A Formally Verified Proof of the Central Limit Theorem

Abstract

We describe a proof of the Central Limit Theorem that has been formally verified in the Isabelle proof assistant. Our formalization builds upon and extends Isabelle’s libraries for analysis and measure-theoretic probability. The proof of the theorem uses characteristic functions, which are a kind of Fourier transform, to demonstrate that, under suitable hypotheses, sums of random variables converge weakly to the standard normal distribution. We also discuss the libraries and infrastructure that supported the formalization, and reflect on some of the lessons we have learned from the effort.

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Notes

  1. 1.

    The probability library in particular can be found at https://isabelle.in.tum.de/dist/library/HOL/HOL-Probability/index.html.

  2. 2.

    https://isabelle.in.tum.de/.

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Acknowledgements

We are grateful to Tobias Nipkow, Lawrence Paulson, Makarius Wenzel, and the entire Isabelle team for the ongoing development of Isabelle. We are especially grateful to Tobias for steadfast encouragement and support. We thank our two anonymous referees for a very careful reading and helpful comments. Avigad and Serafin’s work has been partially supported by NSF Grant DMS-1068829, and Avigad’s work has been partially supported by AFOSR Grants FA9550-12-1-0370 and FA9550-15-1-0053. Hölzl’s work has been partially supported by DFG Projects Ni 491/15-1 and Ni 491/16-1.

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Correspondence to Jeremy Avigad.

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Avigad, J., Hölzl, J. & Serafin, L. A Formally Verified Proof of the Central Limit Theorem. J Autom Reasoning 59, 389–423 (2017). https://doi.org/10.1007/s10817-017-9404-x

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Keywords

  • Interactive theorem proving
  • Measure theory
  • Central limit theorem