Functional CLT for the range of stable random walks

Nikola Sandri\'c; Stjepan \v{S}ebek; Wojciech Cygan

REVIEW 2 major objections 6 minor 1 cited by

Functional CLT for the range of stable random walks

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.07872 v3 pith:PERJBTYW submitted 2019-08-21 math.PR

Wojciech Cygan , Nikola Sandrić , Stjepan Šebek This is my paper · ORCID

classification math.PR MSC 60F1760F0560G5060G52

keywords rangeofarandomwalkcapacityfunctionalcentrallimittheoremstabledomainattractionSkorohodspaceGreenfunctionintersectionpoints

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $d/\alpha=3/2$.

Reading between the lines

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

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

A minor point: the paper is not self-contained, which is normal for this area, but the external dependence is heavier than the abstract suggests.

This 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.

Referee Report

2 major / 6 minor

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].

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 (2)

[Section 2, after (2.11); Section 3, after (3.6)] 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.
[Equations (2.7), (2.11) and (3.4)-(3.6)] 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.

minor comments (6)

[Abstract] There is a typo in the abstract: 'theor em' should be 'theorem'.
[Theorems 1.1 and 1.2] 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.
[Equation (2.4)] 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 2, after (2.11)] 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 3, after (3.6)] In the display following (3.6), 'n h n' should be 'n h_n'.
[Theorem 2.1] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the FCLT is derived from the authors' pointwise CLT and independent variance/intersection estimates, which are weaker inputs, not the target conclusion.

full rationale

The paper's main theorems are functional CLTs, which require both finite-dimensional convergence and tightness. Finite-dimensional convergence is obtained from the pointwise CLT in the same authors' companion paper [7, Theorem 1.1] (for capacity) and from the external Le Gall–Rosen CLT [19, Theorem 4.5] (for cardinality), combined with an independent-increment decomposition supplied by the Markov property. The pointwise CLT is a strictly weaker marginal statement and does not by itself imply joint convergence or tightness. Tightness is then proved from quantitative variance and intersection bounds imported from [7, Lemmas 3.2 and 4.3] and [19, Remark after Cor. 3.2 and Theorem 4.4]. These bounds are fixed estimates, not fitted parameters, and they do not contain the functional convergence being proved. The only same-author citation is [7], which provides independent content: a marginal CLT and variance estimates, not the FCLT. No equation defines its own conclusion, no fitted input is renamed as a prediction, and no uniqueness theorem is imported to force a choice. The skeptical concern that the hypotheses of the imported lemmas may not cover all walks satisfying (A2) is a correctness or hypothesis-checking risk, not a circularity. Accordingly, the paper exhibits no significant circularity; the score reflects only the presence of same-author results as technical inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants, no invented particles, mediators, forces, or dimensions. The constants sigma_d are asserted to exist through the prior CLT and are not fitted to data. The central claim rests on the domain assumptions (A1)-(A4) and on cited theorems, which are entirely standard in this area. The ledger is clean: the paper adds a functional limit theorem, not a new entity or a free parameter.

assumptions (6)

domain assumption Assumptions (A1)-(A4): aperiodicity, domain of attraction of a nondegenerate alpha-stable law, symmetry plus strong transience, and P(xi_1 = 0) > 0.
Theorem 1.1 is proved only under these assumptions. A4 is explicitly technical and excludes simple random walk; A3 is used to control Green function estimates and intersection errors.
domain assumption Assumption P(tau_0^+ = infinity) < 1 in Theorem 1.2.
This excludes the degenerate case where |R_n| = n + 1 almost surely. It is stated as a hypothesis, though it is automatic in many transient stable cases.
standard math Companion CLT and estimate theorems from [7, Theorem 1.1, Lemmas 3.1, 3.2 and 4.3] and from [19, Theorem 4.5 and the estimate after Corollary 3.2].
The finite-dimensional convergence of the capacity process is imported directly from [7]; the cardinality finite-dimensional convergence uses [19, Theorem 4.5]; the intersection bounds are used at (2.7) and (3.4).
standard math Capacity decomposition and subadditivity from [1, Corollary 2.1] and [23, Proposition 25.11].
Used to decompose the range into independent blocks for the Cramer-Wold argument and to derive lower and upper bounds for the capacity and cardinality processes.
standard math Regular variation facts and slow-variation properties from [5] and [19, Lemma 2.2].
Used to conclude that the error terms R_d(n)/sqrt(n) and J_d(n)/sqrt(n) vanish in the regimes d/alpha > 5/2 and d/alpha > 3/2.
standard math Skorohod space tightness criterion and Cramer-Wold theorem from [17, Theorems 16.10-16.11 and Corollary 5.5].
This is the standard two-step scheme for weak convergence in D([0,infinity), R); both conditions (i) and (ii) are checked accordingly.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Functional CLT for the range of stable random walks." pith.science (2026). https://pith.science/paper/PERJBTYW

@misc{pith2026190807872,
  author       = {Pith},
  title        = {Pith review of: Functional CLT for the range of stable random walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PERJBTYW}},
  note         = {Machine review of arXiv:1908.07872}
}

abstract

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3\alpha / 2$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Functional Limit Theorems for the range of stable random walks
math.PR 2025-09 conditional novelty 7.0 of 10

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.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 1 Pith paper

[7]

