Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$

D. Korshunov; I. M. Asymont

REVIEW 1 major objections 4 minor 13 references

Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A unified strong law of large numbers for local-time sums of transient random walks.

desk verdict A genuinely useful unification of SLLNs for functionals of local times, with one missing hypothesis in the main theorem statement that is trivial to fix. read the letter →

arxiv 1908.06611 v1 pith:EGFFVM3L submitted 2019-08-19 math.PR

I. M. Asymont , D. Korshunov This is my paper · ORCID

classification math.PR MSC 60G5060J5560F15

keywords transientrandomwalklocaltimesstronglawoflargenumbersrangeself-intersectionssitesvisitedexactlyjescapeprobability

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

A transient random walk on $\mathbb Z^d$ leaves a footprint of local times $l(n,x)$, and the paper proves that for any function $f$ growing not faster than about $\exp((1/2)\lambda^* k)$ with $\lambda^*=\log(1/(1-\gamma))$, the spatial sum $G_n(f)=\sum_{x\in\mathbb Z^d} f(l(n,x))$ obeys $G_n(f)/n\to \gamma^2\sum_{j\ge 1} f(j)(1-\gamma)^{j-1}$ almost surely and in mean square. Because $f$ is arbitrary, this one theorem supplies strong laws for the number of distinct sites visited, for $\alpha$-fold self-intersections, and for the number of sites visited exactly $j$ times, unifying results that were previously proved case by case. The condition on $f$ is a growth bound that makes the limiting series absolutely convergent; the paper also shows that with stronger tail assumptions on return probabilities the law can hold for $f$ growing like $e^{\lambda^* k}$ up to polynomial corrections. A sympathetic reader should see this as a definitive linear law for additive local-time statistics: the spatial average of any tame function of local time stabilises at a constant determined solely by the escape probability.

What carries the argument

The central representation is $G_n(f)=\sum_{j=1}^n f(j)Q_n(j)$, where $Q_n(j)$ is the number of sites visited exactly $j$ times. Its expectation is a renewal sum: $EQ_n(j)=\sum_{n_0+\cdots+n_j=n}\gamma_{n_0}\,\prod_{i=1}^{j-1}P\{\tau=n_i\}\,\gamma_{n_j}$, with $\gamma_n=P\{$no return by step $n\}$ and $\tau$ the first return time. Since $\gamma_n$ converges to the escape probability $\gamma$, dominated convergence gives $EQ_n(j)/n\to \gamma^2(1-\gamma)^{j-1}$. The variance is controlled by decomposing a general $f$ into non-decreasing parts, writing the covariance as a signed sum over tail events, and bounding the probability of alternating visits $x$-$y$-$x$ by $(1-\gamma)^{i-1}\sum_{r=1}^n r(n-r)P\{S_r=0\}$; transience makes this sum $o(n^2)$.

What would settle it

Take a walk with $\gamma\in(0,1)$ and set $f(k)=c/(1-\gamma)^k$, which violates condition (3). The paper conjectures $G_n(f)/n^2\to c'$ for $d\ge 5$, $G_n(f)/n^{3/2}\to c'$ for $d=3$, and $G_n(f)\log n/n^2\to c'$ for $d=4$. A simulation or rigorous bound showing $G_n(f)/n$ stays bounded, or diverges at a different rate, would falsify that conjecture.

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Extended reading notes

Core claim

Theorem 1 states: if $(S_n)$ is transient with escape probability $\gamma\in(0,1)$, and $f:\mathbb Z_+\to\mathbb R$ satisfies $\sum_{j\ge 1} f^2(j)\, j\,(1-\gamma)^j<\infty$, then $G_n(f)/n$ converges to $\gamma^2\sum_{j\ge 1} f(j)(1-\gamma)^{j-1}$ as $n\to\infty$, both in mean square and with probability $1$. The proof splits into an expectation asymptotics (Lemma 5), valid under the weaker condition $\sum |f(j)|(1-\gamma)^j<\infty$, and a variance bound (Lemma 6) that shows $\operatorname{Var} G_n(f)=o(n^2)$ under the squared condition; a subsequence argument (Lemma 7) upgrades $L^2$ convergence to almost sure convergence using only transience ($\sum_n P\{S_n=0\}<\infty$). The theorem applies to any transient walk in any dimension $d\ge 1$, with no moment or lattice assumptions.

Load-bearing premise

The claim depends on the growth condition $\sum_{j\ge 1} f^2(j)\,j\,(1-\gamma)^j<\infty$; if $f$ grows faster, the linear normalisation $n$ may fail and the stated limit may diverge, so the theorem applies only to functions growing at most exponentially with exponent below $\lambda^*/2$.

