{"id":"84bc3202-d554-4a7c-8b97-2ea9d068bcfa","arxiv_id":"2509.03343","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves functional CLTs for the range of random walks in the domain of attraction of stable laws for d/beta <= 3/2, with limits ranging from Brownian motion to renormalized self-intersection local time.","lead":"Functional limit theorems are proved for the range (number of distinct sites visited) of random walks attracted to stable laws, in the recurrent and weakly transient regimes. The result completes an earlier functional theorem that only covered strongly transient walks, and it gives a tool for functionals such as the energy consumed by a random predator.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's Hölder-exponent split is undefined at d/β=1, leaving the main non-Gaussian FCLT unproved exactly in the planar Brownian / 1D Cauchy boundary case.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing gap: Lemma 2's Case 1 exponent choice collapses at d/β = 1. I read the proof of Theorem 1 and Lemma 2 in the full text and can confirm the mechanism: the two Hölder inequalities in (2.15) require q1 to blow up as d/β → 1+ and q2 to be less than 1, which is incompatible with the conjugate-exponent condition. The paper's own text contains no alternate argument for the boundary, and (A3) is not automatic there, so the advertised FCLT in the planar/1D-Cauchy cases is not established as written. I do not see a fatal error in the rest of the strategy; the result is plausible and likely repairable, but a subcase of the main theorem is currently unsupported. I also noticed a similar exponent breakdown in Lemma 2 Case 4 for d/β < 1, where q1 becomes negative, but I did not elevate it above the d/β = 1 issue because the latter is the central non-Gaussian FCLT. The reader's CONDITIONAL verdict is therefore appropriate, and my stress-test does not change it.","tokens_in":25067,"tokens_out":11079,"duration_ms":118460,"concrete_test":"Re-derive the key estimate (2.9) at d/β = 1 directly, bypassing the two-Hölder split. For the simple symmetric planar random walk (d=2, β=2), use the Fourier representation (2.14) together with the classical asymptotic P(S_k = 0) ∼ c/k and h(n) ∼ (2πσ)^{-1} log n to compute S_{2,2}(n) E[I_{⌊ns⌋,⌊nt⌋}] = (h(n)^2 b(n)^2/n^2) E[I_{⌊ns⌋,⌊nt⌋}] for fixed 0 < s < t. Verify that this quantity is O((s∧t)^{1−η}) as n → ∞, equivalently E[I_{⌊ns⌋,⌊nt⌋}] = O((s∧t) n/(log n)^2). If this holds, the lemma is true and the proof gap is repairable by replacing the exponent split; if it fails, the normalization in (1.4) is wrong for the planar walk.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The tightness proof for the headline regime 1 ≤ d/β < 3/2 rests on Lemma 2, whose Case 1 fixes q1 := β/(d − β) + ηβ and q2 < 1/(1−η), with p_k^{-1} + q_k^{-1} = 1. At d/β = 1, q1 is undefined/infinite and q2 < 1 forces 1/p2 = 1 − 1/q2 < 0, so the required conjugate exponents p2,q2 ≥ 1 do not exist. The Hölder split leading to (2.16) and (2.17) therefore cannot be performed at the boundary. This is not a removable corner case: for d = 2, β = 2 it is the planar Brownian regime, and for d = 1, β = 1 it is the 1D Cauchy case, both explicitly covered by Theorem 1. Moreover, the integrability condition (β−ε)(q1^{-1}+q2^{-1}) > d is unattainable at d = β with q2 ≥ 1, so the fix cannot be merely a limiting choice of the same exponents; a different estimate of the integral in (2.14) is required. As written, the FCLT (1.4) at d/β = 1 is unsupported. A similar exponent breakdown occurs in Case 4 for d/β < 1 (q1 is negative there), but the d/β = 1 gap is the primary load-bearing issue because it protects the central non-Gaussian functional limit theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes functional limit theorems for the range process (R_nt)_{t\\ge0} of a random walk in Z^d that is in the domain of attraction of a non-degenerate \\beta-stable process. Theorem 1 covers three regimes: Gaussian fluctuations for d/\\beta\\ge 3/2, a non-Gaussian limit given by the renormalized self-intersection local time -\\gamma^\\beta for 1\\le