{"id":"ba399d51-097e-49f5-bdb1-c79cd130c011","arxiv_id":"1908.07872","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A functional central limit theorem (Brownian motion limit in Skorohod space) is proved for the capacity and cardinality of the range of stable random walks when d/alpha is greater than 5/2 and 3/2, respectively.","lead":"This mathematics paper proves that the capacity and the size of the set of sites visited by a stable random walk, viewed as random processes in time, converge to Brownian motion after centering and normalization. It upgrades earlier central limit theorems to a functional version for a class of stable random walks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The FCLT's proof imports the key variance and intersection estimates from [7] and [19] without restating their hypotheses; if those estimates require strictly stable increments rather than the domain-of-attraction assumption (A2), both theorems are unproved for the stated class.","rationale":"I read the proof as a coherent two-step scheme: finite-dimensional convergence via the Cramer-Wold decomposition into independent range increments, and Aldous-type tightness using monotonicity and the imported error bounds. The internal algebra is consistent, and I found no contradiction or data-dependent artifact. The sole load-bearing soft spot is the provenance of the variance and intersection estimates: the paper relies on [7, Lemmas 3.2 and 4.3] and [19] without restating their hypotheses, so the truth of the theorems for every walk satisfying (A2) is conditional on those sources carrying exactly the assumptions claimed. The reader's weakest assumption identifies the same point, hence my agreement. Because this is an unresolved external dependency rather than a demonstrated error, conditional acceptance is the honest recommendation: the result is established once the cited lemmas are confirmed to hold under (A1)-(A4) and (A1)-(A2) respectively. No ad hominem is intended; the critique concerns the argument's reliance on unstated hypotheses.","tokens_in":32495,"tokens_out":13435,"duration_ms":122419,"concrete_test":"Download arXiv:1904.05695 and read the statements of [7, Lemma 3.2] and [7, Lemma 4.3], verifying that their hypotheses are exactly 'aperiodic random walk in the domain of attraction of a nondegenerate alpha-stable law with d/alpha > 5/2' (together with A3/A4 where needed) rather than 'increment distribution is itself alpha-stable'. Separately, check [19, Remark after Cor. 3.2] and [19, Theorem 4.4] to confirm they apply under (A1), (A2) with d/alpha > 3/2, including the non-strongly-transient window 3/2 < d/alpha < 2. If the hypotheses match, the concern is resolved and the paper stands; if they are narrower, the theorems must be restated or supplied with proofs.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The capacity argument depends on two imported quantitative inputs: the Green-function intersection bound E[G(R_m, tilde R_m)] <= C_2 R_d(m) from [7, Lemma 3.2], used at (2.7), and the variance bound Var(C_n) <= C_1 n from [7, Lemma 4.3], used in the tightness estimate after (2.11). The cardinality argument similarly relies on E[J_n] <= C I_d(n) from [19, Remark after Cor. 3.2] at (3.4) and Var(|R_n|) <= C_1 n from [19, Theorem 4.4], as well as the fixed-time CLTs [7, Theorem 1.1] and [19, Theorem 4.5]. The manuscript asserts that (A2) makes these results applicable, but it does not quote the full hypotheses of the lemmas from the companion preprint [7]. Since [7] is a same-author preprint, the stated domain-of-attraction generality of Theorems 1.1 and 1.2 is exactly as secure as those external hypotheses. If any imported estimate was proved only for increment distributions that are literally alpha-stable, then the finite-dimensional convergence steps (2.2), (3.1) and the error-term estimates (2.7)-(2.11), (3.4)-(3.6) lose their justification, and the central claims overreach.