Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities
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Bazant_corrections to.pdf
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Author(s)
Lam, Henry
Blanchet, Jose
Burch, Damian
Bazant, Martin Z.
Date Issued
September 2011
Journal
Journal of Theoretical Probability
Publisher
Springer-Verlag
Citation
Lam, Henry et al. “Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities.” Journal of Theoretical Probability 24.4 (2011): 895–927.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Chemical Engineering
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s10959-011-0379-y
