On nonlinear weak law of large numbers

Alexander Veretennikov; Alina Akhmiarova

REVIEW 1 major objections 6 minor 12 references

On nonlinear weak law of large numbers

T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves a nonlinear weak law of large numbers: without any first-moment assumption, nonlinearly pairwise independent smoothed summands keep $S_n/n$ between moving lower and upper means, with lower capacity tending to one.

desk verdict A real, incremental extension of Kolmogorov's WLLN to sublinear expectations without moments; needs a regularity assumption on the expectation and some proof tidying before it is solid. read the letter →

arxiv 2506.07470 v1 pith:YNWPWW5J submitted 2025-06-09 math.PR

Alina Akhmiarova , Alexander Veretennikov This is my paper

classification math.PR MSC 60F15

keywords nonlinearweaklawoflargenumberssublinearexpectationindependenceupperandlowercapacitynofirstmomentsmoothedtruncationuniformintegrabilitytailcondition

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

This paper establishes a weak law of large numbers in a sublinear expectation setting without assuming that any summand has a first moment. The proof works with smooth truncated variables $Y_{n,k}=X_k\chi_n(|X_k|)$ and assumes they are nonlinearly pairwise independent, meaning the sublinear expectation factorizes products of test functions of the two variables. If the capacity tail sums $\psi_n(y)=\sum_{k=1}^n y\,E\mathbf{1}(|X_k|>ny)$ vanish for every $y\in[0,1]$ and the family $\{\psi_n\}$ is uniformly integrable, then for every $\varepsilon>0$ the upper capacity of the average exceeding the moving upper mean, or falling below the moving lower mean, tends to zero. Consequently the lower capacity of the interval between the two moving means tends to one. This matters because in model-ambiguity settings one has upper and lower capacities instead of a single probability, and the result shows the classical weak-law phenomenon survives without moment assumptions.

What carries the argument

The load-bearing device is the smoothed truncation $Y_{n,k}=X_k\chi_n(|X_k|)$ combined with the capacity tail sum $\psi_n(y)=\sum_{k=1}^n y\,E\mathbf{1}(|X_k|>ny)$. Because $\chi_n$ is continuous with compact support, the proof can use a representation of the sublinear expectation of each single variable as a supremum over ordinary countably additive probability measures; this makes available the classical integration-by-parts identity for the truncated second moment $\sigma^\theta_k(t)=t^{-1}\int_{-t}^t x^2\,dF^\theta_{X_k}(x)$, which produces the key estimate $(1/n^2)\sum_k EY_{n,k}^2\to0$. Nonlinear pairwise independence kills the cross terms in the second moment of the smoothed sum $S'_n$, standard tail-probability inequalities transfer the second-moment control to the upper capacity of the bad event, and uniform integrability of $\psi_n$ lets the integral $\int_0^1\psi_n(y)\,dy$ pass to zero after a change of variables. The lower-capacity statement then follows from the duality between upper and lower expectation.

What would settle it

The decisive check is to exhibit a sequence that satisfies the theorem's hypotheses, nonlinear pairwise independence plus $\psi_n(y)\to0$ with $\psi_n$ uniformly integrable, yet for which $V(S_n/n\ge\bar\mu_n+\varepsilon)$ does not tend to zero; such a sequence would disprove the theorem. A natural place to look is a sublinear expectation whose representing measures for the finite collection $(X_1,\dots,X_n)$ are only finitely additive, since the paper itself flags this as the point where its method could break.

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Extended reading notes

Core claim

The central claim is Theorem 1: for each $n$, smooth each summand by $Y_{n,k}=X_k\chi_n(|X_k|)$, where $\chi_n$ is a Lipschitz function equal to $1$ on $[-n,n]$ and $0$ outside $[-(n+1),n+1]$, and assume the $Y_{n,k}$ are nonlinearly pairwise independent. Let $\bar\mu_n=n^{-1}\sum_k EY_{n,k}$ be the averaged upper expectation of the smoothed summands and $\mu_n=n^{-1}\sum_k \underline{E}Y_{n,k}$ the averaged lower expectation, with $E$ the sublinear expectation and $\underline{E}$ its conjugate lower expectation. Under the tail condition $\psi_n(y)\to0$ for all $y\in[0,1]$ and uniform integrability of the family $(\psi_n)$, the theorem concludes $V(S_n/n\ge\bar\mu_n+\varepsilon)\to0$ and $V(S_n/n\le\mu_n-\varepsilon)\to0$ for every $\varepsilon>0$, hence $v(\mu_n-\varepsilon<S_n/n<\bar\mu_n+\varepsilon)\to1$. The statement is an extension: no first moment of any $X_k$ is required, and the boundaries $\bar\mu_n$ and $\mu_n$ are themselves defined through truncated expectations, so they are finite even for heavy-tailed summands.