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves functional central limit theorems for two functionals of the range of random walks in the domain of attraction of a stable law. Theorem 1.1 establishes that, under assumptions (A1)-(A4), 0<α≤2, and d/α>5/2, the centered capacity process (C_{⌊nt⌋}-E C_{⌊nt⌋})/(σ_d√n) converges weakly in D([0,∞),R) with the J1 topology to a standard Brownian motion. Theorem 1.2 establishes the analogous statement for the cardinality of the range under (A1), (A2), d/α>3/2 and P(τ_0^+=∞)<1. Theorem 2.1 gives a corresponding result for symmetric simple random walks in dimension d≥6. The proofs follow the standard two-step scheme: finite-dimensional convergence via the Cramér-Wold theorem together with the existing fixed-time CLTs, and tightness via Aldous' condition, using capacity decompositions and intersection-point estimates imported from the companion paper [7] and from Le Gall-Rosen [19].","tokens_in":32747,"tokens_out":26600,"duration_ms":238866,"significance":"If the proof gaps identified below can be repaired, the paper provides the first functional CLTs for the capacity of the range and for the cardinality of the range of stable random walks in low dimensions. The main ideas—decomposing the range into independent blocks and controlling the error terms via Green-function and intersection-point estimates—are natural and potentially useful. The paper is concise and clearly written, and it includes a useful comparison with the earlier results of Jain-Pruitt and Asselah-Schapira-Sousi. The central claims are plausible and the announced results would be a valuable addition to the literature. However, the manuscript relies heavily on quantitative estimates from [7], a companion preprint by the same authors, and the proof of tightness contains a questionable equality-in-law assertion for stopping times.","major_comments":[{"comment":"The tightness proof asserts that the random variable J(⌊nτ_n⌋,⌊n h_n⌋) has the same law as G(R_{⌊nτ_n⌋}, \\tilde R_{⌊n h_n⌋}) (respectively |R_{⌊nτ_n⌋} ∩ \\tilde R_{⌊n h_n⌋}|), where τ_n is a stopping time. This equality is not valid for arbitrary stopping times. The time-reversal argument used for deterministic times in Lemma 3.1 does not extend to stopping times: for example, if T is the first hitting time of a nonzero level by a one-dimensional simple random walk, then the shifted past range R_T - S_T is confined to a half-space and is not equal in law to R_T. Since the bounds E[J] ≤ C R_d(Kn) and E[J] ≤ C I_d(Kn) are derived from this asserted equality, the verification of condition (ii) (tightness) is incomplete. Please either prove the required bound directly for the shifted past range or replace the stopping-time argument.","section":"Section 2, after (2.11); Section 3, after (3.6)"},{"comment":"The proof imports quantitative estimates from [7, Lemmas 3.2 and 4.3] and [19, Theorem 4.4 and the Remark after Corollary 3.2] without stating the hypotheses under which those results are proved. Assumption (A2) is only a domain-of-attraction condition, while the cited results may require strictly α-stable increments or additional regularity. The manuscript should state the precise assumptions of these lemmas and verify that (A1)-(A4) (respectively (A1), (A2) and d/α>3/2) imply them. This is load-bearing for both the finite-dimensional convergence and the tightness, and the current text leaves the scope of the theorems conditional on the companion preprint.","section":"Equations (2.7), (2.11) and (3.4)-(3.6)"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'theor em' should be 'theorem'.","section":"Abstract"},{"comment":"The constant σ_d is not defined in the theorem statements; it is only described as the constant from [7] or [19]. Please state explicitly that σ_d is the limiting standard deviation so that the normalization is self-contained.","section":"Theorems 1.1 and 1.2"},{"comment":"The display in (2.4) is very involved; a short accompanying explanation of how the lower bound is obtained from the capacity decomposition would improve readability.","section":"Equation (2.4)"},{"comment":"There appears to be a typo where '⌊n h_n⌋' is typeset as '⌊n h h⌋' in the lower-bound display for the tightness proof.","section":"Section 2, after (2.11)"},{"comment":"In the display following (3.6), 'n h n' should be 'n h_n'.","section":"Section 3, after (3.6)"},{"comment":"The proof of Theorem 2.1 is very brief; it would be helpful to note explicitly that the referenced results from [1] apply to simple random walks and that assumption (A4) is not needed there.","section":"Theorem 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper depends substantially on the companion preprint [7] by the same authors, which is not yet published. The editor may wish to verify the status of [7] and, if it is not under review at a refereed journal, consider whether the current manuscript should be required to include the necessary estimates or to wait until [7] is accepted. Independently, the tightness gap concerning the law of J at stopping times is a substantive issue that requires the authors' attention."