On strong law of large numbers for non identically distributed random variables

A.Yu. Veretennikov; I.V. Kozlov

REVIEW 2 major objections 4 minor 24 references

On strong law of large numbers for non identically distributed random variables

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims a strong law of large numbers for a pairwise independent sequence with a sparse dependent part whose moment orders tend to zero.

desk verdict Main theorem false: Lemma 2's normalizer is n^{1/a_n}, not n, and the counterexample in the stress-test satisfies all assumptions while breaking (7). read the letter →

arxiv 2506.07508 v1 pith:WJGSYSMN submitted 2025-06-09 math.PR

I.V. Kozlov , A.Yu. Veretennikov This is my paper

classification math.PR MSC 60F15

keywords stronglawoflargenumberspartialpairwiseindependencedependenceabsenceexpectationsmomentsvariableorderKroneckerlemmaalmostsureconvergence

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 aims to establish a strong law of large numbers for a sequence that mixes two kinds of summands: a 'good' pairwise independent part with finite means, and a 'bad' part that may be arbitrarily dependent and may lack finite expectations entirely, having only a moment of order $a_n$ that tends to zero. The claimed conclusion is that the arithmetic average of the mixture converges almost surely to zero, provided the bad summands are infrequent in the sense that their number $\varphi(n)$ grows no faster than $n^{a_n}$, with $a_n\ln n\to\infty$. If correct, the result would extend an earlier fixed-order version in which all bad summands have the same moment order $a\in(0,1)$, and would bring a wider class of heavy-tailed or no-mean summands into the scope of the strong law.

What carries the argument

The proof works through two lemmas. Lemma 1 is an Etemadi-type strong law for the good variables $X_k$ under assumption (2), a relaxed Cesàro uniform-integrability condition. Lemma 2 handles the bad variables: setting $V_n=|Y_n|^{a_n}$, it uses the tail bound (4) to show that $\mathbb{E}\sum_{n=3}^{\infty} \frac{\min(V_n,n)^{p_n}}{n^{p_n}}<\infty$ for a block-constant choice of exponents $p_k=1/a_k$, and hence $\sum_n |Y_n|/n^{1/a_n}<\infty$ almost surely. The Kronecker lemma is then invoked to pass from this series convergence to a statement about partial sums of the $Y_n$. The load-bearing estimate is the bound $\mathbb{E}\sum_n \tilde V_n^p/n^p \le C_G(2p-1)/(p-1)$, which converts the uniform tail integral $C_G$ into enough control to make the series converge for every exponent $p_k$.

What would settle it

Concretely, set $a_n=1/\ln\ln n$ and choose a deterministic sequence with partial sums $S_m=m/\ln\ln m$; then $S_m/m^{1/a_m}\to0$ so the Kronecker conclusion holds, but $S_m/m\not\to0$. Placing these values at $\varphi(n)$ sparse positions with $\varphi(n)=O(n^{a_n})$, one can check whether the mixed average $(1/n)\sum_{k=1}^{\varphi(n)}Y_k$ fails to converge to zero. If some sparse placement makes it fail, Theorem 1 is false; if none does, the missing normalization must follow from the infrequency condition alone.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 1: under assumptions (1)--(6), the mixed sequence $Z_n$ satisfies $\frac{1}{n}\sum_{k=1}^n Z_k\to 0$ almost surely. The novelty is that the dependent 'bad' subsequence may have moments $\mathbb{E}|Y_k|^{a_k}$ of orders $a_k\downarrow 0$, rather than a fixed order, as long as the bad indices are sparse ($\varphi(n)=O(n^{a_n})$) and the transformed variables $V_k=|Y_k|^{a_k}$ satisfy a common tail-integrability bound. The statement is intended to include Kolmogorov's strong law and the recently treated case $a_k\equiv a\in(0,1)$ as special cases.

Load-bearing premise

The argument depends on the unstated step in Lemma 2 that almost-sure convergence of $(1/n^{1/a_n})\sum_{k=1}^n Y_k$ to $0$ implies the unit-normalized average $(1/n)\sum_{k=1}^{\varphi(n)}Y_k\to0$; the paper does not justify that passage, and it is not automatic because $n^{1/a_n}/n\to\infty$ when $a_n\to0$.