Load-bearing premise

The proof rests on representing the sublinear expectation of each individual summand as a supremum over ordinary countably additive probability measures; if that representation fails and only finitely additive measures are available, the integration-by-parts step controlling the truncated second moment is not justified.

Editorial extensions

If this is right

For a linear expectation and identically distributed summands, the hypotheses reduce to the classical tail condition $t\,\mathbb{P}(|X_1|>t)\to0$, so the classical weak law is a special case.
The theorem applies to summands with infinite first moments whenever the tail sums $\psi_n(y)$ vanish, so heavy-tailed sequences are covered without any moment truncation beyond the smooth cutoff.
The pairwise independence assumption can be relaxed to an averaged negative-correlation condition on the smoothed, centered variables, so the result covers weakly negatively correlated nonlinear sequences as well.
Because the conclusion is stated in upper and lower capacities, it yields bounds that hold simultaneously for every probability measure representing the sublinear expectation, which is the relevant guarantee under model ambiguity.

Reading between the lines

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

If the representation of the sublinear expectation as a supremum over countably additive measures can be proved for finite collections rather than only single variables, the same second-moment argument should extend to nonlinear dependence structures beyond pairwise independence, as long as the covariance-type sum in condition (8) is controlled.
Since all bounds are tracked through $\int_0^1\psi_n(y)\,dy$ and the covariance terms, a quantitative version of the theorem should give explicit rates for the capacity convergence by keeping the constants in the estimates (13)--(15).
An immediate test case is a family of heavy-tailed laws with infinite mean and ambiguous scale parameters: one can compute $\psi_n(y)$ in closed form and check numerically whether the average stays between the moving means with lower capacity approaching one.
The integrated condition (7) mentioned in the paper may be easier to verify in concrete models than pointwise vanishing of $\psi_n$; if it implies the two hypotheses under mild extra assumptions, the theorem's range of application widens.

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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

Quick take: the paper proves a nonlinear WLLN without first moments under pairwise nonlinear independence. The result is a genuine extension of the prior literature but rests on a representation theorem that needs an explicit regularity condition. With that patched, the theorem should survive review.

What is actually new: existing nonlinear LLNs require moment conditions; here the authors drop that entirely and relax independence to a pairwise nonlinear version. The smoothed truncation device is a thoughtful way to stay inside the function class without reproving indicator representations. The proof is a standard truncation-and-second-moment argument, and the main estimates (13)-(15) are valid in substance. The paper also honestly flags possible relaxations of independence in Remarks 5-6.

Soft spots: Proposition 1 is load-bearing. Step 5 integrates by parts under sigma-additive P_theta for each X_k, but the stated Definition 2 does not by itself guarantee sigma-additive representing measures; Peng's lemma typically needs a continuity or regularity condition. The stress-test's concrete counterexample (E=(max+min)/2 on three points) does not actually satisfy subadditivity, so that particular witness fails. The broader concern stands, though: the theorem as stated claims all sublinear expectations satisfying Definition 2, and the proof is not justified in that full generality without an extra assumption. This is fixable by stating the needed regularity hypothesis in Theorem 1 or citing an exact version of Peng's representation that holds under the paper's axioms.

Minor: Steps 6 and 7 write equality where the displayed bound is only one-sided; Step 1 includes an unused and unjustified '= sup_theta' for a joint event; Remark 4 has an unresolved '???'. None of this undermines the main argument.

The bibliography looks appropriate; the self-citation [1] is used as motivation. A careful referee should check the representation assumption and the technical slips.

Recommendation: send to peer review. The core result is plausible and new, and the gaps are repairable.

Referee Report

1 major / 6 minor

Summary. The paper proves a weak law of large numbers for triangular arrays under an upper sublinear expectation E. Smoothed truncations Y_{n,k}=X_k chi_n(|X_k|) are assumed pairwise independent in the nonlinear sense. If psi_n(y)=sum_{k=1}^n y E 1(|X_k|>ny) tends to 0 pointwise on [0,1] and the family {psi_n} is uniformly integrable, Theorem 1 claims V(S_n/n >= bar_mu_n + epsilon) -> 0 and V(S_n/n <= mu_n - epsilon) -> 0, hence v(mu_n - epsilon < S_n/n < bar_mu_n + epsilon) -> 1, where the thresholds are averages of the upper and lower expectations of the truncated variables. The proof follows the classical truncation and second-moment scheme, using a representation of E phi(X_k) by sigma-additive measures (Proposition 1, cited from Peng) and integration by parts. No first moment assumption is imposed.