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does exactly what it says: it upgrades the capacity CLT from the authors' companion paper [7] to a functional CLT, and it proves a low-dimensional cardinality FCLT that Jain–Pruitt's invariance principle did not reach. The real new work is the tightness argument, and it is not trivial — the capacity and range increments are dependent, so the independent-block decomposition plus the error bounds has to be handled carefully. They do that cleanly.\n\nThe statements are precise, the assumptions are explicit, and the conjectures about the boundary cases are clearly separated from the theorems. The simple random walk corollary via [1] is a nice touch, and the proof follows the standard Kallenberg two-step recipe without obvious gaps. The reader's ACCEPT verdict seems right to me.\n\nThe main soft spot is exactly the one the stress test flags: the proof imports variance and intersection estimates from [7] and [19] without restating their full hypotheses. A reader cannot tell from this paper alone whether those estimates hold for every walk in the domain of attraction (A2) or only for literally stable walks. In context, I think the concern is more about presentation than substance — [7] is presumably about the same class of walks, and the authors explicitly say the lemmas hold under (A1)–(A3) — but because [7] is still a preprint, this is a genuine dependence that should be made visible. The fix is easy: quote the lemma statements, or add a sentence confirming they were proved under the same assumptions. I would not call this a load-bearing flaw; it is a completeness issue that a referee should ask about.\n\nA minor point: the paper is not self-contained, which is normal for this area, but the external dependence is heavier than the abstract suggests.\n\nThis paper deserves a serious referee. For a reader working on random walk range or capacity, it is a solid, incremental advance. I would send it to review with a request to clarify the scope of the imported estimates.","headline":"A clean, correct-looking FCLT for the range and capacity of stable random walks; the main caveat is that it leans on a companion preprint, and the authors should make those external hypotheses explicit.","tokens_in":33348,"tokens_out":2754,"would_cite":true,"duration_ms":29827,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F17","60F05","60G50","60G52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that both the capacity and the size of the visited set of a stable random walk, centered and scaled by the square root of time, converge as whole processes to standard Brownian motion.","keywords":["range of a random walk","capacity","functional central limit theorem","stable random walk","domain of attraction","Skorohod space","Green function","intersection points"],"falsifier":"A concrete test is to examine a walk satisfying (A1)-(A4) with $d/\\alpha>5/2$ and compute the capacity variance: if $\\operatorname{Var}(C_n)$ grows faster than linearly in $n$, or if $\\mathbb{E}[G(R_n,\\widetilde R_n)]/\\sqrt{n}$ fails to vanish, the imported tightness estimate is false and the Brownian scaling cannot hold.","tokens_in":32272,"feed_emoji":"📈","tokens_out":12364,"duration_ms":201303,"temperature":0.7,"pith_summary":"The paper proves functional central limit theorems for two functionals of the range of a stable random walk on $\\mathbb{Z}^d$: the capacity of the set of sites visited and the number of distinct sites visited. In both cases the centered process, normalized by $\\sqrt{n}$, converges in the Skorohod path space with the J1 topology to a standard one-dimensional Brownian motion. The capacity result holds for $d/\\alpha > 5/2$ under assumptions of aperiodicity, membership in the domain of attraction of an $\\alpha$-stable law, symmetry, strong transience, and one-step loops; the cardinality result holds for $d/\\alpha > 3/2$ under weaker assumptions, provided the walk has a positive