Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence

Jingwei Liu

REVIEW 3 major objections 5 minor 18 references

Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that deleting an asymptotically negligible number of terms from an i.i.d.

desk verdict The deleting-items LLN/CLT claims are correct for i.i.d. samples only if the deleted indices are chosen independently of the data, which the paper never states; adaptive deletion can break the CLT outright. read the letter →

arxiv 1908.03541 v1 pith:FSSZSMRV submitted 2019-08-09 math.PR math.STstat.TH

Jingwei Liu This is my paper · ORCID

classification math.PRmath.STstat.TH MSC 60F0560F15

keywords deletingitemspartialsumlawoflargenumberscentrallimittheoremconvergenceinprobabilityalmostsureasymptoticbiasestimatorSlutsky'si.i.d.randomvariables

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 asks whether the classical limit theorems of probability survive when one discards a small number of terms from an i.i.d. sample rather than using the whole sum. It proves that when the number of deleted terms $k^*$ grows slowly enough, namely $k^*/n\to 0$ for laws of large numbers and $k^*\!/\sqrt n\to 0$ for the central limit theorem, the deleting-items partial sum has the same asymptotic behaviour as the full sum. The motivation is practical: experiments such as a coin falling off a table or a needle missing the target in Buffon's problem produce a few unusable observations, and the paper asks whether inference can proceed as if those observations were absent. It also writes down the exact large-sample bias of the deleting-items sample mean and of several sample-variance estimators built from the remaining terms. A sympathetic reading of the paper is that the deletion set is fixed or chosen independently of the data, since all proofs rely on identities that fail under data-dependent deletion.

What carries the argument

The central object is the deleting-items partial sum $S_{J\setminus J_{k^*}} = \sum_{i\in J\setminus J_{k^*}}\xi_i$, where $J=\{1,\dots,n\}$ and $J_{k^*}$ is a set of $k^*$ distinct indices removed from the sum. The argument is carried by two asymptotic negligibility conditions, $k^*/n\to 0$ for the laws of large numbers and $k^*\!/\sqrt n\to 0$ for the central limit theorem, together with the algebraic identity that splits a full sum into the kept block and the deleted block. The LLN proofs use Chebyshev's inequality and the variance identity $\operatorname{D}\bigl(\sum_{i\in J\setminus J_{k^*}}\xi_i\bigr)=(n-k^*)\operatorname{D}(\xi_i)$; the CLT proofs reduce the kept block to a classical i.i.d. central limit sum, rescale by $\sqrt{n-k^*}/\sqrt n\to 1$, and use Slutsky's theorem to absorb the deleted-block mean $k^*\mu/(\sqrt n\,\sigma)\to 0$.

What would settle it

Take independent standard normal variables, set $k^*=1$, and let $J_{k^*}$ be the index of the sample maximum $\max_i \xi_i$. Then $\frac{S_{J\setminus J_{k^*}}-n\mu}{\sqrt n\,\sigma} = \frac{\sum_i\xi_i - \max_i\xi_i}{\sqrt n}$, which behaves like the usual standardized sum minus $\max_i\xi_i/\sqrt n$. Since $\max_i\xi_i/\sqrt n\to\infty$ for normal data, the expression diverges in probability rather than converging to $N(0,1)$, directly separating the fixed-deletion-set reading of the theorem from the data-dependent reading.

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Extended reading notes

Core claim

The central claim is a family of deleting-items limit theorems. For i.i.d. variables with mean $\mu$ and finite variance $\sigma^2$, the paper proves that $\frac1n\sum_{i\in J\setminus J_{k^*}}\xi_i \to \mu$ in probability and almost surely when $k^*/n\to 0$, and that $\frac{S_{J\setminus J_{k^*}}-n\mu}{\sqrt n\,\sigma}$ converges in distribution to $N(0,1)$ when $k^*\!/\sqrt n\to 0$. The proofs rewrite the deleting-items sum as $\frac{n-k^*}{n}$ times the average of the un-deleted terms, apply the classical LLN or CLT to that average, and use Slutsky's theorem together with $k^*\mu/(\sqrt n\,\sigma)\to 0$ to discard the deleted block. The same pattern yields uniform WLLN versions, a de Moivre–Laplace CLT, and Lindeberg, Lyapunov and Lindeberg–Feller variants under additional moment and normalization conditions. Section 7 then shows that the deleting-items sample mean is an asymptotically biased estimator of $\mu$, and gives the exact expectations of three variance estimators built from the remaining observations.