Editorial extensions

If this is right

Corollary 2: for every $\alpha\ge 0$, the number of $\alpha$-fold self-intersections $L_n(\alpha)$ satisfies $L_n(\alpha)/n\to \gamma^2\sum_{j\ge 1} j^\alpha(1-\gamma)^{j-1}$ almost surely and in $L^2$.
Corollary 3: for any set $J\subseteq\mathbb N$, the proportion of sites whose local time falls in $J$ converges almost surely to $\gamma^2\sum_{j\in J}(1-\gamma)^{j-1}$, generalising the Erdős–Taylor and Pitt theorems to arbitrary transient walks.
The growth condition (3) is satisfied by all subexponential $f$ and by exponentials up to order $e^{ck}$ with $c<\lambda^*/2$, so the result covers power functions of every degree.
Under return-probability tail conditions, Theorem 4 shows the SLLN persists for $f$ growing like $e^{\lambda^* k}/k^{2+\varepsilon}$ or $e^{\lambda^* k}/(k\log^{2+\varepsilon} k)$.
For recurrent walks the normalisation $n$ fails (e.g., $n/\log n$ in two dimensions), so the transient setting is exactly where the linear law holds.

Reading between the lines

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

The paper's conjectured normalisations for $f(k)\sim c/(1-\gamma)^k$ ($n^2$ in $d\ge 5$, $n^{3/2}$ in $d=3$, $n^2/\log n$ in $d=4$) suggest a phase transition: above the $e^{\lambda^* k/2}$ threshold, a small number of very heavily visited sites dominates $G_n(f)$, so the linear law breaks. Testing this numerically for a specific walk would clarify the boundary of Theorem 1.
The variance-bound method, which avoids Tauberian theorems and monotonicity of $EQ_n(j)$, might extend to additive functionals of other Markov chains with a renewal structure, giving a general law of large numbers for occupation statistics.
Applying the theorem to linear combinations of functions gives a multidimensional SLLN for local-time statistics: the vector $(G_n(f_1)/n,\dots,G_n(f_m)/n)$ converges almost surely to the corresponding deterministic limit vector.
The almost-sure part is proved by a subsequence trick that only needs $\sum_n a_n/n<\infty$; the sharpness of that condition, or a law of the iterated logarithm for $G_n(f)$, is left open.

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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 something real: it proves a single strong law of large numbers for G_n(f) = sum_x f(l(n,x)) over arbitrary transient random walks in any dimension, under a clean growth condition on f. That unifies the range, alpha-fold self-intersections, and exactly-j visits, and it removes the extra moment assumptions that earlier work needed in d=1,2. The proof also fixes a genuine gap in Becker–Koenig by avoiding a Tauberian argument that required the sequence EQ_n(j) to be ultimately increasing; the renewal-decomposition plus dominated-convergence route in Lemma 5 is clear and self-contained. Lemma 6's variance bound works for any non-decreasing f with f(0)=0, which is more general than the power-function-only bound in the literature. The almost-sure argument via the subsequence trick is standard but correctly executed. This is a solid piece of random walk theory.

The main flaw is in the statement of Theorem 1: it omits the necessary hypothesis f(0)=0. As written, G_n(f) sums f(0) over every unvisited site, and there are infinitely many of those, so G_n(f) is infinite for every n whenever f(0) is nonzero. The proof's 'Without loss of generality we assume f(0)=0' is not a lossless reduction; it is a necessary condition for the object to be finite. All the intended applications have f(0)=0, so the content of the paper is unaffected, and adding f(0)=0 to the theorem statement is a one-line fix. Still, as stated, the theorem quantifies over functions for which the conclusion is meaningless. That should be caught in revision.

Two other soft spots, both minor. First, condition (3) restricts f to subexponential or mildly exponential growth, and the paper itself notes this and proves Theorem 4 for faster growth under extra assumptions on return probabilities and a stronger tail condition on f. That is fine, but the boundary is a bit ad hoc. Second, the conjectured normalizations for f growing like (1-gamma)^{-k} are clearly labeled as conjectures, so there is no overclaiming, but they also point to where the theory is unfinished. The citation pattern looks honest: prior work is credited, the gap in Becker–Koenig is explicitly located, and the fixes are real.

Who is this for? People working on random walk local times, range statistics, or self-intersections of transient walks. The new theorem is worth knowing, and the proof technique for the variance is reusable. It deserves a serious referee; the mathematical core is sound, and the one required correction is purely cosmetic in content but necessary in statement. I would send it to review and ask for the f(0)=0 hypothesis to be added.