probability of returning to the origin. A sympathetic reader should care because this upgrades earlier pointwise central limit theorems to statements about the whole trajectory of the fluctuation process, so path-level quantities of the range inherit Brownian limiting behavior.","feed_headline":"Range of stable random walks converges to Brownian motion","feed_subtitle":"Capacity and visited-set size now have whole-path limits, not just pointwise CLTs.","key_machinery":"The carrying mechanism is a decomposition of the range into independent time blocks. For any consecutive intervals covering $[0,n]$, the capacity of the whole range is bounded above by the sum of the capacities of the blocks and below by that sum minus twice a Green-function interaction term; for the cardinality there is an exact identity with the number of intersection points of two independent block ranges. Because the blocks become independent after the strong Markov property, applying this decomposition at finitely many times reduces linear combinations of the process to sums of independent centered increments, which converge to Gaussian via the one-dimensional CLT and the standard linear-combination criterion for joint convergence. Tightness is carried by two quantitative estimates imported from prior work: the variance bound $\\operatorname{Var}(C_n)\\le C_1 n$, and the expected Green-function intersection bound $\\mathbb{E}[G(R_m,\\widetilde R_m)]\\le C_2 R_d(m)$, where $R_d(m)$ is regularly varying with index strictly less than $1/2$ when $d/\\alpha>5/2$; the cardinality version uses the analogous bound for the expected number of intersection points of two independent ranges, whose growth index is below $1/2$ when $d/\\alpha>3/2$. These bounds force the fluctuation over a small time lag to vanish in probability, which is exactly the tightness condition in the two-step weak-convergence criterion.","core_discovery":"Let $R_n$ be the range up to time $n$ and let $C_n = \\operatorname{Cap}(R_n)$, where the capacity of a set $A$ is $\\operatorname{Cap}(A)=\\sum_{x\\in A} P_x(\\tau_A^+=\\infty)$, the sum over visited sites of the probability that the walk, started from that site, never returns to the set. The paper's central claim is that, under (A1)-(A4) with $0<\\alpha\\le 2$ and $d/\\alpha>5/2$, the process $(C_{\\lfloor nt\\rfloor}-\\mathbb{E}C_{\\lfloor nt\\rfloor})/(\\sigma_d\\sqrt{n})$ converges weakly in the Skorohod space $D([0,\\infty),\\mathbb{R})$ with the J1 topology to a standard Brownian motion, where $\\sigma_d>0$ is a constant determined by the walk. The analogous statement for the cardinality process $(|R_{\\lfloor nt\\rfloor}|-\\mathbb{E}|R_{\\lfloor nt\\rfloor}|)/(\\sigma_d\\sqrt{n})$ holds under (A1), (A2), $d/\\alpha>3/2$, and $P(\\tau_0^+=\\infty)<1$. A separate theorem gives the capacity FCLT for the simple random walk in $d\\ge 6$ without the one-step-loop assumption.","pith_inferences":["Since the proof's structure only needs the variance bound, the Green-function intersection bound, and the one-dimensional CLT, a natural extension is to drop the one-step-loop assumption (A4) entirely: the paper's own simple random walk theorem already works without it, suggesting the loop condition is a technical convenience rather than a real boundary.","The same block-decomposition and intersection-point estimates should yield a functional CLT for the intersection local time of two independent copies of the walk in the regime $d/\\alpha>3/2$, because the tightness input for that object is essentially the same intersection-point bound used here.","One testable direction is to push the capacity result to the critical line $d/\\alpha=5/2$: if the variance and intersection bounds with a slowly-varying correction can be established, the same proof scheme would give a Brownian limit with scaling $\\sqrt{n L(n)}$, which the paper states only as a conjecture."],"forward_implications":["By weak convergence in the Skorohod space, any J1-continuous functional of the capacity process—such as its running maximum, hitting times, or occupation times—converges to the corresponding functional of Brownian motion.","The cardinality result applies when $d/\\alpha>3/2$, which includes transient stable walks in dimension $d<3$, a regime not covered by earlier invariance principles for the range cardinality.","The capacity theorem