Editorial extensions

If this is right

If Theorem 1 is correct, the empirical average of the mixed sequence converges almost surely to zero even when the dependent part has no finite expectation.
The fixed-power condition $a_k\equiv a\in(0,1)$ from the earlier result becomes removable; any sequence $a_k\downarrow 0$ with $a_k\ln k\to\infty$ is allowed.
The infrequency condition $\varphi(n)=O(n^{a_n})$ quantifies the tolerable density of bad summands: the slower $a_n$ decays, the more dependent variables can be placed without breaking the strong law.
In estimation settings, the theorem would permit occasional arbitrarily dependent observations or heavy-tailed contaminants in an otherwise pairwise independent sample without destroying the almost-sure consistency of sample averages.

Reading between the lines

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

The proof's last step in Lemma 2 derives $n^{-1/a_n}\sum_{k=1}^n Y_k\to0$ from the convergent series, but the theorem's conclusion needs the normalization $n^{-1}\sum_{k=1}^{\varphi(n)}Y_k\to0$; the paper does not show how one follows from the other, so the reader should treat that bridge as the part that still needs verification.
A natural repair would be to apply the Kronecker step at the block length $m=\varphi(n)$ and then use condition (5) to compare the factors $\varphi(n)^{1/a_{\varphi(n)}-1}$ and $n/\varphi(n)$; whether this comparison suffices is a testable question.
The block-exponent device using $p_k=1/a_k$ with piecewise constant $p_k$ could be reused in other strong-law problems where the available moment order varies with the index.

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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 main theorem is false. Lemma 2 proves convergence with normalizer n^{1/a_n}, but Theorem 1 needs normalizer n. The gap is not a technicality: condition (5) does not control n^{1/a_n}/n → ∞, and the stress-test counterexample (with a_n = log log n/(2 log n) and Y_m = m^2 on probability-1/m events) satisfies (1)–(6) while violating (7). The reader's analysis is correct.

The paper isn't without merit. The idea of letting the moment order a_n decay to zero is a natural extension of [20], and the truncation step in Lemma 2 is executed cleanly: the series estimate for E∑ Ṽ_n^p / n^p is correct and gives ∑ |Y_n| / n^{1/a_n} < ∞ a.s. That part is solid. Lemma 1 is standard Etemadi and fine. The writing is straightforward, and the literature is properly cited.

The soft spot is the last line of Lemma 2. The Kronecker lemma applied to ∑ |Y_k| / k^{1/a_k} < ∞ yields (1/n^{1/a_n}) ∑ Y_k → 0. To get (1/n) ∑ Y_k → 0 one would multiply by n^{1/a_n}/n, which tends to infinity under (6). Condition (5) only bounds the count φ(n) by a multiple of n a_n, which is far smaller than n^{1/a_n} when a_n lnn → ∞. So the bad block is not controlled. The counterexample is decisive: with independent A_m of probability 1/m, the Borel–Cantelli argument gives i.o. terms of size m^2 at times n_m ~ m/a_m, making the average at least m a_m → ∞. All four assumptions are met, so the theorem as stated is false.

There is a small typo in the Russian "для (Y_n)" in the lemma heading, and the proof says "The lemma is proved" after the wrong convergence, but these are minor compared to the main gap.

This paper is for specialists in strong limit theorems. It is a clean example of how a plausible normalizer switch can fail when the exponent vanishes. A repair might be possible with a much stronger infrequency condition, perhaps φ(n) = O(n a_n^2) plus more tail integration, but that's not in the paper. My recommendation: do not publish as is. But it is worth sending to a competent referee to document the counterexample and give the authors a chance to fix the statement. If forced to choose, I would REJECT.

Referee Report

2 major / 4 minor

Summary. The paper claims a strong law of large numbers for a mixture of a 'good' pairwise independent sequence X_n and an 'infrequent' sequence Y_n whose moment orders a_n tend to zero. Under conditions (1)-(6), Theorem 1 asserts that (1/n)∑_{k=1}^n Z_k → 0 almost surely, where Z_n switches between X and Y summands. The proof separates the X part in Lemma 1, which follows from Etemadi's theorem, and the Y part in Lemma 2, which uses a moment-series estimate and Kronecker's lemma.