CLT for the capacity of the range of stable random walks

W . Cygan, N. Sandrić, and S. Šebek. CLT for the capacity of the range of stable random walks. Preprint (2019), arXiv:1904.05695

work page Pith review arXiv 2019
[1]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the ran ge of random walk on Z/u1D451 . Trans. Am. Math. Soc. , 370(11):7627–7645, 2018

work page 2018
[19]

Le Gall and J

J.-F. Le Gall and J. Rosen. The range of stable random wal ks. Ann. Probab., 19(2):650–705, 1991

work page 1991
[2]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the range of random walk onZ4. Ann. Probab., 47(3):1447– 1497, 2019

work page 2019
[3]

R. F. Bass and T. Kumagai. Laws of the iterated logarithm f or the range of random walks in two and three dimensions. Ann. Probab., 30(3):1369–1396, 2002

work page 2002
[4]

R. F. Bass and J. Rosen. An almost sure invariance princip le for the range of planar random walks. Ann. Probab., 33(5):1856–1885, 2005

work page 2005
[5]

N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular Variation. Cambridge University Press, Cambridge, 1989

work page 1989
[6]

Y . Chang. Two observations on the capacity of the range of simple random walks on Z3 and Z4. Electron. Commun. Probab., 22(25):9 pp., 2017

work page 2017

Show all 24 references

[8]

Dvoretzky and P

A. Dvoretzky and P. Erdös. Some problems on random walk in space. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 195 0, pages 353–367. University of California Press, Berkeley and Los Angeles, 1951

1951
[9]

Y . Hamana. An almost sure invariance principle for the ra nge of random walks. Stochastic Process. Appl. , 78(2):131–143, 1998

1998
[10]

N. C. Jain and S. Orey. On the range of random walk. Israel J. Math., 6:373–380 (1969), 1968

1969
[11]

N. C. Jain and W . E. Pruitt. The central limit theorem for the range of transient random walk. Bull. Amer. Math. Soc., 76:758–759, 1970

1970
[12]

N. C. Jain and W . E. Pruitt. The range of recurrent random walk in the plane. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 16:279–292, 1970

1970
[13]

N. C. Jain and W . E. Pruitt. The range of transient random walk. J. Analyse Math., 24:369–393, 1971

1971
[14]

N. C. Jain and W . E. Pruitt. The law of the iterated logarithm for the range of random walk. Ann. Math. Statist., 43:1692–1697, 1972

1972
[15]

N. C. Jain and W . E. Pruitt. The range of random walk. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. Californi a, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 31–50, 1972

1970
[16]

N. C. Jain and W . E. Pruitt. Further limit theorems for the range of random walk. J. Analyse Math., 27:94–117, 1974

1974
[17]

Kallenberg

O. Kallenberg. Foundations of modern probability . Probability and its Applications (New York). Springer- Verlag, New York, second edition, 2002

2002
[18]

J.-F. Le Gall. Propriétés d’intersection des marches a léatoires. I. Convergence vers le temps local d’intersection. Comm. Math. Phys., 104(3):471–507, 1986

1986
[20]

S. C. Port. Limit theorems for transient Markov chains. J. Combinatorial Theory , 2:107–128, 1967

1967
[21]

Sato and T

K. Sato and T. Watanabe. Moments of last exit times for Lé vy processes. Ann. Inst. H. Poincaré Probab. Statist., 40(2):207–225, 2004

2004
[22]

Schapira

B. Schapira. Capacity of the range in dimension 5. Prepr int (2019), arXiv:1904.11183

2019 arXiv
[23]

F. Spitzer. Principles of random walk . Springer-Verlag, New York-Heidelberg, second edition, 1976

1976
[24]

Takeuchi

J. Takeuchi. Moments of the last exit times. Proc. Japan Acad., 43:355–360, 1967. FCLT FOR THE RANGE OF STABLE RANDOM W ALKS 13 (Wojciech Cygan) I NSTITUT FÜR MATHEMATISCHE STOCHASTIK , T ECHNISCHE UNIVERSITÄT DRESDEN , DRESDEN , G ERMANY & I NSTYTUT MATEMATYCZNY , U NIWERSYTE...

1967

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [7]

CLT for the capacity of the range of stable random walks

W . Cygan, N. Sandrić, and S. Šebek. CLT for the capacity of the range of stable random walks. Preprint (2019), arXiv:1904.05695

work page Pith review arXiv 2019

[2] [1]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the ran ge of random walk on Z/u1D451 . Trans. Am. Math. Soc. , 370(11):7627–7645, 2018

work page 2018

[3] [19]

Le Gall and J

J.-F. Le Gall and J. Rosen. The range of stable random wal ks. Ann. Probab., 19(2):650–705, 1991

work page 1991

[4] [2]

Asselah, B

A. Asselah, B. Schapira, and P. Sousi. Capacity of the range of random walk onZ4. Ann. Probab., 47(3):1447– 1497, 2019

work page 2019

[5] [3]

R. F. Bass and T. Kumagai. Laws of the iterated logarithm f or the range of random walks in two and three dimensions. Ann. Probab., 30(3):1369–1396, 2002

work page 2002

[6] [4]

R. F. Bass and J. Rosen. An almost sure invariance princip le for the range of planar random walks. Ann. Probab., 33(5):1856–1885, 2005

work page 2005

[7] [5]

N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular Variation. Cambridge University Press, Cambridge, 1989

work page 1989

[8] [6]

Y . Chang. Two observations on the capacity of the range of simple random walks on Z3 and Z4. Electron. Commun. Probab., 22(25):9 pp., 2017

work page 2017