{ "artifactId": "artifact.library.mathlib.central-limit-theorem.v001", "candidateOnly": false, "category": "Probability and analysis", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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'", "family": "Limit theorems", "id": "library.mathlib.central-limit-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/central-limit-theorem/", "entryData": "/data/library-theorems/central-limit-theorem.json", "evidence": "/proofs/artifact.library.mathlib.central-limit-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.central-limit-theorem.v001.evidence.json", "source": "/sources/upstream/central-limit-theorem/" }, "landmark": { "completeFormalType": "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'", "concepts": [ "independence", "identical distribution", "normalized sums", "convergence in distribution", "Gaussian law" ], "plainLanguage": "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.", "poster": { "alt": "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.", "byteSize": 2769726, "caption": "Inverse-square-root-normalized IID sums converge in distribution to the standard Gaussian law.", "description": "The poster makes the independence, identical-distribution, mean-zero, second-moment-one, normalization, Gaussian endpoint, and absence of a rate or stronger convergence claim explicit.", "derivatives": [ { "byteSize": 104344, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:310a8056c280b8b1925e32541080adffe4490f64f0dd55550678ebeffe64cf24", "width": 640 }, { "byteSize": 246342, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:887d45f4997b4dfe40a0bccc1414be54601b4aec5cc865c60e6c21b3ada619d1", "width": 1024 } ], "generatedAt": "2026-07-21T23:47:07Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-poster-v1.png", "sha256": "sha256:51933cb1a2319704805502b672e1ff9c6295fadeb592d3ec7d5567b468da8503", "title": "Central Limit Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "PROBABILITY · DISTRIBUTIONAL CONVERGENCE\nCENTRAL LIMIT THEOREM\nIndependent · identically distributed\nMean 0 · second moment 1\nSₙ = (√n)⁻¹ ∑ₖ₌₀ⁿ⁻¹ Xₖ\nSₙ converges in distribution\nto the standard Gaussian\nNORMALIZED SUMS BECOME GAUSSIAN IN LAW\nEXACT SCOPE\nNo convergence rate. No almost-sure or pointwise convergence claim." }, "visual": { "alt": "Several identically distributed green inputs pass through a cobalt normalized-sum chamber and converge toward one centered gold Gaussian bell.", "byteSize": 2893094, "caption": "Normalized independent sums approach the standard Gaussian distribution.", "description": "Distinct finite input profiles flow through increasingly aggregated normalized envelopes toward a single centered bell curve, emphasizing distributional convergence rather than finite-stage equality.", "derivatives": [ { "byteSize": 35366, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:99b2a3b8b0da4fccf6ef595bc03c3680f12ad5182139cd06089c2e22e13e3edd", "width": 640 }, { "byteSize": 137390, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:79d2178af5bd6697aac9623dd96b6be176306dd5cf278ed8dec710e0667a63ae", "width": 1200 } ], "generatedAt": "2026-07-21T22:53:46Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/central-limit-theorem/theorem-schematic-v1.png", "sha256": "sha256:64a4ad54953364af885f74391703b99349f579b630507697163b8cb6888e2280", "title": "Central Limit Theorem schematic", "width": 1536 }, "walkthrough": { "introduction": "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.", "pages": [ { "alt": "Equal ivory sample beads pass through forest-green independence separators into one cobalt normalized cloud with a faint gold Gaussian destination contour.", "byteSize": 2966106, "caption": "Centered, second-moment-one identically distributed variables are added and compressed by the inverse square root of the sample count.", "description": "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.", "derivatives": [ { "byteSize": 19766, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/proof-walkthrough-01-v1-640.webp", "sha256": "sha256:c5633a1f701f1fc5047c344dc99856a71253456afc3fc9eb5363ba6437212d1a", "width": 640 }, { "byteSize": 102940, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/central-limit-theorem/proof-walkthrough-01-v1-1200.webp", "sha256": "sha256:3ce88df04492054d438804ddde8abd667df1ef57b4d799cc8ad6c635e89d7622", "width": 1200 } ], "generatedAt": "2026-07-21T23:10:29Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/central-limit-theorem/proof-walkthrough-01-v1.png", "sha256": "sha256:966874b3d74ec6bcaf16fc8220af83c99dd7ccacc668293583c19f6c7d13b963", "title": "Independent copies enter one normalized sum", "width": 1536, "equation": "S_n = (sqrt n)⁻¹ * sum_{k