Significance. The intended contribution is a Kolmogorov-type WLLN for arrays under a sublinear expectation that requires no first moment and only pairwise nonlinear independence. The assumptions are explicit and checkable, the smoothed truncation device is sensible, and the estimates (13)-(15) are transparent and would be valid under a correct representation theorem. However, the central claim is not established for the class of sublinear expectations in Definition 2: the proof depends on Proposition 1, which is false in that generality. If the theorem is restricted to representable expectations (for example, those of Example 1) or supplemented with the regularity assumptions under which Peng's lemma holds, the result would be a useful addition to the nonlinear limit-theorem literature.

major comments (1)

[Section 2, Proposition 1; Section 4, Step 5, Eq. (13)] Proposition 1 is false under the stated Definition 2, and the proof of the central estimate (13) depends on it. Let Omega={1,2,3}, H be all functions, and E[X]=(max X + min X)/2. This functional is monotone, constant preserving, positively homogeneous, and subadditive, so it is admissible under Definition 2. For X(omega)=omega, suppose a family of sigma-additive probability measures satisfied E phi(X)=sup_theta E_{P_theta} phi(X) for every Lipschitz phi. For Lipschitz functions taking the value patterns (1,0,0), (0,1,0), (0,0,1), (1,1,0), and (1,0,1) at the three points, this would force, respectively, sup_theta P_theta({1})=1/2, sup_theta P_theta({2})=1/2, sup_theta P_theta({3})=1/2, sup_theta(P_theta({1})+P_theta({2}))=1/2, and sup_theta(P_theta({1})+P_theta({3}))=1/2. The fourth condition implies P_theta({3})>=1/2 for all theta; combined with sup_theta P_theta({3})=1/2, this gives P_theta({3})=1/2 for all theta. The fifth condition then gives P_theta({1})<=0 for all theta, so P_theta({1})=0 for all theta, contradicting sup_theta P_theta({1})=1/2. Thus no such representation exists. Since Step 5 constructs distribution functions F^theta_k and applies the sigma-additive integration-by-parts formula for sigma^theta_k(t) under these measures, estimate (13) is unsupported in the claimed generality. Remark 2 does not repair the gap, because the failure already occurs for a single random variable. The theorem must be restricted to a class for which such a representation holds (for instance, expectations of the Example 1 form or Peng's framework with the additional regularity conditions needed for Lemma 1.3.4), or the proof must avoid this representation entirely.

minor comments (6)

[Section 4, Step 1, Eq. (9)] The parenthetical equality V(S_n/n >= bar_mu_n + epsilon) = sup_theta P_theta(S_n/n >= bar_mu_n + epsilon) is not justified by Proposition 1, which applies to a single variable and to Lipschitz test functions; for events involving the whole sum only finitely additive measures may exist, as Remark 2 concedes. This equality is not used in the subsequent estimates, so it should be deleted or replaced by an explicit assumption.
[Section 4, Step 7, Eq. (15); Step 6, Eq. (14)] In Eq. (15), the displayed first equality should be an inequality: E(-2 tilde_mu^+_{n,k} Y_{n,k}) <= 2 |tilde_mu^+_{n,k}| E|Y_{n,k}| follows from subadditivity and monotonicity, but equality need not hold. In Eq. (14), the square of sup_theta E_theta(Y_{n,k}) should be written as |sup_theta E_theta(Y_{n,k})|^2 before applying the Cauchy-Schwarz bound, since the supremum may be negative.
[Section 3, Remark 4] Remark 4 contains a literal '???' in the text and states an equivalence between the displayed integral condition and condition (7) without proof; this should be corrected or the remark removed.
[Section 4, Part II] The lower-tail argument is only sketched as 'similar'; writing the mirror calculation for -X_k and the lower means explicitly would make the proof easier to verify, especially because the roles of upper and lower expectations change in that calculation.
[Section 4, Eq. (13)] The convergence of the integral of psi_n to zero from pointwise convergence together with uniform integrability should be justified by an explicit application of Vitali's convergence theorem.
[Throughout] Typos and language issues include 'defnied' in Definition 3, 'estmate' in Step 7, 'enhance' in the Introduction, and the abstract's 'proposed' should be 'is proposed'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; self-citations are contextual and not load-bearing.

