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Canonical static concept record

Central limit theorem

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Summary

For iid real random variables with finite nonzero variance, centered and variance-normalized sums converge in distribution to the standard normal law.

Record metadata

Canonical name: Central limit theorem
Concept ID: central_limit_theorem
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Probability
Domains: Probability, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Limit theorem

Carrier(s)

iid sequence $(X_n)$ of real random variables

Data

centered normalized sums

Axioms / constraints

finite mean
finite nonzero variance

Canonically induces

asymptotic Gaussian distribution

Notes

This is the classical Lindeberg–Lévy form; triangular-array and dependent-variable central limit theorems require different hypotheses.

Concept sources

Wikipedia — Central limit theorem — Central limit theorem · encyclopedic reference · source ID wp-central-limit-theorem
nLab — central limit theorem — central limit theorem · mathematical reference · source ID nlab-central-limit-theorem
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

Incoming relations (arrows to this concept)

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Independent random variables Central limit theorem

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: independent_family_to_central_limit
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: Gaussian limit under iid finite-variance hypotheses

Authored explanation

For iid real variables with finite nonzero variance, centered and normalized sums converge in distribution to a standard normal random variable.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Central limit theorem — Central limit theorem · encyclopedic reference · source ID wp-central-limit-theorem
nLab — central limit theorem — central limit theorem · mathematical reference · source ID nlab-central-limit-theorem
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

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Each relation below starts at this concept.