Khinchin–Kolmogorov theorem
E378995
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Khinchin–Kolmogorov theorem canonical | 2 |
| Kolmogorov three-series theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3677822 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Khinchin–Kolmogorov theorem Context triple: [Aleksandr Khinchin, notableWork, Khinchin–Kolmogorov theorem]
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A.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
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B.
Kolmogorov extension theorem
The Kolmogorov extension theorem is a fundamental result in probability theory that guarantees the existence of a stochastic process with given consistent finite-dimensional distributions.
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C.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
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D.
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.
-
E.
Kolmogorov continuity theorem
The Kolmogorov continuity theorem is a fundamental result in probability theory that provides conditions under which a stochastic process admits a modification with continuous (or Hölder-continuous) sample paths.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Khinchin–Kolmogorov theorem Target entity description: The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
A.
Kolmogorov zero–one law
The Kolmogorov zero–one law is a fundamental result in probability theory stating that certain events determined by the tail behavior of independent random variables must have probability either zero or one.
-
B.
Kolmogorov extension theorem
The Kolmogorov extension theorem is a fundamental result in probability theory that guarantees the existence of a stochastic process with given consistent finite-dimensional distributions.
-
C.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
-
D.
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.
-
E.
Kolmogorov continuity theorem
The Kolmogorov continuity theorem is a fundamental result in probability theory that provides conditions under which a stochastic process admits a modification with continuous (or Hölder-continuous) sample paths.
- F. None of above. chosen
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
probability theorem ⓘ |
| appliesTo | series of independent random variables ⓘ |
| assumes | independence of random variables in the series ⓘ |
| concerns |
almost sure convergence
ⓘ
independent random variables ⓘ series of random variables ⓘ |
| contrastsWith |
convergence in distribution
ⓘ
convergence in probability ⓘ |
| field | probability theory ⓘ |
| hasAspect |
conditions on distributions of summands
ⓘ
conditions on tail probabilities ⓘ conditions on variances or moments ⓘ |
| hasConvergenceMode | almost sure convergence ⓘ |
| hasType | convergence theorem ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies | almost sure convergence of the random series under its conditions ⓘ |
| influenced |
development of strong limit theorems
ⓘ
modern probability theory ⓘ |
| mathematicalDomain |
measure-theoretic probability
ⓘ
real analysis ⓘ |
| namedAfter |
Aleksandr Khinchin
ⓘ
Andrei Kolmogorov ⓘ
surface form:
Andrey Kolmogorov
|
| provides | conditions for almost sure convergence of series of independent random variables ⓘ |
| relatedTo |
Borel–Cantelli lemmas
ⓘ
Khinchin–Kolmogorov theorem self-linksurface differs ⓘ
surface form:
Kolmogorov three-series theorem
convergence of random series ⓘ almost sure limit theorem ⓘ
surface form:
strong law of large numbers
|
| usedFor | establishing almost sure convergence criteria ⓘ |
| usedIn |
limit theorems in probability
ⓘ
stochastic processes ⓘ theory of random series ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Khinchin–Kolmogorov theorem Description of subject: The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Aleksandr Khinchin
this entity surface form:
Kolmogorov three-series theorem