Load-bearing premise

The load-bearing premise is that the index set $J_{k^*}$ removed from the sample is fixed, or is chosen independently of the observed values $\xi_i$; every variance and distribution calculation in the proofs relies on this.

Editorial extensions

If this is right

When $k^*/n\to 0$, the average of the remaining $n-k^*$ terms converges in probability to $\mu$, even though the deleted-block sum itself may be nonzero with probability one.
When $k^*\!/\sqrt n\to 0$, the usual $N(0,1)$ approximation for the normalized sum is unaffected by the deletion; for zero-mean variables, the weaker condition $k^*/n\to 0$ suffices.
The deleting-items sample mean $\frac1n\sum_{i\in J\setminus J_{k^*}}\xi_i$ has expectation $(1-k^*/n)\mu$, and the three variance estimators $\tilde S_1^2$, $\tilde S_2^2$, and $\tilde S_3^2$ have the explicit expectation formulas given in Theorem 24.
The uniform theorem (Theorem 16) implies that any sequence whose centered full average already converges in probability to zero retains that property after an asymptotically negligible block is removed.
The strong-law version (Theorem 17) extends the almost-sure convergence to pairwise i.i.d. variables with finite mean.

Reading between the lines

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

A direct reading of the motivating examples would suggest that the deleted items might be chosen after seeing the data, for instance the coin that falls off the table is detected only because it falls. If 'any $k^*$ different elements' is interpreted that way, the paper's own proof does not apply, and the deleting-items CLT can actually fail: deleting the sample maximum is a data-dependent choice
The same Slutsky-based reduction should carry over to other limiting regimes, such as stable-law limits, with the negligibility condition on $k^*$ adjusted to the tail index; the paper does not state such extensions.
The explicit biases in Theorem 24 suggest simple finite-sample corrections: one could recenter the deleting-items estimators by adding $k^*\mu/n$ or by estimating $\mu$ first, which would make them approximately unbiased under the fixed-deletion-set assumption.

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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

You should know one thing before reading this: the main i.i.d. theorems (12–14, 17, 20) are true, but only when the deletion set J_k* is fixed or otherwise independent of the data. That assumption is never stated. The paper only says J_k* is "any k different elements of J," and the motivating examples (coin falling off the table, needle missing the lines) suggest deletion might be based on the observed values. If deletion is adaptive, Theorem 20 is simply false: take ξ_i ~ N(0,1), delete the single largest observation, and the normalized sum (S_n − M_n)/√n decreases to −∞ in probability, not to N(0,1). So the load-bearing premise is hidden.

That said, the paper does some things well. Identifying the negligibility thresholds k*/n→0 for LLN and k*/√n→0 for CLT with nonzero mean is a legitimate observation. The alternative proof of Theorem 14 is clean. The i.i.d. results really do follow from the classical theorems plus Slutsky, and the variance-bias computations in Theorem 24 are mostly right.

The soft spots are real but fixable. The missing non-adaptivity assumption is the big one; the stress-test counterexample is not a technicality. Theorem 16 overreaches: the proof uses the full-sample WLLN premise to conclude the deleting-items average converges, which does not follow for general sequences. Theorems 21 and 22 are stated without proofs; they should be proved or removed. And there is a verifiable arithmetic slip in E[~S_3²]: the σ² coefficient should be 1 − 1/n − k*/n + k*²/n³, not the expression with k*³/n³ (or k*/n³) that appears in the text. That error propagates into Corollary 1. There is also an imprecision in Theorem 4, which conflates a deterministic expectation going to zero with convergence in probability.