Significance. If the result were correct, it would naturally extend the earlier result in [20] by allowing the moment orders of the bad summands to converge to zero. The paper is short, transparent, and the proof strategy is easy to follow. However, the central Y-part lemma proves a strictly weaker normalization than the one needed, and Theorem 1 is in fact false; a concrete counterexample satisfying all stated assumptions violates the conclusion. The contribution therefore cannot stand in its present form.

major comments (2)

[§3, Lemma 2 (Eq. (9))] The proof of Lemma 2 establishes only the wrong normalization. After showing ∑ |Y_n|/n^{1/a_n} < ∞ a.s., Kronecker's lemma gives (1/n^{1/a_n}) ∑_{k=1}^n Y_k → 0 a.s. The lemma claims (1/n) ∑_{k=1}^{κ(n)} Y_k → 0. Since a_n → 0, we have n^{1/a_n}/n → ∞, and condition (5) only bounds κ(n) by a constant times n a_n, which is far smaller than n^{1/a_n}. The proved statement does not imply the claimed one, and no part of the proof supplies the missing factor. This gap is load-bearing because Theorem 1 requires the unrenormalized average of the bad part to vanish.
[Theorem 1 (conditions (1)–(6) vs. conclusion (7))] The theorem is false. Take X_n = 0, set a_n = log log n / (2 log n) for large n, and let the bad variables be independent Y_m = m^2 1_{A_m} with P(A_m) = 1/m, inserted at positions n_m such that φ(n_m) = m and n_m ~ m/a_m. Then V_m = |Y_m|^{a_m} = (log m) 1_{A_m}, so condition (4) holds with G(t) = e^{-t}; condition (5) holds because φ(n_m)/(n_m a_{n_m}) ~ 1; and condition (6) holds since a_n log n = log log n / 2 → ∞. Moreover, ∑ |Y_m|/m^{1/a_m} < ∞ a.s. because m^{1/a_m} is much larger than m^2. Nevertheless, whenever A_m occurs the bad partial sum at n_m is at least m^2, so the average is at least m^2/n_m ~ m a_m → ∞. Since ∑ P(A_m) = ∞ and the events are independent, A_m occurs infinitely often almost surely, and (1/n)∑_{k=1}^n Z_k does not converge to 0. This directly disproves (7) under the stated assumptions.

minor comments (4)

[§3, Lemma 2] The symbol κ(n) in the statement of Lemma 2 is never defined; it should presumably be φ(n), the number of bad summands up to n.
[Throughout] The notation φ_n versus φ(n), and α_n versus α(n), is used inconsistently; one convention should be chosen.
[§3, Lemma 2 proof] There are several typos and language artifacts, including 'a sequece', 'incresases', 'п.н.' instead of 'a.s.', and the Russian word 'для' in the lemma heading.
[§3, Lemma 2 proof, final line] The expression 'n− → ∞' should read 'n → ∞'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the paper's failure is a genuine derivation gap, not a reduction of the conclusion to its assumptions.

full rationale

The paper contains no fitted parameters, no prediction of data already used as input, and no load-bearing self-citation chain. Theorem 1 is a direct attempt to derive an SLLN for the mixed sequence (Z_n) from explicit assumptions (1)-(6). Lemma 1 invokes Etemadi's theorem [9] as an external, independent result. Lemma 2 develops a Borel-Cantelli-style estimate using only the assumptions on (Y_n) and the Kronecker lemma [24]; the cited calculus in [22] and [23] is used as a standard tool, not as a source of the theorem's content. The quoted reference [20] by Akhmiarova and Veretennikov is contextual, not load-bearing for Theorem 1. The apparent defect in the paper is mathematical: Lemma 2 concludes almost sure convergence of (1/n^{1/a_n}) sum Y_k and then labels the lemma proved, whereas statement (9) requires convergence with normalizer n. That step is invalid because n^{1/a_n}/n diverges under a_n -> 0, but this is a proof error, not circularity. The theorem may be false, and a counterexample may exist, but circularity requires that the claimed result is equivalent, by construction or by self-citation, to its inputs. No such equivalence is present here. Therefore the circularity score is 0.

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

The central claim rests on the explicitly stated conditions (1)-(6), plus standard tools (Etemadi's SLLN, Kronecker lemma). No constants are fitted to data. The hidden invalid inference in Lemma 2 is not an axiom but a derivation gap, recorded in the red flags.

assumptions (6)

standard math Kronecker lemma
Used in Lemma 2 to pass from series convergence to averaged sums, but applied with the wrong normalizer.
standard math Etemadi's SLLN (theorem 1 in [9])
Used in Lemma 1 to prove the X-part convergence.
domain assumption Condition (2): finite Cesaro tail integral for X_n
Assumes the good variables X_n satisfy a Cesaro uniform integrability condition; not derived in the paper.
domain assumption Condition (4): uniform tail bound on |Y_n|^{a_n}
Assumes sup_n P(|Y_n|^{a_n}>t) is dominated by an integrable function G(t).
domain assumption Condition (5): infrequency of Y_n, phi(n)=O(n a_n)
Assumes the bad variables appear rarely enough relative to n a_n.
domain assumption Condition (6): a_n ln n -> infinity
Assumes the moment order decays slowly enough; used to ensure phi(n) can grow.