full rationale

Walked the derivation chain. The theorem is proved directly from the Definition-2 axioms plus Definition-5 pairwise independence and the smooth-truncation conditions (3)/uniform integrability: Step 4 uses pairwise independence exactly to zero out cross terms, Step 5 bounds E(Y^2) via integration by parts under the sigma-additive measures supplied by Proposition 1, and Steps 6-7 combine Markov's inequality. No parameter is fitted to data and then 'predicted'; the moving frontiers \bar\mu_n and \mu_n are non-random means of the truncated variables, not outputs of the theorem. The conclusion (6) is a nontrivial concentration statement about S_n/n around those means and would fail without the second-moment estimate, so it is not equivalent to a definition or to an input assumption. The only self-citations are [1] (the authors' prior LLN paper), used in the Introduction as motivation and in Remark 6 as a comparison; neither enters the proof, so the self-citation is not load-bearing. Proposition 1 is cited from Peng [10] as external support; whether it is valid at the full generality of Definition 2 is a correctness concern, and Remark 2 already acknowledges finite-additivity limitations for collections. A doubt about Proposition 1 would undermine the proof but would not make the derivation circular. Score 2 reflects only the presence of a minor, non-load-bearing self-citation; no circular step was identified.

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

The paper's central claim rests on the standard Peng sublinear expectation framework and on two cited external results: the representation of each single X_k by a family of sigma-additive measures, and Markov/Chebyshev inequalities. No free parameters are fitted and no new entities are postulated; the tail condition and uniform integrability are hypotheses, not fitted constants.

assumptions (5)

domain assumption Sublinear expectation E satisfies monotonicity, constant preservation, positive homogeneity, and subadditivity (Definition 2).
This is the nonlinear expectation framework of Peng [10] adopted as the paper's setting.
domain assumption For each X_k in H, there exists a family of sigma-additive probability measures (P_theta, theta in Theta_k) such that E phi(X_k)=sup_theta E_theta phi(X_k) for all phi in C_l,Lip (Proposition 1, citing Peng [10]).
Used in Step 5 to justify integration by parts under each representing measure; the paper notes that for collections of variables only finite additivity is guaranteed, so this single-variable representation is load-bearing.
standard math Markov and Chebyshev inequalities hold for the upper capacity V (Lemma 1, cited from Peng [10] and Chen et al. [2]).
Used in Step 3 to pass from a probability tail to a second-moment estimate under E.
standard math Uniform integrability of (psi_n) plus pointwise convergence psi_n(y)->0 implies integral_0^1 psi_n(y)dy->0 (Vitali's theorem).
Invoked at the end of Step 5; not proved.
standard math Integration by parts formula sigma_theta(t) = -t hat_gamma(t) + (2/t) integral_0^t x hat_gamma(x)dx from Feller [3].
Used in Step 5 to convert second moments into tail integrals.

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Cite this review

Pith. "Pith review of On nonlinear weak law of large numbers." pith.science (2026). https://pith.science/paper/YNWPWW5J

@misc{pith2026250607470,
  author       = {Pith},
  title        = {Pith review of: On nonlinear weak law of large numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YNWPWW5J}},
  note         = {Machine review of arXiv:2506.07470}
}

read the original abstract

A new version of a weak nonlinear law of large numbers proposed. The existence of the first moment for any summand is not assumed. The assumption of independence is understood in the nonlinear sense, and may be further a little relaxed.

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Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages

[1]

Alina Akhmiarova, Alexander Veretennikov, On the Law of Large Numbers for non-equally distributed weakly dependent random vari- ables, Teoriya veroyatnostey i ee primeneniya, 2025, 70(2), 211 - 227, https://doi.org/10.4213/tvp5725 (in Russian)

work page doi:10.4213/tvp5725 2025
[2]

Z. Chen, P. Wu, B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approxi- mate Reasoning, Volume 54, Issue 3, April 2013, Pages 365-377, https://doi.org/10.1016/j.ijar.2012.06.002

work page doi:10.1016/j.ijar.2012.06.002 2013
[3]

Feller, An Introduction to Probability Theory and Its Applications, 3rd Edition, John Wiley, Hoboken, 1968

W. Feller, An Introduction to Probability Theory and Its Applications, 3rd Edition, John Wiley, Hoboken, 1968

work page 1968
[4]

X. Fang, S. Peng, Q-M. Shao, et al., Limit theorems with rate of con- vergence under sublinear expectations, arXiv:1711.10649 (2017/2018); https://doi.org/10.48550/arXiv.1711.10649

work page Pith review arXiv doi:10.48550/arxiv.1711.10649 2017
[5]

Cheng Hu, Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition, Chin. Ann. Math. Ser. B 39(5), 2018, 791–804 DOI: 10.1007/s11401-018-0096-2

work page doi:10.1007/s11401-018-0096-2 2018
[6]