d/\\beta<3/2 under assumption (A3), and convergence to the Lebesgue measure of the stable range for d/\\beta<1. Theorem 2 applies these results to an energy functional E_t=\\int m(t-s)\\,dR_s arising from a prey-predator model. The proofs use fixed-time moment estimates from Le Gall and Rosen, a new Hölder-type estimate on intersection local times in Lemma 2, Kolmogorov's tightness criterion, and Cramér-Wold for finite-dimensional convergence.","tokens_in":25410,"tokens_out":8194,"duration_ms":87994,"significance":"If correct, the paper fills a real gap by turning known fixed-time CLTs for the range into functional CLTs in the weakly transient and recurrent regimes, and it gives useful by-products: uniform-in-n Hölder regularity of the scaled range processes and a new regularity statement for the renormalized self-intersection local time. The organization is generally careful and the dependence on prior fixed-time results is transparent. However, the central technical estimate Lemma 2 contains a genuine exponent breakdown exactly at the boundary d/\\beta=1, which is included in Theorem 1 and covers important cases such as planar Brownian motion and the one-dimensional Cauchy process. Since Lemma 2 drives the tightness proof, the non-Gaussian FCLT (1.4) is not established as written at that boundary. A similar exponent problem occurs for d/\\beta<1. The application to the energy functional also relies on an unproved and generally false bound |E_t-E_t|\\le 1. These issues are load-bearing and require new arguments.","major_comments":[{"comment":"At d/\\beta=1, the choice q1 := \\beta/(d-\\beta)+\\eta\\beta in Case 1 has a zero denominator, so q1 is undefined. Moreover q2<1/(1-\\eta) with 1/p2+1/q2=1 forces p2<0, contradicting the stated requirement p_k,q_k\\ge 1. Consequently the Hölder split leading to (2.16)-(2.17) and the finiteness condition (\\beta-\\varepsilon)(q1^{-1}+q2^{-1})>d are not available exactly at d/\\beta=1. This boundary is explicitly included in Theorem 1 and includes d=2,\\beta=2 and d=1,\\beta=1. Lemma 2's estimate is used in Lemma 3 via Eq. (4.6) to prove tightness, so the convergence (1.4) at d/\\beta=1 is unsupported. Since the condition cannot hold for q2\\ge 1, a different estimate of the integral in (2.14) is required rather than a limiting choice of the same exponents.","section":"Section 2, Lemma 2, Case 1; used in Eq. (4.6)"},{"comment":"For d/\\beta<1, the same exponent choice gives q1 = \\beta/(d-\\beta)+\\eta\\beta < 0 for every admissible \\eta\\in(1-1/\\beta,1), so q1\\ge 1 fails and the Case-1 Hölder argument is not valid. The sentence claiming that the earlier choice of p1,q1,p2,q2 'is still valid' is therefore incorrect. This affects the estimate (4.20) used in Lemma 6 and hence the tightness proof in Section 4.3. A separate argument is needed for this regime.","section":"Section 2, Lemma 2, Case 4; used in Eq. (4.20)"},{"comment":"The paper asserts 'Since |E_t-E_t|\\le 1' in order to transfer the CLT proved for the linearly interpolated process R to the discrete energy E. This inequality is not true in general. For a decreasing m, the difference is at least of order the total variation of m over intervals of length one, e.g. for m(s)=M-as>0 on [0,2] the difference can exceed 1 for large M. The two processes may still be asymptotically equivalent after the normalization in Theorem 2, but that requires a proof. As written, Theorem 2 does not follow from the convergence established for R alone.","section":"Introduction after Theorem 2 and Section 5.1"}],"minor_comments":[{"comment":"Dvoretzky's name is misspelled as 'Dvoretsky'; also 'stongly transient' should be 'strongly transient'.","section":"Page 2 and reference [9]"},{"comment":"The displayed normalizations h(n)^2 b_\\beta(n)^d / m(n)n^2 would be clearer with explicit parentheses: h(n)^2 b_\\beta(n)^d / (m(n)\\,n^2).","section":"Equations (1.7) and (5.1)"},{"comment":"The notation R^{(i,j)}_n and related intersection local times is compressed; a one-line explanation that these are the range and intersection counts on dyadic blocks would improve