covers the simple random walk in dimension $d\\ge 6$ as a special case (the paper's Theorem 2.1), upgrading the existing pointwise CLT to a functional one.","At the boundary $d/\\alpha=5/2$ the paper leaves the capacity case open and conjectures a Gaussian limit with a slowly-varying correction $\\sqrt{n L(n)}$, mirroring the known cardinality behavior at $d/\\alpha=3/2$."],"supporting_citations":[{"why":"Supplies the one-dimensional CLT for capacity, the variance bound Var(C_n) ≤ C_1 n, and the Green-function intersection estimate used in the finite-dimensional and tightness steps.","marker":"[7]"},{"why":"Supplies the one-dimensional CLT, variance bound, and intersection-point bound for the cardinality process that drive Theorem 1.2.","marker":"[19]"},{"why":"Provides the capacity decomposition used throughout, and supplies the estimates that replace [7] in the simple random walk case of Theorem 2.1.","marker":"[1]"},{"why":"The earlier invariance principle for the range cardinality that this paper's cardinality result extends to the stable domain-of-attraction class and to dimensions below 3.","marker":"[16]"},{"why":"The standard two-step weak-convergence criterion (finite-dimensional distributions plus the stopping-time tightness condition) used to prove both theorems.","marker":"[17]"}],"fun_headline_variants":["Stable walk range converges to Brownian motion in path space","Capacity and visited set size: functional CLT for stable walks","Whole-path limit for range of stable random walks","Brownian limit for stable walk range capacity and size","Stable random walk range: from pointwise to functional CLT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The tightness step depends on imported estimates asserting that the capacity variance grows at most linearly and that expected Green-function intersections between two independent ranges decay at the stated rate; the paper applies these to every walk in the domain of attraction of a stable law, and if the estimates are valid only for strictly stable walks, the theorems do not cover their stated hypotheses.","fun_headline_variants_meta":{"raw":{"variants":["Stable walk range converges to Brownian motion in path space","Capacity and visited set size: functional CLT for stable walks","Whole-path limit for range of stable random walks","Brownian limit for stable walk range capacity and size","Stable random walk range: from pointwise to functional CLT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000617,"raw_usage":{"total_tokens":2826,"prompt_tokens":870,"completion_tokens":1956,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":1874}},"tokens_in":486,"tokens_out":1956,"duration_ms":598703,"temperature":1.0,"reasoning_tokens":1874,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:54:57.081634+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to examine a walk satisfying (A1)-(A4) with $d/\\alpha>5/2$ and compute the capacity variance: if $\\operatorname{Var}(C_n)$ grows faster than linearly in $n$, or if $\\mathbb{E}[G(R_n,\\widetilde R_n)]/\\sqrt{n}$ fails to vanish, the imported tightness estimate is false and the Brownian scaling cannot hold.","supporting_citations":[{"cited_title":"CLT for the capacity of the range of stable random walks","cited_arxiv_id":"1904.05695","evidence_quote":"Supplies the one-dimensional CLT for capacity, the variance bound Var(C_n) ≤ C_1 n, and the Green-function intersection estimate used in the finite-dimensional and tightness steps."},{"cited_title":"Le Gall and J","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional CLT, variance bound, and intersection-point bound for the cardinality process that drive Theorem 1.2."},{"cited_title":"Asselah, B","cited_arxiv_id":null,"evidence_quote":"Provides the capacity decomposition used throughout, and supplies the estimates that replace [7] in the simple random walk case of Theorem 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier invariance principle for the range cardinality that this paper's cardinality result extends to the stable domain-of-attraction class and to dimensions below 3."},{"cited_title":"Kallenberg","cited_arxiv_id":null,"evidence_quote":"The standard two-step weak-convergence criterion (finite-dimensional distributions plus the stopping-time tightness condition) used to prove both theorems."}],"review_version":1}