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Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities

Author

Listed:
  • Henry Lam

    (Boston University)

  • Jose Blanchet

    (Columbia University)

  • Damian Burch

    (Massachusetts Institute of Technology)

  • Martin Z. Bazant

    (Massachusetts Institute of Technology)

Abstract

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

Suggested Citation

  • Henry Lam & Jose Blanchet & Damian Burch & Martin Z. Bazant, 2011. "Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities," Journal of Theoretical Probability, Springer, vol. 24(4), pages 895-927, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-011-0379-y
    DOI: 10.1007/s10959-011-0379-y
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    References listed on IDEAS

    as
    1. Evans, Michael & Swartz, Timothy, 2000. "Approximating Integrals via Monte Carlo and Deterministic Methods," OUP Catalogue, Oxford University Press, number 9780198502784.
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