readability.","section":"Section 4.1, Step 2"},{"comment":"The treatment of the case d/\\beta=2 is very brief; in particular the slowly varying function l(\\lfloor nT\\rfloor) bounding E[I_{\\lfloor nT\\rfloor,\\lfloor n(t-s)\\rfloor}] should be identified explicitly or cited more precisely.","section":"Section 4.2, Step 1'"}],"recommendation":"major_revision","confidential_remarks":"The referee report's central stress-test concern is valid and should be addressed before publication. The paper is potentially publishable, but the d/\\beta=1 gap in Lemma 2 is not a cosmetic corner case; it protects the main non-Gaussian FCLT. The additional gap concerning |E_t-E_t|\\le 1 also needs a rigorous fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real contribution, but the main theorem has a load-bearing gap at exactly the boundary d/beta=1. The rest of the paper is solid, and the gap looks repairable.\n\nThe paper does what the abstract promises: it takes Le Gall and Rosen's fixed-time CLT and moment bounds, plus Cygan, Sandrić and Šebek's J1 FCLT for d/beta>3/2, and produces functional versions in the weakly transient and recurrent regimes, with convergence in C(R+) and uniform Hölder tightness. That is new and not routine. The tightness strategy—Kolmogorov's criterion fed by sharp two-walk intersection estimates—is sensible, and the Hölder regularity for the renormalized self-intersection local time gamma^beta is a nice byproduct. The energy-functional application is a reasonable bonus, though the ecology motivation is light.\n\nThe soft spot is real and load-bearing. In Lemma 2, Case 1, the chosen Hölder exponents q1 := beta/(d−beta) + eta beta and q2 < 1/(1−eta) are supposed to satisfy p_k^{-1}+q_k^{-1}=1 with p_k,q_k>=1. At d/beta=1, q1 is undefined/infinite, and q2<1 forces p2 negative. The integrability condition (beta−epsilon)(q1^{-1}+q2^{-1}) > d also cannot hold with q2>=1 at d=beta. So the estimate (2.17) is not proved at d/beta=1. Lemma 3 depends on Lemma 2, so tightness for the non-Gaussian FCLT in Theorem 1(1.4) is unsupported exactly at d/beta=1—that includes planar Brownian motion (d=2, beta=2) and the one-dimensional Cauchy case (d=1, beta=1), both explicitly claimed. The fix will require a different estimate of the integral in (2.14) at d=beta, not a limiting choice of the same exponents. As written, the boundary case is missing.\n\nThere is also a small typo: the paper defines E_t and E_t identically and then asserts |E_t−E_t|≤1. Presumably one of them involves the linearly interpolated range, and the bound should involve m(0). Minor and easy to fix.\n\nBottom line: I would not rely on this version if I needed the boundary case. For the interior regime 1<d/beta<3/2 and for the strongly transient case, I would use it. The paper deserves a serious referee—send it out and ask for a fix of Lemma 2 at d/beta=1, plus a careful cleanup of the two notations in Section 5. I'd bring it to a reading group because both the tightness mechanism and the gap are instructive.","headline":"Genuine extension of the FCLT for random-walk range, with a real gap at d/beta=1; the interior regimes are likely right, but Theorem 1 as stated is not fully proved.","tokens_in":25935,"tokens_out":3636,"would_cite":true,"duration_ms":37433,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F17","60G52","60J55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves functional limit theorems for the range of a random walk attracted to a β-stable law, in every regime d/β ≤ 3/2, with limits that are Brownian motion, the negative renormalized self-intersection local time of the stable pr","keywords":["functional limit theorem","range of random walk","stable process","self-intersection local time","tightness","Hölder regularity","Young integral","prey–predator model"],"falsifier":"Evaluate Lemma 2's estimate at d/β = 1: take a one-dimensional random walk in the domain of attraction of a symmetric 1-stable law and test whether S_{1,1}(n) E[I_{⌊ns⌋,⌊nt⌋}] stays bounded by C (s∧t)^{χ−η} uniformly in n as s∧t → 0; if the bound fails, or if no such walk satisfies (A3), the middle-case