{"id":"5cc68b43-d707-46be-961a-d4e83e3e87e1","arxiv_id":"1908.03541","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Deleting o(n) terms (for the laws of large numbers) or o(√n) terms (for the central limit theorem) from an i.i.d. sample leaves the classical limit theorems valid, provided the deleted indices are chosen without using the data.","lead":"A probability-theory preprint proves versions of the weak and strong laws of large numbers and the central limit theorem for partial sums with some terms deleted, showing that deleting a vanishing fraction of terms leaves the classical limits unchanged. The main results are easy corollaries of standard theorems, and the paper's bias-estimator application contains a verifiable arithmetic error.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 12-20 are only valid for non-adaptive deletion; the unstated independence of J_k* from the data is the load-bearing assumption, and adaptive deletion breaks the CLT.","rationale":"The reader's weakest_assumption exactly identifies non-adaptive deletion as the load-bearing unstated premise. My independent analysis agrees: all of the paper's central LLN/CLT proofs use the same-law property of the remaining sum, which holds only when the deletion set does not depend on the data. The counterexample with deleting the maximum makes the failure concrete and shows that the issue is not an artifact of proof technique but a genuine false statement in the adaptive regime. Because the paper's formal setup ('any k different elements of J') can be read as arbitrary but non-random, the central claim is defensible under that reading, so the appropriate verdict remains CONDITIONAL rather than REJECT. The Theorem 24 bias-formula error is real but secondary; it affects the application section, not the limit theorems themselves. The concrete simulation test directly settles whether the non-adaptivity assumption is necessary and whether the paper's theorems are correctly interpreted.","tokens_in":23,"tokens_out":5970,"duration_ms":191147,"concrete_test":"Simulate n=10^4 independent N(0,1) samples; for each sample delete the largest observation (k*=1) and record T_n = (S_n - M_n)/√n. Over 10^4 replications the empirical distribution of T_n will be centered near -√(2 log n) ≈ -4.3 and will not match N(0,1), directly refuting Theorem 20 under data-dependent deletion. A sharper analytic check: M_n/√n → ∞ in probability for normal data, so P(T_n ≤ x) → 0 for every finite x. Re-run the same simulation with a fixed deleted index chosen before seeing the data; then T_n will be approximately N(0,1), confirming that non-adaptivity is the decisive condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in Theorems 12-14, 17 and 20 is true for i.i.d. variables when the deleted index set J_k* is fixed, or at least chosen independently of the data: the remaining sum is then a sum of n-k* i.i.d. variables, so the classical LLN/CLT apply, and k*/n→0 or k*/√n→0 makes the missing mean and variance negligible. The paper never states this non-adaptivity requirement. It defines J_k* only as 'any k different elements of J', and the motivating examples ('extreme cases', deleting samples that violate experimental conditions) suggest deletion may be based on observed values. If J_k* is data-dependent, every proof breaks at the same step: the variance identity D(Σ_{J\\J_k*} ξ_i) = (n-k*)Var(ξ_i) and the same-law substitution of Σ_{J\\J_k*} with a sum of i.i.d. variables are no longer valid. The failure is not merely technical. Take ξ_i ~ N(0,1) and delete the single largest observation M_n, so k*=1 and k*/√n→0. Then (S_{J\\J_1})/√n = (S_n - M_n)/√n, which tends to -∞ in probability because M_n/√n → ∞, not to N(0,1). Thus Theorem 20 is false under adaptive deletion. The paper must either explicitly restrict to non-adaptive or independent deletion, or analyze the adaptive regime; as written, the central limit theorem relies on an unstated and motivated-against assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13799,"tokens_out":36050,"duration_ms":373632,"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":[{"comment":"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":"Section 1, Eq. (1); Theorems 12-14, 17, 20"},{"comment":"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":"Section 4, Theorem 16"},{"comment":"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.","section":"Section 7, Eq. (48) and Corollary 1"}],"minor_comments":[{"comment":"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":"Theorems 21-22"},{"comment":"There are several typos, e.g., 'theoretical consideration' should read 'a theoretical consideration', and 'an alysis' should be 'analysis'.","section":"Section 1"},{"comment":"The reference list has a duplicated 'References' heading, and reference [4] misspells 'Edition' as 'Edtion'.","section":"References"},{"comment":"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":"Section 3.3, Theorem 14"},{"comment":"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.","section":"Section 7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main results are correct only under an unstated non-adaptivity assumption on the deletion set, and one stated general theorem has an invalid proof. The application section also contains a demonstrable algebraic error. These issues are fixable within the manuscript's scope, so I recommend major revision rather than rejection. The novelty is modest, however: once the non-adaptivity assumption is made explicit, the main theorems are immediate consequences of classical LLN/CLT applied to the remaining i.i.d. subsample. The self-citation [16] supplies terminology only and is not load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":14584,"tokens_out":7156,"would_cite":false,"duration_ms":69895,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that deleting an asymptotically negligible number of terms from an i.i.d.","keywords":["deleting items partial sum","law of large numbers","central limit theorem","convergence in probability","almost sure convergence","asymptotic bias estimator","Slutsky's theorem","i.i.d. random variables"],"falsifier":"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.","tokens_in":13250,"feed_emoji":"🎲","tokens_out":11216,"duration_ms":112837,"temperature":0.7,"pith_summary":"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.","feed_headline":"Deleting a few terms still leaves the central limit theorem standing","feed_subtitle":"The paper proves LLN and CLT survive when the deleted fraction shrinks to zero, provided deletion is not data-dependent","key_machinery":"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 $k^*\\mu/(\\sqrt n\\,\\sigma)\\to 0$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies all classical WLLN, SLLN and CLT results and convergence lemmas that the deleting-items theorems are built from and compared to.","marker":"[1-13]"},{"why":"Introduces the deleting-items partial sum concept for integrals; the paper transfers that idea to random sequences.","marker":"[16]"},{"why":"Provides the asymptotic-statistics framework (bias and consistency of moment estimators) used to interpret the estimates in Section 7.","marker":"[17-18]"}],"fun_headline_variants":["Deleting a vanishing fraction doesn't break LLN or CLT","Limit laws persist when deletions are negligible","Deleting data? Limit theorems still apply","Sample mean and CLT survive with vanishing deletions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Deleting a vanishing fraction doesn't break LLN or CLT","Limit laws persist when deletions are negligible","Deleting data? Limit theorems still apply","Sample mean and CLT survive with vanishing deletions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001065,"raw_usage":{"total_tokens":4457,"prompt_tokens":929,"completion_tokens":3528,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":3466}},"tokens_in":545,"tokens_out":3528,"duration_ms":27463,"temperature":1.0,"reasoning_tokens":3466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:12:36.114048+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Deleting Items and Disturbing Mesh Theorems for Riemann Definite Integral and Their Applications","cited_arxiv_id":"1702.04464","evidence_quote":"Introduces the deleting-items partial sum concept for integrals; the paper transfers that idea to random sequences."}],"review_version":1}