Who is this for? Someone working on robustness of limit theorems to omitted or discarded data, particularly if they care about the difference between non-adaptive and adaptive deletion. As written, it is not publishable. But the core i.i.d. results are correct under an easy-to-state assumption, the counterexample is instructive, and the flaws are the kind a competent referee can point to and the author can fix. This is not a desk-reject paper; it deserves a real referee, with instructions to have the authors state the independence assumption explicitly, repair Theorem 16, prove or delete Theorems 21–22, and fix the bias formula.

Referee Report

3 major / 5 minor

Summary. The paper develops 'deleting items' limit theorems by replacing the classical partial sum S_n = Σ_{i=1}^n ξ_i with the partial sum over the index set J\J_k*, where J_k* is any k* indices. It states and proves deleting-items versions of Bernoulli, Chebyshev, and Khinchine WLLNs (Theorems 12-14), a general WLLN (Theorems 15-16), SLLNs (Theorems 17-18), and CLTs (Theorems 19-23), with conditions k*/n→0 for LLN results and k*/√n→0 for CLT results. Section 7 applies the deleting-items WLLN to bias computations for estimators of expectation and variance. The core mathematical claim is that, asymptotically, deleting an asymptotically negligible number of terms from an i.i.d. sample does not change the classical LLN/CLT conclusions.

Significance. If the intended setup is that the deleted index set J_k* is non-random or at least independent of the data, then the i.i.d. statements (Theorems 12-14, 17, 19-20) are correct: the remaining variables are still i.i.d., and each theorem follows directly from the classical result applied to the remaining n-k* observations. The paper makes this observation explicit and also computes the biases of several deleted-sample variance estimators. The results are, however, not deep extensions: they are immediate corollaries of classical theorems once the non-adaptivity assumption is stated. The paper does not analyze the adaptive-deletion regime that its motivating examples suggest, and one stated general theorem (Theorem 16) has an unjustified proof. The bias computations in Section 7 also contain algebraic errors. The contribution is therefore modest, but the correct parts could be useful as a reference for 'deleting items' asymptotics.

major comments (3)

[Section 1, Eq. (1); Theorems 12-14, 17, 20] The paper defines J_k* as 'any k different elements of J' without ever requiring that J_k* be non-random or independent of the observed data. Every proof of the i.i.d. theorems uses in an essential way that the remaining variables are i.i.d.: the variance identity D((1/n)Σ_{J\J_k*}ξ_i) = (n-k*)D(ξ_i)/n² in Theorems 12-13, the characteristic-function expression [φ(t/n)]^{n-k*} in Theorem 14, and the direct application of the classical CLT/SLLN to S_{J\J_k*} in Theorems 17 and 20. If J_k* may be chosen using the observed values, these identities fail. The failure is real: for i.i.d. variables with mean 0, finite variance, and tail P(|ξ|>x) ~ c x^{-3}, deleting the n^β largest observations with 1/4 < β < 1/2 gives k*/√n → 0 but S_{J\J_k*}/√n → -∞ in probability, contradicting Theorem 20. The manuscript must either explicitly restrict to non-adaptive (or data-independent) deletion or analyze the adaptive regime; as written, the central CLT theorem relies on an unstated and motivated-against assumption.
[Section 4, Theorem 16] The proof's key step is invalid. From the full-average convergence (1/n)Σ_{J}(ξ_i - Eξ_i) →P 0, the proof concludes that the deleted-average term (1/(n-k*))Σ_{J\J_k*}(ξ_i - Eξ_i) converges to 0 in probability; but this term is not the full average and its convergence is not a consequence of the assumption. The theorem's conclusion requires a separate argument that (1/n)Σ_{J_k*}ξ_i →P 0, which the proof does not supply. As written, Theorem 16 is unproved; it should either be repaired by adding and proving that missing step, or the theorem should be removed or restated under a directly assumed deleted-average WLLN.
[Section 7, Eq. (48) and Corollary 1] The formula for E~S²_3 appears to be algebraically wrong. The correct expansion is E~S²_3 = (1 - 1/n - k*/n + k*²/n³)σ² + (1 - k*/n)(k*²/n²)µ², whereas the manuscript has k*/n³ in the σ² term and k*/n² in the µ² term. The subsequent threshold in Corollary 1 is derived from the incorrect expression and is not valid in general. For example, with n=100, k*=50, σ²=1, µ²=10, the corrected formula gives E~S²_3 ≈ 1.74 > ES² ≈ 0.99, while Corollary 1's condition predicts E~S²_3 ≤ ES². This undermines the claimed bias comparisons for the variance estimators in the applications section.

