Existing in Mathlib · Probability and analysis

Central Limit Theorem

Add more and more independent copies of the same real random variable, whose mean is zero and second moment is one, and divide the sum by the square root of the number of terms. The resulting distributions approach the standard bell curve.

independence
identical distribution
normalized sums
convergence in distribution
Gaussian law

Exact theorem

Exact Mathlib statement

theorem ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum {Ω : Type*} {Ω' : Type*} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {P : MeasureTheory.Measure Ω} {P' : MeasureTheory.Measure Ω'} {X : ℕ → Ω → ℝ} {Y : Ω' → ℝ} [MeasureTheory.IsProbabilityMeasure P] [MeasureTheory.IsProbabilityMeasure P'] (hY : ProbabilityTheory.HasLaw Y (ProbabilityTheory.gaussianReal 0 1) P') (h0 : ∫ x, X 0 x ∂P = 0) (h1 : ∫ x, (X 0 ^ 2) x ∂P = 1) (hindep : ProbabilityTheory.iIndepFun X P) (hident : ∀ i, ProbabilityTheory.IdentDistrib (X i) (X 0) P P) : MeasureTheory.TendstoInDistribution (fun (n : ℕ) ω ↦ (√(n : ℝ))⁻¹ * ∑ k ∈ Finset.range n, X k ω) Filter.atTop Y (fun _ ↦ P) P'

The theorem at a glance

Central Limit Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory probability poster shows blank input beads, the inverse-square-root normalized sum, an unlabeled characteristic-function wave, a gold Gaussian bell, and the exact distributional scope. — Inverse-square-root-normalized IID sums converge in distribution to the standard Gaussian law.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Scientific figure illustrating: Normalized independent sums approach the standard Gaussian distribution. — Normalized independent sums approach the standard Gaussian distribution.

Why it matters

A mathematical landmark

The Central Limit Theorem explains why Gaussian behavior appears throughout probability and statistics. It is a universal limit law for normalized aggregation under exact independence and moment hypotheses.

Proof architecture

How normalized sums become Gaussian in law

4 curated stages

Mathlib proves the selected Central Limit Theorem by forming centered normalized IID sums, factoring their characteristic functions through independence, identifying the Gaussian Fourier limit from the second-order expansion, and applying Lévy convergence to return to probability laws.

These source-anchored stages explain the retained Mathlib proof route. They are generated explanations, not proof evidence or an extracted Lean proof-term dependency graph.

Independent copies enter one normalized sum
Independent copies enter one normalized sum
Centered, second-moment-one identically distributed variables are added and compressed by the inverse square root of the sample count.
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Centered, second-moment-one identically distributed variables are added and compressed by the inverse square root of the sample count.
S_n = (sqrt n)⁻¹ * sum_{k<n} X_k
The selected theorem begins with an independent identically distributed sequence of real random variables, common mean zero, and common second moment one. Its nth object is the partial sum over range n multiplied by the inverse square root of n.
Independence turns the sum into a power
Independence turns the sum into a power
The characteristic function of the normalized sum factors into identical scaled factors, one for each summand.
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The characteristic function of the normalized sum factors into identical scaled factors, one for each summand.
charFun(S_n)(t) = charFun(X_0)(t / sqrt n)^n
charFun_inv_sqrt_mul_sum maps the normalized finite sum to the Fourier side. Independence changes the sum's characteristic function into a product, identical distribution makes every factor the same, and the finite product becomes the nth power of the one-variable characteristic function at the inverse-square-root-scaled frequency.
The centered quadratic term reaches the Gaussian characteristic function
The centered quadratic term reaches the Gaussian characteristic function
Mean zero removes the linear term, while the raw second-moment hypothesis `E[X₀²] = 1` fixes the quadratic term; the scaled nth powers converge to the Gaussian characteristic function.
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Mean zero removes the linear term, while the raw second-moment hypothesis `E[X₀²] = 1` fixes the quadratic term; the scaled nth powers converge to the Gaussian characteristic function.
charFun(X_0)(t / sqrt n)^n → exp(-t^2/2)
`taylor_charFun_two hX h0 h1` gives `φ_X₀(u) = 1 − u²/2 + o(u²)` as `u → 0`. Substituting `u = t/√n` and applying `tendsto_pow_exp_of_isLittleO_sub_add_div` yields pointwise convergence of the nth powers to `exp(−t²/2)`.
Characteristic-function convergence gives the Gaussian law
Characteristic-function convergence gives the Gaussian law
Lévy's theorem converts the pointwise Fourier-side limit into convergence in distribution of the normalized sums to the standard Gaussian.
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Lévy's theorem converts the pointwise Fourier-side limit into convergence in distribution of the normalized sums to the standard Gaussian.
S_n → Y in distribution, with Law(Y) = gaussianReal 0 1
The theorem uses ProbabilityMeasure.tendsto_iff_tendsto_charFun. The declared target random variable has standard Gaussian law, whose characteristic function is the page-3 exponential. Rewriting the normalized sums with the factorization and applying that limit closes the exact convergence-in-distribution statement.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's distributional convergence theorem for independent identically distributed real random variables with mean zero and second moment one, normalized by the inverse square root of n, to a standard Gaussian law. It does not state a rate of convergence or cover arbitrary variance without normalization.

Proof Atlas did not originate the Central Limit Theorem or Mathlib's declaration.
The selected declaration gives neither a quantitative rate nor almost-sure or pointwise convergence.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sum in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.central-limit-theorem.v001
Source: Open the pinned upstream reference