Lyapunov central limit theorem
E683053
The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lyapunov central limit theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7705167 — 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 central limit theorem Context triple: [Lindeberg–Feller central limit theorem, generalizes, Lyapunov central limit theorem]
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A.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
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B.
Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
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C.
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.
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D.
Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
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E.
Freidlin–Wentzell theory
Freidlin–Wentzell theory is a mathematical framework in probability that analyzes the behavior of stochastic dynamical systems under small random perturbations using large deviation principles.
- 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 central limit theorem Target entity description: The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
-
A.
Lindeberg–Feller central limit theorem
The Lindeberg–Feller central limit theorem is a general form of the central limit theorem that provides conditions under which sums of independent, not necessarily identically distributed random variables converge in distribution to a normal law.
-
B.
Limit Laws for Sums of Independent Random Variables
Limit Laws for Sums of Independent Random Variables is a foundational mathematical work that systematically develops the theory of probability limit theorems, including results such as the law of large numbers and central limit behavior for sums of independent random variables.
-
C.
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.
-
D.
Erdős–Rényi law of large numbers
The Erdős–Rényi law of large numbers is a refinement of the classical law of large numbers that provides precise asymptotic behavior and convergence rates for sums of independent random variables, developed by mathematicians Pál Erdős and Alfréd Rényi.
-
E.
Freidlin–Wentzell theory
Freidlin–Wentzell theory is a mathematical framework in probability that analyzes the behavior of stochastic dynamical systems under small random perturbations using large deviation principles.
- F. None of above. chosen
Statements (35)
| Predicate | Object |
|---|---|
| instanceOf |
central limit theorem variant
ⓘ
probability theorem ⓘ |
| appliesTo |
independent random variables
ⓘ
not necessarily identically distributed random variables ⓘ |
| assumption |
Lyapunov condition on higher moments
ⓘ
linked to:
Lyapunov condition
existence of moments of order 2 plus delta ⓘ independence of summands ⓘ |
| category |
asymptotic theorems
ⓘ
theorems in probability theory ⓘ |
| comparedTo | Lindeberg–Feller central limit theorem ⓘ |
| conclusion |
asymptotic normality of sums
ⓘ
normalized sum converges in distribution to a normal distribution ⓘ |
| ensures | Gaussian limit for sums under moment bounds ⓘ |
| field | probability theory ⓘ |
| formalizes | conditions for normal approximation of independent sums ⓘ |
| generalizes | classical central limit theorem for i.i.d. variables ⓘ |
| hasCondition | Lyapunov condition with parameter delta greater than 0 ⓘ |
| hasFormulation | in terms of normalized centered sums ⓘ |
| historicalPeriod | late 19th century mathematics ⓘ |
| implies | standardized sum converges to standard normal distribution ⓘ |
| namedAfter | Aleksandr Lyapunov ⓘ |
| provides | sufficient conditions for central limit behavior ⓘ |
| relatedTo |
Berry–Esseen theorem
ⓘ
law of large numbers ⓘ moment conditions in probability theory ⓘ |
| requires |
finite variance of each summand
ⓘ
non-degenerate limiting variance ⓘ |
| strongerThan | Lindeberg central limit theorem in terms of moment assumptions ⓘ |
| topic |
convergence in distribution
ⓘ
normal approximation ⓘ triangular arrays of random variables ⓘ |
| typeOf | limit theorem ⓘ |
| usedIn |
asymptotic statistics
ⓘ
error analysis of sums of independent variables ⓘ theoretical justification of normal approximations ⓘ |
How these facts were elicited
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Input
Subject: Lyapunov central limit theorem Description of subject: The Lyapunov central limit theorem is a version of the central limit theorem that provides sufficient moment conditions under which the normalized sum of independent (not necessarily identically distributed) random variables converges in distribution to a normal law.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.