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

Henry Lam (), Jose Blanchet (), Damian Burch and Martin Z. Bazant ()
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Henry Lam: Boston University
Jose Blanchet: Columbia University
Damian Burch: Massachusetts Institute of Technology
Martin Z. Bazant: Massachusetts Institute of Technology

Journal of Theoretical Probability, 2011, vol. 24, issue 4, 895-927

Abstract: 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.

Keywords: Edgeworth expansion; Central Limit Theorem; Regular variation; Heavy tail; Power law; Large deviations; 60F05; 60F10 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-011-0379-y

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