functional CLT does not cover the critical endpoint.","tokens_in":24927,"feed_emoji":"🎲","tokens_out":8972,"duration_ms":93253,"temperature":0.7,"pith_summary":"This paper establishes functional central limit theorems for the number of distinct sites visited by a random walk in Z^d that is attracted to a β-stable process, covering the weakly transient and recurrent regimes that earlier fixed-time results and earlier functional results left open. The central message is that the entire range process, linearly interpolated and properly centered and scaled, converges in distribution in the space of continuous paths, with the limit depending only on the ratio d/β: Brownian motion when d/β ≥ 3/2, the negative renormalized self-intersection local time of the stable process when 1 ≤ d/β < 3/2, and the Lebesgue measure of the stable range when d/β < 1. The proof is carried by sharp bounds on moments of the range and of the intersection of two independent walks, which yield tightness in a Hölder space and, as a byproduct, local Hölder regularity of the limiting self-intersection local time. An ecological application gives limit theorems for an energy functional in which a predator's accumulated consumption is a weighted integral of the range process.","feed_headline":"Three regimes, one limit law for the random-walk range","feed_subtitle":"The whole range process converges to Brownian motion, self-intersection local time, or the stable range measure.","key_machinery":"The central object is the range process R_t, the linearly interpolated count of distinct sites visited up to time t, together with its variance scale. The proof mechanism is a decomposition of the range increment into new sites minus intersections with the past, following [12] and [13], plus sharp moment bounds: uniform bounds on moments of the range and, in Lemma 2, a Hölder-type estimate for the expected overlap of two independent walks' ranges, S_{d,β}(n) E[I_{⌊ns⌋,⌊nt⌋}] ≤ C (s∧t)^{χ−η}. These bounds feed Kolmogorov's tightness criterion in a Hölder space, so tightness and finite-dimensional convergence combine to give the functional limit. In the middle regime the limit object is the re","core_discovery":"The paper's central claim, Theorem 1, is that under mild assumptions (A1)–(A2), the scaled centered range process (R_{nt} − E[R_{nt}]) converges in C(R+) in all regimes. If d/β ≥ 3/2, (ng(n))^{-1/2} times the centered range converges to a Brownian motion with variance σ²t; if 1 ≤ d/β < 3/2, under the extra characteristic-function assumption (A3), the scale h(n)²b_β(n)^d/n² produces convergence to −γ^β_t, the renormalized self-intersection local time of the limiting stable process; if d/β < 1, no centering is needed and b_β(n)^{-1}R_{nt} converges to the Lebesgue measure of the range of U^β. The author presents this as completing the picture after the fixed-time CLTs of [13] and the strongly","pith_inferences":["Editorial inference: the critical endpoint d/β = 1 is the first point to check. Lemma 2's stated Hölder bound is not actually proved there—the chosen exponent q₁ becomes undefined and q₂ < 1 contradicts p₂ ≥ 1—and assumption (A3) is not automatic because β ≤ 1. Without an alternate argument, the middle-case theorem does not cover exactly d = β.","Editorial inference: the same Hölder-tightness approach should extend to other additive functionals of the range, such as local times of the range or occupation counts over subsets of sites; the paper does not pursue those extensions.","Editorial inference: in the ecological model, the middle-regime result implies that large-energy fluctuations in two-dimensional-like foraging are controlled by the tail of γ^β, so the model predicts non-Gaussian starvation and mortality events; the paper stops at convergence and does not give tail asymptotics, which would be a natural next step."],"forward_implications":["The complete range process, not just its value at a fixed time, converges, so any continuous functional of the path—suprema, integrals, level crossings—inherits the stated limit.","In the middle regime the fluctuation limit is non-Gaussian