Referee Report

1 major / 4 minor

Summary. The paper proves a strong law of large numbers for the spatial sum G_n(f) = \sum_{x\in\mathbb Z^d} f(l(n,x)) of a function f of the local times of a transient random walk in \mathbb Z^d. Theorem 1 states that if \sum_{j\ge 1} f^2(j) j (1-\gamma)^j < \infty, then G_n(f)/n converges to \gamma^2 \sum_{j\ge 1} f(j)(1-\gamma)^{j-1} in mean square and almost surely, where \gamma is the escape probability. This unifies earlier results for the range, for \alpha-fold self-intersections, and for the number of sites visited exactly j times. The proof proceeds by a renewal decomposition for the expectation (Lemma 5), a variance bound for nondecreasing f with f(0)=0 (Lemma 6), and a subsequence Borel-Cantelli argument for the almost sure convergence. The paper also identifies and fixes gaps in earlier arguments by Becker-K\"onig and by Erd\H{o}s-Taylor, and proves a stronger result, Theorem 4, under weaker growth conditions at the cost of extra technical assumptions on the return probabilities.

Significance. If the missing hypothesis f(0)=0 is added, the main result is a clean, parameter-free limit constant derived from first principles through the renewal equation, with no fitted parameters and no normalization chosen after the fact. The theorem genuinely unifies the range, self-intersection counts, and exact-visit-count statistics for arbitrary transient random walks, and it removes the extra moment conditions needed in earlier treatments for d=1,2. The proof is self-contained and transparent: the variance lemma is of independent interest, and the paper explicitly flags and repairs gaps in prior literature, such as the Tauberian argument in Becker-K\"onig and the maximal-local-time upper bound in Erd\H{o}s-Taylor. The conjectured normalizations for faster-growing f are also a useful contribution to the topic.

major comments (1)

[Theorem 1 (Section 1) and Theorem 4 (Section 5)] Theorems 1 and 4 must include the explicit hypothesis f(0)=0. As stated, for any f with f(0)\neq 0, the quantity G_n(f)=\sum_{x\in\mathbb Z^d} f(l(n,x)) is infinite for every n, because only O(n) sites are visited up to time n while the remaining infinitely many sites contribute the constant f(0) to the sum; the quotient G_n(f)/n is then undefined. The opening of the proof of Theorem 1, 'Without loss of generality we assume f(0)=0,' is therefore not a lossless reduction but the installation of a necessary condition. The intended examples (range, \alpha-fold self-intersections, exactly-j visits) all have f(0)=0, and Lemma 6 already assumes f(0)=0, so adding this hypothesis to Theorems 1 and 4, and to the decomposition in (20), is a local correction that does not affect the substance of the proofs.

minor comments (4)

[Section 1] The phrase 'subexponential functions f(i) of order e^{o(i)}' is informal; condition (3) is a precise growth condition, so the text should say 'functions growing like e^{o(i)}' to avoid confusion with subexponential distributions in probability.
[Section 2, Lemma 5] Lemma 5 does not explicitly state the transience assumption, although the proof uses \gamma_n\to\gamma and the limit is expressed in terms of \gamma; adding 'for a transient random walk' would improve clarity.
[Section 5, Proposition 8] The wording 'for all n \geq N where N is finite with probability 1' should be rephrased as 'there exists a random finite N such that the inequality holds for all n\geq N, almost surely.'
[Section 1] There is a typo in the introduction: 'Bec ker' should be 'Becker.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SLLN limit is derived from first principles via a Markov decomposition, and no fitted parameter or load-bearing self-citation is present.

full rationale

The central derivation of Theorem 1 is self-contained. The limit constant gamma^2 sum f(j)(1-gamma)^{j-1} is obtained by decomposing G_n(f) = sum_j f(j) Q_n(j), writing E Q_n(j) as a convolution of return probabilities via the Markov property (equations (13)-(14)), and then applying dominated convergence using gamma_n -> gamma. Lemma 6 bounds the variance from first principles, and the almost-sure convergence is obtained by a standard Borel-Cantelli subsequence argument. No parameter is fitted to the limit in (4), and no external result is invoked to replace a proof of the target statement. Citations to Dvoretzky-Erdos, Spitzer, Becker-Konig, Erdos-Taylor, and Pitt are contextual comparison or motivating background; the one self-citation, Doney-Korshunov [7], appears only in a conjectural remark about different normalizations for faster-growing f and is not load-bearing for Theorem 1 or Theorem 4. The notable issue in the paper is a correctness flaw, not circularity: Theorem 1 omits the necessary hypothesis f(0)=0, and the proof begins 'Without loss of generality we assume f(0)=0' (Section 4), which is not lossless because G_n(f) is infinite for f(0) != 0. That is a rigor/statement problem, not a reduction of the conclusion to the assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof introduces no free parameters and no new entities. The limit constant is expressed in terms of gamma, the escape probability, which is a fundamental quantity of the walk and is derived, not fitted. All background results are standard probability tools.