minor comments (5)

[Theorems 21-22] Theorems 21 and 22 are stated without proofs. Since they are immediate corollaries of Theorem 23 under the extra conditions (2) and (3), the paper should state this explicitly or supply the missing proofs.
[Section 1] There are several typos, e.g., 'theoretical consideration' should read 'a theoretical consideration', and 'an alysis' should be 'analysis'.
[References] The reference list has a duplicated 'References' heading, and reference [4] misspells 'Edition' as 'Edtion'.
[Section 3.3, Theorem 14] In the first proof of Theorem 14, the final display writes J\J_k instead of J\J_k*, which is a minor notation slip.
[Section 7] The notation ~X, ~S²_1, ~S²_2, ~S²_3 would be clearer if the dependence of k* on n were made explicit, since k* is allowed to vary with n throughout the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the deleting-items limit theorems are direct corollaries of the classical limit theorems and Slutsky-type convergence lemmas, with the self-citation supplying only terminology.

full rationale

The derivation chain is self-contained and non-circular. The paper redefines the partial sum as S_{J\J_k*} = sum over all indices except an arbitrary fixed set of k* terms (Eq. 1), then proves each deleting-items result from classical statements. For example, Theorem 12 uses Chebyshev's inequality and the variance identity D((1/n) sum_{J\J_k*} xi_i) = (n-k*)D(xi_i)/n^2; no target conclusion is imported. Theorem 14 is proved from Theorem 3 by writing (1/n) S_{J\J_k*} = ((n-k*)/n)(1/(n-k*)) S_{J\J_k*} -> 1*a, and Theorem 17 applies Theorem 5 to S_{J\J_k*}/(n-k*) before multiplying by (n-k*)/n. Theorem 20 invokes the classical Lindeberg-Levy CLT for the n-k* retained i.i.d. terms and then uses the negligibility of k*/sqrt(n); this is a corollary, not a circular use, because the classical CLT is an external premise, not the deleting-items claim itself. Theorems 21-23 similarly reduce to Lindeberg/Lyapunov/Feller conditions on the retained terms. Theorem 24 is an explicit bias computation, not a fitted prediction. The only self-citation, [16], supplies the phrase 'deleting items partial sum' and is not load-bearing: no proof step relies on [16], and the limit theorems are derived from classical results and convergence lemmas. Therefore no step reduces by construction to its own inputs. Separate correctness concerns about data-dependent deletion are not circularity under the stated criteria.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The 'deleting items partial sum' is a definition, not a postulated entity; it has no free-standing ontological content and requires no independent evidence. The only free parameter is the deletion count k*, which is constrained by asymptotic rate conditions rather than fitted. The paper's derivation rests on classical theorems, Slutsky's theorem, and two unstated domain assumptions about how the deleted indices are chosen and how subset sums inherit convergence.

free parameters (1)

k* (number of deleted items)
A structural parameter of the theorems constrained only by the asymptotic negligibility conditions k*/n → 0 (LLN) or k*/√n → 0 (CLT). No value is fitted to data, but every theorem's validity depends on these rate conditions, and the paper gives no guidance on choosing k* in practice beyond the limit.

