Lindeberg–Feller central limit theorem

E174594

central limit theorem probability 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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All labels observed (8)

Label	Occurrences
Lindeberg condition	3
Lindeberg–Feller central limit theorem canonical	3
Lindeberg condition for the central limit theorem	1
Lindeberg–Feller theorem	1
Lindeberg–Feller theorem for independent but not identically distributed sequences	1
Lindeberg–Feller theorem for triangular arrays	1
Lyapunov central limit theorem	1
central limit theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1535799 — 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: Lindeberg–Feller central limit theorem
Context triple: [Berry–Esseen theorem, relatedTo, Lindeberg–Feller central limit theorem]

A. 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.
B. central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
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. law of large numbers
The law of large numbers is a fundamental theorem in probability theory stating that as the number of independent trials increases, the sample average converges to the expected value.
E. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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: Lindeberg–Feller central limit theorem
Target entity description: 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.

A. 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.
B. central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
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. law of large numbers
The law of large numbers is a fundamental theorem in probability theory stating that as the number of independent trials increases, the sample average converges to the expected value.
E. Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
F. None of above. chosen

Statements (40)

Predicate	Object
instanceOf	central limit theorem ⓘ probability theorem ⓘ
addresses	heterogeneous variance structures in sums of random variables ⓘ
appliesTo	independent random variables ⓘ not necessarily identically distributed random variables ⓘ
assumes	finite variances of the random variables ⓘ independence within each row of the triangular array ⓘ
characterizes	when the normalized sum converges to a standard normal distribution ⓘ
concerns	standardization by mean and variance ⓘ
conclusion	normalized sums converge in distribution to a normal distribution ⓘ
contrastsWith	central limit theorems requiring identical distribution ⓘ
describes	convergence in distribution of sums of independent random variables ⓘ
ensures	no single term dominates the sum in the limit ⓘ
equivalentTo	Lindeberg condition plus variance normalization for convergence to normal law ⓘ
field	mathematical statistics ⓘ probability theory ⓘ
generalizes	Lyapunov central limit theorem ⓘ classical central limit theorem ⓘ
guarantees	Gaussian limit for properly normalized sums ⓘ
hasVersion	Lindeberg–Feller theorem for independent but not identically distributed sequences ⓘ linked to: Lindeberg–Feller central limit theorem Lindeberg–Feller theorem for triangular arrays ⓘ linked to: Lindeberg–Feller central limit theorem
implies	under Lindeberg condition the central limit theorem holds for the array ⓘ
mathematicalDomain	real-valued random variables ⓘ
namedAfter	Jarl Waldemar Lindeberg ⓘ William Feller ⓘ
provides	necessary and sufficient conditions for central limit behavior in triangular arrays ⓘ
relatedTo	Berry–Esseen theorem ⓘ Lyapunov condition ⓘ law of large numbers ⓘ
requires	variance of the sum to diverge to infinity ⓘ
topic	asymptotic distribution of sums ⓘ
typeOf	limit theorem ⓘ
usedFor	establishing robustness of normal approximation under weak conditions ⓘ proving asymptotic normality of sample means under non-identical distributions ⓘ
usedIn	asymptotic analysis of estimators ⓘ econometrics ⓘ stochastic process theory ⓘ theoretical statistics ⓘ
usesConcept	triangular array of random variables ⓘ
usesCondition	Lindeberg condition ⓘ linked to: Lindeberg–Feller central limit theorem

How these facts were elicited