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 (1)
| Label | Occurrences |
|---|---|
| Lyapunov condition canonical | 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
normalization by variance of the sum ⓘ |
| is | a classical form of central limit theorem hypothesis ⓘ |
| namedAfter | Aleksandr Lyapunov NERFINISHED ⓘ |
| relatedTo | Berry–Esseen bounds NERFINISHED ⓘ |
| requires |
existence of moments of order greater than 2
ⓘ
finite (2 + δ)-th absolute moments for some δ > 0 ⓘ |
| strongerThan | Lindeberg condition NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
central limit theorem NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.