[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"jSVeaUwsWa":3},"# ProbabilityApproximation: Berry–Esseen Bounds for Independent Sums\n\n[![CI](https://github.com/Polarnova/ProbabilityApproximation/actions/workflows/ci.yml/badge.svg)](https://github.com/Polarnova/ProbabilityApproximation/actions/workflows/ci.yml)\n[![Release](https://img.shields.io/github/v/release/Polarnova/ProbabilityApproximation)](https://github.com/Polarnova/ProbabilityApproximation/releases)\n[![Lean 4](https://img.shields.io/badge/Lean-4.32.0-blue)](https://lean-lang.org/)\n[![Blueprint](https://img.shields.io/badge/Verso-Blueprint-5b4b8a)](https://probability.polarnova.site)\n[![License](https://img.shields.io/github/license/Polarnova/ProbabilityApproximation)](LICENSE)\n\nProbabilityApproximation is a Lean 4 and Mathlib library for the Berry–Esseen theorem, quantitative\nnormal approximation, and the multivariate central limit theorem. It formalizes the nonuniform\nscalar bound of Chen and Shao via Stein's method and Bentkus's dimension-dependent Gaussian\napproximation over convex sets.\n\nRead the [interactive Blueprint](https://probability.polarnova.site), or download the\n[complete mathematical text as a PDF](https://probability.polarnova.site/berry-esseen-bounds.pdf).\n\n## Main results\n\n### Nonuniform Berry–Esseen bound\n\nLet $X_1,\\ldots,X_n$ be independent real-valued random variables satisfying\n$\\mathbb E X_i=0$, $\\sum_{i=1}^n\\mathrm{Var}(X_i)=1$, and\n$\\beta_3:=\\sum_{i=1}^n\\mathbb E|X_i|^3\u003C\\infty$. For\n$W:=\\sum_{i=1}^n X_i$, there is a universal constant $C>0$ such that\n\n$$\n\\left|\\Pr(W\\le x)-\\Phi(x)\\right|\n\\le \\frac{C\\beta_3}{1+|x|^3},\n\\qquad x\\in\\mathbb R.\n$$\n\nHere $\\Phi$ denotes the standard normal distribution function.\n\n### Bentkus's multivariate convex-set bound\n\nLet $X_1,\\ldots,X_n$ be independent centered random vectors in $\\mathbb R^d$, set\n$W:=\\sum_{i=1}^n X_i$, and suppose that\n$\\Sigma:=\\mathrm{Cov}(W)$ is positive definite. If\n$Z\\sim\\mathcal N_d(0,\\Sigma)$ and\n$\\beta:=\\sum_{i=1}^n\\mathbb E\\lVert\\Sigma^{-1/2}X_i\\rVert_2^3\u003C\\infty$, then there is a universal\nconstant $C>0$ such that\n\n$$\n\\sup_{A\\in\\mathcal C_d}\n\\left|\\Pr(W\\in A)-\\Pr(Z\\in A)\\right|\n\\le C d^{1/4}\\beta,\n$$\n\nwhere $\\mathcal C_d$ is the class of Borel convex subsets of $\\mathbb R^d$.\n\n## Mathematical infrastructure\n\nThe library also develops reusable foundations for probability, analysis, and convex geometry:\n\n- **Stein's method and scalar probability:** the half-line Stein equation, derivative estimates,\n  leave-one-out concentration, Bennett--Hoeffding bounds, one-sided truncation, residual\n  decompositions, large-jump estimates, and reflection of distribution functions.\n- **Gaussian analysis:** independent Gaussian companions, covariance algebra, multivariate\n  Gaussian pushforwards, whitening, Gaussian-density derivatives, integration by parts, and\n  second-order Taylor remainder estimates.\n- **Convex and geometric measure theory:** convex parallel sets, metric projection, signed\n  distance, smooth approximations of convex indicators, Hausdorff and spherical measure,\n  coarea formulas, Ball's Gaussian perimeter theorem, and Gaussian shell bounds.\n- **Multivariate normal approximation:** smoothing inequalities, rotation and Fubini arguments,\n  small- and large-angle estimates, Gaussian replacement induction, and transport from identity\n  covariance to arbitrary positive-definite covariance.\n\n## Using the library\n\nThe current release is `v0.9.6`, built with Lean and Mathlib `v4.32.0`. Add the tagged package to a\ndownstream `lakefile.toml`:\n\n```toml\n[[require]]\nname = \"ProbabilityApproximation\"\ngit = \"https://github.com/Polarnova/ProbabilityApproximation.git\"\nrev = \"v0.9.6\"\n```\n\nThen update and build:\n\n```bash\nlake update\nlake exe cache get\nlake build @ProbabilityApproximation:release\n```\n\nThe explicit release target downloads the matching precompiled ProbabilityApproximation archive on\nLinux x86-64 and macOS arm64 and fails if that verified platform asset is unavailable; it does not\nsilently replace the download with a local source build.