and governed by self-intersections of the stable process; its sample paths are almost surely locally χ-Hölder for every χ < 2 − d/β, and the paper argues this exponent is likely optimal.","For the energy functional E_t = ∫₀ᵗ m(t−s) dR_s, the paper obtains functional CLTs: Gaussian for d/β ≥ 3/2, self-intersection-driven for 1 ≤ d/β < 3/2, and range-measure-driven for d/β < 1, with Young integrals as the limiting objects.","Convergence holds in the uniform-on-compacts topology of C(R+), which is stronger than the previously available J1-topology result in the strongly transient regime and is needed for the energy application."],"supporting_citations":[{"why":"Supplies the fixed-time CLT, the moment inequalities (4.2) and (4.10), and the intersection estimates that the paper's tightness proof adapts and extends to the functional setting.","marker":"[13]"},{"why":"Provides the decomposition of the range into new sites minus past intersections and the construction of renormalized self-intersection local time that the middle-regime limit is built on.","marker":"[12]"},{"why":"The earlier functional CLT in the strongly transient regime d/β > 3/2, which this paper's Gaussian regime extends to d/β ≥ 3/2 and to uniform-on-compacts topology.","marker":"[8]"},{"why":"Establishes existence and the order of the singularity of the self-intersection local time of stable processes, from which the renormalized process γ^β_t is obtained.","marker":"[21]"},{"why":"Provides the characteristic-function decay estimate used in Lemma 2 to control the expected intersection of two independent scaled walks.","marker":"[22]"},{"why":"Supplies regular variation and Potter bounds used throughout to move between the scale functions bβ, h and g.","marker":"[4]"},{"why":"Gives Skorokhod's construction used in the d/β < 1 case to couple the rescaled walk with the stable process and read off convergence of the range.","marker":"[23]"},{"why":"Defines the Young integral and its integration-by-parts formula, which give meaning to the energy limits involving powers (t−s)^χ against Brownian motion, γ^β, or the range measure.","marker":"[28]"}],"fun_headline_variants":["Range of stable walks: Brownian, local time, or Lebesgue","Random-walk range converges in all three regimes","Three limit laws for the range of stable walks","From Brownian to local time: range limits solved","Stable walk range: three regimes, three limit processes"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof assumes that a uniform Hölder bound on the expected overlap of two independent walks (Lemma 2) holds at the critical ratio d/β = 1; at that point the chosen Hölder exponents no longer exist, so the theorem's coverage of the case d = β is not actually established.","fun_headline_variants_meta":{"raw":{"variants":["Range of stable walks: Brownian, local time, or Lebesgue","Random-walk range converges in all three regimes","Three limit laws for the range of stable walks","From Brownian to local time: range limits solved","Stable walk range: three regimes, three limit processes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1142,"prompt_tokens":716,"completion_tokens":426,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":346}},"tokens_in":460,"tokens_out":426,"duration_ms":5180,"temperature":1.0,"reasoning_tokens":346,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:58:35.688397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Lemma 2's estimate at d/β = 1: take a one-dimensional random walk in the domain of attraction of a symmetric 1-stable law and test whether S_{1,1}(n) E[I_{⌊ns⌋,⌊nt⌋}] stays bounded by C (s∧t)^{χ−η} uniformly in n as s∧t → 0; if the bound fails, or if no such walk satisfies (A3), the middle-case functional CLT does not cover the critical endpoint.","supporting_citations":[{"cited_title":"Functional CLT for the range of stable random walks","cited_arxiv_id":"1908.07872","evidence_quote":"The earlier functional CLT in the strongly transient regime d/β > 3/2, which this paper's Gaussian regime extends to d/β ≥ 3/2 and to uniform-on-compacts topology."}],"review_version":1}