_meta":{"raw":{"variants":["Two-walk convex hull limits turn non-Gaussian in all exceptional cases","Drift polygon decides Gaussian vs non-Gaussian hull fluctuations","Non-Gaussian limits for every exceptional two-walk hull","Explicit non-Gaussian limits complete two-walk hull picture","Two-walk convex hulls: non-Gaussian limits in all edge cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1560,"prompt_tokens":917,"completion_tokens":643,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":554}},"tokens_in":533,"tokens_out":643,"duration_ms":5335,"temperature":1.0,"reasoning_tokens":554,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:48:48.732362+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate two independent planar random walks with identical non-zero drift and identity increment covariance, measure $n^{{-1/2}}$(L_n - E[L_n]) at large n, and check that the limiting sample variance equals 6 - pi (about 2.8584); a different value would falsify Theorem 7.1(ii).","supporting_citations":[{"cited_title":"Ivanković, T","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier two-walk Gaussian-limit results whose exceptional cases this paper completes."},{"cited_title":"McRedmond and A","cited_arxiv_id":null,"evidence_quote":"Provides the single-walk diameter and perimeter limit theorems and the line-segment L2 approximation that the multi-walk framework extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the martingale-difference method and perimeter CLT for one planar random walk with drift."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Brownian-hull scaling limits used for the zero-drift components of the perimeter and diameter limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Wald's maximal central limit theorem, the one-dimensional prototype for replacing maxima by endpoints in the L2 approximations."},{"cited_title":"Jacod and A","cited_arxiv_id":null,"evidence_quote":"Provides the semimartingale convergence criteria used to prove convergence to the Itô-integral limit in Theorem 7.1."},{"cited_title":"Dellacherie and P.-A","cited_arxiv_id":null,"evidence_quote":"Supplies the moment-function bound used in the uniform Berry-Esseen approximation that carries the equal-drift perimeter proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the correlation formula for maxima of correlated Brownian motions used in the explicit variance computations."}],"review_version":2}