\n\nImport all public declarations with:\n\n```lean\nimport ProbabilityApproximation\n```\n\nThe principal declarations are:\n\n```lean\n#check ProbabilityTheory.nonuniformBerryEsseen\n#check ProbabilityTheory.exists_bentkus_convex_set_constant\n```\n\nFull repository, Blueprint, and PDF builds run in GitHub Actions. For local proof development,\nfetch Mathlib's cache and build only the affected production module, for example:\n\n```bash\ngit clone https://github.com/Polarnova/ProbabilityApproximation.git\ncd ProbabilityApproximation\nlake exe cache get\nlake build ProbabilityApproximation.BerryEsseen\n```\n\n## Blueprint\n\nThe [Verso Blueprint](https://github.com/leanprover/verso-blueprint) presents both proofs as a\nmathematical text with complete statements, source citations, Lean declarations, and dependency\ngraphs. The same text is available as a\n[downloadable PDF](https://probability.polarnova.site/berry-esseen-bounds.pdf).\n\nAfter the root production artifacts are current, `./scripts/site.sh serve dev` inside\n`blueprint-verso` retains fidelity metadata for optional local review. The Blueprint workspace\nshares the root dependencies and refuses to rebuild a missing or stale production library. Public\nbuilds run in GitHub Actions with the default `release` profile and omit fidelity tags from the\nreading view.\n\n## Contributing\n\nRead [`AGENTS.md`](AGENTS.md) for the source-fidelity, Mathlib-reuse, proof, and Blueprint\nconventions.\n\n## Citation\n\nIf you use the Lean formalization, cite the specific software release through GitHub's\n**Cite this repository** menu or [`CITATION.cff`](CITATION.cff). Cite the original mathematical\nsource as well: Bikelis (1966) for the classical finite-third-moment nonuniform bound,\nChen--Shao (2001) for the stronger truncated-moment theorem, Chen--Shao (2005) for the proof\npresentation used here, or Bentkus (2004) for the multivariate convex-set theorem. Work using the\nauxiliary geometric or explicit-constant results should additionally cite the corresponding source\nlisted below.\n\n## Authors\n\nAsher Yan with ChatGPT 5.6.\n\n## References\n\n- A. Bikelis,\n  [*On estimates of the remainder term in the central limit theorem*](https://doi.org/10.15388/LMJ.1966.19732),\n  *Lithuanian Mathematical Journal* 6(3) (1966), 323--346.\n- Herbert Federer,\n  [*Geometric Measure Theory*](https://doi.org/10.1007/978-3-642-62010-2),\n  Springer, Grundlehren der mathematischen Wissenschaften 153 (1969).\n- Keith Ball,\n  [*The reverse isoperimetric problem for Gaussian measure*](https://doi.org/10.1007/BF02573986),\n  *Discrete and Computational Geometry* 10(4) (1993), 411--420.\n- Louis H. Y. Chen and Qi-Man Shao,\n  [*A non-uniform Berry--Esseen bound via Stein's method*](https://doi.org/10.1007/PL00008782),\n  *Probability Theory and Related Fields* 120(2) (2001), 236--254.\n- Louis H. Y. Chen and Qi-Man Shao,\n  [*Stein's method for normal approximation*](https://blog.nus.edu.sg/louischen/files/2019/01/IMS4-pp-1-59-2hma1np.pdf),\n  in *An Introduction to Stein's Method*, Lecture Notes Series 4, Institute for Mathematical\n  Sciences, National University of Singapore (2005), 1--59.\n- Vidmantas Bentkus,\n  [*On the dependence of the Berry--Esseen bound on dimension*](https://doi.org/10.1016/S0378-3758(02)00094-0),\n  *Journal of Statistical Planning and Inference* 113(2) (2003), 385--402.\n- Vidmantas Bentkus,\n  [*A Lyapunov-type bound in R^d*](https://www.mathnet.ru/eng/tvp230),\n  *Teoriya Veroyatnostei i ee Primeneniya* 49(2) (2004), 400--410.\n- Martin Raič,\n  [*A multivariate Berry--Esseen theorem with explicit constants*](https://doi.org/10.3150/18-BEJ1072),\n  *Bernoulli* 25(4A) (2019), 2824--2853.\n\nThe Blueprint bibliography is generated from\n[`references.bib`](blueprint-verso/ProbabilityApproximationBlueprint/references.bib).\n\n## Related projects\n\n[Mathlib](https://github.com/leanprover-community/mathlib4) supplies the mathematical foundation.\n[FABL](https://github.com/Polarnova/FABL) is the initial downstream consumer of the scalar theorem.\n\n## License\n\nProbabilityApproximation is released under the Apache License 2.0. See [`LICENSE`](LICENSE).\n",1784995017396]