assumptions (4)

standard math Classical WLLN, SLLN and CLT theorems (Theorems 1-11) are accepted as given, including Kolmogorov's SLLN for i.i.d. variables.
Invoked as black boxes in nearly every proof (e.g., Theorem 14 'According to Theorem 3', Theorem 17 'Applying Theorem 5'). Note Theorem 5 is labeled 'pairwise independent identically distributed', which is weaker than the known sufficient condition for the SLLN; the paper does not prove these background results.
standard math Slutsky-type algebra for limits (Lemma 4) and the convergence relationships (Lemma 1).
Used to multiply converging terms and pass through scaling factors such as (n−k*)/n → 1; stated without proof but standard.
domain assumption The deletion set J_k* is non-adaptive: its choice does not depend on the values of the observed variables.
Unstated. The paper defines J_k* as 'any k different elements of J' (Section 1) yet all proofs use distributional equalities and variance formulas (e.g., D(Σ_{J\Jk*} ξi) = (n−k*)D(ξi) in Theorems 12-13, and the same-law substitution in Theorems 17 and 20) that fail for data-dependent deletion. This is the paper's weakest premise.
domain assumption Subset sums of i.i.d. variables behave like initial-segment sums of length n−k*.
Used implicitly in Theorem 14's alternative proof, Theorem 17, Theorem 20 and Theorem 23. Distributionally true for i.i.d. variables, but for a.s. claims (Theorems 17-18) or for general dependent sequences (Theorem 16) the paper does not justify transferring convergence from initial segments to arbitrary subsets.

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Cite this review

Pith. "Pith review of Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence." pith.science (2026). https://pith.science/paper/FSSZSMRV

@misc{pith2026190803541,
  author       = {Pith},
  title        = {Pith review of: Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FSSZSMRV}},
  note         = {Machine review of arXiv:1908.03541}
}

read the original abstract

The deleting items theorems of weak law of large numbers (WLLN),strong law of large numbers (SLLN) and central limit theorem (CLT) are derived by substituting partial sum of random variable sequence with deleting items partial sum. We address the background of deleting items limit theory of random variable sequence, discuss the classical limit theory of Chebyshev WLLN, Bernoulli WLLN and Khinchine WLLN with standard mathematical analytical technique, then develop the deleting items theorems of WLLN, SLLN and CLT based on convergence theorems and Slutsky's theorem. Our theorems extend the classical limit theory of random variable sequence and provide the construction of some asymptotic bias estimators of sample expectation and variance.

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Pith tools

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

[1] [1]

E. Seneta. A Tricentenary History of the Law of Large Numbers . Bernoulli.19(4),2013,1088-1121

work page 2013

[2] [2]

Kolmogorov

A.N. Kolmogorov. Foundations of the Theory of Probability. Tran slated by Nathan Morrison,(2nd English Edition). Chelsea Publishing Com- pany. New York. 1956

work page 1956

[3] [3]

M. Loeve. Probability Theory (3rd Edition). D. Van Nostrand Co. Inc.. Princeton, N.J., 1963

work page 1963

[4] [4]

W. Feller. An Introduction to Probability Theory and Its Applicatio ns (Vol.1) (3rd Edtion). John Wiley & Sons, Inc. New York. 1968

work page 1968

[5] [5]

W. Feller. An Introduction to Probability Theory and Its Applica- tions,(Vol.2),(3rd Edition). John Wiley & Sons, Inc. New York. 1971

work page 1971

[6] [6]

Billingsley

P. Billingsley. Probability and Measure.(3rd Edition). John Wiley& Sons, New York. 1995

work page 1995

[7] [7]

Shiryayev

A.N. Shiryayev. Probability. Springer, 1996

work page 1996

[8] [8]

Kallenberg

O. Kallenberg. Foundations of Modern Probability. Springer.1997

work page 1997