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An Extension to the Tangent Sequence Martingale Inequality

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Stephen Montgomery-Smith, Shih-Chi Shen

Probability · Submitted 17 Jul 2001

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||_p le c_p || \sum_{k=1}^n d_k ||_p .

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arXiv IDmath/0107120 (version 2)
Pith IDNKX66GSG, Pith's permanent record of this paper and its review
CitationShih-Chi Shen, Stephen Montgomery-Smith. "An Extension to the Tangent Sequence Martingale Inequality." arXiv:math/0107120 (2001). https://pith.science/paper/NKX66GSG
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,043 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.73 · compares with prior results 0.05 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||_p le c_p || \sum_{k=1}^n d_k ||_p .

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. result 0.79 0.64 0.07 asserted 1.00
s1 Let (d_k), (e_k) be real-valued martingale difference sequences. background 0.85 0.10 0.02 unclear 0.61
s2 If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||_p le c_p || \sum_{k=1}^n d_k ||_p . result 0.91 0.77 0.93 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • arXiv listing abstract v2
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||_p le c_p || \sum_{k=1}^n d_k ||_p .

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