assumptions (5)

domain assumption The random walk is transient, meaning gamma = P{S_n != 0 for all n >= 1} > 0, and gamma < 1 to exclude the trivial case of no returns.
The entire paper is set in this regime; transience gives finite expected number of returns and makes the renewal decomposition in Lemma 5 valid.
standard math Transience is equivalent to convergence of the return probability series sum_{n>=1} P{S_n=0} < infinity.
Used in the proof of Theorem 1 at equation (22) to show the variance bound tends to zero and to apply Kronecker's lemma.
standard math Markov property of the random walk and independence of the increments.
Used throughout, in particular in the renewal decomposition of EQ_n(j) in equation (13) and in the variance bound Lemma 6.
standard math Every function f: Z+ -> R decomposes as a difference of two non-decreasing functions with f_k(0)=0 via positive and negative parts of its increments.
Used in the proof of Theorem 1 to reduce to non-decreasing f; the reduction preserves condition (3) as shown in the inequalities preceding equation (21).
standard math Dominated convergence, Borel-Cantelli lemma, Chebyshev's inequality, and standard subsequence arguments are valid.
These tools are used in the expectation limit (Lemma 5) and in the almost sure convergence step of Theorem 1.

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Cite this review

Pith. "Pith review of Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$." pith.science (2026). https://pith.science/paper/EGFFVM3L

@misc{pith2026190806611,
  author       = {Pith},
  title        = {Pith review of: Strong law of large numbers for a function of the local times of a transient random walk in $\mathbb Z^d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EGFFVM3L}},
  note         = {Machine review of arXiv:1908.06611}
}

abstract

For an arbitrary transient random walk $(S_n)_{n\ge 0}$ in $\mathbb Z^d$, $d\ge 1$, we prove a strong law of large numbers for the spatial sum $\sum_{x\in\mathbb Z^d}f(l(n,x))$ of a function $f$ of the local times $l(n,x)=\sum_{i=0}^n\mathbb I\{S_i=x\}$. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and Erd\H{o}s), which corresponds to a function $f(i)=\mathbb I\{i\ge 1\}$; (b) $\alpha$-fold self-intersections of the random walk (studied by Becker and K\"{o}nig), which corresponds to $f(i)=i^\alpha$; (c) sites visited by the random walk exactly $j$ times (considered by Erd\H{o}s and Taylor and by Pitt), where $f(i)=\mathbb I\{i=j\}$.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Becker, M., K¨ onig, W.: Moments and distribution of the lo cal times of a transient random walk on Zd . J. Theor . Probab.22, 365–374 (2009)

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Bingham, N.H., Goldie, C.M., and Teugels, J.L.: Regular V ariation.Cambridge University Press, Cam- bridge (1987)

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Doney, R., Korshunov, D.: Local asymptotics for the time o f ﬁrst return to the origin of transient random walk. Stat. Probab. Lett. 81, 1419–1424 (2011)

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Dvoretzky, A., Erd˝ os, P .: Some problems on random walk inspace. In: Proc. 2nd Berkeley Symp. Math. Stat. Probab., 353–367 (1951)

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Pith tools

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

[1] [1]

Becker, M., K¨ onig, W.: Moments and distribution of the lo cal times of a transient random walk on Zd . J. Theor . Probab.22, 365–374 (2009)

work page 2009

[2] [2]

Bingham, N.H., Goldie, C.M., and Teugels, J.L.: Regular V ariation.Cambridge University Press, Cam- bridge (1987)

work page 1987

[3] [3]

ˇCern´ y, J.: Moments and distribution of the local time of a two-dimensional random walk. Stoch. Proc. Appl. 117, 262–270 (2007)

work page 2007

[4] [4]

Acta Math

Erd˝ os, P ., Taylor, S.J.: Some problems concerning the st ructure of random walk paths. Acta Math. Acad. Sci. Hungar .11, 137–162 (1960)

work page 1960

[5] [5]

B.: Strong renewal theorems with inﬁnite mea n

Erickson, K. B.: Strong renewal theorems with inﬁnite mea n. Trans. Amer . Math. Soc. 151, 263–291 (1970)

work page 1970

[6] [6]

Esseen, C.G.: On the concentration function of a sum of ind ependent random variables. Z. W ahrschein- lichkeitstheorie und V erw. Gebiete9 290–308 (1968)

work page 1968

[7] [7]

Doney, R., Korshunov, D.: Local asymptotics for the time o f ﬁrst return to the origin of transient random walk. Stat. Probab. Lett. 81, 1419–1424 (2011)

work page 2011

[8] [8]

In: Proc

Dvoretzky, A., Erd˝ os, P .: Some problems on random walk inspace. In: Proc. 2nd Berkeley Symp. Math. Stat. Probab., 353–367 (1951)

work page 1951