Kolmogorov, Sur la loi de grands nombres, Atti Accad

A.N. Kolmogorov, Sur la loi de grands nombres, Atti Accad. naz. Lincei. Rend. 1929, 9(6), 470–474

work page 1929
[7]

Kolmogorov, Foundations of the Theory of Probability, 2nd Rus- sian edition, Moscow, Nauka, 1974 (in Russian)

A.N. Kolmogorov, Foundations of the Theory of Probability, 2nd Rus- sian edition, Moscow, Nauka, 1974 (in Russian)

work page 1974
[8]

A. A. Lebedev and V. A. Lebedev, Generalization of the Expectation for a Unified Description of Random and Indefinite Factors, Theory Probab. Appl., 38:3 (1993), 470–478, DOI: https://doi.org/10.1137/1138043

work page doi:10.1137/1138043 1993

Show all 12 references

[9]

W. Li, C. Sun, Two laws of large numbers for sub- linear expectations, arXiv.1511.05631; (2015/2023) https://doi.org/10.48550/arXiv.1511.05631

2015 doi arXiv
[10]

Peng, Nonlinear Expectations and Stochastic Calculus under Uncer- tainty — with Robust Central Limit Theorem and G-Brownian Motion, arXiv:1002.4546; https://arxiv.org/pdf/1002.4546

S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncer- tainty — with Robust Central Limit Theorem and G-Brownian Motion, arXiv:1002.4546; https://arxiv.org/pdf/1002.4546

arXiv
[11]

Peng, Nonlinear Expectations and Stochastic Calculus under Un- certainty, with Robust CLT and G-Brownian Motion

S. Peng, Nonlinear Expectations and Stochastic Calculus under Un- certainty, with Robust CLT and G-Brownian Motion. Berlin, Springer, 2019; https://doi.org/10.1007/978-3-662-59903-7 12

2019 doi
[12]

Song, Stein’s Method for Law of Large Numbers under Sublinear Expectations, arXiv:1904.04674 (2019); https://doi.org/10.48550/arXiv.1904.04674 13

Y. Song, Stein’s Method for Law of Large Numbers under Sublinear Expectations, arXiv:1904.04674 (2019); https://doi.org/10.48550/arXiv.1904.04674 13

2019 doi arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Alina Akhmiarova, Alexander Veretennikov, On the Law of Large Numbers for non-equally distributed weakly dependent random vari- ables, Teoriya veroyatnostey i ee primeneniya, 2025, 70(2), 211 - 227, https://doi.org/10.4213/tvp5725 (in Russian)

work page doi:10.4213/tvp5725 2025

[2] [2]

Z. Chen, P. Wu, B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approxi- mate Reasoning, Volume 54, Issue 3, April 2013, Pages 365-377, https://doi.org/10.1016/j.ijar.2012.06.002

work page doi:10.1016/j.ijar.2012.06.002 2013

[3] [3]

Feller, An Introduction to Probability Theory and Its Applications, 3rd Edition, John Wiley, Hoboken, 1968

W. Feller, An Introduction to Probability Theory and Its Applications, 3rd Edition, John Wiley, Hoboken, 1968

work page 1968

[4] [4]

X. Fang, S. Peng, Q-M. Shao, et al., Limit theorems with rate of con- vergence under sublinear expectations, arXiv:1711.10649 (2017/2018); https://doi.org/10.48550/arXiv.1711.10649

work page Pith review arXiv doi:10.48550/arxiv.1711.10649 2017

[5] [5]

Cheng Hu, Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition, Chin. Ann. Math. Ser. B 39(5), 2018, 791–804 DOI: 10.1007/s11401-018-0096-2

work page doi:10.1007/s11401-018-0096-2 2018

[6] [6]

Kolmogorov, Sur la loi de grands nombres, Atti Accad

A.N. Kolmogorov, Sur la loi de grands nombres, Atti Accad. naz. Lincei. Rend. 1929, 9(6), 470–474

work page 1929

[7] [7]

Kolmogorov, Foundations of the Theory of Probability, 2nd Rus- sian edition, Moscow, Nauka, 1974 (in Russian)

A.N. Kolmogorov, Foundations of the Theory of Probability, 2nd Rus- sian edition, Moscow, Nauka, 1974 (in Russian)

work page 1974

[8] [8]

A. A. Lebedev and V. A. Lebedev, Generalization of the Expectation for a Unified Description of Random and Indefinite Factors, Theory Probab. Appl., 38:3 (1993), 470–478, DOI: https://doi.org/10.1137/1138043

work page doi:10.1137/1138043 1993