{ "artifactId": "artifact.library.mathlib.strong-law-large-numbers.v001", "candidateOnly": false, "category": "Probability and analysis", "claimBoundary": "This page indexes Mathlib's real-valued almost-sure strong law for an integrable, identically distributed sequence with pairwise independence. It concludes almost-everywhere convergence of empirical averages to the integral of X 0. It does not state a convergence rate, finite-sample concentration bound, Lp convergence, convergence in distribution only, or the more general Banach-valued theorem.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem ProbabilityTheory.strong_law_ae_real {Ω : Type*} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) μ) (hindep : Pairwise ((· ⟂ᵢ[μ] ·) on X)) (hident : ∀ i, ProbabilityTheory.IdentDistrib (X i) (X 0) μ μ) : ∀ᵐ ω ∂μ, Filter.Tendsto (fun n : ℕ => (∑ i ∈ Finset.range n, X i ω) / n) Filter.atTop (𝓝 μ[X 0])", "family": "Limit theorems", "id": "library.mathlib.strong-law-large-numbers.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/strong-law-large-numbers/", "entryData": "/data/library-theorems/strong-law-large-numbers.json", "evidence": "/proofs/artifact.library.mathlib.strong-law-large-numbers.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.strong-law-large-numbers.v001.evidence.json", "source": "/sources/upstream/strong-law-large-numbers/" }, "landmark": { "completeFormalType": "theorem ProbabilityTheory.strong_law_ae_real {Ω : Type*} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) μ) (hindep : Pairwise ((· ⟂ᵢ[μ] ·) on X)) (hident : ∀ i, ProbabilityTheory.IdentDistrib (X i) (X 0) μ μ) : ∀ᵐ ω ∂μ, Filter.Tendsto (fun n : ℕ => (∑ i ∈ Finset.range n, X i ω) / n) Filter.atTop (𝓝 μ[X 0])", "concepts": [ "pairwise independence", "identical distribution", "integrability", "empirical averages", "almost-sure convergence" ], "plainLanguage": "For almost every outcome, the running average of pairwise independent, identically distributed integrable real random variables settles to their common expected value.", "poster": { "alt": "An ivory probability poster shows pairwise-independent observation streams feeding running averages that settle toward a common horizontal expectation line, with an almost-everywhere scope marker.", "byteSize": 2515943, "caption": "Pairwise-independent identically distributed integrable real variables have empirical averages converging almost everywhere to their common expectation.", "description": "The poster separates the sequence hypotheses, the empirical-average expression, the almost-everywhere convergence conclusion, and a four-stage proof route through truncation, sparse geometric subsequences, interpolation to all indices, and positive/negative decomposition.", "derivatives": [ { "byteSize": 114228, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-poster-v1-640.webp", "sha256": "sha256:1adb21ad3b59a3d983b277beb4ce7f7387b74b35a7ba30890832c805abe80634", "width": 640 }, { "byteSize": 229926, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-poster-v1-1024.webp", "sha256": "sha256:5b39ec1f3610f4c1e4c52e8607fc73b5a69dce79c13af5a96d6ac55042bab0b7", "width": 1024 } ], "generatedAt": "2026-07-25T01:31:58.173493907-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-poster-v1.png", "sha256": "sha256:3d280bd41d16222e5a8b813fdfd0c98919b333406cd94a286584977f99d90612", "title": "Strong Law of Large Numbers at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "PROBABILITY · ALMOST-SURE CONVERGENCE\nSTRONG LAW OF LARGE NUMBERS\nIntegrable · identically distributed · pairwise independent\nAₙ(ω) = (1/n) ∑ᵢ₌₀ⁿ⁻¹ Xᵢ(ω)\nFor almost every ω\nAₙ(ω) → μ[X₀]\nEMPIRICAL AVERAGES SETTLE ALMOST SURELY\nHOW THE PROOF MOVES\n1 · Truncate large values and control exceptional events\n2 · Prove convergence along geometric subsequences\n3 · Interpolate from sparse subsequences to every n\n4 · Recombine positive and negative parts\nEXACT SCOPE\nReal-valued almost-sure convergence. No rate or finite-sample bound." }, "visual": { "alt": "Many pairwise-linked random observations feed a running-average ribbon whose fluctuations narrow around one gold expectation level while a negligible set fades away.", "byteSize": 3274358, "caption": "Along almost every outcome, the empirical-average path settles onto the common expectation.", "description": "A wide forest-green probability field shows repeated real-valued observations with pairwise rather than all-at-once connections. Their successive averages form a narrowing emerald ribbon around a single antique-gold level; a small translucent exceptional region remains visually separate.", "derivatives": [ { "byteSize": 44504, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-schematic-v1-640.webp", "sha256": "sha256:64cc5f7d1409bc8c049004d48406ef1c534ed77de2b2c42517556d6295b3feab", "width": 640 }, { "byteSize": 204508, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-schematic-v1-1200.webp", "sha256": "sha256:22565102ac7a0a9381c8479f5d92c911ef013651ecf487af6e2459bf79108829", "width": 1200 } ], "generatedAt": "2026-07-25T01:23:03.038730838-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/strong-law-large-numbers/theorem-schematic-v1.png", "sha256": "sha256:8ea78fdd9eaf3df91ecb7fb7a8796eae4ca2204019519b5a047773662158d20c", "title": "Almost-sure stabilization of empirical averages", "width": 1536 }, "walkthrough": null, "whyLandmark": "The strong law is probability's fundamental pathwise stabilization theorem. It connects repeated observation with long-run average behavior and complements the collection's distributional Central Limit Theorem with a genuinely almost-sure conclusion." }, "nonClaims": [ "The selected declaration is real-valued; Mathlib's more general Banach-valued strong law is a separate declaration.", "It provides no convergence rate, concentration inequality, or finite-sample threshold.", "ProofAtlas indexes an existing Mathlib theorem; generated explanations and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.strong-law-large-numbers.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Empirical averages of pairwise independent identically distributed integrable real random variables converge almost everywhere to their common expectation.", "targetId": "target.library.mathlib.strong-law-large-numbers.v001", "title": "Strong Law of Large Numbers", "upstreamOrigin": { "declarationName": "ProbabilityTheory.strong_law_ae_real", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Probability/StrongLaw.lean#L598", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:b74c93434df44eb75b2567f43a58b9e0353138660ad07b99263d8019bcf4f1c6", "sourceByteLength": 46891, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Probability/StrongLaw.lean", "sourceLine": 598, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }