Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
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Authors: | Henry Lam Jose Blanchet Damian Burch Martin Z Bazant |
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Institution: | (1) Department of Mathematics, University of California, San Diego, USA;(2) Department of Mathematics, Osaka University, Osaka, Japan; |
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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. |
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