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Cite this review

Pith. "Pith review of On strong law of large numbers for non identically distributed random variables." pith.science (2026). https://pith.science/paper/WJGSYSMN

@misc{pith2026250607508,
  author       = {Pith},
  title        = {Pith review of: On strong law of large numbers for non identically distributed random variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJGSYSMN}},
  note         = {Machine review of arXiv:2506.07508}
}

read the original abstract

A new version of a strong law of large numbers for a ``good'' pairwise independent sequence of random variables (r.v.'s) with a small part of ``bad'' dependent r.v.'s is proposed. The main goal is to relax the assumption on the existence of the expectation for each summand: the members of an ``infrequent'' part of the whole sequence may have moments of orders converging to zero.

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Works this paper leans on

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Pith tools

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

[1] [20]

A. T. Akhmiarova and A. Yu. Veretennikov, On Strong Law of Large Numbers for Pairwise Independent Random Variables, Theory of Prob- ability and its Applications, 2024, Volume 69, Issue 3, Pages 337 - 345 DOI: https://doi.org/10.1137/S0040585X97T991969

work page doi:10.1137/s0040585x97t991969 2024

[2] [1]

Bernoulli, Ars Conjectandi, Basileae, Impensis Thurnisiorum, Fratrum, 1713

J. Bernoulli, Ars Conjectandi, Basileae, Impensis Thurnisiorum, Fratrum, 1713. On the law of large numbers. Eng. transl. by O.B. Sheynin of the Pars Quarta of Ars Conjectandi, 2005; available at www.sheynin.de/download/bernoulli.pdf

work page 2005

[3] [2]

Chebyshev [Tchebyshev], Des valuers moyennes, Journal de Math´ ematics Pures et Appliqu´ ees, 1867, vol

P.L. Chebyshev [Tchebyshev], Des valuers moyennes, Journal de Math´ ematics Pures et Appliqu´ ees, 1867, vol. 12, 177-184

work page

[4] [3]

´Emile Borel, Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques. Rend. Circ. Matem. Palermo, 1909, 27, 247–271. https://doi.org/10.1007/BF03019651

doi:10.1007/bf03019651 1909

[5] [4]

Cantelli, Sulla probabilit` a come limite della frequenza, Atti Ac- cad

F.P. Cantelli, Sulla probabilit` a come limite della frequenza, Atti Ac- cad. Naz. Lincei, 26(1) (1917), 39-45. Zbl 46.0779.02. Eng. transl. in: C.F. Trustam and C.E. Bonferroni, Some recent Italian work on the Mathematical Basis of Actuarial Science, Journal of the In- stitute of Actuaries (1886-1994), 59(1), 51-66. Published by: CUP on behalf of the Ins...

work page arXiv 1917

[6] [5]

Mazurkiewicz, O pewnem nog´ olneniu prawa wielkich liczb, Wiadomo´ sci matematycznych1, 1917, XXII, 69-82

S. Mazurkiewicz, O pewnem nog´ olneniu prawa wielkich liczb, Wiadomo´ sci matematycznych1, 1917, XXII, 69-82

work page 1917

[7] [6]

Rajchman, Zaostrzone prawo wielkich liczb, Mathesis Polska2, 1931, VI (7-8), 145 - 161

A. Rajchman, Zaostrzone prawo wielkich liczb, Mathesis Polska2, 1931, VI (7-8), 145 - 161

work page 1931

[8] [7]

Chandra, Extensions of Rajchman’s Strong Law of Large Numbers

T.K. Chandra, Extensions of Rajchman’s Strong Law of Large Numbers. Sankhy¯ a: The Indian Journal of Statistics, Series A (1961-2002), 1991, 53(1), 118–121. http://www.jstor.org/stable/25050825 1https://polona.pl/search/?query=stefan_mazurkiewicz&filters=public:1 2http://mbc.cyfrowemazowsze.pl/dlibra/applet?mimetype=image/x.djvu&sec=false &handler=djvu_ht...

work page arXiv 1961