Lyapunov condition
E683054
condition for central limit theorem
condition in probability theory
moment condition
sufficient condition
The Lyapunov condition is a sufficient moment condition on sums of independent random variables that guarantees convergence in distribution to a normal law in central limit theorems.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Lyapunov condition canonical | 1 |
| Lyapunov condition on higher moments | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7705186 — 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: Lyapunov condition Context triple: [Lindeberg–Feller central limit theorem, relatedTo, Lyapunov condition]
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A.
Lyapunov vector
A Lyapunov vector is a mathematical construct in dynamical systems theory that characterizes the directions in phase space associated with exponential growth or decay rates quantified by Lyapunov exponents.
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B.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
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C.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
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D.
Lyapunov equation
The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
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E.
Lipschitz continuity condition
The Lipschitz continuity condition is a mathematical regularity criterion that bounds how fast a function can change, ensuring controlled variation and playing a key role in analysis and differential equations.
- 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: Lyapunov condition Target entity description: The Lyapunov condition is a sufficient moment condition on sums of independent random variables that guarantees convergence in distribution to a normal law in central limit theorems.
-
A.
Lyapunov vector
A Lyapunov vector is a mathematical construct in dynamical systems theory that characterizes the directions in phase space associated with exponential growth or decay rates quantified by Lyapunov exponents.
-
B.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
-
C.
Lyapunov stability theory
Lyapunov stability theory is a fundamental framework in dynamical systems and control theory that uses energy-like functions to assess the stability of equilibrium points without explicitly solving differential equations.
-
D.
Lyapunov equation
The Lyapunov equation is a fundamental matrix equation in control theory and dynamical systems used to analyze the stability of equilibrium points and design stable controllers.
-
E.
Lipschitz continuity condition
The Lipschitz continuity condition is a mathematical regularity criterion that bounds how fast a function can change, ensuring controlled variation and playing a key role in analysis and differential equations.
- F. None of above. chosen
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
condition for central limit theorem
ⓘ
condition in probability theory ⓘ moment condition ⓘ sufficient condition ⓘ |
| appliesTo |
sums of independent random variables
ⓘ
triangular arrays of independent random variables ⓘ |
| assumes |
independence of summands
ⓘ
non-degenerate variance of the sum ⓘ |
| comparedTo |
Lindeberg–Feller condition
ⓘ
linked to:
Lindeberg–Feller central limit theorem
|
| domain |
asymptotic distribution theory
ⓘ
limit theorems ⓘ |
| ensures |
convergence in distribution to a normal law
ⓘ
convergence of characteristic functions to that of a normal distribution ⓘ negligibility of large individual summands ⓘ |
| field |
mathematical statistics
ⓘ
probability theory ⓘ |
| guarantees | asymptotic normality of standardized sums ⓘ |
| historicalContext | introduced in early 20th century ⓘ |
| implies | Lindeberg condition ⓘ |
| involves |
Lyapunov fraction
ⓘ
normalization by variance of the sum ⓘ |
| is | a classical form of central limit theorem hypothesis ⓘ |
| namedAfter | Aleksandr Lyapunov ⓘ |
| relatedTo |
Berry–Esseen bounds
ⓘ
linked to:
Berry–Esseen theorem
|
| requires |
existence of moments of order greater than 2
ⓘ
finite (2 + δ)-th absolute moments for some δ > 0 ⓘ |
| strongerThan |
Lindeberg condition
ⓘ
linked to:
Lindeberg–Feller central limit theorem
|
| typeOf | sufficient central limit theorem condition ⓘ |
| usedFor |
controlling tail behavior of summands
ⓘ
proving central limit theorems for non-identically distributed variables ⓘ |
| usedIn |
Lyapunov central limit theorem
ⓘ
central limit theorem ⓘ |
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: Lyapunov condition Description of subject: The Lyapunov condition is a sufficient moment condition on sums of independent random variables that guarantees convergence in distribution to a normal law in central limit theorems.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lyapunov condition