Triple

T4092334
Position Surface form Disambiguated ID Type / Status
Subject Khinchin–Kahane type inequalities E87730 entity
Predicate generalizes P2372 FINISHED
Object Khinchin inequality
The Khinchin inequality is a fundamental result in probability theory and functional analysis that bounds the moments of sums of independent Rademacher (±1) random variables in terms of their ℓ² norm.
E87730 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Khinchin inequality | Statement: [Khinchin–Kahane type inequalities, generalizes, Khinchin inequality]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Khinchin inequality
Context triple: [Khinchin–Kahane type inequalities, generalizes, Khinchin inequality]
  • A. Khinchin–Kahane type inequalities
    Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
  • B. Khinchin's law of the iterated logarithm
    Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
  • C. Khinchin–Kolmogorov theorem
    The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
  • D. Khinchin's representation theorem
    Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
  • E. Berry–Esseen theorem
    The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Khinchin inequality
Triple: [Khinchin–Kahane type inequalities, generalizes, Khinchin inequality]
Generated description
The Khinchin inequality is a fundamental result in probability theory and functional analysis that bounds the moments of sums of independent Rademacher (±1) random variables in terms of their ℓ² norm.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Khinchin inequality
Target entity description: The Khinchin inequality is a fundamental result in probability theory and functional analysis that bounds the moments of sums of independent Rademacher (±1) random variables in terms of their ℓ² norm.
  • A. Khinchin–Kahane type inequalities chosen
    Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
  • B. Khinchin's law of the iterated logarithm
    Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.
  • C. Khinchin–Kolmogorov theorem
    The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
  • D. Khinchin's representation theorem
    Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
  • E. Berry–Esseen theorem
    The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
  • F. None of above.

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69aed94425148190be337845d56fac22 completed March 9, 2026, 2:29 p.m.
NER Named-entity recognition batch_69aefcae22a081908af65a960306b78c completed March 9, 2026, 5 p.m.
NED1 Entity disambiguation (via context triple) batch_69b56b6cfb288190ac08c3a37327ac9a completed March 14, 2026, 2:06 p.m.
NEDg Description generation batch_69b56cd11b5c8190b7e7c9c91b6564b6 completed March 14, 2026, 2:12 p.m.
NED2 Entity disambiguation (via description) batch_69b56d3ff45881909f8b2c21ce51e0f0 completed March 14, 2026, 2:14 p.m.
Created at: March 9, 2026, 3:40 p.m.